Accepted Manuscript Existence, Uniqueness and Stability of Solutions to Second Order Nonlinear Differential Equations with Non-instantaneous Impulses Malik Muslim, Avadhesh Kumar, Michal Feckan PII: DOI: Reference:
S1018-3647(16)30551-1 http://dx.doi.org/10.1016/j.jksus.2016.11.005 JKSUS 423
To appear in:
Journal of King Saud University - Science
Received Date: Accepted Date:
23 September 2016 21 November 2016
Please cite this article as: M. Muslim, A. Kumar, M. Feckan, Existence, Uniqueness and Stability of Solutions to Second Order Nonlinear Differential Equations with Non-instantaneous Impulses, Journal of King Saud University - Science (2016), doi: http://dx.doi.org/10.1016/j.jksus.2016.11.005
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Existence, Uniqueness and Stability of Solutions to Second Order Nonlinear Differential Equations with Non-instantaneous Impulses Malik Muslim, Avadhesh Kumar School of Basic Sciences, Indian Institute of Technology Mandi, Kamand (H.P.) - 175 005, India emails:
[email protected],
[email protected] Michal Feˇckan∗† Department of Mathematical Analysis and Numerical Mathematics Faculty of Mathematics, Physics and Informatics Comenius University in Bratislava Mlynsk´a dolina, 842 48 Bratislava, Slovakia email:
[email protected]
∗ †
corresponding author Partially supported by the Slovak Research and Development Agency under the contract No. APVV-14-0378.
1
Existence, Uniqueness and Stability of Solutions to Second Order Nonlinear Differential Equations with Non-instantaneous Impulses
Abstract: In this paper, we consider a non-instantaneous impulsive system represented by second order nonlinear differential equation with deviated argument in a Banach space X. We used the strongly continuous cosine family of linear operators and Banach fixed point method to study the existence and uniqueness of the solution of the non-instantaneous impulsive system. Also, we study the existence and uniqueness of the solution of the nonlocal problem and stability of the non-instantaneous impulsive system. Finally, we give examples to illustrate the application of these abstract results. Key Words: Non-instantaneous impulses, Deviated argument, Banach fixed point theorem. AMS Subject Classification: 34K30, 34K45, 34G20, 47D09.
1
Introduction
The dynamics of many evolving processes are subject to abrupt changes, such as shocks, harvesting and natural disaster. These phenomena involve short term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. Sometimes time abrupt changes may stay for time intervals such impulses are called non-instantaneous impulses. The importance of the study of non-instantaneous impulsive differential equations lies in its diverse fields of applications such as in the theory of stage by stage rocket combustion, maintaining hemodynamical equilibrium etc. A very well known application of non-instantaneous impulses is the introduction of insulin in the bloodstream which is abrupt change and the consequent absorption which is a gradual process as it remains active for a finite interval of time. The theory of impulsive differential equations has found enormous applications in realistic mathematical modeling of a wide range of practical situations. It has emerged as an important area of research such as modeling of impulsive problems in physics, population dynamics, ecology, biological systems, biotechnology and so forth. Recently, Hern´andez et. al. [1] studied mild and classical solutions for the impulsive differ-
2
ential equation with non-instantaneous impulses which is of the form 0 x (t) = Ax(t) + f (t, x(t)), t ∈ (si , ti+1 ] i = 0, 1, ..., m, x(t) = gi (t, x(t)), t ∈ (ti , si ], i = 1, 2, ..., m, x(0) = x0 ∈ X
(1.1)
In [2] Wang and Feˇckan have a remark on the conditions in (1.1): x(t) = gi (t, x(t)),
t ∈ (ti , si ], i = 1, 2, ..., m,
(1.2)
where gi ∈ C([ti , si ] × X, X) and there are positive constants Lgi , i = 1, ..., m such that kgi (t, x1 ) − gi (t, x2 )k ≤ Lgi kx1 − x2 k, ∀ x1 , x2 ∈ X, t ∈ [ti , si ] It follows from Theorem 2.1 in [1] that max{Lgi : i = 1, ..., m} < 1 is a necessary condition. Then Banach fixed point theorem gives a unique zi ∈ C([ti , si ], X) so that z = gi (t, z) if and only if z = zi (t). So (1.2) is equivalent to x(t) = zi (t),
t ∈ (ti , si ], i = 1, 2, ..., m,
(1.3)
which does not depend on the state x(.). Thus, it is necessary to modify (1.2) and consider the condition x(t) = gi (t, x(t− i )),
t ∈ (ti , si ], i = 1, 2, ..., m.
(1.4)
− + Of course then x(t+ i ) = gi (ti , x(ti )), i = 1, 2, ..., m. The symbols x(ti ) := lim→0+ x(ti + ) and x(t− i ) := lim→0− x(ti + ) represent the right and left limits of x(t) at t = ti respectively. Motivated by above remark, Wang and Feˇckan [2] have shown existence, uniqueness and stability of solutions of such general class of impulsive differential equations. In this paper, we continue in this direction to study the second order nonlinear differential equation with non-instantaneous impulses and deviated argument in a Banach Space X 00 x (t) = Ax(t) + f (t, x(t), x[h(x(t), t)]), t ∈ (si , ti+1 ), i = 0, 1, ..., m, t ∈ (ti , si ], i = 1, 2, ..., m, x(t) = Ji1 (t, x(t− i )), (1.5) 0 (t) = J 2 (t, x(t− )), x t ∈ (ti , si ], i = 1, 2, ..., m, i i 0 x(0) = x0 , x (0) = y0 ,
where x(t) be a state function, 0 = s0 = t0 < t1 < s1 < t2 , ..., tm < sm < tm+1 = T < ∞. We consider in (1.5) that x ∈ C((ti , ti+1 ], X), i = 0, 1, ..., m and there exist x(t− i ) and − 1 (t, x(t− )) and J 2 (t, x(t− )) represent ) = x(t ). The functions J ), i = 1, 2, ..., m with x(t x(t+ i i i i i i i noninstantaneous impulses during the intervals (ti , si ] , i = 1, 2, ..., m, so impulses at t− i have some duration, namely on intervals (ti , si ]. A is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators (C(t))t∈R on X. Ji1 , Ji2 , h and f are suitable functions and they will be specified later. Many partial differential equations that arise in several problems connected with the transverse motion of an extensible beam, the vibration of hinged bars, and many other physical phenomena can be formulated as the second order abstract differential equations in the infinite 3
dimensional spaces. A useful tool for the study of second-order abstract differential equations is the theory of strongly continuous cosine families of operators. Existence and uniqueness of the solution of second-order nonlinear systems and controllability of these systems in Banach spaces have been investigated extensively by many authors [3, 4, 5, 6, 7]. In certain real world problems, delay depends not only on the time but also on the unknown quantity. The differential equations with deviated arguments are generalization of delay differential equations. Gal C. G [8] has consider a nonlinear abstract differential equations with deviated arguments and study the existence and uniqueness of solutions. Recently, Muslim et. al. [9] studied exact and trajectory controllability of second order impulsive nonlinear systems with deviated argument. There are only few papers discussing the second order differential equations with deviated arguments in infinite dimensional spaces. As per my knowledge, there is no paper discussing the existence, uniqueness and stability of the mild solution of the second order differential equation with non-instantaneous impulses and deviated argument in Banach space. In order to fill this gap, we consider a nonlinear second order differential equation with deviated argument. Moreover, the study of second order differential equations with noninstantaneous impulses has not only mathematical significance but also it has applications such as harmonic oscillator with impulses and forced string equation, which we present in examples.
2
Preliminaries and Assumptions
We briefly review definitions and some useful properties of the theory of cosine family. Definition 2.1. (see, [12]) A one parameter family (C(t))t∈R of bounded linear operators mapping the Banach space X into itself is called a strongly continuous cosine family if and only if (i) C(s + t) + C(s − t) = 2C(s)C(t) for all s, t ∈ R, (ii) C(0) = I, (iii) C(t)x is continuous in t on R for each fixed point x ∈ X. (S(t))t∈R is the sine function associated to the strongly continuous cosine family, (C(t))t∈R which is defined by Z t S(t)x = C(s)x ds, x ∈ X, t ∈ R. 0
D(A) be the domain of the operator A which is defined by
D(A) = {x ∈ X : C(t)x is twice continuously differentiable in t}. D(A) is the Banach space endowed with the graph norm kxkA = kxk + kAxk for all x ∈ D(A). We define a set E = {x ∈ X : C(t)x is once continuously differentiable in t} which is a Banach space endowed with norm kxkE = kxk + sup0≤t≤1 kAS(t)xk for all x ∈ E. With the help of C(t) and S(t), we define a operator valued function C(t) S(t) ¯ h(t) = . AS(t) C(t) 4
¯ Operator valued function h(t) is a strongly continuous group of bounded linear operators on the space E × X generated by the operator 0 I ¯ A = A 0 defined on D(A) × E. It follows that AS(t) : E → X is a bounded linear operator and that AS(t)x → 0 as t → 0, for each x ∈ E. If x : [0, ∞) → X is locally integrable function then Z t y(t) = S(t − s)x(s)ds 0
defines an E valued continuous function which is a consequence of the fact that " R # Z t t S(t − s)x(s)ds 0 ¯ − s) h(t ds = R 0t x(s) 0 0 C(t − s)x(s)ds
defines an (E × X) valued continuous function. Proposition 2.1:(see, [12]) Let (C(t))t∈R be a strongly continuous cosine family in X. The following are true: (i) there exist constants K ≥ 1 and ω ≥ 0 such that |C(t)| ≤ Keω|t| for all t ∈ R. Rt (ii) |S(t2 ) − S(t1 )| ≤ K| t12 eω|s| ds| for all t1 , t2 ∈ R.
For more details on cosine family theory, we refer to Travis, Webb & Fattorini [10, 11, 12]. Let P C([0, T ], X) be the space of piecewise continuous functions. P C([0, T ], X) = {x : J = [0, T ] → X : x ∈ C((tk , tk+1 ], X), k = 0, 1, ..., m and there exist + − x(t− k ) and x(tk ), k = 1, 2, ..., m with x(tk ) = x(tk )}. It can be seen easily that P C([0, T ], X) for all t ∈ [0, T ], is a Banach space endowed with the supremum norm, kψkP CB := sup{kψ(η)ke−Ωη ,
0 ≤ η ≤ t}
for some Ω > 0. We set, CL (J, X) = {y ∈ P C([0, T ], X) : ky(t)−y(s)k ≤ L|t−s|, ∀ t, s ∈ [0, T ]}, where L is a suitable positive constant. Clearly CL (J, X) is a Banach space endowed with P CB norm. In order to prove the existence, uniqueness and stability of the solution for the problem (1.5), we need the following assumptions: (A1) A be the infinitesimal generator of a strongly continuous cosine family, (C(t))t∈R of bounded linear operators. S (A2) f : J1 × X × X → X, J1 = m i=0 [si , ti+1 ] is a continuous function and there exist positive constants K1 and K2 such that kf (t, x1 , y1 ) − f (t, x2 , y2 )k ≤ K1 (kx1 − x2 k + ky1 − y2 k) for every x1 , x2 , y1 , y2 ∈ X, t ∈ J1 . Also there exists a positive constant N such that kf (t, x, y)k ≤ N, ∀ t ∈ J1 and x, y ∈ X. 5
(A3) h : X × J1 → J is continuous and there exists a positive constant Lh such that |h(x1 , s) − h(x2 , s)| ≤ Lh kx1 − x2 k, ∀ x1 , x2 ∈ X, t ∈ J1 and it holds h(., 0) = 0. (A4) Jil ∈ C(Ii × X, X), Ii = [ti , si ] and there are positive constants LJ l , i = 1, 2, ..., m, l = i 1, 2, such that kJil (t, x1 ) − Jil (s, x2 )k ≤ LJ l (|t − s| + kx1 − x2 k), ∀ t, s ∈ Ii and x1 , x2 ∈ X. i
(A5) There exist positive constants CJ 1 and CJ 2 , i = 1, 2, ..., m such that i
i
kJi1 (t, x)k ≤ CJ 1 and kJi2 (t, x)k ≤ CJ 2 , ∀ t ∈ Ii and x ∈ X. i
i
In the following definition, we introduce the concept of mild solution for the problem (1.5). Definition 2.2. A function x ∈ CL (J, X) is called a mild solution of the impulsive problem (1.5) if it satisfies the following relations: x(0) = x0 , x0 (0) = y0 , the non-instantaneous impulse conditions x(t) = Ji1 (t, x(t− t ∈ (ti , si ], i = 1, 2, ..., m, i )), − 0 2 x (t) = Ji (t, x(ti )), t ∈ (ti , si ], i = 1, 2, ..., m and x(t) is the solution of the following integral equations x(t) x(t)
=
C(t)x0 + S(t)y0 +
Z
t
S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [0, t1 ],
0 si )(Ji1 (si , x(t− i )))
+ S(t − si )(Ji2 (si , x(t− C(t − i ))) Z t S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [si , ti+1 ], i = 1, 2, .., m. +
=
si
3
Existence and Uniqueness Result
Theorem 3.1. Let x0 ∈ D(A), y0 ∈ E. If all the assumptions (A1)-(A5) are satisfied, then the second order problem (1.5) has a unique mild solution x ∈ CL (J, X). Proof: Since AS(t) is a bounded linear operator therefore, we set ρ = supt∈J kAS(t)k. For more details on kAS(t)k, we refer [4, 7, 13]. By choosing ( N KT ωT −Ωsi ωT ωT e , e δ = max Ke CJ 1 + KT e CJ 2 + i i 1≤i≤m ω ) N KT KeωT kx0 k + KT eωT ky0 k + eωT e−Ωti CJ 1 , i ω 6
we set W = {x ∈ CL (J, X) : kxkP CB ≤ δ}. We define a map F : W → W given by (Fx)(t)
− 2 Ji1 t, C(ti − si−1 )(Ji1 (si−1 , x(t− i−1 ))) + S(ti − si−1 )(Ji (si−1 , x(ti−1 )))
= +
Z
!
ti
si−1
S(ti − s)f (s, x(s), x[h(x(s), s)])ds , t ∈ (ti , si ] i = 1, 2, ..., m;
(Fx)(t) = C(t)x0 + S(t)y0 +
Z
t
S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [0, t1 ];
0
(Fx)(t)
− 2 C(t − si )(Ji1 (si , x(t− i ))) + S(t − si )(Ji (si , x(ti ))) Z t + S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ (si , ti+1 ] i = 1, 2, ..., m.
=
si
First, we need to show that Fx ∈ CL (J, X) for any x ∈ CL (J, X) and some L > 0. If ti+1 ≥ t˜2 > t˜1 > si , then we get k(Fx)(t˜2 ) − (Fx)(t˜1 )k
k(C(t˜2 − si ) − C(t˜1 − si ))(Ji1 (si , x(t− i )))k − 2 + k(S(t˜2 − si ) − S(t˜1 − si ))(Ji (si , x(ti )))k Z t˜1 k(S(t˜2 − s) − S(t˜1 − s))kds + N
≤
si t˜2
+ N ≤
Z
t˜1
kS(t˜2 − s)kds
I1 + I2 + I3 + I4 .
(3.1)
We have, I1 = k(C(t˜2 − si ) − C(t˜1 − si ))(Ji1 (si , x(t− i )))k Z t˜2 −si = k AS(τ )(Ji1 (si , x(t− i )))dτ k t˜1 −si
≤ C1 (t˜2 − t˜1 ),
(3.2)
where C1 = ρCJ 1 . i Similarlly, we have I2 = k(S(t˜2 − si ) − S(t˜1 − si ))(Ji2 (si , x(t− i )))k Z t˜2 −si eωτ (Ji2 (si , x(t− = kK i )))dτ k t˜1 −si
≤ C2 (t˜2 − t˜1 ),
7
(3.3)
where C2 = Keωti+1 CJ 2 . i Similarly, we calculate third and fourth part of inequality (3.1) as follows I3 = N ≤ N
Z
t˜1
si Z t˜1 si
k(S(t˜2 − s) − S(t˜1 − s))kds kK
t˜2 −s
Z
≤ C3 (t˜2 − t˜1 ),
t˜1 −s
eωτ dτ kds (3.4)
where C3 = KN ti+1 eωti+1 and I4 = N
Z
t˜2
t˜1
kS(t˜2 − s)kds ≤ C4 (t˜2 − t¯1 ),
(3.5)
where C4 = KN ti+1 eωti+1 . We use the inequalities (3.2), (3.3), (3.4) and (3.5)) in inequality (3.1) and get the following inequality k(Fx)(t˜2 ) − (Fx)(t˜1 )k ≤ L|t˜2 − t˜1 |,
(3.6)
where L ≥ C1 + C2 + C3 + C4 . If t1 ≥ t˜2 > t˜1 ≥ 0, then we get k(Fx)(t˜2 ) − (Fx)(t˜1 )k
k(C(t˜2 ) − C(t˜1 ))x0 k + k(S(t˜2 ) − S(t˜1 ))y0 k Z t˜1 k(S(t˜2 − s) − S(t˜1 − s))kds + N
≤
0
+ N ≤
Z
t˜2
t˜1
kS(t¯2 − s)kds
I5 + I6 + I7 + I8 .
(3.7)
We have, I5 = k(C(t˜2 ) − C(t˜1 ))x0 k
= k
Z
t˜2
AS(τ )x0 dτ k
t˜1
≤ C5 (t˜2 − t˜1 ),
(3.8)
where C5 = ρkx0 k. Similarly, we have I6 = k(S(t˜2 ) − S(t˜1 ))y0 k
=
kK
Z
t˜2
t˜1
eωτ y0 dτ k
≤ C6 (t˜2 − t˜1 ), 8
(3.9)
where C6 = Keωt1 ky0 k. Similarly, we calculate third and fourth part of inequality (3.7) as follows I7 = N
t˜1
Z
0
≤ N
Z
k(S(t˜2 − s) − S(t˜1 − s))kds
t˜1
0
kK
Z
≤ C7 (t˜2 − t˜1 ),
t˜2 −s
t˜1 −s
eωτ dτ kds (3.10)
where C7 = KN t1 eωt1 and I8 = N
Z
t˜2
t˜1
kS(t˜2 − s)kds ≤ C8 (t˜2 − t˜1 ),
(3.11)
where C8 = KN t1 eωt1 . We use the inequalities (3.8), (3.9), (3.10) and (3.11)) in inequality (3.7) and get the following inequality k(Fx)(t˜2 ) − (Fx)(t˜1 )k ≤ L|t˜2 − t˜1 |,
(3.12)
where L ≥ C5 + C6 + C7 + C8 . Finally, if si ≥ t˜2 > t˜1 > ti , then we get k(Fx)(t˜2 ) − (Fx)(t˜1 )k
≤ LJ 1 .(t˜2 − t˜1 ). i
Summarizing, we see that Fx ∈ CL (J, X) for any x ∈ CL (J, X) and some L > 0. Next, we need to show that F : W → W. Now for t ∈ (si , ti+1 ] and x ∈ W, we have k(Fx)(t)k
− 2 kC(t − si )(Ji1 (si , x(t− i )))k + kS(t − si )(Ji (si , x(ti )))k Z t kS(t − s)f (s, x(s), x[h(x(s), s)])kds +
≤
si
Keω(t−si ) CJ 1 + K(t − si )eω(t−si ) CJ 2 i i Z t K(t − s)eω(t−s) ds. + N
≤
si
Hence, k(Fx)kP CB ≤
Ke
ωT
CJ 1 + KT e i
ωT
CJ 2 i
N KT ωT −Ωsi + e e . ω
Now for t ∈ [0, t1 ] and x ∈ W, we have Z
t
kS(t − s)kds Z t ≤ Keωt kx0 k + Kteωt ky0 k + N K(t − s)eω(t−s) ds.
k(Fx)(t)k ≤ kC(t)x0 k + kS(t)y0 k + N
0
0
9
(3.13)
Hence, k(Fx)kP CB ≤ KeωT kx0 k + KT eωT ky0 k +
N KT ωT e . ω
Similarly for t ∈ (ti , si ] and x ∈ W, we have k(Fx)kP CB ≤ e−Ωti CJ 1 . i
After summarizing the above inequalities, we get k(Fx)kP CB ≤ δ. Therefore F : W → W. For any x, y ∈ W, t ∈ (si , ti+1 ], i = 1, 2, ..., m, we have k(Fx)(t) − (Fy)(t)k
≤
− Keω(t−si ) LJ 1 kx(t− i ) − y(ti )k i
− LJ 2 kx(t− i ) − y(ti )k i Z t + KK1 (2 + LLh ) (t − s)eω(t−s) kx(s) − y(s)kds ω(t−si )
+ K(t − si )e
si
≤
Ke
ω(t−si )+Ωti
LJ 1 kx − ykP CB i
ω(t−si )+Ωti
+ K(t − si )e
+ KK1 (2 + LLh )
t
si
≤
(t − s)eω(t−s)+Ωs dskx − ykP CB
Keω(t−si )+Ωti LJ 1 kx − ykP CB i
ω(t−si )+Ωti
+ K(t − si )e +
LJ 2 kx − ykP CB i
Z
LJ 2 kx − ykP CB i
KK1 (2 + LLh (Ω − ω)
)teΩt
kx − ykP CB .
Hence, k(Fx)(t) − (Fy)(t)ke−Ωt
≤
Keω(t−si )+Ω(ti −t) LJ 1 kx − ykP CB i
+ K(t − si )eω(t−si )+Ω(ti −t) LJ 2 kx − ykP CB i
+ ≤ +
KK1 (2 + LLh )ti+1 kx − ykP CB (Ω − ω)
KeΩ(ti −si ) LJ 1 + Kti+1 eΩ(ti −si ) LJ 2 i
i
!
KK1 (2 + LLh )ti+1 kx − ykP CB . (Ω − ω)
For t ∈ [0, t1 ], we obtain k(Fx)(t) − (Fy)(t)k ≤ KK1 (2 + LLh ) 10
Z
0
t
(t − s)eω(t−s) kx(s) − y(s)kds
≤ KK1 (2 + LLh ) ≤ Hence,
Similarly, for t ∈ (ti , si ], i = 1, 2, ..., m, we have ≤
0
t
(t − s)eω(t−s)+Ωs dskx − ykP CB
KK1 (2 + LLh )teΩt kx − ykP CB . (Ω − ω)
k(Fx)(t) − (Fy)(t)ke−Ωt ≤
k(Fx)(t) − (Fy)(t)k
Z
KK1 (2 + LLh )t1 kx − ykP CB . (Ω − ω)
− LJ 1 Ke(ti −si−1 )ω LJ 1 kx(t− i−1 ) − y(ti−1 )k i
i
− + K(ti − si−1 )e(ti −si−1 )ω LJ 2 kx(t− i−1 ) − y(ti−1 )k i ! Z ti (ti − s)e(ti −s)ω kx(s) − y(s)kds + KK1 (2 + LLh ) si−1
≤
LJ 1 Ke(ti −si−1 )ω+Ωti−1 LJ 1 kx − ykP CB i
i
+ K(ti − si−1 )e(ti −si−1 )ω+Ωti−1 LJ 2 kx − ykP CB i ! Z ti (ti − s)e(ti −s)ω+Ωs dskx − ykP CB + KK1 (2 + LLh ) si−1
≤
LJ 1 Ke(ti −si−1 )ω+Ωti−1 LJ 1 kx − ykP CB i
i
+ K(ti − si−1 )e(ti −si−1 )ω+Ωti−1 LJ 2 kx − ykP CB i ! Ωt i KK1 (2 + LLh )ti e + kx − ykP CB . (Ω − ω) Therefore, we get k(Fx)(t) − (Fy)(t)ke−Ωt
≤
LJ 1 Ke(ti −si−1 )ω+(ti−1 −ti )Ω LJ 1 i
i
+ K(ti − si−1 )e(ti −si−1 )ω+(ti−1 −ti )Ω LJ 2 i ! KK1 (2 + LLh )ti kx − ykP CB + (Ω − ω) ≤ +
LJ 1 Ke(ti−1 −si−1 )Ω LJ 1 + Ksi e(ti−1 −si−1 )Ω LJ 2 i
i
!
KK1 (2 + LLh )ti kx − ykP CB . (Ω − ω) 11
i
After summarizing the above inequalities, we have the following k(Fx) − (Fy)kP CB ≤ LF kx − ykP CB , where LF = max
1≤i≤m
(
Ke
Ω(ti −si )
Ω(ti −si )
LJ 1 + Kti+1 e i
LJ 2 i
! KK1 (2 + LLh )ti+1 , + (Ω − ω)
KK1 (2 + LLh )t1 KK1 (2 + LLh )ti , LJ 1 Ke(ti−1 −si−1 )Ω LJ 1 + Ksi e(ti−1 −si−1 )Ω LJ 2 + i i i (Ω − ω) (Ω − ω)
!)
.
Hence, F is a strict contraction mapping for sufficiently large Ω > ω. Application of Banach fixed point theorem immediately gives a unique mild solution to the problem (1.5).
4
Nonlocal Problems
The nonlocal condition is a generalization of the classical initial condition. The study of nonlocal initial value problems are important because they appears in many physical systems. Byszewski [14] was the first author who studied the existence and uniqueness of mild solutions to the Cauchy problems with nonlocal conditions. In this section, we investigate the existence and uniqueness of mild solution (1.5) with nonlocal conditions. We consider the following nonlocal differential problem with deviated argument in a Banach space X : x00 (t) = Ax(t) + f (t, x(t), x[h(x(t), t)]), x(t) = Ji1 (t, x(t− i )), 0 2 x (t) = Ji (t, x(t− i )),
t ∈ (si , ti+1 ) i = 0, 1, ..., m,
t ∈ (ti , si ], i = 1, 2, ..., m, 0
(4.1)
t ∈ (ti , si ], i = 1, 2, ..., m,
x(0) = x0 + p(x), x (0) = y0 + q(x),
where x(t) be a state function, 0 = s0 < t1 < s1 < t2 , ..., tm < sm < tm+1 = T < ∞. − 2 The functions Ji1 (t, x(t− i )) and Ji (t, x(ti )) represent non-instantaneous impulses same as in system (1.5). A is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators (C(t))t∈R on X. The functions p(x) and q(x) will be suitably specified later. Definition 4.1. A function x ∈ CL (J, X) is called a mild solution of the impulsive problem (4.1) if it satisfies the following relations: x(0) = x0 + p(x), x0 (0) = y0 + q(x), the non-instantaneous impulse conditions x(t) = Ji1 (t, x(t− t ∈ (ti , si ], i = 1, 2, ..., m, i )), x0 (t) = Ji2 (t, x(t− )), t ∈ (ti , si ], i = 1, 2, ..., m i and x(t) is the solution of the following integral equations x(t)
=
C(t)(x0 + p(x)) + S(t)(y0 + q(x)) +
Z
0
12
t
S(t − s)f (s, x(s), x[h(x(s), s)])ds,
x(t)
t ∈ [0, t1 ],
− 2 C(t − si )(Ji1 (si , x(t− i ))) + S(t − si )(Ji (si , x(ti ))) Z t S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [si , ti+1 ], i = 1, 2, .., m. +
=
si
Further, we need assumptions on the functions p and q to show the existence and uniqueness of the solution for the problem (4.1) (A6) The functions p, q : C(J, X) → X are continuous and there exist positive constants cp and cq such that (i) kp(x1 ) − p(x2 )k ≤ cp ||x1 − x2 ||, (ii) ||q(x1 ) − q(x2 )|| ≤ cq ||x1 − x2 ||. Theorem 4.2. Let x0 ∈ D(A), y0 ∈ E. If all the assumptions (A1)-(A6) are satisfied, then the second order nonlocal problem (4.1) has a unique mild solution x ∈ CL (J, X) provided that Keωt1 cp + K
(eωt1 − 1) cq < 1. ω
Proof: By choosing (
N KT ωT −Ωsi e e , δ = max Ke CJ 1 + KT e CJ 2 + i i 1≤i≤m ω ) N KT ωT −Ωti ωT ωT e , e CJ 1 , Ke kx0 + p(x)k + KT e ky0 + q(x)k + i ω 0
ωT
ωT
we set W 0 = {x ∈ CL (J, X) : kxkP CB ≤ δ 0 }.
We define a map F : W 0 → W 0 given by (Fx)(t)
= +
− 2 Ji1 t, C(ti − si−1 )(Ji1 (si−1 , x(t− i−1 ))) + S(ti − si−1 )(Ji (si−1 , x(ti−1 )))
Z
si−1
(Fx)(t)
=
!
ti
S(ti − s)f (s, x(s), x[h(x(s), s)])ds , t ∈ (ti , si ] i = 1, 2, ..., m;
C(t)(x0 + p(x)) + S(t)(y0 + q(x)) +
Z
0
∀ t ∈ [0, t1 ]; 13
t
S(t − s)f (s, x(s), x[h(x(s), s)])ds,
− 2 C(t − si )(Ji1 (si , x(t− i ))) + S(t − si )(Ji (si , x(ti ))) Z t S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ (si , ti+1 ] i = 1, 2, ..., m. +
(Fx)(t)
=
si
We have, k(Fx) − (Fy)kP CB ≤ L0F kx − ykP CB ,
where L0F
= max
1≤i≤m
(
Ke
Ω(ti −si )
Ω(ti −si )
LJ 1 + Kti+1 e i
LJ 2 i
! KK1 (2 + LLh )ti+1 + , (Ω − ω)
(eωt1 − 1) KK1 (2 + LLh )t1 cq + , ω (Ω − ω) !) KK1 (2 + LLh )ti (ti−1 −si−1 )Ω . + Ksi e LJ 2 + i (Ω − ω)
Keωt1 cp + K LJ 1 Ke(ti−1 −si−1 )Ω LJ 1 i
i
Thus, F is a strict contraction mapping for sufficiently large Ω > ω. Application of Banach fixed point theorem immediately gives a unique mild solution to the problem (4.1). The proof of this theorem is the consequence of the theorem (3.1).
5
Ulam’s type stability
In this section, we show Ulam’s type stability for the system (1.5). Let > 0, ψ ≥ 0 and φ ∈ P C(J, R+ ) be the nondecreasing. We consider the following inequalities 00 ky (t) − Ay(t) − f (t, y(t), y[h(y(t), t)])k ≤ , t ∈ (si , ti+1 ) i = 0, 1, ..., m, ky(t) − Ji1 (t, y(t− t ∈ (ti , si ], i = 1, 2, ..., m, (5.1) i ))k ≤ , 0 ))k ≤ , t ∈ (t , s ], i = 1, 2, ..., m ky (t) − Ji2 (t, y(t− i i i
and
and
00 ky (t) − Ay(t) − f (t, y(t), y[h(y(t), t)])k ≤ φ(t), t ∈ (si , ti+1 ) i = 0, 1, ..., m, ky(t) − Ji1 (t, y(t− t ∈ (ti , si ], i = 1, 2, ..., m, i ))k ≤ ψ, 0 ky (t) − Ji2 (t, y(t− ))k ≤ ψ, t ∈ (ti , si ], i = 1, 2, ..., m i 00 ky (t) − Ay(t) − f (t, y(t), y[h(y(t), t)])k ≤ φ(t), t ∈ (si , ti+1 ) i = 0, 1, ..., m, t ∈ (ti , si ], i = 1, 2, ..., m. ky(t) − Ji1 (t, y(t− i ))k ≤ ψ, 0 ky (t) − Ji2 (t, y(t− ))k ≤ ψ, t ∈ (ti , si ], i = 1, 2, ..., m. i
Now, we take the vector space
2 m Z = CL (J, X) ∩m i=0 C ((si , ti+1 ), X) ∩i=0 C((si , ti+1 ), D(A)).
The following definitions are inspired by Wang et al. [2]. 14
(5.2)
(5.3)
Definition 5.1. The equation (1.5) is Ulam-Hyers stable with if there exists c(K1 Lh LJ m) > 0 such that for each > 0 and for each solution y ∈ Z of the inequality (5.1), there exists a mild solution x ∈ CL (J, X) of the equation (1.5) with ky(t) − x(t)k ≤ c(K1 Lh LJ m), t ∈ J.
(5.4)
Definition 5.2. The equation (1.5) is generalized Ulam-Hyers stable if there exists θK1 ,Lh ,LJ ,m ∈ C(R+ , R+ ), θ(0) = 0 such that for each > 0 and for each solution y ∈ Z of the inequality (5.1), there exists a mild solution x ∈ CL (J, X) of the equation (1.5) with ky(t) − x(t)k ≤ θK1 ,Lh ,LJ ,m , t ∈ J.
(5.5)
Definition 5.3. The equation (1.5) is Ulam-Hyers-Rassias stable with respect to (φ, ψ) if there exists c(K1 Lh LJ mφ) > 0 such that for each > 0 and for each solution y ∈ Z of the inequality (5.3), there exists a mild solution x ∈ CL (J, X) of the equation (1.5) with ky(t) − x(t)k ≤ c(K1 Lh LJ mφ)(ψ + φ(t)), t ∈ J.
(5.6)
Definition 5.4. The equation (1.5) is generalized Ulam-Hyers-Rassias stable with respect to (φ, ψ) if there exists c(K1 Lh LJ mφ) > 0 such that for each solution y ∈ Z of the inequality (5.2), there exists a mild solution x ∈ CL (J, X) of the equation (1.5) with ky(t) − x(t)k ≤ c(K1 Lh LJ mφ)(ψ + φ(t)), t ∈ J.
(5.7)
Remark 5.1 A function y ∈ Z is a solution of the inequality (5.3) if and only if there is G ∈ m 1 g1 ∈ ∩m i=1 C([ti , si ], X) and g2 ∈ ∩i=1 C ([ti , si ], X) such that:
2 m ∩m i=0 C ((si , ti+1 ), X)∩i=0 C((si , ti+1 ), D(A)),
m (a) kG(t)k ≤ φ(t), t ∈ ∩m i=0 (si , ti+1 ), kg1 (t)k ≤ ψ and kg2 (t)k ≤ ψ, t ∈ ∩i=0 [ti , si ];
(b) y 00 (t) = Ay(t) + f (t, y(t), y[h(y(t), t)]) + G(t), t ∈ (si , ti+1 ) i = 0, 1, ..., m; (c) y(t) = Ji1 (t, y(t− i )) + g1 (t), t ∈ (ti , si ], i = 1, 2, ..., m. (d) y 0 (t) = Ji2 (t, y(t− i )) + g2 (t), t ∈ (ti , si ], i = 1, 2, ..., m. Easily, we can have similar remarks for the inequalities (5.1) and (5.2). Remark 5.2 A function y ∈ Z is a solution of the inequality (5.3) then y is a solution of the following integral inequality ky(t) − Ji1 (t, y(t− t ∈ (ti , si ], i = 1, 2, ..., m; i ))k ≤ ψ, 0 (t) − J 2 (t, y(t− ))k ≤ ψ, ky t ∈ (ti , si ], i = 1, 2, ..., m; i i Rt ky(t) − C(t)x0 − S(t)y0 − 0 S(t − s)f (s, y(s), y[h(y(s), s)])dsk R t ω(t−s) − 1]φ(s)ds, t ∈ [0, t1 ]; ≤ K ω 0 [e (5.8) 1 (s , y(t− ))) − S(t − s )(J 2 (s , y(t− ))) ky(t) − C(t − s )(J i i i i i i i i R t − si S(t − s)f (s, y(s), y[h(y(s), s)])dsk R t ω(t−s) [eω(t−si ) − 1] + K − 1]φ(s)ds, ≤ ψKeω(t−si ) + ψK ω ω si [e t ∈ [si , ti+1 ], i = 1, 2, ..., m. 15
By Remark (5.1), we have 00 y (t) = Ay(t) + f (t, y(t), y[h(y(t), t)]) + G(t), t ∈ (si , ti+1 ) i = 0, 1, ..., m; y(t) = Ji1 (t, y(t− i )) + g1 (t), t ∈ (ti , si ], i = 1, 2, ..., m; 0 t ∈ (ti , si ], i = 1, 2, ..., m. y (t) = Ji2 (t, y(t− i )) + g2 (t),
(5.9)
The solution y ∈ Z with y(0) = x0 and y 0 (0) = y0 of the equation (5.9) is given by y(t) = Ji1 (t, y(t− t ∈ (ti , si ], i = 1, 2, ..., m; i )) + g1 (t), 0 (t) = J 2 (t, y(t− )) + g (t), y t ∈ (ti , si ], i = 1, 2, ..., m; 2 i i Rt y(t) = C(t)x0 + S(t)y0 + 0 S(t − s)[f (s, y(s), y[h(y(s), s)]) + G(s)]ds, t ∈ [0, t1 ]; (5.10) − 2 y(t) = C(t − si )((Ji1 (si , y(t− i ))) + g1 (si )) + S(t − si )((Ji (si , y(ti ))) + g2 (si )) + R t S(t − s)[f (s, y(s), y[h(y(s), s)]) + G(s)]ds, t ∈ [s , t ], i = 1, 2, ..., m. i i+1 si
Easily, we can have similar remarks for the solution of the inequalities (5.1) and (5.2). In order to discuss the stability of the problem (1.5), we need the following additional assumption:
(A7) Let φ ∈ C(J, R+ ) be a nondecreasing function. There exists cφ > 0 such that Z
0
t
φ(s)ds ≤ cφ φ(t), ∀ t ∈ J.
Lemma 5.1(Impulsive Gronwall inequality)(see Theorem 16.4, [15]). Let M0 = M ∪ {0}, where M = {1, ..., m} and the following inequality holds u(t) ≤ a(t) +
Z
t
b(s)u(s)ds +
0
X
0 0, k ∈ M. Then for t ∈ R+ , Z t k u(t) ≤ a(t)(1 + β) exp b(s)ds , t ∈ (tk , tk+1 ], k ∈ M0 , (5.12) 0
where β = supk∈M {βk } and t0 = 0. Theorem 5.5. Let x0 ∈ D(A), y0 ∈ E. If the assumptions (A1)- (A4) and (A7) are satisfied. Then, the equation (1.5) is Ulam-Hyers-Rassias stable with respect to (φ, ψ). Proof: Let y ∈ CL (J, D(A)) ∩ C 2 ((si , ti+1 ], X) be a solution of the inequality (5.3) and x is the unique mild solution of the problem (1.5) which is given by x(t) = Ji1 (t, x(t− t ∈ (ti , si ], i = 1, 2, ..., m; i )), 0 2 (t, x(t− )), x (t) = J t ∈ (ti , si ], i = 1, 2, ..., m; i i Rt x(t) = C(t)x0 + S(t)y0 + 0 S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [0, t1 ]; (5.13) 1 (s , x(t− ))) + S(t − s )(J 2 (s , x(t− ))) x(t) = C(t − s )(J i i i i i i i i + R t S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [s , t ], i = 1, 2, ..., m. i i+1 si 16
For t ∈ [si , ti+1 ], i = 1, 2, ..., m. By inequality (5.8), we have − 2 ky(t) − C(t − si )(Ji1 (si , y(t− i ))) − S(t − si )(Ji (si , y(ti ))) Z t S(t − s)f (s, y(s), y[h(y(s), s)])dsk − si
K ψK ω(t−si ) [e − 1] + ω ω Z t K ωT φ(s)ds + e ω 0
≤ ψKeω(t−si ) +
ψK ωT e ω K ωT e (ψ + cφ φ(t)). + ω
≤ ψKeωT + ≤ ψKeωT
Z
t
si
[eω(t−s) − 1]φ(s)ds
For t ∈ (ti , si ], i = 1, 2, ..., m, we have ky(t) − Ji1 (t, y(t− i ))k ≤ ψ. For t ∈ [0, t1 ], we have Z
t
ky(t) − C(t)x0 − S(t)y0 − S(t − s)f (s, y(s), y[h(y(s), s)])dsk 0 Z t K ≤ [eω(t−s) − 1]φ(s)ds ω 0 K ωT e cφ φ(t). ≤ ω Hence, for t ∈ [si , ti+1 ], i = 1, 2, ..., m, we have − 2 ky(t) − x(t)k ≤ ky(t) − C(t − si )(Ji1 (si , y(t− i ))) − S(t − si )(Ji (si , y(ti ))) Z t S(t − s)f (s, y(s), y[h(y(s), s)])dsk − si
− − 1 ωT 2 +KeωT kJi1 (si , y(t− kJi2 (si , y(t− i )) − Ji (si , x(ti ))k + KT e i )) − Ji (si , x(ti ))k Z t kf (s, y(s), y[h(y(s), s)]) − f (s, x(s), x[h(x(s), s)])dsk +KT eωT si
K ωT − e (ψ + cφ φ(t)) + (KeωT LJ 1 + KT eωT LJ 2 )k(y(t− i ) − (x(ti )k i i ω Z t ωT ky(s) − x(s)kds +K1 (2 + LLh )KT e 0 Z t K ωT ωT ky(s) − x(s)kds e [(2 + cφ )(ψ + φ(t))] + K1 (2 + LLh )KT e ≤ ω 0 i X − (KeωT LJ 1 + KT eωT LJ 2 )k(y(t− (5.14) + j ) − (x(tj )k. ≤ ψKeωT +
j
j
j=1
17
For t ∈ (ti , si ], i = 1, 2, ..., m, we have ky(t) − x(t)k
≤
− − 1 1 ky(t) − Ji1 (t, y(t− i ))k + kJi (t, y(ti )) − Ji (t, x(ti ))k
≤
ψ +
i X j=1
− LJ 1 k(y(t− j ) − (x(tj )k j
K ωT e [(2 + cφ )(ψ + φ(t))] ω Z t ky(s) − x(s)kds + K1 (2 + LLh )KT eωT
≤
0
+
i X j=1
− (KeωT LJ 1 + KT eωT LJ 2 )k(y(t− j ) − (x(tj )k. j
j
(5.15)
Now, for t ∈ [0, t1 ], we have ky(t) − x(t)k
Z t K ωT ωT ky(s) − x(s)kds e cφ φ(t) + K1 (2 + LLh )KT e ≤ ω 0 K ωT ≤ e [(2 + cφ )(ψ + φ(t))] ω Z t ky(s) − x(s)kds + K1 (2 + LLh )KT eωT 0
+
i X j=1
− (KeωT LJ 1 + KT eωT LJ 2 )k(y(t− j ) − (x(tj )k. j
j
(5.16)
We observe that inequalities (5.14), (5.15) and (5.16) give together an impulsive Gronwall inequality of a form of (5.11) on J. Therefore, we can apply impulsive Gronwall inequality (5.12) for t ∈ J, since t ∈ (ti , ti+1 ] for some i ∈ M0 . Consequently, we have K ωT ωT e (2 + cφ )(1 + KeωT LJ )i eK1 (2+LLh )KT e t (ψ + φ(t)) ω K ωT ωT e (2 + cφ )(1 + KeωT LJ )m eK1 (2+LLh )KT e T (ψ + φ(t)) ≤ ω ≤ c(K1 Lh LJ mφ)(ψ + φ(t)),
ky(t) − x(t)k ≤
for any t ∈ J, where LJ = supi∈M {LJ 1 + T LJ 2 } and c(K1 Lh LJ mφ) is a constant depending i i on K1 , Lh , LJ , m, φ. Hence, the equation (1.5) is Ulam-Hyers-Rassias stable with respect to (φ, ψ). Theorem 5.6. If the assumptions (A1)- (A4) and (A7) are satisfied. Then, the equation (1.5) is generalized Ulam-Hyers-Rassias stable with respect to (φ, ψ). Proof: It can be easily proved by applying same procedure of the theorem (5.5) and taking inequality (5.2). 18
Theorem 5.7. If the assumptions (A1)- (A4) and (A7) are satisfied. Then, the equation (1.5) is Ulam-Hyers stable. Proof: It can be easily proved by applying same procedure of the theorem (5.5) and taking inequality (5.1).
6
Application
Example 1: Let X = L2 (0, π). We consider the following partial differential equations with deviated argument ∂tt Z(t, y) = ∂yy Z(t, y) + f2 (y, Z(h(t), y)) + f3 (t, y, Z(t, y)), y ∈ (0, π), t ∈ (2i, 2i + 1], i ∈ {0} ∪ N, Z(t, 0) = Z(t, π) = 0, t ∈ [0, T ], 0 < T < ∞, Z(0, y) = x0 , y ∈ (0, π), (6.1) ∂ Z(0, y) = y , y ∈ (0, π), t 0 Z(t)(y) = (sin it)Z((2i − 1)− , y), y ∈ (0, π), t ∈ (2i − 1, 2i], i ∈ N, ∂t Z(t)(y) = (i cos it)Z((2i − 1)− , y), y ∈ (0, π), t ∈ (2i − 1, 2i], i ∈ N, where 0 = s0 < t1 < s1 < t2 , ..., tm < sm < tm+1 = T < ∞ with ti = 2i − 1, si = 2i and Z y ¯ f3 (t, y, Z(t, y)) = K(y, s)(a1 |Z(t, s)| + b1 |Z(t, s)|))ds. 0
We assume that a1 , b1 ≥ 0, (a1 , b1 ) 6= (0, 0), h : J1 → [0, T ] is locally H¨older continuous in t ¯ : [0, π] × [0, π] → R. with h(0) = 0 and K We define an operator A, as follows, Ax = x00
with D(A) = {x ∈ X : x00 ∈ X and x(0) = x(π) = 0}.
(6.2)
Here, clearly the operator A is the infinitesimal generator of a strongly continuous cosine family of operators on X. A has infinite series representation Ax =
∞ X
n=1
−n2 (x, xn )xn ,
x ∈ D(A),
p where xn (s) = 2/π sin ns, n = 1, 2, 3... is the orthonormal set of eigenfunctions of A. Moreover, the operator A is the infinitesimal generator of a strongly continuous cosine family C(t)t∈R on X which is given by ∞ X cos nt(x, xn )xn , x ∈ X, C(t)x = n=1
and the associated sine family S(t)t∈R on X which is given by S(t)x =
∞ X 1 sin nt(x, xn )xn , n
n=1
19
x ∈ X.
The equation (6.1) can be reformulated as the following abstract differential equation in X: 00 x (t) = Ax(t) + f (t, x(t), x[h(x(t), t)]), t ∈ (si , ti+1 ), i ∈ {0} ∪ N, x(t) = Ji1 (t, x(t− t ∈ (ti , si ], i ∈ N, i )), (6.3) 0 (t) = J 2 (t, x(t− )), t ∈ (ti , si ], i ∈ N, x i i 0 x(0) = x0 , x (0) = y0 ,
where x(t) = Z(t, .), that is x(t)(y) = Z(t, y), y ∈ (0, π). Functions Ji1 (t, x(t− i )) = (sin it)Z((2i− − − − 2 1) , y) and Ji (t, x(ti )) = i(cos it)Z((2i − 1) , y) represent noninstantaneous impulses during intervals (ti , si ]. The operator A is same as in equation (6.2). The function f : J1 × X × X → X, is given by f (t, ψ, ξ)(y) = f2 (y, ξ) + f3 (t, y, ψ), where f2 : [0, π] × X → H01 (0, π) is given by Z f2 (y, ξ) =
y
¯ K(y, x)ξ(x)dx,
0
and kf3 (t, y, ψ)k ≤ V (y, t)(1 + kψkH 2 (0,π) ) with V (., t) ∈ X and V is continuous in its second argument. For more details see [7, 8]. Thus, the theorem (3.1) can be applied to the problem (6.1). We can choose the functions p(x) and q(x) as given below p(x) = q(x) =
n X
αk x(tk ), tk ∈ J for all k = 1, 2, 3, · · · , n,
k n X k
βk x(tk ), tk ∈ J for all k = 1, 2, 3, · · · , n,
where αk and βk are constants. Example 2: We consider particular linear case of the abstract differential equation (6.3) in the space X = R. A forced string equation 00 x (t) + a1 x(t) + a2−sin x(c1 t) = g(t), t ∈ (si , ti+1 ) i = 0, 1, ..., m, x(t) = a3 tanh(x(ti ))r(t), t ∈ (ti , si ], i = 1, 2, ..., m, (6.4) 0 (t) = a tanh(x(t− ))r 0 (t), x t ∈ (ti , si ], i = 1, 2, ..., m, 3 i x(0) = x0 , x0 (0) = y0 ,
where a1 ∈ R+ , a2 , a3 ∈ R, c1 ∈ (0, 1], g ∈ C(J1 , R) and r ∈ C 1 (J2 , R) for J2 = ∪m i=1 Ii . We define A, as follows Ax = −a1 x with D(A) = R.
Here, clearly the value −a1 behave like infinitesimal generator of a strongly continuous cosine √ √ family C(t) = cos a1 t. The associated sine family is given by S(t) = √1a1 sin a1 t. Deviated 20
argument in the abstract differential equation (6.3) is represented by the term a2 x(c1 t) of the dif− 0 ferential equation (6.4). Noninstantaneous impulses a3 tanh(x(t− i ))r(t) and a3 tanh(x(ti ))r (t) are created when bob of the string is extremely pushed on each intervals (ti , si ]. Example 3: We generalize the above example to consider a coupled system of strings or pendulums x00n (t) + an1 sin xn (t) + an2 sin xn (cn t) = bn1 xn−1 (t) + bn2 xn+1 (t) + gn (t), xn (t) = an3 tanh(xn (t− i ))rn (t), 0 x0n (t) = an3 tanh(xn (t− i ))rn (t), xn (0) = xn0 , x0n (0) = yn0 ,
i = 0, 1, ..., m, n ∈ Z,
t ∈ (ti , si ], i = 1, 2, ..., m,
t ∈ (si , ti+1 ), (6.5)
t ∈ (ti , si ], i = 1, 2, ..., m,
where an1 , an2 , an3 , bn1 , bn2 ∈ R, cn ∈ (0, 1], gn ∈ C(J1 , R) and rn ∈ C 1 (J2 , R). Moreover we suppose supn |ank | < ∞, k = 1, 2, 3, supn (|bn1 | + |bn2 |) < ∞ and supn (kgn k + ||rn k + krn0 k) < ∞. Then we consider (6.5) on `∞ and use Exercise 1 on p. 39 from [10]. The lattice ODE (6.5) is a generalization of the discrete sine-Gordon equation [16] and xn (cn t) represent pantograph-like terms [17].
7
Conclusion
The research presented in this paper focuses on the existence, uniqueness and stability of solutions to the impulsive systems represented by second order nonlinear differential equations with noninstantaneous impulses and deviated argument. We used strongly continuous cosine family of bounded linear operators and Banach’s fixed point theorem to get the existence and uniqueness of the solutions. Moreover, Ulam’s type stability is established by using impulsive Gronwall inequality.
References [1] E. Hern´andez, D. O’Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641–1649. [2] J. Wang, M. Feˇckan, A general class of impulsive evolution equations, Nonlinear Anal., 46 (2015),915–933.
Topol. Methods
[3] D. N. Chalishajar, Controllability of damped second order initial value problem for a class of differential inclusions with nonlocal conditions on non compact intervals, Nonlinear functional analysis and applications (Korea), 14 (1) (2009). [4] D. N. Pandey, S. Das and N. Sukavanam, Existence of solution for a second-order neutral differential equation with state dependent delay and non-instantaneous impulses, International Journal of Nonlinear Science, vol. 18.(2014)145–155.
21
[5] F. S. Acharya, Controllability of Second order semilinear impulsive partial neutral functional differential equations with infinite delay , International Journal of Mathematical Sciences and Applications. 3 (1)(2013) 207-218. [6] G. Arthi, K. Balachandran, Controllability of second order impulsive evolution systems with infinite delay,Nonlinear Anal. : Hybrid Systems, 11 (2014), 139–153. [7] R. Sakthivel, N. I. Mahmudov and J. H. Kim, On controllability of second order nonlinear impulsive differential systems, Nonlinear Anal., 71 (2009), 45–52. [8] C. G. Gal, Nonlinear abstract differential equations with deviated argument, J. Math. Anal and Appl.,(2007), 177-189. [9] M. Muslim, Avadhesh Kumar and R. P. Agarwal, Exact and trajectory controllability of second order nonlinear impulsive systems with deviated argument, Functional Differential Equations, vol. 23 (2016),No. 1-2, 27 - 41. [10] H. O. Fattorini, Second order linear differential equations in Banach spaces, North Holland Mathematics Studies 108, Elsevier Science, North Holland, 1985. [11] C. C. Travis, G. F. Webb, Compactness, regularity and uniform continuity properties of strongly continuous cosine family, Houstan Journal of Mathematics, 3(1977), 555–567. [12] C. C. Travis, G. F. Webb, Cosine families and abstract nonlinear second order differential equations Acta mathematica Academia Scientia Hungaricae., 32 (1978), 76–96. [13] E. Hern´andez, M. A. McKibben, Some comments on: Existence of solutions of abstract nonlinear second-order neutral functional integrodifferential equations, Computers with Mathematics with Applications, (2005), 1–15. [14] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear nonlocal Cauchy problem, J. Math. Anal and Appl.,162 (1991), 494 - 505. [15] D. D. Bainov, P. S. Simeonov, Integral inequalities and applications, Kluwer Academic Publishers, Dordrecht, 1992. [16] A. Scott, Nonlinear Sciences: Emergence and Dynamics of Coherent Structures, Oxford Univ. Press, 2nd edition, 2003. [17] C. Derfel and A. Iserles, The pantograph equation in the complex plane, J. Math. Anal. Appl. 213 (1997), 117 - 132.
22
Existence, Uniqueness and Stability of Solutions to Second Order Nonlinear Differential Equations with Non-instantaneous Impulses
Abstract: In this paper, we consider a non-instantaneous impulsive system represented by second order nonlinear differential equation with deviated argument in a Banach space X. We used the strongly continuous cosine family of linear operators and Banach fixed point method to study the existence and uniqueness of the solution of the non-instantaneous impulsive system. Also, we study the existence and uniqueness of the solution of the nonlocal problem and stability of the non-instantaneous impulsive system. Finally, we give examples to illustrate the application of these abstract results. Key Words: Non-instantaneous impulses, Deviated argument, Banach fixed point theorem. AMS Subject Classification: 34K30, 34K45, 34G20, 47D09.
1
Introduction
The dynamics of many evolving processes are subject to abrupt changes, such as shocks, harvesting and natural disaster. These phenomena involve short term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. Sometimes time abrupt changes may stay for time intervals such impulses are called non-instantaneous impulses. The importance of the study of non-instantaneous impulsive differential equations lies in its diverse fields of applications such as in the theory of stage by stage rocket combustion, maintaining hemodynamical equilibrium etc. A very well known application of non-instantaneous impulses is the introduction of insulin in the bloodstream which is abrupt change and the consequent absorption which is a gradual process as it remains active for a finite interval of time. The theory of impulsive differential equations has found enormous applications in realistic mathematical modeling of a wide range of practical situations. It has emerged as an important area of research such as modeling of impulsive problems in physics, population dynamics, ecology, biological systems, biotechnology and so forth. Recently, Hern´andez et. al. [1] studied mild and classical solutions for the impulsive differ-
2
ential equation with non-instantaneous impulses which is of the form 0 x (t) = Ax(t) + f (t, x(t)), t ∈ (si , ti+1 ] i = 0, 1, ..., m, x(t) = gi (t, x(t)), t ∈ (ti , si ], i = 1, 2, ..., m, x(0) = x0 ∈ X
(1.1)
In [2] Wang and Feˇckan have a remark on the conditions in (1.1): x(t) = gi (t, x(t)),
t ∈ (ti , si ], i = 1, 2, ..., m,
(1.2)
where gi ∈ C([ti , si ] × X, X) and there are positive constants Lgi , i = 1, ..., m such that kgi (t, x1 ) − gi (t, x2 )k ≤ Lgi kx1 − x2 k, ∀ x1 , x2 ∈ X, t ∈ [ti , si ] It follows from Theorem 2.1 in [1] that max{Lgi : i = 1, ..., m} < 1 is a necessary condition. Then Banach fixed point theorem gives a unique zi ∈ C([ti , si ], X) so that z = gi (t, z) if and only if z = zi (t). So (1.2) is equivalent to x(t) = zi (t),
t ∈ (ti , si ], i = 1, 2, ..., m,
(1.3)
which does not depend on the state x(.). Thus, it is necessary to modify (1.2) and consider the condition x(t) = gi (t, x(t− i )),
t ∈ (ti , si ], i = 1, 2, ..., m.
(1.4)
− + Of course then x(t+ i ) = gi (ti , x(ti )), i = 1, 2, ..., m. The symbols x(ti ) := lim→0+ x(ti + ) and x(t− i ) := lim→0− x(ti + ) represent the right and left limits of x(t) at t = ti respectively. Motivated by above remark, Wang and Feˇckan [2] have shown existence, uniqueness and stability of solutions of such general class of impulsive differential equations. In this paper, we continue in this direction to study the second order nonlinear differential equation with non-instantaneous impulses and deviated argument in a Banach Space X 00 x (t) = Ax(t) + f (t, x(t), x[h(x(t), t)]), t ∈ (si , ti+1 ), i = 0, 1, ..., m, t ∈ (ti , si ], i = 1, 2, ..., m, x(t) = Ji1 (t, x(t− i )), (1.5) 0 (t) = J 2 (t, x(t− )), x t ∈ (ti , si ], i = 1, 2, ..., m, i i 0 x(0) = x0 , x (0) = y0 ,
where x(t) be a state function, 0 = s0 = t0 < t1 < s1 < t2 , ..., tm < sm < tm+1 = T < ∞. We consider in (1.5) that x ∈ C((ti , ti+1 ], X), i = 0, 1, ..., m and there exist x(t− i ) and − 1 (t, x(t− )) and J 2 (t, x(t− )) represent ) = x(t ). The functions J ), i = 1, 2, ..., m with x(t x(t+ i i i i i i i noninstantaneous impulses during the intervals (ti , si ] , i = 1, 2, ..., m, so impulses at t− i have some duration, namely on intervals (ti , si ]. A is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators (C(t))t∈R on X. Ji1 , Ji2 , h and f are suitable functions and they will be specified later. Many partial differential equations that arise in several problems connected with the transverse motion of an extensible beam, the vibration of hinged bars, and many other physical phenomena can be formulated as the second order abstract differential equations in the infinite 3
dimensional spaces. A useful tool for the study of second-order abstract differential equations is the theory of strongly continuous cosine families of operators. Existence and uniqueness of the solution of second-order nonlinear systems and controllability of these systems in Banach spaces have been investigated extensively by many authors [3, 4, 5, 6, 7]. In certain real world problems, delay depends not only on the time but also on the unknown quantity. The differential equations with deviated arguments are generalization of delay differential equations. Gal C. G [8] has consider a nonlinear abstract differential equations with deviated arguments and study the existence and uniqueness of solutions. Recently, Muslim et. al. [9] studied exact and trajectory controllability of second order impulsive nonlinear systems with deviated argument. There are only few papers discussing the second order differential equations with deviated arguments in infinite dimensional spaces. As per my knowledge, there is no paper discussing the existence, uniqueness and stability of the mild solution of the second order differential equation with non-instantaneous impulses and deviated argument in Banach space. In order to fill this gap, we consider a nonlinear second order differential equation with deviated argument. Moreover, the study of second order differential equations with noninstantaneous impulses has not only mathematical significance but also it has applications such as harmonic oscillator with impulses and forced string equation, which we present in examples.
2
Preliminaries and Assumptions
We briefly review definitions and some useful properties of the theory of cosine family. Definition 2.1. (see, [12]) A one parameter family (C(t))t∈R of bounded linear operators mapping the Banach space X into itself is called a strongly continuous cosine family if and only if (i) C(s + t) + C(s − t) = 2C(s)C(t) for all s, t ∈ R, (ii) C(0) = I, (iii) C(t)x is continuous in t on R for each fixed point x ∈ X. (S(t))t∈R is the sine function associated to the strongly continuous cosine family, (C(t))t∈R which is defined by Z t S(t)x = C(s)x ds, x ∈ X, t ∈ R. 0
D(A) be the domain of the operator A which is defined by
D(A) = {x ∈ X : C(t)x is twice continuously differentiable in t}. D(A) is the Banach space endowed with the graph norm kxkA = kxk + kAxk for all x ∈ D(A). We define a set E = {x ∈ X : C(t)x is once continuously differentiable in t} which is a Banach space endowed with norm kxkE = kxk + sup0≤t≤1 kAS(t)xk for all x ∈ E. With the help of C(t) and S(t), we define a operator valued function C(t) S(t) ¯ h(t) = . AS(t) C(t) 4
¯ Operator valued function h(t) is a strongly continuous group of bounded linear operators on the space E × X generated by the operator 0 I ¯ A = A 0 defined on D(A) × E. It follows that AS(t) : E → X is a bounded linear operator and that AS(t)x → 0 as t → 0, for each x ∈ E. If x : [0, ∞) → X is locally integrable function then Z t y(t) = S(t − s)x(s)ds 0
defines an E valued continuous function which is a consequence of the fact that " R # Z t t S(t − s)x(s)ds 0 ¯ − s) h(t ds = R 0t x(s) 0 0 C(t − s)x(s)ds
defines an (E × X) valued continuous function. Proposition 2.1:(see, [12]) Let (C(t))t∈R be a strongly continuous cosine family in X. The following are true: (i) there exist constants K ≥ 1 and ω ≥ 0 such that |C(t)| ≤ Keω|t| for all t ∈ R. Rt (ii) |S(t2 ) − S(t1 )| ≤ K| t12 eω|s| ds| for all t1 , t2 ∈ R.
For more details on cosine family theory, we refer to Travis, Webb & Fattorini [10, 11, 12]. Let P C([0, T ], X) be the space of piecewise continuous functions. P C([0, T ], X) = {x : J = [0, T ] → X : x ∈ C((tk , tk+1 ], X), k = 0, 1, ..., m and there exist + − x(t− k ) and x(tk ), k = 1, 2, ..., m with x(tk ) = x(tk )}. It can be seen easily that P C([0, T ], X) for all t ∈ [0, T ], is a Banach space endowed with the supremum norm, kψkP CB := sup{kψ(η)ke−Ωη ,
0 ≤ η ≤ t}
for some Ω > 0. We set, CL (J, X) = {y ∈ P C([0, T ], X) : ky(t)−y(s)k ≤ L|t−s|, ∀ t, s ∈ [0, T ]}, where L is a suitable positive constant. Clearly CL (J, X) is a Banach space endowed with P CB norm. In order to prove the existence, uniqueness and stability of the solution for the problem (1.5), we need the following assumptions: (A1) A be the infinitesimal generator of a strongly continuous cosine family, (C(t))t∈R of bounded linear operators. S (A2) f : J1 × X × X → X, J1 = m i=0 [si , ti+1 ] is a continuous function and there exist positive constants K1 and K2 such that kf (t, x1 , y1 ) − f (t, x2 , y2 )k ≤ K1 (kx1 − x2 k + ky1 − y2 k) for every x1 , x2 , y1 , y2 ∈ X, t ∈ J1 . Also there exists a positive constant N such that kf (t, x, y)k ≤ N, ∀ t ∈ J1 and x, y ∈ X. 5
(A3) h : X × J1 → J is continuous and there exists a positive constant Lh such that |h(x1 , s) − h(x2 , s)| ≤ Lh kx1 − x2 k, ∀ x1 , x2 ∈ X, t ∈ J1 and it holds h(., 0) = 0. (A4) Jil ∈ C(Ii × X, X), Ii = [ti , si ] and there are positive constants LJ l , i = 1, 2, ..., m, l = i 1, 2, such that kJil (t, x1 ) − Jil (s, x2 )k ≤ LJ l (|t − s| + kx1 − x2 k), ∀ t, s ∈ Ii and x1 , x2 ∈ X. i
(A5) There exist positive constants CJ 1 and CJ 2 , i = 1, 2, ..., m such that i
i
kJi1 (t, x)k ≤ CJ 1 and kJi2 (t, x)k ≤ CJ 2 , ∀ t ∈ Ii and x ∈ X. i
i
In the following definition, we introduce the concept of mild solution for the problem (1.5). Definition 2.2. A function x ∈ CL (J, X) is called a mild solution of the impulsive problem (1.5) if it satisfies the following relations: x(0) = x0 , x0 (0) = y0 , the non-instantaneous impulse conditions x(t) = Ji1 (t, x(t− t ∈ (ti , si ], i = 1, 2, ..., m, i )), − 0 2 x (t) = Ji (t, x(ti )), t ∈ (ti , si ], i = 1, 2, ..., m and x(t) is the solution of the following integral equations x(t) x(t)
=
C(t)x0 + S(t)y0 +
Z
t
S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [0, t1 ],
0 si )(Ji1 (si , x(t− i )))
+ S(t − si )(Ji2 (si , x(t− C(t − i ))) Z t S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [si , ti+1 ], i = 1, 2, .., m. +
=
si
3
Existence and Uniqueness Result
Theorem 3.1. Let x0 ∈ D(A), y0 ∈ E. If all the assumptions (A1)-(A5) are satisfied, then the second order problem (1.5) has a unique mild solution x ∈ CL (J, X). Proof: Since AS(t) is a bounded linear operator therefore, we set ρ = supt∈J kAS(t)k. For more details on kAS(t)k, we refer [4, 7, 13]. By choosing ( N KT ωT −Ωsi ωT ωT e , e δ = max Ke CJ 1 + KT e CJ 2 + i i 1≤i≤m ω ) N KT KeωT kx0 k + KT eωT ky0 k + eωT e−Ωti CJ 1 , i ω 6
we set W = {x ∈ CL (J, X) : kxkP CB ≤ δ}. We define a map F : W → W given by (Fx)(t)
− 2 Ji1 t, C(ti − si−1 )(Ji1 (si−1 , x(t− i−1 ))) + S(ti − si−1 )(Ji (si−1 , x(ti−1 )))
= +
Z
!
ti
si−1
S(ti − s)f (s, x(s), x[h(x(s), s)])ds , t ∈ (ti , si ] i = 1, 2, ..., m;
(Fx)(t) = C(t)x0 + S(t)y0 +
Z
t
S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [0, t1 ];
0
(Fx)(t)
− 2 C(t − si )(Ji1 (si , x(t− i ))) + S(t − si )(Ji (si , x(ti ))) Z t + S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ (si , ti+1 ] i = 1, 2, ..., m.
=
si
First, we need to show that Fx ∈ CL (J, X) for any x ∈ CL (J, X) and some L > 0. If ti+1 ≥ t˜2 > t˜1 > si , then we get k(Fx)(t˜2 ) − (Fx)(t˜1 )k
k(C(t˜2 − si ) − C(t˜1 − si ))(Ji1 (si , x(t− i )))k − 2 + k(S(t˜2 − si ) − S(t˜1 − si ))(Ji (si , x(ti )))k Z t˜1 k(S(t˜2 − s) − S(t˜1 − s))kds + N
≤
si t˜2
+ N ≤
Z
t˜1
kS(t˜2 − s)kds
I1 + I2 + I3 + I4 .
(3.1)
We have, I1 = k(C(t˜2 − si ) − C(t˜1 − si ))(Ji1 (si , x(t− i )))k Z t˜2 −si = k AS(τ )(Ji1 (si , x(t− i )))dτ k t˜1 −si
≤ C1 (t˜2 − t˜1 ),
(3.2)
where C1 = ρCJ 1 . i Similarlly, we have I2 = k(S(t˜2 − si ) − S(t˜1 − si ))(Ji2 (si , x(t− i )))k Z t˜2 −si eωτ (Ji2 (si , x(t− = kK i )))dτ k t˜1 −si
≤ C2 (t˜2 − t˜1 ),
7
(3.3)
where C2 = Keωti+1 CJ 2 . i Similarly, we calculate third and fourth part of inequality (3.1) as follows I3 = N ≤ N
Z
t˜1
si Z t˜1 si
k(S(t˜2 − s) − S(t˜1 − s))kds kK
t˜2 −s
Z
≤ C3 (t˜2 − t˜1 ),
t˜1 −s
eωτ dτ kds (3.4)
where C3 = KN ti+1 eωti+1 and I4 = N
Z
t˜2
t˜1
kS(t˜2 − s)kds ≤ C4 (t˜2 − t¯1 ),
(3.5)
where C4 = KN ti+1 eωti+1 . We use the inequalities (3.2), (3.3), (3.4) and (3.5)) in inequality (3.1) and get the following inequality k(Fx)(t˜2 ) − (Fx)(t˜1 )k ≤ L|t˜2 − t˜1 |,
(3.6)
where L ≥ C1 + C2 + C3 + C4 . If t1 ≥ t˜2 > t˜1 ≥ 0, then we get k(Fx)(t˜2 ) − (Fx)(t˜1 )k
k(C(t˜2 ) − C(t˜1 ))x0 k + k(S(t˜2 ) − S(t˜1 ))y0 k Z t˜1 k(S(t˜2 − s) − S(t˜1 − s))kds + N
≤
0
+ N ≤
Z
t˜2
t˜1
kS(t¯2 − s)kds
I5 + I6 + I7 + I8 .
(3.7)
We have, I5 = k(C(t˜2 ) − C(t˜1 ))x0 k
= k
Z
t˜2
AS(τ )x0 dτ k
t˜1
≤ C5 (t˜2 − t˜1 ),
(3.8)
where C5 = ρkx0 k. Similarly, we have I6 = k(S(t˜2 ) − S(t˜1 ))y0 k
=
kK
Z
t˜2
t˜1
eωτ y0 dτ k
≤ C6 (t˜2 − t˜1 ), 8
(3.9)
where C6 = Keωt1 ky0 k. Similarly, we calculate third and fourth part of inequality (3.7) as follows I7 = N
t˜1
Z
0
≤ N
Z
k(S(t˜2 − s) − S(t˜1 − s))kds
t˜1
0
kK
Z
≤ C7 (t˜2 − t˜1 ),
t˜2 −s
t˜1 −s
eωτ dτ kds (3.10)
where C7 = KN t1 eωt1 and I8 = N
Z
t˜2
t˜1
kS(t˜2 − s)kds ≤ C8 (t˜2 − t˜1 ),
(3.11)
where C8 = KN t1 eωt1 . We use the inequalities (3.8), (3.9), (3.10) and (3.11)) in inequality (3.7) and get the following inequality k(Fx)(t˜2 ) − (Fx)(t˜1 )k ≤ L|t˜2 − t˜1 |,
(3.12)
where L ≥ C5 + C6 + C7 + C8 . Finally, if si ≥ t˜2 > t˜1 > ti , then we get k(Fx)(t˜2 ) − (Fx)(t˜1 )k
≤ LJ 1 .(t˜2 − t˜1 ). i
Summarizing, we see that Fx ∈ CL (J, X) for any x ∈ CL (J, X) and some L > 0. Next, we need to show that F : W → W. Now for t ∈ (si , ti+1 ] and x ∈ W, we have k(Fx)(t)k
− 2 kC(t − si )(Ji1 (si , x(t− i )))k + kS(t − si )(Ji (si , x(ti )))k Z t kS(t − s)f (s, x(s), x[h(x(s), s)])kds +
≤
si
Keω(t−si ) CJ 1 + K(t − si )eω(t−si ) CJ 2 i i Z t K(t − s)eω(t−s) ds. + N
≤
si
Hence, k(Fx)kP CB ≤
Ke
ωT
CJ 1 + KT e i
ωT
CJ 2 i
N KT ωT −Ωsi + e e . ω
Now for t ∈ [0, t1 ] and x ∈ W, we have Z
t
kS(t − s)kds Z t ≤ Keωt kx0 k + Kteωt ky0 k + N K(t − s)eω(t−s) ds.
k(Fx)(t)k ≤ kC(t)x0 k + kS(t)y0 k + N
0
0
9
(3.13)
Hence, k(Fx)kP CB ≤ KeωT kx0 k + KT eωT ky0 k +
N KT ωT e . ω
Similarly for t ∈ (ti , si ] and x ∈ W, we have k(Fx)kP CB ≤ e−Ωti CJ 1 . i
After summarizing the above inequalities, we get k(Fx)kP CB ≤ δ. Therefore F : W → W. For any x, y ∈ W, t ∈ (si , ti+1 ], i = 1, 2, ..., m, we have k(Fx)(t) − (Fy)(t)k
≤
− Keω(t−si ) LJ 1 kx(t− i ) − y(ti )k i
− LJ 2 kx(t− i ) − y(ti )k i Z t + KK1 (2 + LLh ) (t − s)eω(t−s) kx(s) − y(s)kds ω(t−si )
+ K(t − si )e
si
≤
Ke
ω(t−si )+Ωti
LJ 1 kx − ykP CB i
ω(t−si )+Ωti
+ K(t − si )e
+ KK1 (2 + LLh )
t
si
≤
(t − s)eω(t−s)+Ωs dskx − ykP CB
Keω(t−si )+Ωti LJ 1 kx − ykP CB i
ω(t−si )+Ωti
+ K(t − si )e +
LJ 2 kx − ykP CB i
Z
LJ 2 kx − ykP CB i
KK1 (2 + LLh (Ω − ω)
)teΩt
kx − ykP CB .
Hence, k(Fx)(t) − (Fy)(t)ke−Ωt
≤
Keω(t−si )+Ω(ti −t) LJ 1 kx − ykP CB i
+ K(t − si )eω(t−si )+Ω(ti −t) LJ 2 kx − ykP CB i
+ ≤ +
KK1 (2 + LLh )ti+1 kx − ykP CB (Ω − ω)
KeΩ(ti −si ) LJ 1 + Kti+1 eΩ(ti −si ) LJ 2 i
i
!
KK1 (2 + LLh )ti+1 kx − ykP CB . (Ω − ω)
For t ∈ [0, t1 ], we obtain k(Fx)(t) − (Fy)(t)k ≤ KK1 (2 + LLh ) 10
Z
0
t
(t − s)eω(t−s) kx(s) − y(s)kds
≤ KK1 (2 + LLh ) ≤ Hence,
Similarly, for t ∈ (ti , si ], i = 1, 2, ..., m, we have ≤
0
t
(t − s)eω(t−s)+Ωs dskx − ykP CB
KK1 (2 + LLh )teΩt kx − ykP CB . (Ω − ω)
k(Fx)(t) − (Fy)(t)ke−Ωt ≤
k(Fx)(t) − (Fy)(t)k
Z
KK1 (2 + LLh )t1 kx − ykP CB . (Ω − ω)
− LJ 1 Ke(ti −si−1 )ω LJ 1 kx(t− i−1 ) − y(ti−1 )k i
i
− + K(ti − si−1 )e(ti −si−1 )ω LJ 2 kx(t− i−1 ) − y(ti−1 )k i ! Z ti (ti − s)e(ti −s)ω kx(s) − y(s)kds + KK1 (2 + LLh ) si−1
≤
LJ 1 Ke(ti −si−1 )ω+Ωti−1 LJ 1 kx − ykP CB i
i
+ K(ti − si−1 )e(ti −si−1 )ω+Ωti−1 LJ 2 kx − ykP CB i ! Z ti (ti − s)e(ti −s)ω+Ωs dskx − ykP CB + KK1 (2 + LLh ) si−1
≤
LJ 1 Ke(ti −si−1 )ω+Ωti−1 LJ 1 kx − ykP CB i
i
+ K(ti − si−1 )e(ti −si−1 )ω+Ωti−1 LJ 2 kx − ykP CB i ! Ωt i KK1 (2 + LLh )ti e + kx − ykP CB . (Ω − ω) Therefore, we get k(Fx)(t) − (Fy)(t)ke−Ωt
≤
LJ 1 Ke(ti −si−1 )ω+(ti−1 −ti )Ω LJ 1 i
i
+ K(ti − si−1 )e(ti −si−1 )ω+(ti−1 −ti )Ω LJ 2 i ! KK1 (2 + LLh )ti kx − ykP CB + (Ω − ω) ≤ +
LJ 1 Ke(ti−1 −si−1 )Ω LJ 1 + Ksi e(ti−1 −si−1 )Ω LJ 2 i
i
!
KK1 (2 + LLh )ti kx − ykP CB . (Ω − ω) 11
i
After summarizing the above inequalities, we have the following k(Fx) − (Fy)kP CB ≤ LF kx − ykP CB , where LF = max
1≤i≤m
(
Ke
Ω(ti −si )
Ω(ti −si )
LJ 1 + Kti+1 e i
LJ 2 i
! KK1 (2 + LLh )ti+1 , + (Ω − ω)
KK1 (2 + LLh )t1 KK1 (2 + LLh )ti , LJ 1 Ke(ti−1 −si−1 )Ω LJ 1 + Ksi e(ti−1 −si−1 )Ω LJ 2 + i i i (Ω − ω) (Ω − ω)
!)
.
Hence, F is a strict contraction mapping for sufficiently large Ω > ω. Application of Banach fixed point theorem immediately gives a unique mild solution to the problem (1.5).
4
Nonlocal Problems
The nonlocal condition is a generalization of the classical initial condition. The study of nonlocal initial value problems are important because they appears in many physical systems. Byszewski [14] was the first author who studied the existence and uniqueness of mild solutions to the Cauchy problems with nonlocal conditions. In this section, we investigate the existence and uniqueness of mild solution (1.5) with nonlocal conditions. We consider the following nonlocal differential problem with deviated argument in a Banach space X : x00 (t) = Ax(t) + f (t, x(t), x[h(x(t), t)]), x(t) = Ji1 (t, x(t− i )), 0 2 x (t) = Ji (t, x(t− i )),
t ∈ (si , ti+1 ) i = 0, 1, ..., m,
t ∈ (ti , si ], i = 1, 2, ..., m, 0
(4.1)
t ∈ (ti , si ], i = 1, 2, ..., m,
x(0) = x0 + p(x), x (0) = y0 + q(x),
where x(t) be a state function, 0 = s0 < t1 < s1 < t2 , ..., tm < sm < tm+1 = T < ∞. − 2 The functions Ji1 (t, x(t− i )) and Ji (t, x(ti )) represent non-instantaneous impulses same as in system (1.5). A is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators (C(t))t∈R on X. The functions p(x) and q(x) will be suitably specified later. Definition 4.1. A function x ∈ CL (J, X) is called a mild solution of the impulsive problem (4.1) if it satisfies the following relations: x(0) = x0 + p(x), x0 (0) = y0 + q(x), the non-instantaneous impulse conditions x(t) = Ji1 (t, x(t− t ∈ (ti , si ], i = 1, 2, ..., m, i )), x0 (t) = Ji2 (t, x(t− )), t ∈ (ti , si ], i = 1, 2, ..., m i and x(t) is the solution of the following integral equations x(t)
=
C(t)(x0 + p(x)) + S(t)(y0 + q(x)) +
Z
0
12
t
S(t − s)f (s, x(s), x[h(x(s), s)])ds,
x(t)
t ∈ [0, t1 ],
− 2 C(t − si )(Ji1 (si , x(t− i ))) + S(t − si )(Ji (si , x(ti ))) Z t S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [si , ti+1 ], i = 1, 2, .., m. +
=
si
Further, we need assumptions on the functions p and q to show the existence and uniqueness of the solution for the problem (4.1) (A6) The functions p, q : C(J, X) → X are continuous and there exist positive constants cp and cq such that (i) kp(x1 ) − p(x2 )k ≤ cp ||x1 − x2 ||, (ii) ||q(x1 ) − q(x2 )|| ≤ cq ||x1 − x2 ||. Theorem 4.2. Let x0 ∈ D(A), y0 ∈ E. If all the assumptions (A1)-(A6) are satisfied, then the second order nonlocal problem (4.1) has a unique mild solution x ∈ CL (J, X) provided that Keωt1 cp + K
(eωt1 − 1) cq < 1. ω
Proof: By choosing (
N KT ωT −Ωsi e e , δ = max Ke CJ 1 + KT e CJ 2 + i i 1≤i≤m ω ) N KT ωT −Ωti ωT ωT e , e CJ 1 , Ke kx0 + p(x)k + KT e ky0 + q(x)k + i ω 0
ωT
ωT
we set W 0 = {x ∈ CL (J, X) : kxkP CB ≤ δ 0 }.
We define a map F : W 0 → W 0 given by (Fx)(t)
= +
− 2 Ji1 t, C(ti − si−1 )(Ji1 (si−1 , x(t− i−1 ))) + S(ti − si−1 )(Ji (si−1 , x(ti−1 )))
Z
si−1
(Fx)(t)
=
!
ti
S(ti − s)f (s, x(s), x[h(x(s), s)])ds , t ∈ (ti , si ] i = 1, 2, ..., m;
C(t)(x0 + p(x)) + S(t)(y0 + q(x)) +
Z
0
∀ t ∈ [0, t1 ]; 13
t
S(t − s)f (s, x(s), x[h(x(s), s)])ds,
− 2 C(t − si )(Ji1 (si , x(t− i ))) + S(t − si )(Ji (si , x(ti ))) Z t S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ (si , ti+1 ] i = 1, 2, ..., m. +
(Fx)(t)
=
si
We have, k(Fx) − (Fy)kP CB ≤ L0F kx − ykP CB ,
where L0F
= max
1≤i≤m
(
Ke
Ω(ti −si )
Ω(ti −si )
LJ 1 + Kti+1 e i
LJ 2 i
! KK1 (2 + LLh )ti+1 + , (Ω − ω)
(eωt1 − 1) KK1 (2 + LLh )t1 cq + , ω (Ω − ω) !) KK1 (2 + LLh )ti (ti−1 −si−1 )Ω . + Ksi e LJ 2 + i (Ω − ω)
Keωt1 cp + K LJ 1 Ke(ti−1 −si−1 )Ω LJ 1 i
i
Thus, F is a strict contraction mapping for sufficiently large Ω > ω. Application of Banach fixed point theorem immediately gives a unique mild solution to the problem (4.1). The proof of this theorem is the consequence of the theorem (3.1).
5
Ulam’s type stability
In this section, we show Ulam’s type stability for the system (1.5). Let > 0, ψ ≥ 0 and φ ∈ P C(J, R+ ) be the nondecreasing. We consider the following inequalities 00 ky (t) − Ay(t) − f (t, y(t), y[h(y(t), t)])k ≤ , t ∈ (si , ti+1 ) i = 0, 1, ..., m, ky(t) − Ji1 (t, y(t− t ∈ (ti , si ], i = 1, 2, ..., m, (5.1) i ))k ≤ , 0 ))k ≤ , t ∈ (t , s ], i = 1, 2, ..., m ky (t) − Ji2 (t, y(t− i i i
and
and
00 ky (t) − Ay(t) − f (t, y(t), y[h(y(t), t)])k ≤ φ(t), t ∈ (si , ti+1 ) i = 0, 1, ..., m, ky(t) − Ji1 (t, y(t− t ∈ (ti , si ], i = 1, 2, ..., m, i ))k ≤ ψ, 0 ky (t) − Ji2 (t, y(t− ))k ≤ ψ, t ∈ (ti , si ], i = 1, 2, ..., m i 00 ky (t) − Ay(t) − f (t, y(t), y[h(y(t), t)])k ≤ φ(t), t ∈ (si , ti+1 ) i = 0, 1, ..., m, t ∈ (ti , si ], i = 1, 2, ..., m. ky(t) − Ji1 (t, y(t− i ))k ≤ ψ, 0 ky (t) − Ji2 (t, y(t− ))k ≤ ψ, t ∈ (ti , si ], i = 1, 2, ..., m. i
Now, we take the vector space
2 m Z = CL (J, X) ∩m i=0 C ((si , ti+1 ), X) ∩i=0 C((si , ti+1 ), D(A)).
The following definitions are inspired by Wang et al. [2]. 14
(5.2)
(5.3)
Definition 5.1. The equation (1.5) is Ulam-Hyers stable with if there exists c(K1 Lh LJ m) > 0 such that for each > 0 and for each solution y ∈ Z of the inequality (5.1), there exists a mild solution x ∈ CL (J, X) of the equation (1.5) with ky(t) − x(t)k ≤ c(K1 Lh LJ m), t ∈ J.
(5.4)
Definition 5.2. The equation (1.5) is generalized Ulam-Hyers stable if there exists θK1 ,Lh ,LJ ,m ∈ C(R+ , R+ ), θ(0) = 0 such that for each > 0 and for each solution y ∈ Z of the inequality (5.1), there exists a mild solution x ∈ CL (J, X) of the equation (1.5) with ky(t) − x(t)k ≤ θK1 ,Lh ,LJ ,m , t ∈ J.
(5.5)
Definition 5.3. The equation (1.5) is Ulam-Hyers-Rassias stable with respect to (φ, ψ) if there exists c(K1 Lh LJ mφ) > 0 such that for each > 0 and for each solution y ∈ Z of the inequality (5.3), there exists a mild solution x ∈ CL (J, X) of the equation (1.5) with ky(t) − x(t)k ≤ c(K1 Lh LJ mφ)(ψ + φ(t)), t ∈ J.
(5.6)
Definition 5.4. The equation (1.5) is generalized Ulam-Hyers-Rassias stable with respect to (φ, ψ) if there exists c(K1 Lh LJ mφ) > 0 such that for each solution y ∈ Z of the inequality (5.2), there exists a mild solution x ∈ CL (J, X) of the equation (1.5) with ky(t) − x(t)k ≤ c(K1 Lh LJ mφ)(ψ + φ(t)), t ∈ J.
(5.7)
Remark 5.1 A function y ∈ Z is a solution of the inequality (5.3) if and only if there is G ∈ m 1 g1 ∈ ∩m i=1 C([ti , si ], X) and g2 ∈ ∩i=1 C ([ti , si ], X) such that:
2 m ∩m i=0 C ((si , ti+1 ), X)∩i=0 C((si , ti+1 ), D(A)),
m (a) kG(t)k ≤ φ(t), t ∈ ∩m i=0 (si , ti+1 ), kg1 (t)k ≤ ψ and kg2 (t)k ≤ ψ, t ∈ ∩i=0 [ti , si ];
(b) y 00 (t) = Ay(t) + f (t, y(t), y[h(y(t), t)]) + G(t), t ∈ (si , ti+1 ) i = 0, 1, ..., m; (c) y(t) = Ji1 (t, y(t− i )) + g1 (t), t ∈ (ti , si ], i = 1, 2, ..., m. (d) y 0 (t) = Ji2 (t, y(t− i )) + g2 (t), t ∈ (ti , si ], i = 1, 2, ..., m. Easily, we can have similar remarks for the inequalities (5.1) and (5.2). Remark 5.2 A function y ∈ Z is a solution of the inequality (5.3) then y is a solution of the following integral inequality ky(t) − Ji1 (t, y(t− t ∈ (ti , si ], i = 1, 2, ..., m; i ))k ≤ ψ, 0 (t) − J 2 (t, y(t− ))k ≤ ψ, ky t ∈ (ti , si ], i = 1, 2, ..., m; i i Rt ky(t) − C(t)x0 − S(t)y0 − 0 S(t − s)f (s, y(s), y[h(y(s), s)])dsk R t ω(t−s) − 1]φ(s)ds, t ∈ [0, t1 ]; ≤ K ω 0 [e (5.8) 1 (s , y(t− ))) − S(t − s )(J 2 (s , y(t− ))) ky(t) − C(t − s )(J i i i i i i i i R t − si S(t − s)f (s, y(s), y[h(y(s), s)])dsk R t ω(t−s) [eω(t−si ) − 1] + K − 1]φ(s)ds, ≤ ψKeω(t−si ) + ψK ω ω si [e t ∈ [si , ti+1 ], i = 1, 2, ..., m. 15
By Remark (5.1), we have 00 y (t) = Ay(t) + f (t, y(t), y[h(y(t), t)]) + G(t), t ∈ (si , ti+1 ) i = 0, 1, ..., m; y(t) = Ji1 (t, y(t− i )) + g1 (t), t ∈ (ti , si ], i = 1, 2, ..., m; 0 t ∈ (ti , si ], i = 1, 2, ..., m. y (t) = Ji2 (t, y(t− i )) + g2 (t),
(5.9)
The solution y ∈ Z with y(0) = x0 and y 0 (0) = y0 of the equation (5.9) is given by y(t) = Ji1 (t, y(t− t ∈ (ti , si ], i = 1, 2, ..., m; i )) + g1 (t), 0 (t) = J 2 (t, y(t− )) + g (t), y t ∈ (ti , si ], i = 1, 2, ..., m; 2 i i Rt y(t) = C(t)x0 + S(t)y0 + 0 S(t − s)[f (s, y(s), y[h(y(s), s)]) + G(s)]ds, t ∈ [0, t1 ]; (5.10) − 2 y(t) = C(t − si )((Ji1 (si , y(t− i ))) + g1 (si )) + S(t − si )((Ji (si , y(ti ))) + g2 (si )) + R t S(t − s)[f (s, y(s), y[h(y(s), s)]) + G(s)]ds, t ∈ [s , t ], i = 1, 2, ..., m. i i+1 si
Easily, we can have similar remarks for the solution of the inequalities (5.1) and (5.2). In order to discuss the stability of the problem (1.5), we need the following additional assumption:
(A7) Let φ ∈ C(J, R+ ) be a nondecreasing function. There exists cφ > 0 such that Z
0
t
φ(s)ds ≤ cφ φ(t), ∀ t ∈ J.
Lemma 5.1(Impulsive Gronwall inequality)(see Theorem 16.4, [15]). Let M0 = M ∪ {0}, where M = {1, ..., m} and the following inequality holds u(t) ≤ a(t) +
Z
t
b(s)u(s)ds +
0
X
0 0, k ∈ M. Then for t ∈ R+ , Z t k u(t) ≤ a(t)(1 + β) exp b(s)ds , t ∈ (tk , tk+1 ], k ∈ M0 , (5.12) 0
where β = supk∈M {βk } and t0 = 0. Theorem 5.5. Let x0 ∈ D(A), y0 ∈ E. If the assumptions (A1)- (A4) and (A7) are satisfied. Then, the equation (1.5) is Ulam-Hyers-Rassias stable with respect to (φ, ψ). Proof: Let y ∈ CL (J, D(A)) ∩ C 2 ((si , ti+1 ], X) be a solution of the inequality (5.3) and x is the unique mild solution of the problem (1.5) which is given by x(t) = Ji1 (t, x(t− t ∈ (ti , si ], i = 1, 2, ..., m; i )), 0 2 (t, x(t− )), x (t) = J t ∈ (ti , si ], i = 1, 2, ..., m; i i Rt x(t) = C(t)x0 + S(t)y0 + 0 S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [0, t1 ]; (5.13) 1 (s , x(t− ))) + S(t − s )(J 2 (s , x(t− ))) x(t) = C(t − s )(J i i i i i i i i + R t S(t − s)f (s, x(s), x[h(x(s), s)])ds, t ∈ [s , t ], i = 1, 2, ..., m. i i+1 si 16
For t ∈ [si , ti+1 ], i = 1, 2, ..., m. By inequality (5.8), we have − 2 ky(t) − C(t − si )(Ji1 (si , y(t− i ))) − S(t − si )(Ji (si , y(ti ))) Z t S(t − s)f (s, y(s), y[h(y(s), s)])dsk − si
K ψK ω(t−si ) [e − 1] + ω ω Z t K ωT φ(s)ds + e ω 0
≤ ψKeω(t−si ) +
ψK ωT e ω K ωT e (ψ + cφ φ(t)). + ω
≤ ψKeωT + ≤ ψKeωT
Z
t
si
[eω(t−s) − 1]φ(s)ds
For t ∈ (ti , si ], i = 1, 2, ..., m, we have ky(t) − Ji1 (t, y(t− i ))k ≤ ψ. For t ∈ [0, t1 ], we have Z
t
ky(t) − C(t)x0 − S(t)y0 − S(t − s)f (s, y(s), y[h(y(s), s)])dsk 0 Z t K ≤ [eω(t−s) − 1]φ(s)ds ω 0 K ωT e cφ φ(t). ≤ ω Hence, for t ∈ [si , ti+1 ], i = 1, 2, ..., m, we have − 2 ky(t) − x(t)k ≤ ky(t) − C(t − si )(Ji1 (si , y(t− i ))) − S(t − si )(Ji (si , y(ti ))) Z t S(t − s)f (s, y(s), y[h(y(s), s)])dsk − si
− − 1 ωT 2 +KeωT kJi1 (si , y(t− kJi2 (si , y(t− i )) − Ji (si , x(ti ))k + KT e i )) − Ji (si , x(ti ))k Z t kf (s, y(s), y[h(y(s), s)]) − f (s, x(s), x[h(x(s), s)])dsk +KT eωT si
K ωT − e (ψ + cφ φ(t)) + (KeωT LJ 1 + KT eωT LJ 2 )k(y(t− i ) − (x(ti )k i i ω Z t ωT ky(s) − x(s)kds +K1 (2 + LLh )KT e 0 Z t K ωT ωT ky(s) − x(s)kds e [(2 + cφ )(ψ + φ(t))] + K1 (2 + LLh )KT e ≤ ω 0 i X − (KeωT LJ 1 + KT eωT LJ 2 )k(y(t− (5.14) + j ) − (x(tj )k. ≤ ψKeωT +
j
j
j=1
17
For t ∈ (ti , si ], i = 1, 2, ..., m, we have ky(t) − x(t)k
≤
− − 1 1 ky(t) − Ji1 (t, y(t− i ))k + kJi (t, y(ti )) − Ji (t, x(ti ))k
≤
ψ +
i X j=1
− LJ 1 k(y(t− j ) − (x(tj )k j
K ωT e [(2 + cφ )(ψ + φ(t))] ω Z t ky(s) − x(s)kds + K1 (2 + LLh )KT eωT
≤
0
+
i X j=1
− (KeωT LJ 1 + KT eωT LJ 2 )k(y(t− j ) − (x(tj )k. j
j
(5.15)
Now, for t ∈ [0, t1 ], we have ky(t) − x(t)k
Z t K ωT ωT ky(s) − x(s)kds e cφ φ(t) + K1 (2 + LLh )KT e ≤ ω 0 K ωT ≤ e [(2 + cφ )(ψ + φ(t))] ω Z t ky(s) − x(s)kds + K1 (2 + LLh )KT eωT 0
+
i X j=1
− (KeωT LJ 1 + KT eωT LJ 2 )k(y(t− j ) − (x(tj )k. j
j
(5.16)
We observe that inequalities (5.14), (5.15) and (5.16) give together an impulsive Gronwall inequality of a form of (5.11) on J. Therefore, we can apply impulsive Gronwall inequality (5.12) for t ∈ J, since t ∈ (ti , ti+1 ] for some i ∈ M0 . Consequently, we have K ωT ωT e (2 + cφ )(1 + KeωT LJ )i eK1 (2+LLh )KT e t (ψ + φ(t)) ω K ωT ωT e (2 + cφ )(1 + KeωT LJ )m eK1 (2+LLh )KT e T (ψ + φ(t)) ≤ ω ≤ c(K1 Lh LJ mφ)(ψ + φ(t)),
ky(t) − x(t)k ≤
for any t ∈ J, where LJ = supi∈M {LJ 1 + T LJ 2 } and c(K1 Lh LJ mφ) is a constant depending i i on K1 , Lh , LJ , m, φ. Hence, the equation (1.5) is Ulam-Hyers-Rassias stable with respect to (φ, ψ). Theorem 5.6. If the assumptions (A1)- (A4) and (A7) are satisfied. Then, the equation (1.5) is generalized Ulam-Hyers-Rassias stable with respect to (φ, ψ). Proof: It can be easily proved by applying same procedure of the theorem (5.5) and taking inequality (5.2). 18
Theorem 5.7. If the assumptions (A1)- (A4) and (A7) are satisfied. Then, the equation (1.5) is Ulam-Hyers stable. Proof: It can be easily proved by applying same procedure of the theorem (5.5) and taking inequality (5.1).
6
Application
Example 1: Let X = L2 (0, π). We consider the following partial differential equations with deviated argument ∂tt Z(t, y) = ∂yy Z(t, y) + f2 (y, Z(h(t), y)) + f3 (t, y, Z(t, y)), y ∈ (0, π), t ∈ (2i, 2i + 1], i ∈ {0} ∪ N, Z(t, 0) = Z(t, π) = 0, t ∈ [0, T ], 0 < T < ∞, Z(0, y) = x0 , y ∈ (0, π), (6.1) ∂ Z(0, y) = y , y ∈ (0, π), t 0 Z(t)(y) = (sin it)Z((2i − 1)− , y), y ∈ (0, π), t ∈ (2i − 1, 2i], i ∈ N, ∂t Z(t)(y) = (i cos it)Z((2i − 1)− , y), y ∈ (0, π), t ∈ (2i − 1, 2i], i ∈ N, where 0 = s0 < t1 < s1 < t2 , ..., tm < sm < tm+1 = T < ∞ with ti = 2i − 1, si = 2i and Z y ¯ f3 (t, y, Z(t, y)) = K(y, s)(a1 |Z(t, s)| + b1 |Z(t, s)|))ds. 0
We assume that a1 , b1 ≥ 0, (a1 , b1 ) 6= (0, 0), h : J1 → [0, T ] is locally H¨older continuous in t ¯ : [0, π] × [0, π] → R. with h(0) = 0 and K We define an operator A, as follows, Ax = x00
with D(A) = {x ∈ X : x00 ∈ X and x(0) = x(π) = 0}.
(6.2)
Here, clearly the operator A is the infinitesimal generator of a strongly continuous cosine family of operators on X. A has infinite series representation Ax =
∞ X
n=1
−n2 (x, xn )xn ,
x ∈ D(A),
p where xn (s) = 2/π sin ns, n = 1, 2, 3... is the orthonormal set of eigenfunctions of A. Moreover, the operator A is the infinitesimal generator of a strongly continuous cosine family C(t)t∈R on X which is given by ∞ X cos nt(x, xn )xn , x ∈ X, C(t)x = n=1
and the associated sine family S(t)t∈R on X which is given by S(t)x =
∞ X 1 sin nt(x, xn )xn , n
n=1
19
x ∈ X.
The equation (6.1) can be reformulated as the following abstract differential equation in X: 00 x (t) = Ax(t) + f (t, x(t), x[h(x(t), t)]), t ∈ (si , ti+1 ), i ∈ {0} ∪ N, x(t) = Ji1 (t, x(t− t ∈ (ti , si ], i ∈ N, i )), (6.3) 0 (t) = J 2 (t, x(t− )), t ∈ (ti , si ], i ∈ N, x i i 0 x(0) = x0 , x (0) = y0 ,
where x(t) = Z(t, .), that is x(t)(y) = Z(t, y), y ∈ (0, π). Functions Ji1 (t, x(t− i )) = (sin it)Z((2i− − − − 2 1) , y) and Ji (t, x(ti )) = i(cos it)Z((2i − 1) , y) represent noninstantaneous impulses during intervals (ti , si ]. The operator A is same as in equation (6.2). The function f : J1 × X × X → X, is given by f (t, ψ, ξ)(y) = f2 (y, ξ) + f3 (t, y, ψ), where f2 : [0, π] × X → H01 (0, π) is given by Z f2 (y, ξ) =
y
¯ K(y, x)ξ(x)dx,
0
and kf3 (t, y, ψ)k ≤ V (y, t)(1 + kψkH 2 (0,π) ) with V (., t) ∈ X and V is continuous in its second argument. For more details see [7, 8]. Thus, the theorem (3.1) can be applied to the problem (6.1). We can choose the functions p(x) and q(x) as given below p(x) = q(x) =
n X
αk x(tk ), tk ∈ J for all k = 1, 2, 3, · · · , n,
k n X k
βk x(tk ), tk ∈ J for all k = 1, 2, 3, · · · , n,
where αk and βk are constants. Example 2: We consider particular linear case of the abstract differential equation (6.3) in the space X = R. A forced string equation 00 x (t) + a1 x(t) + a2−sin x(c1 t) = g(t), t ∈ (si , ti+1 ) i = 0, 1, ..., m, x(t) = a3 tanh(x(ti ))r(t), t ∈ (ti , si ], i = 1, 2, ..., m, (6.4) 0 (t) = a tanh(x(t− ))r 0 (t), x t ∈ (ti , si ], i = 1, 2, ..., m, 3 i x(0) = x0 , x0 (0) = y0 ,
where a1 ∈ R+ , a2 , a3 ∈ R, c1 ∈ (0, 1], g ∈ C(J1 , R) and r ∈ C 1 (J2 , R) for J2 = ∪m i=1 Ii . We define A, as follows Ax = −a1 x with D(A) = R.
Here, clearly the value −a1 behave like infinitesimal generator of a strongly continuous cosine √ √ family C(t) = cos a1 t. The associated sine family is given by S(t) = √1a1 sin a1 t. Deviated 20
argument in the abstract differential equation (6.3) is represented by the term a2 x(c1 t) of the dif− 0 ferential equation (6.4). Noninstantaneous impulses a3 tanh(x(t− i ))r(t) and a3 tanh(x(ti ))r (t) are created when bob of the string is extremely pushed on each intervals (ti , si ]. Example 3: We generalize the above example to consider a coupled system of strings or pendulums x00n (t) + an1 sin xn (t) + an2 sin xn (cn t) = bn1 xn−1 (t) + bn2 xn+1 (t) + gn (t), xn (t) = an3 tanh(xn (t− i ))rn (t), 0 x0n (t) = an3 tanh(xn (t− i ))rn (t), xn (0) = xn0 , x0n (0) = yn0 ,
i = 0, 1, ..., m, n ∈ Z,
t ∈ (ti , si ], i = 1, 2, ..., m,
t ∈ (si , ti+1 ), (6.5)
t ∈ (ti , si ], i = 1, 2, ..., m,
where an1 , an2 , an3 , bn1 , bn2 ∈ R, cn ∈ (0, 1], gn ∈ C(J1 , R) and rn ∈ C 1 (J2 , R). Moreover we suppose supn |ank | < ∞, k = 1, 2, 3, supn (|bn1 | + |bn2 |) < ∞ and supn (kgn k + ||rn k + krn0 k) < ∞. Then we consider (6.5) on `∞ and use Exercise 1 on p. 39 from [10]. The lattice ODE (6.5) is a generalization of the discrete sine-Gordon equation [16] and xn (cn t) represent pantograph-like terms [17].
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Conclusion
The research presented in this paper focuses on the existence, uniqueness and stability of solutions to the impulsive systems represented by second order nonlinear differential equations with noninstantaneous impulses and deviated argument. We used strongly continuous cosine family of bounded linear operators and Banach’s fixed point theorem to get the existence and uniqueness of the solutions. Moreover, Ulam’s type stability is established by using impulsive Gronwall inequality.
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