Proceedings of the 6th International Conference on Differential Equations and Dynamical Systems, (2009) 162–166 c 2009 Watam Press DCDIS A Supplement, Copyright
NONLOCAL INITIAL VALUE PROBLEM FOR FIRST-ORDER DYNAMIC EQUATIONS ON TIME SCALES Douglas R. Anderson1 and Abdelkader Boucherif2 1 Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562 USA, Email:
[email protected] 2
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia, Email:
[email protected]
Abstract. In this study, conditions for the existence of at least one solution to a nonlinear first-order nonlocal initial value problem on time scales are discussed. The results extend previous work in the continuous case to the discrete, quantum, and general time scales setting, and are based on the Leray-Schauder fixed point theorem. Keywords. Time scales; Nonlinear dynamic equations; LeraySchauder fixed point theorem; Initial value problems; Existence. AMS (MOS) subject classification: 34B05, 39A10.
and Z
b
Z
∞
p(s)∆s < tm
R0∗
dη , q(η)
(1.5)
−1 Pm Pm where α := 1 + j=1 γj , A := 1 + |α| j=1 |γj |, and Rt R0∗ := A a m w(s, R0 )∆s.
Problem (1.1) extends to general time scales the special case T = R; see Boucherif and Precup [5]. There has of late been interest in first-order problems on time We are interested in the first-order nonlocal time-scale scales. Anderson [1], Cabada and Vivero [6], Dai and Tisinitial value problem dell [7], Otero-Espinar and Vivero [9], Sun [11], Sun and Li [12], and Tian and Ge [13] all recently consider first ∆ σ order boundary value problems on time scales, but none x (t) =mf t, x (t) , t ∈ (a, b)T , X (1.1) of them consider the nonlocal problem. For more general γj x(tj ) = 0 x(a) + information concerning dynamic equations on time scales, j=1 introduced by Aulbach and Hilger [2] and Hilger [8], see κ where m ≥ 1 and the points tj ∈ T for j ∈ {1, 2, . . . , m} the excellent text by Bohner and Peterson [3]. with a ≤ t1 ≤ · · · ≤ tm < b; It is straightforward to check that problem (1.1) is equivalent to the following integral equation in C[a, b]T m X 1+ γj 6= 0, γj ∈ R, j ∈ {1, . . . , m}; (1.2) Z t Z tj m X j=1 x(t) = f s, xσ (s) ∆s−α γj f s, xσ (s) ∆s. a a j=1 the function f : [a, b]T × R → R is continuous and satisfies ( w t, |x| : t ∈ [a, tm ]T |f (t, x)| ≤ (1.3) This can be viewed as a fixed point problem in C[a, b]T p(t)q |x| : t ∈ [tm , b]T , for the completely continuous operator L : C[a, b]T → C[a, b]T given by where w : [a, tm ]T × R+ → R+ is integrable and nondecreasing in its second argument; p : [tm , b]T → R+ is Z t Z tj m X right-dense continuous; and q : R+ → R+ is nondecreas- Lx(t) = f s, xσ (s) ∆s − α γj f s, xσ (s) ∆s. ing with 1/q integrable on R+ . Moreover, we assume that a a j=1 there exists R0 > 0 such that Z Notice that L appears as a sum of two integral operators, 1 tm 1 η > R0 implies w(s, η)∆s < (1.4) one, say of Fredholm type, whose values depend only on η a A
1
Introduction
162
the restrictions of functions to [a, tm ]T , Z tj m X −α γ f s, xσ (s) ∆s j a j=1 Rt + a f s, xσ (s) ∆s : t < tm , LF x(t) = Z tj m X −α γj f s, xσ (s) ∆s a j=1 Rt : t ≥ tm ; + a m f s, xσ (s) ∆s
t ∈ [a, tm ]T , we have Z tj m X γj f s, xσ (s) ∆s |x(t)| = λ −α a j=1 Z t σ + f s, x (s) ∆s a Z tm m X f s, xσ (s) ∆s ≤ 1+|α| |γj | j=1
and the other, a Voltera type operator,
Z ≤
a
tm
A
w s, kxka,tm ∆s.
a
0 Z LV x(t) =
: t < tm , t
f s, xσ (s) ∆s : t ≥ tm ,
Now we take the maximum over [a, tm ]T to obtain Z
tm
kxka,tm ≤ A
tm
w s, kxka,tm ∆s.
a
depending on the restrictions of functions to [tm , b]T . Thus the growth conditions on the nonlinearity f in the This, according to (1.4), guarantees that sequel will be split into two parts, one for the subinterval kxka,tm ≤ R0 . containing the points involved by the nonlocal condition, and the other for the rest of the domain of definition. Next, consider t ∈ [tm , b]T . Then To emphasize the generality and flexibility of the time scale setting, we note the following corollary equations. If Z tj m X T = Z, then for t ∈ {a + 1, a + 2, · · · , b − 1}, the nonlocal |x(t)| = λ −α γj f s, xσ (s) ∆s a initial value problem (1.1) takes the form j=1 Z t σ + f s, x (s) ∆s x(t + 1) m= x(t) + f t, x(t + 1) , a X Z tm γj x(tj ) = 0, tj ∈ {a, a + 1, · · · , b}. x(a) + ≤ A w s, R0 ∆s j=1
(2.1)
a
Z
t
+ p(s)q (|x(s)|) ∆s := ϕ(t), For a discrete domain with nonconstant step size, tm take q > 1 and consider the quantum time scale T = {0, · · · , q −2 , q −1 , 1, q, q 2 , · · · }. Then for t ∈ and for t ∈ [tm , b]T we have ϕ∆ (t) = p(t)q(|x(t)|) ≤ {qa, q 2 a, · · · , q −2 b, q −1 b}, the nonlocal initial value prob- p(t)q(ϕ(t)). Since ϕ is continuous and strictly increaslem (1.1) takes the form ing on [tm , b]T , we have by change of variable (Theorem 5.40 in Bohner and Peterson [4]) that x(qt) = x(t) + (q − 1)tf t, x(qt) , Z t ∆ Z t Z ϕ(t) m ϕ (s) dη X = ∆s ≤ p(s)∆s. γj x(tj ) = 0, tj ∈ {a, qa, · · · , b}. x(a) + tm q(ϕ(s)) tm ϕ(tm ) q(η) j=1
2
Existence Results
This, together with (1.5), guarantees that ϕ(t) ≤ R1 for all t ∈ [tm , b]T for some R1 > 0. Consequently, |x(t)| ≤ R1 for all t ∈ [tm , b]T , so that kxktm ,b ≤ R1 .
In what follows, set kxka,b = maxt∈[a,b]T |x(t)|.
(2.2)
If we set R := max{R0 , R1 }, then estimates (2.1) and QED Theorem 2.1 Assume (1.2)−(1.5). Then the nonlocal (2.2) yield kxka,b ≤ R. initial value problem (1.1) has at least one solution. Remark 2.2 If t = a, that is if we have the local inim
tial condition x(a) = 0 in (1.1), then (1.5) is simply the R R dη Proof: The result will follow from the Leray-Schauder condition ab p(s)∆s < 0∞ q(η) . fixed point theorem once we have proven the boundedness In the next theorem we modify assumption (1.3) to get of the set of all solutions to equations of the form x = λLx for λ ∈ [0, 1]. Let x be one such solution. Then for a related existence result. 163
Theorem 2.3 Assume that f : [a, b]T × R → R is continuous and satisfies for all t ∈ [a, b]T and all x ∈ R the inequality |f (t, x)| ≤ p(t)q (|x|) , Rb with p0 := a p(s)∆s < +∞ and q : R+ → R+ is continuous and nondecreasing such that lim sup η→∞
η > Ap0 . q (η)
(2.3)
Then the nonlocal initial value problem (1.1) has at least one solution. Proof: First we show that the set S := {x ∈ C[a, b]T : x = λLx for some λ ∈ (0, 1)} is bounded. For any u ∈ S, we have Z tj m X γj f s, uσ (s) ∆s |u(t)| = λ −α a j=1 Z t + f s, uσ (s) ∆s a Z tj m X ≤ |α| |γj | p(s)q (|uσ (s)|) ∆s
≤
3
Some Particular Cases
3.1
Nonlinearities with at most linear growth
In this subsection we show that the existence of solutions to problem (1.1) follows directly from the Schauder fixed point theorem in the particular case that (1.2) again holds, and the nonlinearity f satisfies the following growth condition in x, ( |f (t, x)| ≤
k|x| + d : t ∈ [a, tm ]T c|x| + d : t ∈ [tm , b]T
(3.1)
a
j=1 b
Z +
This shows that the set S is bounded. By the Schaefer fixed point theorem (see Smart [10]) the equation x = λLx has a solution for λ = 1, which is a solution of (1.1). QED
p(s)q (|uσ (s)|) ∆s
for all x ∈ R, provided that k(tm − a)A < 1. In this case (1.3) holds, with w(t, η) = kη + d, p(t) = 1, q(η) = cη + d, and
a
|α|
m X
Z
p(s)∆s a
j=1
+
k(tm − a)A . 1 − k(tm − a)A
R0 =
!
b
Z
tj
|γj |
p(s)∆s q (kuka,b ) . a
Let η0 = kuka,b . Then Z m X |γj | η0 ≤ |α|
+
tj
p(s)∆s
a
j=1
Z
In order to apply the Schauder fixed point theorem, we seek a nonempty, bounded, closed and convex subset B of C[a, b]T with L(B) ⊂ B. Let x be any element of C[a, b]T . For t ∈ [a, tm ]T , we have
!
b
Z tj m X |Lx(t)| = −α γj f s, xσ (s) ∆s a j=1 Z t σ + f s, x (s) ∆s
p(s)∆s q (η0 ) a
≤ 1 + |α|
m X
Z |γj |
j=1
b
! p(s)∆s q (η0 )
a
= Ap0 q (η0 ) . Hence
η0 ≤ Ap0 . q (η0 )
Z
a tm
≤ A
f s, xσ (s) ∆s
a
≤ k(tm − a)Akxka,tm + d(tm − a)A.
(2.4)
On the other hand, the conditions on q and p0 imply that As a result, there exists η ∗ > 0 such that for all η > η ∗ we have η > Ap0 . q (η)
(2.5)
Comparing the last two inequalities we see that η0 ≤ η ∗ . Consequently, kuka,b ≤ η ∗ .
kLxka,tm ≤ k(tm − a)Akxka,tm + d(tm − a)A.
(3.2)
For any constant β > 0, recall the delta exponential function (see Definition 2.30 in Bohner and Peterson [3]) given
164
by eβ (t, tm ); then for any t ∈ [tm , b]T we have Z tj m X γj f s, xσ (s) ∆s |Lx(t)| = −α a j=1 Z t σ + f s, x (s) ∆s
Theorem 3.1 Assume (1.2) and that f (·, x) is measurable for all x ∈ R. Assume moreover that f (·, 0) is bounded, and that there exists a bounded function ` such that (3.6) holds, with A(tm − a)k`ka,tm < 1. Then (1.1) has a unique solution x∗ ∈ C[a, b]T such that kxn − x∗ ka,b → 0 as n → ∞, where x0 is any function in C[a, b]T and xn = Lxn−1 for each n ∈ N.
a
≤ k(tm − a)Akxka,tm + d(tm − a)A Z t + (c|x(s)| + d) ∆s
Proof: Let x, y ∈ C[a, b]T . For t ∈ [a, tm ]T we have Z tm |x(s) − y(s)|∆s |Lx(t) − Ly(t)| ≤ k`ka,tm A
tm
≤
k(tm − a)Akxka,tm +d (tm − a)A + b − tm Z t +c e β (s, tm )|x(s)|eβ (s, tm )∆s
a
≤
tm
≤ k(tm − a)Akxka,tm + d0 +
c eβ (t, tm )kxkβ , β
k`ka,tm A(tm − a)kx − yka,tm .
It follows that kLx − Lyka,tm ≤ k`ka,tm A(tm − a)kx − yka,tm .
(3.7)
where d0 := d (tm − a)A + b − tm , and the Bielecki-type norm introduced by Tisdell and Zaidi [14] on time scales denoted kxkβ is given by kxkβ :=
sup
For t ∈ [tm , b]T we have |Lx(t) − Ly(t)|
≤
k`ka,tm A(tm − a)kx − yka,tm Z t +k`ktm ,b |x(s) − y(s)|∆s
≤
k`ka,tm A(tm − a)kx − yka,tm k`ktm ,b eβ (t, tm )kx − ykβ . + β
e β (t, tm )|x(t)|.
t∈[tm ,b]T
tm
Multiplying by e β (t, tm ) and taking the supremum over [tm , b]T , we arrive at kLxkβ ≤ k(tm − a)Akxka,tm + d0 +
c kxkβ . β
(3.3)
Consequently, kLx − Lykβ
If we consider an equivalent norm on C[a, b]T given by
≤
kxk := max {kxka,tm , kxkβ } , then from (3.2) and (3.3) we have
k`ka,tm A(tm − a)kx − yka,tm k`ktm ,b + kx − ykβ , β
so that
kLxk ≤ k(tm − a)A + c/β kxk + d1 ,
(3.4)
kLx − Lyk ≤
k`ktm ,b k`ka,tm A(tm − a) + β
kx − yk.
where d1 := max{d0 , d(tm − a)A}. Since k(tm − a)A < 1, there exists a β > 0 large enough such that k(tm − a)A + Finally, we choose any β > 0 such that c/β < 1. Hence there exists a number R > 0 with k`ka,tm A(tm − a) + k`ktm ,b /β < 1, k(tm − a)A + c/β R + d1 ≤ R. (3.5) and we apply the Banach contraction principle.
QED
Now we take B = {x ∈ C[a, b]T : kxk ≤ R}. Inequalities Remark 3.2 Under the assumptions of Theorem 3.1, (3.4) and (3.5) guarantee that L(B) ⊂ B, and thus the condition (3.1) is satisfied with k = k`k a,tm , c = k`ktm ,b , Schauder fixed point theorem can be applied. and d = kf (·, 0)ka,b .
3.2
Lipschitz nonlinearities
We deal in this subsection with problem (1.1) when the nonlinearity f satisfies a Lipschitz condition in x of the form |f (t, x) − f (t, y)| ≤ `(t)|x − y|,
4
Example
In this section we present an example applying Theorem 2.1.
t ∈ [a, b]T ,
Example 4.1 For T = R and [a, b] = [0, 1], the nonlocal x, y ∈ R, (3.6) initial value problem ( where ` : [a, b]T → R is a bounded function. The following x0 (t) = f t, x(t) , t ∈ (0, 1), existence and uniqueness result is based on the Banach (4.1) contraction principle. x(0) + 21 x (1/3) + 14 x (2/3) = 0, 165
where
[4] M. Bohner and A. Peterson, editors, Advances in Dynamic Equations on Time Scales, Birkh¨ auser, Boston, 2003.
(
: t ∈ [0, 2/3], : t ∈ [2/3, 1],
1 f (t, x) = sin 6πt x2
[5] A. Boucherif and R. Precup, On the nonlocal initial value problem for first order differential equations, Fixed Point Theory, 4:2 (2003) 205–212. [6] A. Cabada and D. R. Vivero, Existence of solutions of firstorder dynamic equations with nonlinear functional boundary value conditions, Nonlinear Anal., 63 (2005) e697–e706.
has at least one solution. Proof: Considering the function f and its bounds, ( w t, |x| = | sin(6πt)| · 1 : t ∈ [0, 2/3], |f (t, x)| ≤ 2 p(t)q |x| = | sin(6πt)| · x : t ∈ [2/3, 1].
[7] Q. Y. Dai and C. C. Tisdell, Existence of solutions to firstorder dynamic boundary value problems, Inter. J. Difference Equations., 1:1 (2006) 1—17. [8] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56.
It is straightforward to calculate the constant values −1
α = (1 + 1/2 + 1/4)
25 ∗ A = 10 7 , and R0 = 14π . Clearly if we take R0 = it follows that η > R0 implies
1 η
Z
2/3
w(s, η)ds = 0
< and Z
1
Z
[9] V. Otero-Espinar and D.R. Vivero, The existence and approximation of extremal solutions to several first-order discontinuous dynamic equations with nonlinear boundary value conditions, Nonlinear Anal., 68 (2008) 2027–2037.
= 4/7, 40 21π
> 0,
[10] D. R. Smart, Fixed Point Theorems, Cambridge University Press, 1974.
Z 1 2/3 4 | sin(6πs)|ds = η 0 3πη 1 7 = , A 10
[11] J.-P. Sun, Twin positive solutions of nonlinear first-order boundary value problems on time scales, Nonlinear Anal., 68 (2008) 1754–1758. [12] J.-P. Sun and W.-T. Li, Existence and multiplicity of positive solutions to nonlinear first-order PBVPs on time scales, Computers and Math. Appl., 54 (2007) 861–871. [13] Y. Tian and W. G. Ge, Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales, Nonlinear Anal., 69 (2008) 2833–2842.
1
2 p(s)ds = | sin(6πs)|ds = 3π 2/3 2/3 Z ∞ Z ∞ dη dη 1 14π < = = ∗ = . 2 ∗ ∗ q(η) η R 25 R0 R0 0
[14] C. C. Tisdell and A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal., 68:11 (2008) 3504–3524.
Thus (1.2)−(1.5) are all satisfied, so that by Theorem 2.1, Received June 2008; revised October 2008. the nonlocal initial value problem (4.1) has at least one solution. QED
5
Acknowledgements
D. Anderson wishes to thank the organizers for the invitation to speak at the 6th International Conference on Differential Equations and Dynamical Systems, Morgan State University, Baltimore, Maryland. A. Boucherif is visiting the Division of Applied Mathematics, Brown University through a grant from the Arab Fund For Economic And Social Development, Kuwait. He is grateful to both institutions and Professor John Mallet-Paret for making the visit possible.
References [1] D. R. Anderson, Existence of solutions for first-order multipoint problems with changing-sign nonlinearity, Journal of Difference Equations and Applications, 14:6 (2008) 657–666. [2] B. Aulbach and S. Hilger, Linear dynamic processes with inhomogeneous time scale, Nonlinear Dyn. Quantum Dyn. Sys., (Gaussig, 1990) volume 59 of Math. Res., 9–20. Akademie Verlag, Berlin, 1990. [3] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkh¨ auser, Boston, 2001.
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