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Fixed Point Theory, 15(2014), No. 2, 475-486 http://www.math.ubbcluj.ro/∼ nodeacj/sfptcj.html

MULTIPLE POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR P-LAPLACIAN IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES ILKAY YASLAN KARACA∗ , OZLEM BATIT OZEN∗∗ AND FATMA TOKMAK∗∗∗ ∗ Department

of Mathematics, Ege University 35100 Bornova, Izmir, Turkey E-mail: [email protected]

∗∗ Department


∗∗∗ Department


Abstract. In this paper, by using the classical fixed-point index theorem for compact maps and Leggett-Williams fixed point theorem, some sufficient conditions for the existence of multiple positive solutions for a class of second-order p-Laplacian boundary value problem with impulse on time scales are established. We also give an example to illustrate our results Key Words and Phrases: Impulsive dynamic equation, p-Laplacian, positive solutions, fixed point theorems, time scales. 2010 Mathematics Subject Classification: 34B18, 34B37, 34K10, 47H10.

1. Introduction It is known that the theory of impulsive differential equations have become more important in recent years in some mathematical models of real processes and phenomena. For the introduction of the basic theory of impulsive equations, see [1, 12] and the references therein. The theory of dynamic equations on time scales has been developing rapidly and have received much attention in recent years. The study unifies existing results from the theory of differential and finite difference equations and provides powerful new tools for exploring connections between the traditionally separated fields. We refer to the books by Bohner and Peterson [3, 4]. Recently, the boundary value problems for impulsive differential equations have been studied extensively. To identify a few, we refer to the reader to see [7, 8, 9, 11, 18, 20]. However, the corresponding theory of such equations is still in the beginning stages of its development, especially the impulsive dynamic equations on time scales, see [2, 15, 19]. There is not so much work on impulsive boundary value problems 475

476

ILKAY YASLAN KARACA, OZLEM BATIT OZEN AND FATMA TOKMAK

with p-Laplacian on time scales except that in [5, 6, 10, 16, 17]. To our knowledge, no paper has considered the second-order BVP with integral boundary conditions for p-Laplacian impulsive dynamic equations on time scales. This paper fills this gap in the literature. In this paper, we are concerned with the existence of many positive solutions of the following boundary value problem for p-Laplacian impulsive dynamic equations on time scales −(φp (u4 ))∇ (t) = f (t, u(t)), u(t+ k)

−

t ∈ [0, 1]T ,

u(t− k)

t 6= tk ,

= Ik (u(tk )), k = 1, 2, ..., m Z 1 αu(0) − βu4 (0) = u(s)4s, u4 (1) = 0

k = 1, 2, ..., m

(1.1) (1.2) (1.3)

0

where T is a time scale, 0, 1 ∈ T, [0, 1]T = [0, 1] ∩ T, tk ∈ (0, 1) ∩ T, k = 1, 2, ..., m with 0 < t1 < t2 < ... < tm < 1, α > 1, β > 0, φp (s) is a p-Laplacian function, i.e., 1 1 φp (s) = |s|p−2 s for p > 1, (φp )−1 (s) = φq (s) where + = 1. p q We assume that the following conditions are satisfied: (H1) f ∈ C([0, 1] × [0, ∞), [0, ∞)); − (H2) Ik ∈ C([0, ∞), [0, ∞)), tk ∈ [0, 1]T and u(t+ k ) = limh→0 u(tk + h), u(tk ) = limh→0 u(tk − h) represent the right and left limits of u(t) at t = tk , k = 1, 2, ..., m. We remark that by a solution u of (1.1)-(1.3) we mean u : T → R is delta differentiable, u4 : Tk → R is nabla differentiable on Tk ∩ Tk and u4∇ : Tk ∩ Tk → R is continuous, and satisfies the impulsive and boundary value conditions (1.2)-(1.3). By using the fixed point index theory in the cone [13], we get the existence of at least two or more positive solutions for the BVP (1.1)-(1.3). By using LeggettWilliams fixed point theorem [14], we also establish the existence of triple positive solutions for BVP (1.1)-(1.3). The organization of this paper is as follows. In section 2, we provide some necessary background. In section 3, the main results for problem (1.1)-(1.3) are given. In section 4, we give an example. 2. Preliminaries In this section, we list some background material from the theory of cones in Banach spaces. Definition 2.1. Let (E, k.k) be a real Banach space. A nonempty, closed set K ⊂ E is said to be a cone provided the following are satisfied: (i) if x, y ∈ K and a, b ≥ 0, then ax + by ∈ K; (ii)if y ∈ K and −y ∈ K, then y = 0. If K ⊂ E is a cone, then K can induce a partially order relation ≤ on E by x ≤ y, if and only if y − x ∈ K, for all x, y ∈ E. Definition 2.2. A function f : T → R is called concave on IT = I ∩ T, if f (λt + (1 − λ)s) ≥ λf (t)+(1−λ)f (s), for all t, s ∈ IT and all λ ∈ [0, 1] such that λt+(1−λ)s ∈ IT .

P-LAPLACIAN IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES

477

Remark 2.3. If u4∇ ≤ 0 on [0, 1]Tk ∩Tk , then we say that u is concave on [0, 1]T . Theorem 2.4. [13] Let K be a cone in a real Banach space E. Let D be an open ¯ K 6= K. Assume that T : D ¯K → K bounded subset of E with DK = D ∩ K 6= ∅ and D is completely continuous such that x 6= T x for x ∈ ∂DK . Then the following results hold: (i) If kT xk ≤ kxk, x ∈ ∂DK , then iK (T, DK ) = 1. (ii) If there exists e ∈ K \ {0} such that x 6= T x + λe for all x ∈ ∂DK and all λ > 0, then iK (T, DK ) = 0. ¯ ⊂ DK . If iK (T, DK ) = 1 and iK (T, UK ) = 0, (iii) Let U be open in K such that U ¯K . The same result holds if iK (T, DK ) = 0 then T has a fixed point in DK \ U and iK (T, UK ) = 1. Let E be a real Banach space with cone K, a map θ : K → [0, ∞) is said to be nonnegative continuous concave functional on K if θ is continuous and θ(tx + (1 − t)y) ≥ tθ(x) + (1 − t)θ(y) for all x, y ∈ K and t ∈ [0, 1]T . Let a, b be two numbers such that 0 < a < b and θ a nonnegative continuous concave functional on K. We define the following convex sets: Ka = {x ∈ K : kxk < a}, K(θ, a, b) = {x ∈ K : a ≤ θ(x), kxk < b}. ¯c → K ¯ c be Theorem 2.5. [14] (Leggett-Williams fixed point theorem) Let T : K completely continuous and θ be a nonnegative continuous concave functional on P ¯ c . Suppose there exists 0 < d < a < b ≤ c such such that θ(x) ≤ kxk for all x ∈ K that (i) {x ∈ K(θ, a, b) : θ(x) > a} = 6 ∅ and θ(T x) > a for x ∈ K(θ, a, b), (ii) kT xk < d for kxk ≤ d, (iii) θ(T x) > a for x ∈ K(θ, a, c) with kT xk > b. ¯ c such that kx1 k < d, a < θ(x2 ) Then T has at least three fixed points x1 , x2 , x3 in K and kx3 k > d with θ(x3 ) < a. 3. Main results In this section, by defining an appropriate Banach space and cone, we impose the growth conditions on f , Ik which allow us to apply the theorems in section 2 to establish the existence results of the positive solutions for the BVP (1.1)-(1.3). Let J 0 = [0, 1]T \ {t1 , t2 , ..., tm } . We define E = u : [0, 1]T → R is continuous at t 6= tk , there exist u(t− k ) and − u(t+ ) with u(t ) = u(t ) f or k = 1, 2, ..., m , k k k which is a Banach space with the norm kuk = sup |u(t)|. t∈[0,1]T

478


By a solution of (1.1)-(1.3), we mean a function u ∈ E ∩ C 2 (J 0 ) which satisfies (1.1)-(1.3). We define a cone K ⊂ E as K = u ∈ E : u is a concave, nonnegative and nondecreasing function, 4

1

Z

αu(0) − βu (0) =

u(s)4s .

0

Lemma 3.1. Suppose that (H1) and (H2) are satisfied. Then u ∈ E ∩ C 2 (J 0 ) is a positive solution of the impulsive boundary value problem (1.1)-(1.3) if and only if u(t) is a solution of the following integral equation Z 1 Z 1 Z 1 β 1 f (r, u(r))∇r ∆τ u(t) = f (s, u(s))∇s + (1 − σ(τ ))φq φq α−1 α−1 0 τ 0 Z t Z 1 m X 1 X f (r, u(r))∇r ∆s + φq + Ik (utk )(1 − tk ) + Ik (utk ). (3.1) α−1 s 0 t 0 such that u0 = T u0 + λ0 e.

(3.5)

482


Then, Lemma 3.2, Lemma 3.4 (d) and (3.5) imply that for t ∈ [0, 1]T u0

= ≥ > =

T u0 + λ0 e ≥ γkT u0 k + λ0 Z 1 β f (s, u(s))∇s + λ0 γ φq α−1 0 Z 1 β φp (Lρ2 )∇s + λ0 φq γ α−1 0 β γ Lρ2 + λ0 = γρ2 + λ0 , α−1

i.e., γρ2 > γρ2 + λ0 , which is a contradiction. Hence by (ii) of Theorem 2.4, it follows that iK (T, Ωρ2 ) = 0. Finally, similar to the proof of iK (T, Kρ1 ) = 1, we can prove that iK (T, Kρ3 ) = 1. ¯ ρ ⊂ Kγρ ⊂ Ωρ . Similarly with Since ρ1 < γρ2 and Lemma 3.4 (b), we have K 1 2 2 ¯ ρ ⊂ Kρ ⊂ Kρ . Therefore (iii) of Theorem ρ2 < ρ3 and Lemma 3.4 (b), we have Ω 2 2 3 2.4 implies that BVP (1.1)-(1.3) has at least two positive solutions u1 , u2 with u1 ∈ ¯ ρ , u2 ∈ Kρ \ Ω ¯ρ . Ωρ2 \ K 1 3 2 Theorem 3.5 can be generalized to obtain many solutions. Theorem 3.6. Suppose (H1) and (H2) hold. Then we have the following assertions. 0 +1 (H5) If there exists {ρi }2m ⊂ (0, ∞) with ρ1 < γρ2 < ρ2 < ρ3 < γρ4 < ... < i=1 γρ2m0 < ρ2m0 < ρ2m0 +1 such that ρ

ρ

f0 2m−1 < φp (l), I0 2m−1 < lρ2m−1 , (m = 1, 2, ..., m0 , m0 + 1), ρ2 m fγρ > φp (L), (m = 1, 2, ..., m0 ), 2m

then problem (1.1)-(1.3) has at least 2m0 solutions in K. 0 (H6) If there exists {ρi }2m i=1 ⊂ (0, ∞) with ρ1 < γρ2 < ρ2 < ρ3 < γρ4 < ... < γρ2m0 < ρ2m0 such that ρ

ρ

ρ2 m f0 2m−1 < φp (l), I0 2m−1 < lρ2m−1 , fγρ > φp (L), (m = 1, 2, ..., m0 ), 2m

then problem (1.1)-(1.3) has at least 2m0 − 1 solutions in K. Theorem 3.7. Suppose (H1) and (H2) hold. Then we have the following assertions. 0 +1 (H7) If there exists {ρi }2m ⊂ (0, ∞) with ρ1 < ρ2 < γρ3 < ρ3 < ... < ρ2m0 < i=1 γρ2m0 +1 < ρ2m0 +1 such that ρ2m−1 fγρ > φp (L), (m = 1, 2, ..., m0 , m0 + 1), 2m−1

f0ρ2 m < φp (l), I0ρ2m < lρ2m , (m = 1, 2, ..., m0 ), then problem (1.1)-(1.3) has at least 2m0 solutions in K. 0 (H8) If there exists {ρi }2m i=1 ⊂ (0, ∞) with ρ1 < ρ2 < γρ3 < ρ3 < ... < γρ2m0 −1 < ρ2m0 −1 < ρ2m0 such that ρ2m−1 fγρ > φp (L), f0ρ2m < φp (l), I0ρ2m < lρ2m , (m = 1, 2, ..., m0 ), 2m−1

then problem (1.1)-(1.3) has at least 2m0 − 1 solutions in K.


483

Theorem 3.8. Let 0 < d < a < γb < b ≤ c and assume that conditions (H1), (H2) hold, and the following conditions hold: (H9) There exists a constant ck such that Ik (y) ≤ ck for k = 1, 2, ..., m; (H10) fab > φp (L); m X α ck α−1 α − 1 k=1 . , d> (H11) f0c ≤ φp (R), f0d < φp (R) where 0 < R < β+α 1 − R β+α α−1

Then the BVP (1.1)-(1.3) has at least three nonnegative solutions u1 , u2 , and u3 in ¯ c such that K ku1 k < d, a < β(u2 ) and ku3 k > d with β(u3 ) < a. Proof. By (H1) and (H2), T : K → K is completely continuous. For u ∈ K, let θ(u) = min u(t), t∈[0,1]T

then it is easy to check that θ is a nonnegative continuous concave functional on K with θ(u) ≤ kuk for u ∈ K. ¯ c , by (H9) and (H11) we have Let u ∈ K kT uk

max T u(t) ( Z 1 Z t Z 1 β φq f (s, u(s))∇s + φq = max f (r, u(r))∇r 4s α−1 t∈[0,1]T 0 0 s Z 1 Z 1 1 (1 − σ(s))φq f (r, u(r))∇r ∆s + α−1 0 s ) m X 1 X Ik (u(tk ))(1 − tk ) + Ik (u(tk )) + α−1 tk a}. Also, u ∈ K(θ, a, b) and (H10) condition imply that θ(T u)

=

min (T u)(t) = T u(0) t∈[0,1]T

= + + > >

Z 1 β φq f (s, u(s))∇s α−1 0 Z 1 Z 1 1 (1 − σ(s))φq f (r, u(r))∇r ∆s α−1 0 s m 1 X Ik (u(tk ))(1 − tk ) α−1 k=1 Z 1 β φq f (s, u(s))∇s α−1 0 Z 1 β a(α − 1) φq φp ∇s = a. α−1 β 0

Finally, we show that θ(T u) > a for u ∈ K(θ, a, c) with kT uk > b. From Lemma 3.2, we have θ(T u) = min (T u)(t) ≥ γkT uk > γb > a. t∈[0,1]T


485

So θ(T u) > a is satisfied. Therefore all of the conditions of Theorem 2.5 hold. Hence BVP (1.1)-(1.3) has at least three positive solutions u1 , u2 and u3 such that ku1 k < d, a < θ(u2 ) and ku3 k > d with

θ(u3 ) < a.

4. Example Example 4.1. Consider the following second-order impulsive p-Laplacian boundary value problem, −(φ2 (u4 (t)))∇ = f (t, u(t)), + u( 12 )

t ∈ [0, 1]T \ { 21 },

− u( 12 )

= I1 (u( 12 )), Z 1 u∆ (1) = 0, 2u(0) − u∆ (0) = u(s)4s, −

(4.1) (4.2) (4.3)

0

where f (t, u) =

 

1 26 (1

− u), 51(u − 1),  7 11 (u − 4) + 153,

u ∈ [0, 1]; u ∈ (1, 4]; u ∈ (4, ∞);

and u . 10 Taking ρ1 = 1, ρ2 = 4, ρ3 = 70, α = 2 and β = 1, we have l = 15 , L = 1, γ = 13 . We can obtain that I1 (u) =

ρ1 < γρ2

and ρ2 < ρ3 .

Now, we show that (H3) is satisfied: 1 1 1 51 < φ2 ( ) = , f 44 = > φ2 (1) = 1, f01 = 3 26 5 25 48 1 1 1 and I070 = 7 < lρ3 = 14. f070 < 0.039 < φ2 ( ) = 0.04, I01 = ( ) < lρ1 = 5 10 5 Then, all conditions of Theorem 3.5 hold. Hence, we get the BVP (4.1)-(4.3) has at least two positive solutions. References [1] M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclusions, New York, 2006. [2] M. Benchohra, S.K. Ntouyas, A. Ouahab, Extremal solutions of second order impulsive dynamic equations on time scales, J. Math. Anal. Appl., 324(2006), 425-434. [3] M. Bohner, A. Peterson, Dynamic Equations on Time Scales:An Introduction with Applications, Birkh¨ auser, Boston, 2001. [4] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨ auser, Boston, 2003. [5] H. Chen, H. Wang, Q. Zhang, T. Zhou, Double positive solutions of boundary value problems for p-Laplacian impulsive functional dynamic equations on time scales, Comput. Math. Appl., 53(2007), 1473-1480.

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Received: December 28, 2012; Accepted: September 19, 2013