MULTIPLE POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR P-LAPLACIAN IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES

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1 Fixed Point Theory, 5(24, No. 2, nodeacj/fptcj.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 Mathematic, Ege Univerity 35 Bornova, Izmir, Turkey Department of Mathematic, Ege Univerity 35 Bornova, Izmir, Turkey Department of Mathematic, Ege Univerity 35 Bornova, Izmir, Turkey Abtract. In thi paper, by uing the claical fixed-point index theorem for compact map and Leggett-William fixed point theorem, ome ufficient condition for the exitence of multiple poitive olution for a cla of econd-order p-laplacian boundary value problem with impule on time cale are etablihed. We alo give an example to illutrate our reult Key Word and Phrae: Impulive dynamic equation, p-laplacian, poitive olution, fixed point theorem, time cale. 2 Mathematic Subject Claification: 34B8, 34B37, 34K, 47H.. Introduction It i known that the theory of impulive differential equation have become more important in recent year in ome mathematical model of real procee and phenomena. For the introduction of the baic theory of impulive equation, ee [, 2] and the reference therein. The theory of dynamic equation on time cale ha been developing rapidly and have received much attention in recent year. The tudy unifie exiting reult from the theory of differential and finite difference equation and provide powerful new tool for exploring connection between the traditionally eparated field. We refer to the book by Bohner and Peteron [3, 4]. Recently, the boundary value problem for impulive differential equation have been tudied extenively. To identify a few, we refer to the reader to ee [7, 8, 9,, 8, 2]. However, the correponding theory of uch equation i till in the beginning tage of it development, epecially the impulive dynamic equation on time cale, ee [2, 5, 9]. There i not o much work on impulive boundary value problem 475

2 476 ILKAY YASLAN KARACA, OZLEM BATIT OZEN AND FATMA TOKMAK with p-laplacian on time cale except that in [5, 6,, 6, 7]. To our knowledge, no paper ha conidered the econd-order BVP with integral boundary condition for p-laplacian impulive dynamic equation on time cale. Thi paper fill thi gap in the literature. In thi paper, we are concerned with the exitence of many poitive olution of the following boundary value problem for p-laplacian impulive dynamic equation on time cale (φ p (u (t = f(t, u(t, t [, ] T, t t k, k =, 2,..., m (. u(t k u(t k = I k(u(t k, k =, 2,..., m (.2 αu( u ( = u(, u ( = (.3 where T i a time cale,, T, [, ] T = [, ] T, t k (, T, k =, 2,..., m with < t < t 2 <... < t m <, α >, >, φ p ( i a p-laplacian function, i.e., φ p ( = p 2 for p >, (φ p ( = φ q ( where p q =. We aume that the following condition are atified: (H f C([, ] [,, [, ; (H2 I k C([,, [,, t k [, ] T and u(t k = lim h u(t k h, u(t k = lim h u(t k h repreent the right and left limit of u(t at t = t k, k =, 2,..., m. We remark that by a olution u of (.-(.3 we mean u : T R i delta differentiable, u : T k R i nabla differentiable on T k T k and u : T k T k R i continuou, and atifie the impulive and boundary value condition (.2-(.3. By uing the fixed point index theory in the cone [3], we get the exitence of at leat two or more poitive olution for the BVP (.-(.3. By uing Leggett- William fixed point theorem [4], we alo etablih the exitence of triple poitive olution for BVP (.-(.3. The organization of thi paper i a follow. In ection 2, we provide ome neceary background. In ection 3, the main reult for problem (.-(.3 are given. In ection 4, we give an example. 2. Preliminarie In thi ection, we lit ome background material from the theory of cone in Banach pace. Definition 2.. Let (E,. be a real Banach pace. A nonempty, cloed et K E i aid to be a cone provided the following are atified: (i if x, y K and a, b, then ax by K; (iiif y K and y K, then y =. If K E i 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 i called concave on I T = I T, if f(λt ( λ λf(t( λf(, for all t, I T and all λ [, ] uch that λt( λ I T.

3 P-LAPLACIAN IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES 477 Remark 2.3. If u on [, ] T k T k, then we ay that u i concave on [, ] T. Theorem 2.4. [3] Let K be a cone in a real Banach pace E. Let D be an open bounded ubet of E with D K = D K and D K K. Aume that T : D K K i completely continuou uch that x T x for x D K. Then the following reult hold: (i If T x x, x D K, then i K (T, D K =. (ii If there exit e K \ uch that x T x λe for all x D K and all λ >, then i K (T, D K =. (iii Let U be open in K uch that Ū D K. If i K (T, D K = and i K (T, U K =, then T ha a fixed point in D K \ ŪK. The ame reult hold if i K (T, D K = and i K (T, U K =. Let E be a real Banach pace with cone K, a map θ : K [, i aid to be nonnegative continuou concave functional on K if θ i continuou and θ(tx ( ty tθ(x ( tθ(y for all x, y K and t [, ] T. Let a, b be two number uch that < a < b and θ a nonnegative continuou concave functional on K. We define the following convex et: K a = x K : x < a, K(θ, a, b = x K : a θ(x, x < b. Theorem 2.5. [4] (Leggett-William fixed point theorem Let T : Kc K c be completely continuou and θ be a nonnegative continuou concave functional on P uch that θ(x x for all x K c. Suppoe there exit < d < a < b c uch that (i x K(θ, a, b : θ(x > a and θ(t x > a for x K(θ, a, b, (ii T x < d for x d, (iii θ(t x > a for x K(θ, a, c with T x > b. Then T ha at leat three fixed point x, x 2, x 3 in K c uch that x < d, a < θ(x 2 and x 3 > d with θ(x 3 < a. 3. Main reult In thi ection, by defining an appropriate Banach pace and cone, we impoe the growth condition on f, I k which allow u to apply the theorem in ection 2 to etablih the exitence reult of the poitive olution for the BVP (.-(.3. Let J = [, ] T \ t, t 2,..., t m. We define E = u : [, ] T R i continuou at t t k, there exit u(t k and u(t k with u(t k = u(t k for k =, 2,..., m, which i a Banach pace with the norm u = up u(t. t [,] T

4 478 ILKAY YASLAN KARACA, OZLEM BATIT OZEN AND FATMA TOKMAK By a olution of (.-(.3, we mean a function u E C 2 (J which atifie (.-(.3. We define a cone K E a K = u E : u i a concave, nonnegative and nondecreaing function, αu( u ( = u(. Lemma 3.. Suppoe that (H and (H2 are atified. Then u E C 2 (J i a poitive olution of the impulive boundary value problem (.-(.3 if and only if u(t i a olution of the following integral equation u(t = ( f(, u( ( σ(τφ q ( τ f(r, u(r r τ t ( φ q f(r, u(r r I k (u tk ( t k I k (u tk. (3. t k <t Proof. Integrating of (. from t to, one ha φ p (u ( φ p (u (t = t f(, u(. By the boundary condition (.3, we have ( u (t = φ q f(, u(. t Integrating the differential equation above from to t, we get t ( u(t = u( φ q f(r, u(r r I k (u tk. (3.2 t k <t Applying the boundary condition (.3, one ha u( = ( f(, u( ( ( σ(τφ q f(r, u(r r τ I k (u tk ( t k. (3.3 Therefore, by (3.2 and (3.3, we have u(t = ( f(, u( t φ q ( f(r, u(r r Then, ufficient proof i complete. ( σ(τφ q ( I k (u tk ( t k t k <t τ τ f(r, u(r r τ I k (u tk.

5 P-LAPLACIAN IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES 479 Converely, let u be a in (3.. If we take the delta derivative of both ide of (3., then ( u (t = φ q f(, u(, i.e., φ p (u (t = t t f(, u(. So u ( =. It i eay to ee that u(t atify (.2 and (.3. Furthermore, from (H, (H2 and (3., it i clear that u(t. The proof i complete. Lemma 3.2. If u K, then min u(t γ u, where γ =. t [,] T Proof. Since u K, nonnegative and nondecreaing u = u(, min u(t = u(. t [,] T On the other hand, u(t i concave on [, ] T \ t,..., t m. So, for every t [, ] T, we have u(t u( u( u(, t i.e., Therefore, u( tu( ( tu( u(t. Thi together with αu( u ( = The Lemma i proved. Define T : K E by u( ( u( u(. u(, implie that (T u(t = ( φq f(, u( t φ q ( f(r, u(r r u(. ( ( σ(τφ q τ I k (u tk ( t k t k <t f(r, u(r r τ I k (u tk. (3.4

6 48 ILKAY YASLAN KARACA, OZLEM BATIT OZEN AND FATMA TOKMAK From (3.4 and Lemma 3., it i eay to obtain the following reult. Lemma 3.3. Aume that (H and (H2 hold. Then T : K K i completely continuou. and We define Ω ρ = u K : min t [,] T u(t < γρ = u K : γ u min t [,] T u(t < γρ K ρ = u K : u < ρ. Lemma 3.4. [3] Ω ρ ha the following propertie: (a Ω ρ i open relative to K. (b K γρ Ω ρ K ρ. (c u Ω ρ if and only if min t [,]T u(t = γρ. (d If u Ω ρ, then γρ u(t ρ for t [, ] T. Now for convenience we introduce the following notation. Let fa b f(t, u = min min : u [a, b], t [,] T φ p (a fγρ ρ f(t, u = min min : u [γρ, ρ], t [,] T φ p (ρ f ρ = max f(t, u max : u [, ρ], t [,] T φ p (ρ I ρ (k = max I k(u : u [, ρ], l ( mα =, L =. Theorem 3.5. Suppoe (H and (H2 hold. (H3 If there exit ρ, ρ 2, ρ 3 (, with ρ < γρ 2 and ρ 2 < ρ 3 uch that f ρ < φ p (l, I ρ (k < lρ, fγρ ρ2 2 > φ p (L, f ρ3 < φ p (l, I ρ3 (k < lρ 3, then problem (.-(.3 ha at leat two poitive olution u, u 2 with u Ω ρ2 \ K ρ, u 2 K ρ3 \ Ω ρ2. (H4 If there exit ρ, ρ 2, ρ 3 (, with ρ < ρ 2 < γρ 3 < ρ 3 uch that fγρ ρ > φ p (L, f ρ2 < φ p (l, I ρ2 (k < lρ 2, fγρ ρ3 3 > φ p (L, then problem (.-(.3 ha at leat two poitive olution u, u 2 with u K ρ2 \ Ω ρ, u 2 Ω ρ3 \ K ρ2.

7 P-LAPLACIAN IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES 48 Proof. We only conider the condition (H3. If (H4 hold, then the proof i imilar to that of the cae when (H3 hold. By Lemma 3.3, we know that the operator T : K K i completely continuou. Firt, we how that i K (T, K ρ =. In fact, by (3.4, f ρ < φ p (l and I ρ (k < lρ, we have for u K ρ, (T u(t = = = < = t ( f(, u( φ q ( ( σ(φ q f(r, u(r r I k (u(t k ( t k I k (u(t k t k <t ( f(, u( φ q ( φ q f(r, u(r r I k (u(t k α α ( ( ( ( α lρ = lρ [ (m α ( f(, u( φ q f(r, u(r r f(r, u(r r I k (u(t k f(r, u(r r f(r, u(r r α I k (u(t k f(, u( α I k (u(t k φ p (lρ α lρ α lρ m ] = ρ, i.e., T u < u for u K ρ. By (i of Theorem 2.4, we obtain that i K (T, K ρ =. Secondly, we how that i K (T, Ω ρ2 =. Let e(t. Then e K. We claim that u T u λe, u Ω ρ2, λ >. Suppoe that there exit u Ω ρ2 and λ > uch that u = T u λ e. (3.5

8 482 ILKAY YASLAN KARACA, OZLEM BATIT OZEN AND FATMA TOKMAK Then, Lemma 3.2, Lemma 3.4 (d and (3.5 imply that for t [, ] T u = T u λ e γ T u λ γ f(, u( λ > γ φ p (Lρ 2 λ = γ Lρ 2 λ = γρ 2 λ, i.e., γρ 2 > γρ 2 λ, which i a contradiction. Hence by (ii of Theorem 2.4, it follow that i K (T, Ω ρ2 =. Finally, imilar to the proof of i K (T, K ρ =, we can prove that i K (T, K ρ3 =. Since ρ < γρ 2 and Lemma 3.4 (b, we have K ρ K γρ2 Ω ρ2. Similarly with ρ 2 < ρ 3 and Lemma 3.4 (b, we have Ω ρ2 K ρ2 K ρ3. Therefore (iii of Theorem 2.4 implie that BVP (.-(.3 ha at leat two poitive olution u, u 2 with u Ω ρ2 \ K ρ, u 2 K ρ3 \ Ω ρ2. Theorem 3.5 can be generalized to obtain many olution. Theorem 3.6. Suppoe (H and (H2 hold. Then we have the following aertion. (H5 If there exit ρ i 2m i= (, with ρ < γρ 2 < ρ 2 < ρ 3 < γρ 4 <... < γρ 2m < ρ 2m < ρ 2m uch that f ρ2m < φ p (l, I ρ2m < lρ 2m, (m =, 2,..., m, m, fγρ ρ2m > φ p (L, (m =, 2,..., m, then problem (.-(.3 ha at leat 2m olution in K. (H6 If there exit ρ i 2m i= (, with ρ < γρ 2 < ρ 2 < ρ 3 < γρ 4 <... < γρ 2m < ρ 2m uch that f ρ2m < φ p (l, I ρ2m < lρ 2m, f ρ2m γρ 2m > φ p (L, (m =, 2,..., m, then problem (.-(.3 ha at leat 2m olution in K. Theorem 3.7. Suppoe (H and (H2 hold. Then we have the following aertion. (H7 If there exit ρ i 2m i= (, with ρ < ρ 2 < γρ 3 < ρ 3 <... < ρ 2m < γρ 2m < ρ 2m uch that f ρ2m γρ 2m > φ p (L, (m =, 2,..., m, m, f ρ2m < φ p (l, I ρ2m < lρ 2m, (m =, 2,..., m, then problem (.-(.3 ha at leat 2m olution in K. (H8 If there exit ρ i 2m i= (, with ρ < ρ 2 < γρ 3 < ρ 3 <... < γρ 2m < ρ 2m < ρ 2m uch that fγρ ρ2m 2m > φ p (L, f ρ2m < φ p (l, I ρ2m < lρ 2m, (m =, 2,..., m, then problem (.-(.3 ha at leat 2m olution in K.

9 P-LAPLACIAN IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES 483 Theorem 3.8. Let < d < a < γb < b c and aume that condition (H, (H2 hold, and the following condition hold: (H9 There exit a contant c k uch that I k (y c k for k =, 2,..., m; (H fa b > φ p (L; (H f c φ p (R, f d < φ p (R where < R < α, d > α α R c k ( α α Then the BVP (.-(.3 ha at leat three nonnegative olution u, u 2, and u 3 in K c uch that. u < d, a < (u 2 and u 3 > d with (u 3 < a. Proof. By (H and (H2, T : K K i completely continuou. For u K, let θ(u = min u(t, t [,] T then it i eay to check that θ i a nonnegative continuou concave functional on K with θ(u u for u K. Let u K c, by (H9 and (H we have T u = max T u(t t [,] T = max t [,] T ( f(, u( t ( ( σ(φ q f(r, u(r r I k (u(t k I k (u(t k ( t k t k <t ( φ p (Rc φ q ( φ q φ p (Rc r ( α = Rc α c k < c, that i T u < c. φ q ( φ p (Rc r c k f(r, u(r r c k

10 484 ILKAY YASLAN KARACA, OZLEM BATIT OZEN AND FATMA TOKMAK Now, we how condition (ii of Theorem 2.5 hold. Let u d, it follow from (H that T u = max T u(t t [,] T = max t [,] T ( f(, u( t ( ( σ(φ q f(r, u(r r I k (u(t k I k (u(t k ( t k t k <t ( φ p (Rd φ q ( φ q φ p (Rd r ( α = Rd α c k < d. φ q ( φ p (Rd r c k f(r, u(r r So, condition (ii of Theorem 2.5 hold. Next, we how that u K(θ, a, b : θ(u > a = and θ(t u > a for u K(θ, a, b. In fact, take u(t = ab 2 > a, o u u K(θ, a, b : θ(u > a. Alo, u K(θ, a, b and (H condition imply that θ(t u = min (T u(t = T u( t [,] T = f(, u( > > ( σ(φ q ( I k (u(t k ( t k f(, u( c k f(r, u(r r ( a( φ p = a. Finally, we how that θ(t u > a for u K(θ, a, c with T u > b. From Lemma 3.2, we have θ(t u = min (T u(t γ T u > γb > a. t [,] T

11 P-LAPLACIAN IMPULSIVE DYNAMIC EQUATIONS ON TIME SCALES 485 So θ(t u > a i atified. Therefore all of the condition of Theorem 2.5 hold. Hence BVP (.-(.3 ha at leat three poitive olution u, u 2 and u 3 uch that u < d, a < θ(u 2 and u 3 > d with θ(u 3 < a. 4. Example Example 4.. Conider the following econd-order impulive p-laplacian boundary value problem, where and (φ 2 (u (t = f(t, u(t, t [, ] T \ 2, (4. u( 2 u( 2 = I (u( 2, (4.2 u ( =, 2u( u ( = f(t, u = 26 ( u, u [, ]; 5(u, u (, 4]; (u 4 53, u (4, ; 7 I (u = u. u(, (4.3 Taking ρ =, ρ 2 = 4, ρ 3 = 7, α = 2 and =, we have l = 5, L =, γ = 3. We can obtain that Now, we how that (H3 i atified: ρ < γρ 2 and ρ 2 < ρ 3. f = 26 < φ 2( 5 = 25, f 4 4 = > φ 2( =, f 7 <.39 < φ 2 ( 5 =.4, I = ( < lρ = 5 and I 7 = 7 < lρ 3 = 4. Then, all condition of Theorem 3.5 hold. Hence, we get the BVP (4.-(4.3 ha at leat two poitive olution. Reference [] M. Benchohra, J. Henderon, S. Ntouya, Impulive Differential Equation and Incluion, New York, 26. [2] M. Benchohra, S.K. Ntouya, A. Ouahab, Extremal olution of econd order impulive dynamic equation on time cale, J. Math. Anal. Appl., 324(26, [3] M. Bohner, A. Peteron, Dynamic Equation on Time Scale:An Introduction with Application, Birkhäuer, Boton, 2. [4] M. Bohner, A. Peteron, Advance in Dynamic Equation on Time Scale, Birkhäuer, Boton, 23. [5] H. Chen, H. Wang, Q. Zhang, T. Zhou, Double poitive olution of boundary value problem for p-laplacian impulive functional dynamic equation on time cale, Comput. Math. Appl., 53(27,

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