Research Article Positive Solutions for Neumann Boundary Value Problems of Second-Order Impulsive Differential Equations in Banach Spaces

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1 Abstract and Applied Analysis Volume 212, Article ID 41923, 14 pages doi:1.1155/212/41923 Research Article Positive Solutions for Neumann Boundary Value Problems of Second-Order Impulsive Differential Equations in Banach Spaces Xiaoya Liu and Yongxiang Li Department of athematics, Northwest Normal University, Lanzhou 737, China Correspondence should be addressed to Xiaoya Liu, Received 8 September 211; Revised 28 December 211; Accepted 6 January 212 Academic Editor: Yuriy Rogovchenko Copyright q 212 X. Liu and Y. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The existence of positive solutions for Neumann boundary value problem of second-order impulsive differential equations u t u t f t, u t, t J, t / t k, Δu t tk I k u t k, k 1, 2,...,m, u u 1 θ, in an ordered Banach space E was discussed by employing the fixed point index theory of condensing mapping, where > is a constant, J, 1, f C J K, K, I k C K, K, k 1, 2,...,m,andK is the cone of positive elements in E.oreover, an application is given to illustrate the main result. 1. Introduction The theory of impulsive differential equations is a new and important branch of differential equation theory, which has an extensive physical, chemical, biological, engineering background and realistic mathematical model, and hence has been emerging as an important area of investigation in the last few decades; see 1. Correspondingly, boundary value problems of second-order impulsive differential equations have been considered by many authors, and some basic results have been obtained; see 2 7. But many of them obtained extremal solutions by monotone iterative technique coupled with the method of upper and lower solutions; see 2 4. The research on positive solutions is seldom and most in real space R; see 5 7. In this paper, we consider the existence of positive solutions to the second-order impulsive differential equation Neumann boundary value problem in an ordered Banach space E: u t u t f t, u t, t J, t / t k, Δu t tk I k u t k, k 1, 2,...,m, u u 1 θ, 1.1

2 2 Abstract and Applied Analysis where >isaconstant, f C J E, E, J, 1 ; <t 1 <t 2 < <t m < 1; I k C E, E is an impulsive function, k 1, 2,...,m. Δu t tk denotes the jump of u t at t t k, that is, Δu t tk u t k u t k, where u t k and u t k represent the right and left limits of u t at t t k, respectively. In the special case where E R,, I k, k 1, 2,...,m,NBVP 1.1 has been proved to have positive solutions; see 8, 9. otivated by the aforementioned facts, our aim is to study the positive solutions for NBVP 1.1 in a Banach space by fixed point index theory of condensing mapping. oreover, an application is given to illustrate the main result. As far as we know, no work has been done for the existence of positive solutions for NBVP 1.1 in Banach spaces. 2. Preliminaries Let E be an ordered Banach space with the norm and partial order, whose positive cone K {x E x θ} is normal with normal constant N. LetJ, 1 ; t <t 1 < t 2 < < t m < t m 1 1; J k t k 1,t k, k 1, 2,...,m 1, J J \{t 1,t 2,...,t m }.Let PC 1 J, E {u C J, E u t is continuous at t / t k, and left continuous at t t k,and u t k exists, k 1, 2,...,m}. Evidently, PC1 J, E is a Banach space with the norm u PC 1 max{ u t C, u PC }, where u C sup t J u t, u PC sup t J u t ;see 2. An abstract function u PC 1 J, E C 2 J,E is called a solution of NBVP 1.1 if u t satisfies all the equalities of 1.1. Let C J, E denote the Banach space of all continuous E-value functions on interval J with the norm u C sup t J u t. Letα denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see 1, 11. For any B C J, E and t J, setb t {u t u B} E. IfB is bounded in C J, E, then B t is bounded in E, andα B t α B. Now, we first give the following lemmas in order to prove our main results. Lemma 2.1 see 12. Let B C J, E be equicontinuous. Then α B t is continuous on J, and α B max t J α B t α B J. 2.1 Lemma 2.2 see 13. Let B {u n } C J, E be a bounded and countable set. Then α B t is Lebesgue integral on J, and α { J u n t dt n N }) 2 α B t dt. J 2.2 Lemma 2.3 see 14. Let D E be bounded. Then there exists a countable set D D, such that α D 2α D.

3 Abstract and Applied Analysis 3 To prove our main results, for any h C J, E, we consider the Neumann boundary value problem NBVP of linear impulsive differential equation in E: u t u t h t, t J, Δu t tk y k, k 1, 2,...,m, 2.3 u u 1 θ, where >, y k E, k 1, 2,...,m. Lemma 2.4. For any h C J, E, >, and y k E, k 1, 2,...,m, the linear NBVP 2.3 has a unique solution u PC 1 J, E C 2 J,E given by u t G t, s h s ds G t, t k y k, 2.4 where G t, s cosh 1 t cosh s sinh, s t 1, cosh t cosh 1 s sinh, t<s Proof. Suppose that u t is a solution of 2.3 ; then u t u t h t, [ e 2 t e ] u t ) t e t u t e t u t e t h t. 2.6 Let y t e 2 t e t u t ; then y t e t h t, Δy t tk e t k y k. 2.7 Integrating 2.7 from to t 1, we have 1 y t 1 y e s h s ds. 2.8 Again, integrating 2.7 from t 1 to t, where t t 1,t 2, then y t y t ) t 1 e t s h s ds y e s h s ds e t 1 y 1. t 1 2.9

4 4 Abstract and Applied Analysis Repeating the aforementioned procession, for t J, we have y t y e s h s ds e t k y k. <t k <t 2.1 Hence, ) t e u t e 2 t y e s h s ds e t k y k ) <t k <t For t J, integrating 2.11 from to t, we have u t e t u s e 2s y ds e 2 s e ) 2 s e t k y k ds <t k <s { [ e t 1 ) u 2 y e 2t 1 e 2 t e τ h τ dτ ds e s h s ds e s h s ds ) e 2t e 2 t k e t k y k ]}. <t k <t 2.12 Notice that y u u ; thus,fort J, we have [ 1 u t 2 e t 2 ) u u u e 1 2 ) u u e t e t t s e h s ds t e t e s h s ds ] ) e 2t e 2 t k e t t k y k <t k <t [ ) ) u u e t u u e t

5 Abstract and Applied Analysis 5 e t <t k <t s e h s ds e t e t tk y k e tk t y k ) ], e s h s ds u t 1 2 [ ) ) u u e t u u e t 2.13 e t <t k <t s e h s ds e t e t tk y k e t t k y k ) ]. e s h s ds 2.14 In view of that u u 1 θ, we have u e 1 s e 1 s e 1 t k e 1 t k e ) h s ds e e ) e cosh 1 s sinh h s ds cosh 1 t k sinh y k Substituting 2.15 into 2.13, fort J, weobtain u t e 1 t e 1 t ) e s e s ) <t k <t 2 e ) h s ds e e 1 t e 1 t ) e t k e t k ) 2 e ) y k e t t t k <1 cosh t cosh 1 s h s ds sinh cosh t cosh 1 t k y k sinh G t, s h s ds G t, t k y k Inversely, we can verify directly that the function u PC 1 J, E C 2 J,E defined by 2.4 is a solution of the linear NBVP 2.3. Therefore, the conclusion of Lemma 2.4 holds.

6 6 Abstract and Applied Analysis By 2.5, itiseasytoverifythatg t, s has the following property: 1 cosh 2 G t, s. sinh sinh 2.17 Evidently, C J, E is also an ordered Banach space with the partial order reduced by the positive cone C J, K {u C J, E u t θ, t J}. C J, K is also normal with the same normal constant N. Define an operator A : C J, K C J, K as follows: Au t G t, s f s, u s ds G t, t k I k u t k Clearly, A : C J, K C J, K is continuous, and the positive solution of NBVP 1.1 is the nontrivial fixed point of operator A. However, the integral operator A is noncompactness in general Banach space. In order to employ the topological degree theory and the fixed point theory of condensing mapping, there demands that the nonlinear f and impulsive function I k satisfy some noncompactness measure condition. Thus, we suppose the following. P For any R>, f J K R and I k K R are bounded and α f t, D ) Lα D, α I k D k α D, k 1, 2,...,m, 2.19 where K R K B θ, R, D K is arbitrarily countable set, L>and k are counsants and satisfy 4L/ 2cosh 2 m k/ sinh < 1. Lemma 2.5. Suppose that condition P is satisfied; then A : C J, K C J, K is condensing. Proof. Since A B is bounded and equicontinuous for any bounded and nonrelative compact set B C J, K, by Lemma 2.3, there exists a countable set B 1 {u n } B, such that α A B 2α A B By assumption P and Lemma 2.1, { }) α A B 1 t α G t, s f s, u n s ds G t, t k I k u n t k { }) { }) α G t, s f s, u n s ds α G t, t k I k u n t k 2 G t, s α f s, B 1 s ) ds G t, t k α I k B 1 t k

7 Abstract and Applied Analysis 7 2 2L G t, s Lα B 1 s ds G t, s dsα B 1 G t, t k k α B 1 t k k G t, t k α B 1 2L α B 1 cosh2 m k α B 1 sinh 2L cosh2 m k )α B 1. sinh 2.21 Since A B 1 is equicontinuous, by Lemma 2.1, we have α A B 1 max t J 2L α A B 1 t cosh2 m k )α B sinh Combining 2.2 and P, we have 4L α A B 2α A B 1 2cosh2 m k )α B sinh Hence, A : C J, K C J, K is condensing. Let P be a cone in C J, K defined by P {u C J, K u t σu τ, t, τ J}, 2.24 where σ 1/cosh 2. Lemma 2.6. For any f J, K K, A C J, K P. Proof. For any u C J, K, t,τ J,by 2.18 and the second inequality of 2.17, we have A u τ G τ, s f s, u s ds G τ, t k I k u t k cosh 2 1 f s, u s ds sinh ) I k u t k. 2.25

8 8 Abstract and Applied Analysis By this, 2.18, and the first inequality of 2.17, we have A u t G t, s f s, u s ds G t, t k I k u t k 1 1 f s, u s ds sinh ) I k u t k 2.26 σa u τ. Hence, A C J, K P. Thus, for any f J, K K, A : P P is condensing mapping; the positive solution of NBVP 1.1 is equivalent to the nontrivial fixed point of A in P. For <r<r<, let P r {u P u C <r}, and P r {u P u C r}, which is the relative boundary bound of P r in P. Denote that P r,r P R \ P r ; then the fixed point of A in P r,r is the positive solution of NBVP 1.1. We will use the fixed point theory of condensing mapping to find the fixed point of A in P r,r. Let X be a Banach space and let P X be a cone in X. Assume that Ω is a bounded open subset of X and let Ω be its bound. Let Q : P Ω P be a condensing mapping. If Qu / u for every u P Ω, then the fixed point index i Q, P Ω,P is defined. If i Q, P Ω,P /, then Q has a fixed point in P Ω. As the fixed point index theory of completely continuous mapping, see 1, 11, we have the following lemmas that are needed in our argument for condensing mapping. Lemma 2.7. Let Q : P P be condensing mapping; if u / λau, u P r, <λ 1, 2.27 then i Q, P r,p 1. Lemma 2.8. Let Q : P P be condensing mapping; if there exists v P, v / θ, such that u Au / τv, u P r,τ, 2.28 then i Q, P r,p. 3. ain Results P1 i There exist δ >, a, a k >, such that for all x P δ and t J, f t, x ax, I k x a k x,anda m a k/σ 2 <. ii There exist b, b k >, h C J, K, andy k K, such that for all x P and t J, f t, x bx h t, I k x b k x y k,andb σ 2 m b k >. P2 i There exist δ>, b, b k >, such that for all x P δ and t J, f t, x bx, I k x b k x,andb σ 2 m b k >. ii There exist a, a k >, h C J, K, andy k K, such that for all x P and t J, f t, x ax h t, I k x a k x y k,anda m a k/σ 2 <.

9 Abstract and Applied Analysis 9 Theorem 3.1. Let E be an ordered Banach space, whose positive cone K is normal, f C J K, K, and I k C K, K, k 1, 2,...,m. Suppose that conditions P and P1 or P2 are satisfied; then the NBVP 1.1 has at least one positive solution. Proof. We show, respectively, that the operator A defined by 2.18 has a nontrivial fixed point in two cases that P1 is satisfied and P2 is satisfied. Case 1. Assume that P1 is satisfied; let <r<δ, where δ is the constant in condition P1, to prove that A satisfies u / λau, u P r, <λ If 3.1 is not true, then there exist u P r and <λ 1, such that u λ Au ;bythe definition of A, u t satisfies u t u t λ f t, u t, t J, t / t k, Δu t t k λ I k u t k, k 1, 2,...,m, 3.2 u u 1 θ. Integrating 3.2 from to 1, using i of assumption P1, we have a u t dt a k u t k. 3.3 Since u P, for any t, s J, by the definition of P, we have u t σu s, u t k 1/σ u s,andthus m σ a u s a k u s, σ 3.4 that is; a m a k/σ 2 u s θ. Soweobtainthatu s θ in J, which contracts with u P r. Hence 3.1 is satisfied; by Lemma 2.7, we have i A, P r,p Let e C J, K, e 1, v t e, and obviously v P. We show that if R is large enough, then u Au / τv, u P R,τ. 3.6

10 1 Abstract and Applied Analysis In fact, if there exist u P R, τ such that u Au τ v, then Au u τ v ;bythe definition of A, u t satisfies u t u t τ v f t, u t, t J, t / t k, Δu t t k I k u t k, k 1, 2,...,m, 3.7 u u 1 θ. By ii of assumption P1, we have u t u t f t, u t τ v bu t h t, t J. 3.8 Integrating on J and using ii of assumption P1, we have b u t dt b k u t k h t dt y k. 3.9 If b>, for any t, s J, by the definition of P, we have u t σu s ; u t k σu s ;thus σ b σ b k )u s h t dt y k. 3.1 By σ b σ m b k > and the normality of cone K, we have u C N h C m y k ) σ b σ m b R 1. k 3.11 If b, then for any t, s J, by the definition of P, we have u t 1/σ u s, u t k σu s ;thus b σ b k )u s σ h t dt y k By b σ 2 m b k >, and the normality of K, we have u C N h C m y k ) b /σ σ m b k R Let R>max{R 1,R 2,r}; then 3.6 is satisfied; by Lemma 2.8, we have i A, P R,P. 3.14

11 Abstract and Applied Analysis 11 Combining 3.5, 3.14, and the additivity of fixed point index, we have i A, P r,r,p i A, P R,P i A, P r,p 1 / Therefore A has a fixed point in P r,r, which is the positive solution of NBVP 1.1. Case 2. Assume that P2 is satisfied; let <r<δ, where δ is the constant in condition P2, to proof that A satisfies u Au / τv, u P r,τ, 3.16 where v t e P, e / θ. In fact, if there exists u P r and τ, such that u Au τ v, then u satisfies 3.7 and i of condition P2, and we have u t u t f t, u t τ v bu t, t J Integrating on J and using i of assumption P2, we have b u t dt b k u t k θ If b>, for any t, s J, by the definition of P, we have u t σu s, u t k σu s, for all t, s J, andthus σ b σ b k )u s θ By σ b σ m b k >, we obtain that u s θ, which contracts with u P r. Hence 3.16 is satisfied. If b, for any t, s J, by the definition of P, we have u t 1/σ u s, u t k σu s, for all t, s J, andthus 1 σ b σ b k )u s θ. 3.2 By b σ 2 m b k >,weobtainthatu s θ, which contracts with u P r. Hence 3.16 is satisfied. Hence, by Lemma 2.8, we have i A, P r,p Next, we show that if R is large enough, then u / λau, u P R, <λ

12 12 Abstract and Applied Analysis In fact, if there exists u P R and <λ 1 such that u λ Au, then u satisfies 3.2. Integrating 3.2 on J, andusing ii of P2, we have a u t dt a k u t k h t dt y k For any t, s J, by the definition of P, we have u t σu s, u t k 1/σ u s, for all t, s J, andthus σ a 1 σ a k )u s h t dt y k By a m a k/σ 2 <, we have u s h t dt m y k σ a 1/σ m a k By the normality of K, we have u s N h t dt m y k σ a 1/σ m a k Thus u C N h C m ) y k σ a 1/σ m a k R Let R>max{R 3,r}; then 3.22 is satisfied; by Lemma 2.7, we have i A, P R,P Combining 3.21, 3.28, and the additivity of fixed index, we have i A, P r,r,p i A, P R,P i A, P r,p Therefore A has a fixed point in P r,r, which is the positive solution of NBVP 1.1. Remark 3.2. The conditions P1 and P2 are a natural extension of suplinear condition and sublinear condition in Banach space E. Hence if I k θ, then Theorem 3.1 improves and extends the main results in 8, 9.

13 Abstract and Applied Analysis Example We provide an example to illustrate our main result. Example 4.1. Consider the following problem: 2 w t, x w t, x t2 e t s w 2 s, x ds 1 1 w t, x, t J, x I, ) 1 2,x, x I, Δ t w t, x t 1/2 1 1 w t w,x w 1,x, x I, t 4.1 where J, 1, J J \{1/2}, I,T,and T> is a constant. Conclusion Problem 4.1 has at least one positive solution. Proof. Let E C I, and K {w C I w x, x I}; then E is a Banach space with norm w max t I w x, andk is a positive cone of E. Letu t w t, ; then the problem 4.1 can be transformed into the form of NBVP 1.1, where 1, f t, u e t s u 2 s ds 1/1 u t,and I 1 u 1/2 1/1 u 1/2. Evidently C J, E is a Banach space with norm u C max t J u t, andc J, K is positive cone of C J, E. LetP {u C J, K u t σu s, t,s J}, where σ 1/cosh 2 1; then P is a cone in C J, K, and for any u P, t J, we have σ u C u t u C. Next, we will verify that the conditions P and P1 in Theorem 3.1 are satisfied. It is easy to verify that for any R>, f J K R and I 1 K R are bounded. Let g t, u e t s u 2 s ds; then g is completely continuous. So for any countable bounded set D K, we have α f t, D ) α g t, D ) 1 1 α D 1 1 α D, α I 1 D 1 α D, and for L 1/1, and 1 1/1, by simple calculations, 4L/ 2cosh 2 1 / sinh < 1. So condition P is satisfied. Let δ 1/1; then for u P δ, we have f t, u δ e t s ds u t 1 δ e 1 σ 1 u t 1 ) u t, σ 1 )) 1 I 1 u 1 ) u 1 2 1σ u t. 4.3 Let a δ e 1 /σ 1/1, a 1 1/1 σ; by simple calculations, we have a a 1 /σ 2 < 1. So i of condition P1 is satisfied.

14 14 Abstract and Applied Analysis Let R 1, for u P, u C R; we have f t, u 1/2 σ 2 R 1/1 u t. For u P, u C R, we have f t, u R 2 e t 1. Hence, let h t 1e t 1, y 1 ; then for any u P, we have 1 f t, u 2 σ2 R 1 1 )) 1 I 1 u 2 ) u t h t 1 1 σu t. 4.4 Let b 1/2 σ 2 R 1/1,and b 1 1/1 σ; by simple calculations, we have b σ 2 b 1 > 1, so ii of condition P1 is satisfied. By Theorem 3.1, Problem 4.1 has at least one positive solution. Acknowledgment This work was supported by NNSF of China no , References 1 V. Lakshmikantham, D. D. Baĭnov,andP.S.Simeonov,Theory of Impulsive Differential Equations, vol. 6 of Series in odern Applied athematics, World Scientific, Teaneck, NJ, USA, X. Liu and D. Guo, ethod of upper and lower solutions for second-order impulsive integrodifferential equations in a Banach space, Computers & athematics with Applications, vol. 38, no. 3-4, pp , W. Ding and. Han, Periodic boundary value problem for the second order impulsive functional differential equations, Applied athematics and Computation, vol. 155, no. 3, pp , Yao, A. Zhao, and J. Yan, Periodic boundary value problems of second-order impulsive differential equations, Nonlinear Analysis: Theory, ethods & Applications, vol. 7, no. 1, pp , Y. Tian, D. Jiang, and W. Ge, ultiple positive solutions of periodic boundary value problems for second order impulsive differential equations, Applied athematics and Computation, vol. 2, no. 1, pp , X. Lin and D. Jiang, ultiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations, athematical Analysis and Applications, vol. 321, no. 2, pp , Q. Li, F. Cong, and D. Jiang, ultiplicity of positive solutions to second order Neumann boundary value problems with impulse actions, Applied athematics and Computation, vol. 26, no. 2, pp , D.-q. Jiang and H.-z. Liu, Existence of positive solutions to second order Neumann boundary value problems, athematical Research and Exposition, vol. 2, no. 3, pp , 2. 9 J.-P. Sun and W.-T. Li, ultiple positive solutions to second-order Neumann boundary value problems, Applied athematics and Computation, vol. 146, no. 1, pp , K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology, Jinan, China, D. J. Guo and J. X. Sun, Ordinary Differential Equations in Abstract Spaces, Shandong Science and Technology, Jinan, China, H.-P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Analysis: Theory, ethods & Applications, vol. 7, no. 12, pp , Y. X. Li, Existence of solutions to initial value problems for abstract semilinear evolution equations, Acta athematica Sinica, vol. 48, no. 6, pp , 25 Chinese.

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