Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 21, Article ID 49741, 14 pages doi:1.1155/21/49741 Research Article Nonlinear Boundary Value Problem of First-Order Impulsive Functional Differential Equations Kexue Zhang 1 and Xinzhi Liu 2 1 School of Control Science and Engineering, Shandong University, Jinan, Shandong 2561, China 2 Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1 Correspondence should be addressed to Xinzhi Liu, xzliu@math.uwaterloo.ca Received 8 December 29; Accepted 3 January 21 Academic Editor: Shusen Ding Copyright q 21 K. Zhang and X. Liu. 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. This paper investigates the nonlinear boundary value problem for a class of first-order impulsive functional differential equations. By establishing a comparison result and utilizing the method of upper and lower solutions, some criteria on the existence of extremal solutions as well as the unique solution are obtained. Examples are discussed to illustrate the validity of the obtained results. 1. Introduction It is now realized that the theory of impulsive differential equations provides a general framework for mathematical modelling of many real world phenomena. In particular, it serves as an adequate mathematical tool for studying evolution processes that are subjected to abrupt changes in their states. Some typical physical systems that exhibit impulsive behaviour include the action of a pendulum clock, mechanical systems subject to impacts, the maintenance of a species through periodic stocking or harvesting, the thrust impulse maneuver of a spacecraft, and the function of the heart. For an introduction to the theory of impulsive differential equations, refer to 1. It is also known that the method of upper and lower solutions coupled with the monotone iterative technique is a powerful tool for obtaining existence results of nonlinear differential equations 2. There are numerous papers devoted to the applications of this method to nonlinear differential equations in the literature, see 3 9 and references therein. The existence of extremal solutions of impulsive differential equations is considered in papers 3 11. However, only a few papers have implemented the technique in nonlinear boundary value problem of impulsive differential equations 5, 12. In this paper, we will investigate
2 Journal of Inequalities and Applications nonlinear boundary value problem of a class of first-order impulsive functional differential equations. Such equations include the retarded impulsive differential equations as special cases 5, 12 14. The rest of this paper is organized as follows. In Section 2, we establish a new comparison principle and discuss the existence and uniqueness of the solution for first order impulsive functional differential equations with linear boundary condition. We then obtain existence results for extremal solutions and unique solution in Section 3 by using the method of upper and lower solutions coupled with monotone iterative technique. To illustrate the obtained results, two examples are discussed in Section 4. 2. Preliminaries Let J,T, T >, J J {t 1,t 2,...,t p } with <t 1 <t 2 < <t p <T. We define that PC J {x : J R : x is continuous for any t J ; x t k and x t k exist and x t k x t k }, PC 1 J {x : J R : x is continuously differentiable for any t J ; x t k, x t exist and k x t k x t k }. It is clear that PC J and PC 1 J are Banach spaces with respective norms x PC sup x t, x PC 1 x PC x PC. 2.1 t J Let us consider the following nonlinear boundary value problem NBVP : x t f t, x t, [ ϕx ] t, t J J { t 1,t 2,...,t p }, Δx t I k t, x t, [ ϕx ] t, t tk,k 1, 2,...,p, 2.2 g x,x T, where f : J R 2 R is continuous in the second and the third variables, and for fixed x, y R, f,x,y PC J, g C R 2, R, I k C R 3, R, k 1, 2,...,p and ϕ : PC J PC J is continuous. A function x PC 1 J is called a solutions of NBVP 2.2 if it satisfies 2.2. Remark 2.1. i If ϕx t x t and the impulses I k depend only on x t k, the equation of NBVP 2.2 reduces to the simpler case of impulsive differential equations: x t f t, x t, t J, Δx t k I k x t k, k 1, 2,...,p 2.3 which have been studied in many papers. In some situation, the impulse I k depends also on some other parameters e.g., the control of the amount of drug ingested by a patient at certain moments in the model for drug distribution 1, 3. ii If ϕx t x θ t, where θ C J, J, the equation of NBVP 2.2 can be regarded as retarded differential equation which has been considered in 5, 12 14. We will need the following lemma.
Journal of Inequalities and Applications 3 Lemma 2.2 see 1. Asumme that B the sequence {t k } satisfies t <t 1 <t 2 < <t k < with lim k t k, B 1 m PC 1 R is left continous at t k for k 1, 2,..., B 2 for k 1, 2,..., t t, m t p t m t q t, m t k dk m t k b k, t/ t k 2.4 Then where p, q C R, R, d k and b k are real constants. m t m t t d k exp p s ds t t <t k <t t d k exp t s<t k <t t <t k <t t k <t j <t t d j exp p σ dσ q s ds s t p s ds b k. t k 2.5 In order to establish a comparison result and some lemmas, we will make the following assumptions on the function ϕ. H1 There exists a constant R> such that [ ] ϕx t R inf x t, for any x PC J, t J. 2.6 t J H2 The function ϕ satisfies Lipschitz condition, that is, there exists a L> such that ϕx ϕy PC L x y PC, x, y PC J. 2.7 Inspired by the ideas in 5, 6, we shall establish the following comparison result. Theorem 2.3. Let m PC 1 J such that m t Mm t N [ ϕm ] t, t J, Δm t k L k m t k, k 1, 2,...,p, 2.8 m μm T, where M>, N, L k < 1, k 1, 2,...,p, and <μe MT 1.
4 Journal of Inequalities and Applications Suppose in addition that condition H1 holds and NR e MT μ μ p 1 L k e Ms ds 1 L k 2 2.9 then m t, for t J. Proof. For simplicity, we let 1 L k, k 1, 2,...,p.Setv t m t e Mt, then we have v t Ne Mt[ ϕm ] t, t J, v t k ck v t k, k 1, 2,...,p, 2.1 v μe MT v T. Obviously, v t implies m t. To show v t, we suppose, on the contrary, that v t > for some t J. It is enough to consider the following cases. i there exists a t J, such that v t >, and v t for all t J; ii there exist t,t J, such that v t <, v t >. Case i. By 2.1, we have v t fort / t k and Δv t k, k 1, 2,...,m, hence v t is nonincreasing in J,thatis,v T v.ifμ<e MT, then v <v T, which is a contradiction. If μ e MT, then v v T which implies v t C>. But from 2.1, wegetv t < for t J. Hence, v T <v. It is again a contradiction. Case ii. Let inf t J v t λ, then λ>. For some i {1, 2,...,p}, there exists t t i,t i 1 such that v t λ or v t λ. We only consider v t λ, as for the case v t λ,the proof is similar. From 2.1 and condition H1, weget v t Ne Mt[ ϕm ] t Ne Mt[ ϕ v t e Mt] t { NRe Mt inf v t e Mt} NRe Mt inf {v t } t J t J 2.11 λnre Mt, t J. Consider the inequalities v t λnre Mt, t J, v t k ck v t k, k 1, 2,...,p. 2.12
Journal of Inequalities and Applications 5 By Lemma 2.2, we have v t v t t <t k <t t t λnre Ms ds, 2.13 s<t k <t that is v t λ λnr t <t k <t t t e Ms ds. 2.14 s<t k <t First, we assume that t >t.lett t in 2.14, then v t λ t <t k <t λnr t t s<t k <t e Ms ds. 2.15 Noting that v t >, we have t <t k <t <NR t t s<t k <t e Ms ds. 2.16 Hence p 2 p <NR e Ms ds 2.17 which is a contradiction. Next, we assume that t <t.bylemma 2.2 and 2.1, we have <v t v μe MT v T <t k <t <t k <t t λnre Ms ds s<t k <t t λnr e Ms ds, s<t k <t 2.18 then p <μe MT v T λnr e Ms ds. 2.19
6 Journal of Inequalities and Applications Setting t T in 2.14, we have v T v t t <t k <T t λnre Ms ds λ λnr t <t k <T t e Ms ds 2.2 λ p λnr e Ms ds with 2.19, weobtainthat p <μe MT v T λnr e Ms ds [ p ] p λ λnr e Ms ds μe MT λnr p 2 p μλe MT μλnre MT e Ms ds e Ms ds λnr e Ms ds, 2.21 that is, p 2 [ p ] μe MT μnre MT NR e Ms ds T <NR μe MT 1 e Ms ds. 2.22 Therefore, p 1 L k 2 < NR e MT μ μ 1 L k e Ms ds 2.23 which is a contradiction. The proof of Theorem 2.3 is complete. The following corollary is an easy consequence of Theorem 2.3.
Journal of Inequalities and Applications 7 Corollary 2.4. Assume that there exist M>, N, L k < 1, fork 1, 2,...,p such that m PC 1 J satisfies 2.8 with <μe MT 1 and RN e MT μ e MT μ p 1 L k 2 1 L k ds 2.24 then m t, fort J. Remark 2.5. Setting μ 1, Corollary 2.4 reduces to the Theorem 2.3 of Li and Shen 6. Therefore, Theorem 2.3 and Corollary 2.4 develops and generalizes the result in 6. Remark 2.6. We show some examples of function ϕ satisfying H1. i ϕx t x θ t, where θ C J J, satisfies H1 with R 1, [ ] ϕx t x θ t inf x t, for t J. 2.25 t J ii ϕx t t T x s ds,satisfies H1 with R T, [ ϕx ] t t T x s ds t T inf t J x t Tinf x t, for t J. 2.26 t J Consider the linear boundary value problem LBVP y t My t N [ ϕy ] t σ t, t J, Δy t k L k y t k I k tk,η t k, [ ϕη ] t k L k η t k, k 1, 2,...,p, 2.27 g η,η T M 1 y η M2 y T η T, where M>, N, L k < 1, k 1, 2,...,p,andη, σ PC J. By direct computation, we have the following result. Lemma 2.7. y PC 1 J is a solution of LBVP 2.27 if and only if y is a solution of the impulsive integral equation y t Ce Mt Bη G t, s { σ s N [ ϕy ] s } ds G t, t k { L k y t k I k tk,η t k, [ ϕη ] t k L k η t k }, t J, <t k <T 2.28 where Bη g η,η T M 1 η M 2 η T, C M 1 M 2 e MT 1, M 1 / M 2 e MT and CM 2 e M s t T e M s t, s<t T G t, s 2.29 CM 2 e M s t T, t s T
8 Journal of Inequalities and Applications Lemma 2.8. Let H2 hold. Suppose further p NLT L k r<1, r max{ CM 1, CM 2 }, C M 1 M 2 e MT 1, 2.3 where M>, N, M 1 / M 2 e MT,thenLBVP 2.27 has a unique solution. By Lemma 2.7 and Banach fixed point theorem, the proof of Lemma 2.8 is apparent, so we omit the details. 3. Main Results In this section, we use monotone iterative technique to obtain the existence results of extremal solutions and the unique solution of NBVP 2.2. We shall need the following definition. Definition 3.1. A function α PC 1 J is said to be a lower solution of NBVP 2.2 if it satisfies α t f t, α t, [ ϕα ] t, t J, Δα t k I k tk,α t k, [ ϕα ] t k, g α,α T. k 1, 2,...,p, 3.1 Analogously, β PC 1 J is an upper solution of NBVP 2.2 if β t f t, β t, [ ϕβ ] t, t J, Δβ t k I k tk,β t k, [ ϕβ ] t k, g β,β T. k 1, 2,...,p, 3.2 For convenience, let us list the following conditions. H3 There exist constants M>, N such that f t, x, ϕx f t, x, ϕx M x x N ϕx ϕx 3.3 wherever α t x x β t. H4 There exist constants L k < 1fork 1, 2,...,psuch that I k tk,x,ϕx I k tk, x, ϕx L k x x, k 1, 2,...,p 3.4 wherever α t k x x β t k. H5 The function ϕ satisfies ϕx ϕx ϕ x x, for x, x PC J, x x. 3.5
Journal of Inequalities and Applications 9 H6 There exist constants M 1, M 2 with <M 2 e MT M 1 such that g x, y g x, y M 1 x x M 2 y y 3.6 wherever α x x β,andα T y y β T. Let α,β {x PC 1 J : α t x t β t, for all t J}. Now we are in the position to establish the main results of this paper. Theorem 3.2. Let H 1 H 6 and inequalities 2.9 and 2.3 hold. Assume further that there exist lower and upper solutions α and β of NBVP 2.2, respectively, such that α β on J. Then there exist monotone sequences {α n }, {β n } PC 1 J with α α n β n β,such that lim n α n x t, lim n β n x t uniformly on J. Moreover, x t, x t are minimal and maximal solutions of NBVP 2.2 in α,β, respectively. Proof. For any η α,β, consider LVBP 2.27 with σ t f t, η t, [ ϕη ] t Mη t N [ ϕη ] t. 3.7 By Lemma 2.8, we know that LBVP 2.27 has a unique solution y PC 1 J. Define an operator A : PC J PC J by y Aη, then the operator A has the following properties: a α Aα, Aβ β, b Aη 1 Aη 2,ifα η 1 η 2 β. To prove a,letα 1 Aα and m t α t α 1 t. m t α t α 1 t f t, α t, [ ] ϕα t { Mα 1 t N [ ] ϕα 1 t f t, α t, [ ] ϕα t Mα t N [ ] } ϕα t Mm t N [ ϕm ] t, Δm t k Δα t k Δα 1 t k I k tk,α t k, [ ] ϕα tk { L k α 1 t k I k tk,α t k, [ ] ϕα tk L k α t k } 3.8 L k m t k, m α α 1 { α 1 g α,α T α M } 2 α 1 T α T M 1 M 1 M 2 M 1 m T. By Theorem 2.3, wegetm t fort J, thatis,α Aα. Similarly, we can show that Aβ β.
1 Journal of Inequalities and Applications To prove b, setm t x 1 t x 2 t, where x 1 Aη 1 and x 2 Aη 2.Using H3, H4 and H6,weget m t x 1 t x 2 t M η 1 t x 1 t N [ ϕη 1 ] t [ ϕx1 ] t f t, η1 t, [ ϕη 1 ] t M η 2 t x 2 t N [ ϕη 2 ] t [ ϕx2 ] t f t, η2 t, [ ϕη 2 ] t Mm t N [ ϕm ] t, Δm t k Δx 1 t k Δx 2 t k L k η1 t k x 1 t k I k tk,η 1 t k, [ ] ϕη 1 tk L k η2 t k x 2 t k I k tk,η 2 t k, [ ] ϕη 2 tk L k m t k, 3.9 m x 1 x 2 1 M 1 g η 1,η 1 T η 1 M 2 M 1 x1 T η 1 T 1 M 1 g η 2,η 2 T η 2 M 2 M 1 x2 T η 2 T M 2 M 1 m T. By Theorem 2.3, wegetm t fort J, thatis,aη 1 Aη 2, then b is proved. Let α n Aα n 1 and β n Aβ n 1 for n 1, 2, 3,... By the properties a and b, we have α α 1 α n β n β 1 β. 3.1 By the definition of operator A, we have that {α n} and {β n} are uniformly bounded in α,β. Thus {α n } and {β n } are uniformly bounded and equicontinuous in α,β. By Arzela-Ascoli Theorem and 3.1, we know that there exist x, x in α,β such that lim n α n t x t, lim n β n t x t uniformly on J 3.11 Moreover, x t, x t are solutions of NBVP 2.2 in α,β. To prove that x, x are extremal solutions of NBVP 2.2, letu t α,β be any solution of NBVP 2.2, thatis, u t f t, u t, [ ϕu ] t, t J, Δu t k I k tk,u t k, [ ϕu ] t k, g u,u T. k 1, 2,...,p, 3.12
Journal of Inequalities and Applications 11 By Theorem 2.3 and Induction, we get α n t u t β n t with t J and n 1, 2, 3,...which implies that x t u t x t,thatis,x and x are minimal and maximal solution of NBVP 2.2 in α,β, respectively. The proof is complete. Theorem 3.3. Let the assumptions of Theorem 3.2 hold and assume the following. H7 There exist constants M >, Ñ such that f t, x, ϕx f t, x,ϕx M x x Ñ ϕx ϕx, 3.13 where α t x x β t. H8 There exist constants L k < 1, k 1, 2,...,psuch that I k tk,x,ϕx I k tk, x, ϕx L k x x, k 1, 2,...,p, 3.14 where α t k x x β t k. H9 There exist constants M 1, M 2 with < M 2 e MT < M 1 such that g x, y g x, y M 1 x x M 2 y y 3.15 whenever α x x β, and α T y y β T. Then NBVP 2.2 has a unique solution in α,β. Proof. By Theorem 3.2, we know that there exist x,x α,β, which are minimal and maximal solutions of NBVP 2.2 with x t x t, t J. Let m t x t x t.using H7, H8,and H9, weget m t x t x t f t, x t, [ ϕx ] t f t, x t, [ ϕx ] t Mm t Ñ [ ϕm ] t, Δm t k Δx t k Δx t k I k tk,x t k, [ ϕx ] t k I k tk,x t k, [ ] ϕx tk L k m t k, 3.16 m x x M 2 M 1 x T x T M 2 M 1 m T. By Theorem 2.3, we have that m t <, t J, thatis,x t x t. Hence x t x t, this completes the proof.
12 Journal of Inequalities and Applications 4. Examples To illustrate our main results, we shall discuss in this section some examples. Example 4.1. Consider the problem x t 1 t 1 1 sin t x 2 5 e 2π x s ds sin t 2 4 3 t 4 3 e 2π, t,t, t/ t k, Δx t k 1 2 e π/2 x t k 3 tk 1 t k x s ds sin t k 4 1/7 4 17 eπ/2 cos t k, k 1, 4.1 x 1 2 x T 1 6π x s ds, where T 2π, k 1, t 1 π. Let [ ] t 1 ϕx t x s ds sin t 4, t f t, x, y 1 1 sin t x 2 5 e 2π 3 y2 4 3 e 2π, 4.2 I 1 t, x, y 1 2 e π/2 x 3 y 1/7 4 17 eπ/2 cos t. Setting α t 2andβ t 3, it is easy to verify that α t is a lower solution, and β t is an upper solution with α t β t. For t J, and2 x t x t 3, we have ϕx 1 1 3 inf t J x t, ϕx ϕx ϕ x x, 4.3 t 1 ϕx ϕx x s x s ds x x. t
Journal of Inequalities and Applications 13 Setting M 1/2, N e 2π /6, L 1, R 1/3, L 1 1/2 e π/2 and M 1 1, M 2 1/2, then conditions H1 H6 are all satisfied: 2π s<t k <2π 1 L k e Ms ds π p 1 L k 2 1 L 1 2 1 L 1 e Ms ds 2π π e Ms ds 43.4893, 1 1 2 e π/2 2.829, NR e MT M 2 /M 1 2eπ 1.49, M 2 /M 1 18e2π NR e MT M 2 /M 1 2π 1 L k e Ms ds.2131 <.829, M 2 /M 1 s<t k <2π p NLT L k r NLT L 1 CM 1.182 < 1, 4.4 then inequalities 2.9 and 2.3 are satisfied. By Theorem 3.2, problem 4.1 has extremal solutions x,x α,β. Example 4.2. Consider the problem x t 1 e 3π x t 2 e π e 2x 1 3 1 2, t,t, t / t k, 1/5 Δx t k 2e π x t k 2 e 2x tk 1 cos tk, k 1, 4.5 x 1 2 x T, where T 2π, k 1, t 1 π. Let ϕx e 2x 1, f t, x, y 1 1 sin t x 2 5 e 3π 16 e π 1 y2 e 4 1 2 16e 3π e π 1, 4.6 I 1 t, x, y 2e π x 2 y 1/5 cos t. Setting α t 2andβ t 3, then α t is a lower solution, and β t is an upper solution with α t β t. For t J, and2 x t x t 3, we have ϕx e 2x 1 x, ϕx ϕx ϕ x x, and ϕx ϕx e 2x e 2x e x e x e x e x 2e 6 e x x 1 14e 6 x x. Setting M 1/2,
14 Journal of Inequalities and Applications N e 3π / e π 1, L 14e 6, R 1, L 1 2e π and M 1 1, M 2 1/2, then conditions H1 H6 are all satisfied: p 1 L k 2 2π s<t k <2π 1 L k ds eπ 2 2 2πe π e π 1.1388, NR e MT M 2 /M 1 e MT 2e π 1.37 <.1388, M 2 /M 1 e 2π e π 1 p NLT L k r NLT L 1 CM 1.295 < 1, 4.7 then inequalities 2.24 and 2.3 are satisfied. By Corollary 2.4 and Theorem 3.2, problem 4.5 has extremal solutions x,x α,β. Moreover, let M 1/2, Ñ e 3π / e π 1, L 1 2e π and M 1 1, M 2 1/2. It is easy to see that conditions H7 H9 are satisfied. By Corollary 2.4 and Theorem 3.3, problem 4.5 has an unique solution in α,β. References 1 V. Lakshmikanthan, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. 2 D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988. 3 D. Franco, E. Liz, J. J. Nieto, and Y. V. Rogovchenko, A contribution to the study of functional differential equations with impulses, Mathematische Nachrichten, vol. 218, pp. 49 6, 2. 4 J. J. Nieto and R. Rodríguez-López, Boundary value problems for a class of impulsive functional equations, Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2715 2731, 28. 5 L. Chen and J. Sun, Nonlinear boundary value problem of first order impulsive functional differential equations, Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 726 741, 26. 6 L. Li and J. Shen, Periodic boundary value problems for functional differential equations with impulses, Mathematica Scientia, vol. 25A, pp. 237 244, 25. 7 X. Liu and D. Guo, Periodic boundary value problems for a class of second-order impulsive integrodifferential equations in Banach spaces, Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 284 32, 1997. 8 D. Guo and X. Liu, Periodic boundary value problems for impulsive integro-differential equations in Banach spaces, Nonlinear World, vol. 3, no. 3, pp. 427 441, 1996. 9 X. Liu and D. Guo, Initial value problems for first order impulsive integro-differential equations in Banach spaces, Communications on Applied Nonlinear Analysis, vol. 2, no. 1, pp. 65 83, 1995. 1 D. Guo and X. Liu, First order impulsive integro-differential equations on unbounded domain in a Banach space, Dynamics of Continuous, Discrete and Impulsive Systems, vol. 2, no. 3, pp. 381 394, 1996. 11 D. Franco and J. J. Nieto, First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, vol. 42, no. 2, pp. 163 173, 2. 12 X. Yang and J. Shen, Nonlinear boundary value problems for first order impulsive functional differential equations, Applied Mathematics and Computation, vol. 189, no. 2, pp. 1943 1952, 27. 13 W. Ding, J. Mi, and M. Han, Periodic boundary value problems for the first order impulsive functional differential equations, Applied Mathematics and Computation, vol. 165, no. 2, pp. 433 446, 25. 14 Z. Luo and Z. Jing, Periodic boundary value problem for first-order impulsive functional differential equations, Computers & Mathematics with Applications, vol. 55, no. 9, pp. 294 217, 28.