A New Analytical Method for Solutions of Nonlinear Impulsive Volterra-Fredholm Integral Equations

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1 Nonlinear Analysis and Differential Equations, Vol. 6, 218, no. 4, HIKARI Ltd, A New Analytical Method for Solutions of Nonlinear Impulsive Volterra-Fredholm Integral Equations Haiyong Qin School of Mathematics, Qilu Normal University, Jinan 2513, China Copyright c 218 Haiyong Qin. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The main purpose of this paper is to demonstrate the use of a new comparison result and Mönch fixed point theorem for solutions of nonlinear impulsive Volterra-Fredholm integral equations in Banach space. The benefit of this method is that it overcomes the difficulties caused by Fredholm operator without any restrictive conditions on estimate of noncompactness measure. The results obtained here improve and develop some previous works largely. Mathematics Subject Classification: 45B5, 45D5, 45G15 Keywords: Volterra-Fredholm integral equations, Comparison result, Fixed point theorem 1 Introduction One of the fundamental classes of equations is Volterra-Fredholm integral ones. Nonlinear Volterra-Fredholm integral equations arise in the fields of parabolic boundary value problems, the spatio-temporal development of an epidemic and various scientific problems. This type of equations has been applied to a wide range of science and engineering. For this reason, Volterra-Fredholm integral equations have been attracted more attention in recent decades. Maleknejad and Hadizadeh [8] investigated this type of systems by means of Adomian s decomposition method. Maleknejad and Fadaei Yami [9] extended the method used in [8] to solve nonlinear Volterra-Fredholm integral systems.

2 122 Haiyong Qin Wazwaz [1] provided a modified decomposition method for handing nonlinear Volterra-Fredholm integral equation. Yousefi and Razzaghi [15] presented a numerical method for solving nonlinear Volterra-Fredholm integral equations based on Legendre wavelets method. Babolian et al.[3] proposed a new direct method for numerical solutions of nonlinear Volterra-Fredholm integral and integro-differential equations. Mahmoudi [16] studied the nonlinear Volterra and Fredholm integral equation of the second kind using continues Legendre wavelets. Negarchi and Nouri [13] used a Müntz-Legendre collocation and the least squares approximation method to consider numerical behavior of a class of Volterra-Fredholm integral equation. Maleknejad and Dehbozorgi [1] established a new computational method for nonlinear Volterra integral equation of the second kind. Impulsive equations describe processes which experience a sudden change of their state at certain moments. Because it can explain well various natural phenomenon, these problems have aroused people s widespread attention, see [2] and [7]. In recent decades, many authors have studied various kinds of differential, integral and integro-differential equations, particularly in the fields of existence and controllability results of many systems, see [4],[5],[6],[11],[12] and [14]. In [2], when the nonlinear term f does not include the first derivative term x and integral term Sx, by monotone iterative technique, Guo investigated a class of initial value problems. Inspired by the above literatures, consider the following problem for nonlinear impulsive Volterra-Fredholm integral equations in Banach space x(t = p(t + + <t k <t (t sf(s, x(s, x (s, (T x(s, (Sx(sds [I k (x(t k, x (t k + q(t, t k Īk(x(t k, x (t k ], t J, (1 where p, q C 1 [, a], f C(J X X X X, X, I k, Īk C(X X, X, (T x(t = k(t, sx(sds, (Sx(t = h(t, sx(sds. (2 X is a real Banach space, J = [, a] (a >, = t < t 1 < t 2 < < t m < t m+1 = a. F = {(t, s J J t s}. In (2, k C(F, R +, h C(J J, R +, R + =[, +, R + =(, +. Let J = [, t 1 ], J 1 = (t 1, t 2 ],, J m 1 = (t m 1, t m ], J m = (t m, a], k = max{k(t, s (t, s F }, h = max{h(t, s (t, s J J}, I k and Īk denotes the jump of x(t at t = t k. Applying a new comparison result, the Mönch fixed theorem and estimate step by step, we obtain the existence of solutions for problem (1. Compared to previous results, this paper has three advantages: Firstly, we present a new analytical method which will be used to overcome difficulties when conditions

3 Nonlinear impulsive Volterra-Fredholm integral equations 123 of norm include the norm of Sx, and in this way we obtain the existence of solutions for nonlinear impulsive Volterra-Fredholm integral equations (1. Secondly, the restrictive conditions on estimate of noncompactness measure are omitted. Thirdly, the condition (H 2 in this paper is weaker than previous works in this fields. 2 Preliminaries This section states some necessary definitions and preliminary results. Lemma 2.1 ([11] Let D = {x n } L[J, X] and there exists g L[J, R + ] such that x n (t g(t, a.e.t J for all x n D, Then α(d(t L[J, R + ] and ({ } α x n (sds n N 2 α(d(tds, t J. Lemma 2.2 ([11] Let B C[J, X] is equicontinuous bounded set, then α(b(t C[J, R + ] and α ( B(sds α(b(sds. Lemma 2.3 ([11].(Mönch Let E be a Banach space and Ω E be a bounded open set, θ Ω, A : Ω E is continuous and satisfies the following conditions (i x λax, λ [, 1], x Ω; (iid Ω is countable and D co({θ} (AD imply that D is a relatively compact set. Then A has at least one fixed point in Ω. Lemma 2.4 ([6] Suppose that u, f, g C([t, T, R +. Let w C(R +, R + be nondecreasing with w(u > for u >, and α, β C 1 ([t, T, [t, T be nondecreasing with α(t, β(t t on [t, T. If α(t β(t T u(t M 1 +M 2 f(sw(u(sds+m 3 g(sw(u(sds+m 4 α(t β(t t [u(s] λ ds, (3 where M i (i = 1, 2, 3, 4 are nonnegative constants, λ < 1 and 1 1 ds = w(s. Then for t t < T, we have u(t m ( (4 where m(s = (2s MM 1 1 λ s 1 λ (1 λmm 4 T, M is a positive constant and m ( represents the inverse of m(.

4 124 Haiyong Qin 3 Main Results We are concerned with the existence of solutions of problem (1. Let P C[J, X] := {x : J X x C[(t k, t k+1, X], and there exist x(t k and x(t+ k, k = 1, 2,..., m with x(t k = x(t k}. Endowed with the norm x P C = sup t J x(t, (P C[J, X], P C is a Banach space. T r = {x X x r} and B r = {x P C[J, X] x P C r}. α( represents noncompactness measure. For this purpose, we also need the following assumptions hold (H 1 For any r >, f is uniformly continuous on J T r T r T r T r, I k and Īk (k = 1, 2,, m are bounded on T r T r. (H 2 There exist b 1, b 5 L[J, R + ] and b 2, b 3, b 4 C[J, R + ], λ 1 < 1 such that f(s, x, y, u, v b 1 (s + b 2 (s x + b 3 (s y + b 4 (s u + b 5 (s v λ 1, (5 s J, x, y, u, v X. (H 3 Suppose l i C[J, R + ] (i = 1, 2, 3, 4 are bounded, and for any equicontinuous set D B r, (t, s F and λ 2 < 1 such that α(f(t, s, D(s, (T D(s, (SD(s l 1 (tα(d(s + l 2 (tα(d (s + l 3 (tα((t D(s + l 3 (t[α((sd(s] λ 2. (6 Theorem 3.1 Assume that (H 1 -(H 3 are satisfied. Then the Problem (1 has at least one solution in P C[J, X]. Proof. x P C[J, X] is one solution of equation (1 if and only if x P C[J, E] is the solution of nonlinear impulsive integral equation where x(t = Ax(t, (7 Then (Ax(t = p(t + (t sf(s, x(s, x (s, (T x(s, (Sx(sds + [I k (x(t k, x (t k + q(t, t k Īk(x(t k, x (t k ], t J. (8 <t k <t (Ax (t = p (t + + <t k <t f(s, x(s, x (s, (T x(s, (Sx(sds q (t, t k Īk(x(t k, x (t k. (9

5 Nonlinear impulsive Volterra-Fredholm integral equations 125 Let x Ω, where Ω = {x P C[J, X] x = λax, λ 1}. Then there exists λ [, 1] such that x(t = λ (A x(t, x (t = λ (A x (t, t J. (i t J = [, t 1 ]. From (8 and (9 we have x(t = λ (p(t + x (t = λ (p (t + (t sf(s, x(s, x (s, (T x(s, (S x(sds, (1 f(s, x(s, x (s, (T x(s, (S x(sds. (11 Since p C 1 [, a] and (H 2, there exists a constant M such that x(t M+a ( b1 (s+b 2 (s x(s +b 3 (s x (s +b 4 (s (T x(s +b 5 (s (S x(s λ 1 ds, M+a b 1 (sds+a +a(h λ 1 b 5(sds x(s λ 1 ds. ( b2 (s+ak b 4 (s x(s ds+a b 3 (s x (s ds (12 x (t M + b 1 (sds + +(h λ 1 ( b2 (s + ak b 4 (s x(s ds + b 3 (s x (s ds b 5 (sds x(s λ 1 ds. (13 Set u(t = x(t + x (t, thus u(t C([, t 1, R +, Adding the above inequalities, we know u(t 2M + (a + 1 b 1 (sds + (a + 1 ( b2 (s + ak b 4 (s u(s ds +(a + 1 b 3 (s u(s ds + (a + 1(h λ 1 b 5 (sds u(s λ 1 ds. (14 Then by Lemma 2.4 and continuity of x(t on J, there exists a constant C independent of u such that u(t C, t J. The above inequality implies that x(t + x (t C. From (H 1,there also exists a constant β > independent of x such that f(s, x(s, x (s, (T x(s, (Sx(s β, (15 I 1 (x, y β, Ī1(x, y β, (16

6 126 Haiyong Qin t J, x(t, x (t C. (17 Thus x(t + 1 = x(t 1 + I 1 ( x(t 1, x (t 1 C + β, (18 x (t + 1 = x (t 1 + Ī1( x(t 1, x (t 1 C + β. (19 (ii t (t 1, t 2 ], Let ū(t = x(t, t (t 1, t 2 ]; ū(t = x(t + 1, t = t 1. (2 ū (t = x (t, t (t 1, t 2 ]; ū (t = x (t + 1, t = t 1. (21 Then ū(t, ū (t C([t 1, t 2 ], X and 1 ū(t = λ (x + tx 1 + λ (t sf(s, x(s, x (s, (T x(s, (S x(sds +λ I 1 ( x(t 1, x (t 1 + λ (t t 1 Ī1( x(t 1, x (t 1, (22 ū t1 (t = λ x 1 + λ f(s, x(s, x (s, (T x(s, (S x(sds +λ Ī 1 ( x(t 1, x (t 1. (23 Consequently, similar to the proof of (i, we can know that there exists C 1 > that does not depend on x such that ū(t + ū (t C 1, t [t 1, t 2 ]. So x(t C 1, x (t C 1, t [t 1, t 2 ]. By the same method as above, we can show that there exists a constant C m > that does not depend on x such that x(t C m, x (t C m, t J m = [t m, a]. Let C = max{c i i m}. Then x(t C, x (t C, t J. Thus Ω is a bounded set in P C[J, X]. Take R > C, let Ω = {x P C[J, X] x P C < R}. Then Ω is a bounded open set in P C[J, X] and θ Ω. From the chose of R we know that if x Ω and λ [, 1], then x λax, x λ(ax, That is, (i of in Lemma 2.3 is satisfied. Let H Ω be a countable set and H co({θ} A(H. By (H 1, it is easy to see that A(H is equicontinuous on each J k (k =, 1, 2,..., m. Since T H, SH are equicontinuous bounded sets on each J k (k =, 1, 2,..., m, by Lemma 2.2 we get α((t H(s α((sh(s Therefore we have s s α(k(s, τh(τdτ k α(h(τdτ, s J, (24 α(h(s, τh(τdτ h α(h(τdτ, s J. (25

7 Nonlinear impulsive Volterra-Fredholm integral equations 127 α(h(t α((ah(t ( = α (t sf(s, x(s, x (s, (T x(s, (Sx(sds x H 2al 1 α(h(sds + 2al 2 α(h (sds +2a 2 l 3 k α(h(sds + 2a 2 l 4 [h ] λ 2 α(h (t α((ah (t [α(h(s] λ 2 ds, (26 ( = α f(s, x(s, x (s, (T x(s, (Sx(sds x H 2l 1 α(h(sds + 2l 2 α(h (sds +2al 3 k α(h(sds + 2al 4 [h ] λ 2 [α(h(s] λ 2 ds. (27 where l i = max{l i (t}(i = 1, 2, 3, 4. Let m(t = α(h(t + α(h (t, t J, then m(t C[J, R + ]. Thus (26 and (27 imply ( m(t 2(a+1 (l 1 + l 2 m(sds + l 3 k a m(sds + l 4 [h ] λ 2 a [m(s] λ 2 ds 2(a + 1(l 1 + l 2 + l 3 k a m(sds + 2(a + 1l 4 [h ] λ 2 a [m(s] λ 2 ds (28 By Lemma 2.4, m(t, t J. Therefore α(h(t = α(h (t =, that is, H, H is relatively compact sets in P C[J, X], Thus α(h(t 1 = α(h (t 1 =, Since I 1, Ī1 C[X X, X], α(i 1 (H(t 1, H (t 1 = α(ī1(h(t 1, H (t 1 =. H(t 1, H (t 1 are relatively compact sets in X, For t (t 1, t 2 ], we know α(h(t 2a (l 1 α(h(sds + l 2 α(h (sds + al 3 k α(h(sds +al 4 [h ] λ 2 [α(h(s] λ 2 ds +α(i 1 (H(t 1, H (t 1 +(t t 1 α(ī1(h(t 1, H (t 1 = 2a (l 1 α(h(sds + l 2 α(h (sds +al 3 k α(h(sds +al 4 [h ] λ 2 ( α(h (t α [α(h(s] λ 2 ds f(s, x(s, x (s, (T x(s, (Sx(sds x H (29

8 128 Haiyong Qin +α ( (I 1 (x(t 1, x (t 1 x H 2 (l 1 α(h(sds + l 2 +al 3 k α(h(sds +al 4 [h ] λ 2 α(h (sds [α(h(s] λ 2 ds (3 From (29 and (3 we obtain m(t 2(a + 1[l 1 + l 2 + al 3 k ] m(sds + 2al 4 [h ] λ 2 (a + 1 m(sds. (31 From Lemma 2.4 we get m(t, t J 1. Therefore α(h(t = α(h (t =, t J 1. Especially, α(h(t 2 = α(h (t 2 =, so H, H are relatively compact sets in C[J 1, E]. Similarly, we can show that H, H are relatively compact sets in C[J k, E](k = 2, 3,, m. thus H, H are relatively compact sets in P C[J, E], H is relatively compact set in P C[J, E]. The condition (ii in Lemma 2.3 is satisfied, so A has at least one fixed point in Ω, that is, system (1 has at least one fixed point in P C[J, E]. Remark 3.1: When conditions of norm include the norm of Sx, it is difficult to obtain the existence of solutions of system (1, thus the new comparison result we present in this paper is important and interesting. Remark 3.2: Condition (H 2 require b 1 and b 5 are only Lebesgue integrable in this paper. Remark 3.3: we do not need the restrictive conditions on estimate of noncompact measure in this paper. Acknowledgements. This research is supported by a Project of Shandong Province Higher Educational Science and Technology Program(Grant No. J17KB121, Shandong Provincial Natural Science Foundation (Grant No. ZR216AB4, Foundation for Young Teachers of Qilu Normal University (Grant Nos. 216L65, 215L63, 217JX2311 and 217JX2312, Scientific Research Foundation for University Students of Qilu Normal University (Grant Nos. XS217L1 and XS217L5. References [1] Abdul-Majid Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Applied Mathematics and Computation, 127 (22, [2] D. Guo, Initial value problems for nonlinear second order impulsive integro-differential equations in Banach spaces, Journal of Mathematical Analysis and Application, 2 (1996,

9 Nonlinear impulsive Volterra-Fredholm integral equations 129 [3] E. Babolian, Z. Masouri and S. Hatamzadeh Varmazyar, New direct method to solve nonlinear Volterra-Fredholm integral and integrodifferential equations using operational matrix with block-pulse functions, Progress In Electromagnetics Research B, 8 (28, [4] H. D. Heinz, On the behavior of measure of noncompactness with respect to differentiation and integration of vectorvalued functions, Nonlinear Analysis: Theory, Methods & Applications, 7 (1983, [5] H. Qin, C. Zhang, T. Li and Y. Chen, Controllability of abstract fractional differential evolution equations with nonlocal conditions, Journal of Mathematics and Computer Science(JMCS, 17 (217, [6] H. Qin, X. Zuo and J. Liu, Some new generalized retarded Gronwall-Like inequalities and their applications in nonlinear systems, Journal of Control Science and Engineering, 216 (216, 1-8, Article ID [7] H. Qin, Z. Gu, Y. Fu and T. Li, Existence of mild solutions and controllability of fractional impulsive integrodifferential systems with nonlocal conditions, Journal of Function Spaces, 217 (217, 1-11, Article ID [8] K. Maleknejad and M. Hadizadeh, A new Computational method for Volterra-Fredholm integral equations, Computers and Mathematics with Applications, 37 (1999, [9] K. Maleknejad and M. R. Fadaei Yami, A computational method for system of Volterra-Fredholm integral equations, Applied Mathematics and Computation, 183 (26, [1] K. Maleknejad and R. Dehbozorgi, Adaptive numerical approach based upon Chebyshev operational vector for nonlinear Volterra integral equations and its convergence analysis, Journal of Computational and Applied Mathematics, 344 (218, [11] L. Liu, Iterative method for solutions and coupled quasi-solutions of nonlinear integro-differential equations of mixed type in Banach spaces, Nonlinear Analysis: Theory, Methods and Applications, 42 (2,

10 13 Haiyong Qin [12] Lakshmi Narayan Mishra, H. M. Srivastava and Mausumi Sen, Existence rusults for some nonlinear functional-integral equations in Banach algebra with applications, International Journal of Analysis and Applications, 11 (216, 1-1. [13] Neda Negarchi and Kazem Nouri, Numerical solution of Volterra- Fredholm integral equations using the collocation method based on a special form of the Müntz-Legendre polynomials, Journal of Computational and Applied Mathematics, 344 (218, [14] Saïd Abbas, Wafaa A. Albarakati, Mouffak Benchohra and Johnny Henderson, Existence and Ulam stabilities for Hadamard fractional integral euqations with random efffects, Electronic Journal of Differential Equations, 216 (216, no. 25, [15] S. Yousefi and M. Razzaghi, Legendre wavelets method for nonlinear Volterra-Fredholm integral equations, Mathematics and Computers in Simulation, 7 (25, [16] Y. Mahmoudi, Wavelet Galerkin method for numerical solution of nonlinear integral equations, Applied Mathematics and Computation, 167 (25, Received: October 28, 218; Published: December 3, 218