Existence results for impulsive neutral functional integrodifferential equation with infinite delay

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1 Available online at J. Nonlinear Sci. Appl , Research Article Existence results for impulsive neutral functional integrodifferential equation with infinite delay T. Gunasekar a,, F. Paul Samuel b, M. Mallika Arjunan c a School of Advanced Sciences, Fluid Dynamics Division, VIT University, Vellore , Tamil Nadu. b Department of Mathematics and Physics, University of Eastern Africa, Baraton, Eldoret 25-31, Kenya. c Department of Mathematics, C.B.M College of Arts and Science, Kovaipudur, Coimbatore , Tamil Nadu, India. Communicated by Choonkil Park Abstract In this paper, we study the existence of mild solutions for a impulsive semilinear neutral functional integrodifferential equations with infinite delay in Banach spaces. The results are obtained by using the Hausdorff measure of noncompactness. Examples are provided to illustrate the theory. c 213 All rights reserved. Keywords: Impulsive differential equation, Neutral functional differential equation, Mild solution, Hausdorff measures of noncompactness. 21 MSC: 34K3, 34K4, 47D3. 1. Introduction Measures of noncompactness are a very useful tool in many branches of mathematics. They are used in the fixed point theory, linear operators theory, theory of differential and integral equations and others [5]. There are two measures which are the most important ones. The Kuratowski measure of noncompactness σx of a bounded set X in a metric space is defined as infimum of numbers r > such that X can be covered with a finite number of sets of diameter smaller than r. The Hausdorff measure of noncompactness χx defined as infimum of numbers r > such that X can be covered with a finite number of balls of radii smaller than r. The Hausdorff measure is convenient in applications. The notion of a measure of noncompactness turns out to be a very important and useful tool in many branches of mathematical analysis. The notion of a measure of weak compactness was introduced by De Blasi Corresponding author addresses: gunasekarphd@yahoo.com T. Gunasekar, paulsamuelphd@yahoo.com F. Paul Samuel, arjunphd7@yahoo.co.in M. Mallika Arjunan Received

2 T. Gunasekar, F. Paul Samuel, M. Mallika Arjunan, J. Nonlinear Sci. Appl , [12] and was subsequently used in numerous branches of functional analysis and the theory of differential and integral equations. Several authors have studied the measures of noncompactness in Banach spaces [3, 5, 6, 7, 8, 9]. The study of the impulsive differential equations has attracted a great deal of attention. The theory of impulsive differential and integrodifferential equations become an important area of invetigation in recent years. In [13, 18, 19], the authors studied the existence of solutions for first-order impulsive partial neutral functional differential equations with infinite delay in Banach space with the compactness assumption on associated semigroups. Now impulsive partial neutral functional differential equations have become an important object of investigation in recent years stimulated by their numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, etc. With regard to this matter, we refer the reader to [1, 13, 17, 18, 19, 25, 26, 28, 29, 32, 33]. On the other hand, study of the existence and stability of the differential equations with delay was initiated by Travis and Webb [3] and Webb [31]. Since such equations are often more realistic to describe natural phenomena than those without delay, they have been investigated in variant aspects by many authors. Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention in the past decades; see, for example, [4, 1, 11, 14, 15, 19, 2, 21, 22, 23] and the references therein. In this paper, we study the following impulsive neutral functional integrodifferential equations with infinite delay d dt xt gt, x t = Axt gt, x t + ft, x t, x = ϕ B, et, s, x s ds, t J = [, b] xt i = I i x ti, i = 1, 2,..., n, < t 1 < t 2 <... < t n < b, 1.3 where A is the infinitesimal generator of an analytic semigroup of linear operators defined on a Banach space X. The history x t :, ] X, x t θ = xt + θ, belongs to some abstract phase space B defined axiomatically; g : J B X, f : J B X X, e : J J B X, I i : X X, i = 1, 2,..., p are appropriate functions; < t 1 < t 2 <... < t n < b are fixed numbers and the symbol ξt represent the jump of the function ξ at t, which is defined by ξt = ξt + ξt. We give the existence of mild solution of the initial value problem under the condition in respect of Hausdorff s measure of non-compactness. The results obtained in this paper are generlizations of the results given by Arjunan [24, 27], Banas and Goebel [5] and Hernandez [18, 19, 21, 22]. 2. Preliminaries Let X be a Banach space and A : DA X X be the infinitesimal generator of an analytic semigroup of linear operators T t t on X, ρa. M is a constant such that T t M for every t J = [, b]. The notation A α, α, 1 is a closed linear operator on its domain D A α. Furthermore, the subspace D A α is dense in X and the expression x α = A α x, x D A α defines a norm on D A α. Hereafter we denote by X α the Banach space D A α normed with X α. Then for each < α 1, X α is a Banach space. For semigroup {T t : t }, the following properties will be used. Lemma 2.1 [19]. Let < β < α 1, then the following properties hold : 1 X β is a Banach space and X β X α and the imbedding is compact. 2 there exists M 1 such that T t M, for all t b;

3 T. Gunasekar, F. Paul Samuel, M. Mallika Arjunan, J. Nonlinear Sci. Appl , for any α 1, there exists a positive constant C α such that A α T t Cα t α, < t b. To describe appropriately our problems we say that a function u : [σ, τ] X is a normalized piecewise continuous function on [σ, τ] if u is piecewise continuous and left continuous on σ, τ]. We denothe by PC [σ, τ]; X the space formed by the normalized piecewise continuous fuction from [σ, τ] into X. In particular, we introduce the space PC fromed by the all functions u : [, b] X such that u is continuous at t t i, ut i = ut i and ut + i exists, for all i = 1,..., n. It is clear that PC endowed with the norm of the uniform convergence is a Banach space. In this work, we will employ an axiomatic definition of the phase space B which is similar to that introduced by Hale and Kato [16] and it is appropiate to treat retarded impulsive differential equations. Definition 2.1 [16]. Let B be a linear space of functions mapping, ] into X endowed with a seminorm B and we will assume that B satisfies the following axioms: A If x :, σ + b] X, b >, such that x σ B and x [σ,σ+b] PC[σ, σ + b] : X, then for every t [σ, σ + b the following conditions hold: i x t is in B, ii xt H x t B, iii x t B Kt σ sup{ xs : σ s t} + Mt + σ x σ B, where H > is a constant; K, M : [, [1,, K is continuous, M is locally bounded and H, K, M are independent of x. B The space B is complete. Definition 2.2. The Hausdorff s measures of noncompactness χ Y is defined by χ Y B = inf {r >, B can be covered by finite number of balls with radii r} for bounded set B in any Banach space Y. Lemma 2.2 [5]. Let Y be a real Banach space and B, C Y be bounded, the following properties are satisfied: 1 B is pre-compact if and only if χ Y B = ; 2 χ Y B = χ Y B = χ Y convb where B and convb mean the closure and convex hull of B respectively; 3 χ Y B χ Y C when B C; 4 χ Y B + C χ Y B + χ Y C, where B + C = {x + y; x B, y C}; 5 χ Y B C max {χ Y B, χ Y C}; 6 χ Y λb = λ χ Y B for any λ R; 7 If the map Q : DQ Y Z is Lipschitsz continuous with constant k then χ Z QB kχ Y B for any bounded subset B DQ, where Z be a Banach space; 8 If {W n } + n=1 is a decreasing sequence of bounded closed nonempty subsets of Y and lim χ Y W n =, then + n + n=1 W n is nonempty and compact in Y. Definition 2.3 [33]. The map Q : W Y Y is said to be a χ Y contraction if there exists a positive constant k < 1 such that χ Y QC kχ Y C for any bounded closed subset C W where Y is a Banach space.

4 T. Gunasekar, F. Paul Samuel, M. Mallika Arjunan, J. Nonlinear Sci. Appl , Lemma 2.3 [1] Darbo. If Q : W Y is closed and convex and W, the continuous map Q : W W is a χ Y -contraction, if the set {x W : x = λγx} is bounded for < λ < 1, then the map Q has atleast one fixed point in W. Lemma 2.4 [5] Darbo-Sadovskii. If W Y is bounded closed and convex, the continuous map Q : W W is a χ Y contraction, then the map Q has at least one fixed point in W. In this paper, we denote χ by the Hausdorff s measure of noncompactness of X and denote χ c by the Hausdorff s measure of noncompactness of C[; b]; X. To discuss the existence we need the following lemmas in this paper. Lemma 2.5 [5]. 1 If W C[a, b]; X is bounded, then χw t χ c W, for any t [a, b], where W t = {ut; u W } X. 2 If W is equicontinuous on [a, b], then χw t is continuous for t [a, b] and χ c W = sup{χw t, t [a, b]}; 3 If W C[a, b]; X is bounded and equicontinuous, then χw t is continuous for t [a, b] and χ a W sds a χw sds, for all t [a, b], where a W sds = { a xsds : x W }. Lemma 2.6 [29]. 1 If W PC[a, b]; X is bounded, then χw t χ PC W, for any t [a, b], where W t = {ut; u W } X. 2 If W is piecewise equicontinuous on [a, b], then χw t is piecesise continuous for t [a, b] and χ PC W = sup{χw t, t [a, b]}; 3 If W C[a, b]; X is bounded and piecewise equicontinuous, then χw t is piecewise continuous for t [a, b], and χ a W sds a χw sds, for all t [a, b], where a W sds = { a xsds : x W }. Lemma 2.7. If the semigroup T t is equicontinuous η L[, b]; R +, then the set { T t susds : us ηsfora.e. s [, b]} is equcontinuous for t [, b]. 3. Existence Results Definition 3.4. A function x :, b] Xis a mild solution of the initial value problem if x = ϕ, x. J PC and xt =T tϕ g, ϕ + gt, x t + T t sfs, x s, + T t t i I i x ti, t J. <t i <t es, τ, x τ dτds For the system , for some α, 1, we assume that the following hypotheses are satisfied: Hf The function f : J B X X satisfies the following conditions: 1 For each x :, b] X, x = ϕ B and x J PC, the function t ft, x t, et, s, x sds is strongly measurable and ft,.,. is continuous for a.e. t J; 2 There exists an integrable function α : J [, + and a monotone continuous dondecreasing function Ω : [, +, + such that ft, v, w αtω v B + w, t J, v, w B X;

5 T. Gunasekar, F. Paul Samuel, M. Mallika Arjunan, J. Nonlinear Sci. Appl , He Hg HI H 1 3 There exists an integrable function η : J [, + such that χt sft, D 1, D 2 ηt sup χd 1 θ + χd 2 θ for a.e. s, t J, and any bounded subset D 1 PC[, ]; X and D 2 X, where D 1 θ = {vθ : v D 1 }. 4 There exists a positive constant L f such that ft, v 1, w 1 ft, v 2, w 2 L f v 1 v 2 ϕ + w 1 w 2, where < L f < 1, t, v i, w i J B X, i = 1, 2. 1 There exists a constant N 1 > such that t [et, s, x et, s, y]ds N1 x y B, for t, s J, x, y B 2 For each t, s J J, the function et, s,. : B X is continuous and for each x B, the function e.,., x : J J X is strongly measurable. There exists an integrable function m : J [, and constant γ >, such that es, τ, x γ mτφ x, where φ : [, +, + is a continuous nondecreasing function. Assume that the finite bound of γ msds is L. 1 There exists < β < 1, such that gt, v X β = D A β, A β g. is continuous for all t, v J B and there exist positive constants c 1 and C 2, such that A β gt, v C1 v B + C 2 2 There exists a positive constant L g such that, A β gt, v 1 A β gt, v 2 L g v 1 v 2 B, v 1, v 2 B. 1 There exist positive constants L i such that I i u = I i v u v B, u, v B. 2 There exist positive constants C j i, i = 1,..., n, j = 1, 2, such tht I i v = Ci 1 v B + C2 i, v B. b ˆmsds c ds Ωs+φs where µ 1 = K b MH + M b + M A β C 1 K b ϕ B + K b A β C 2 M K b M and µ 2 = K b A β C 1 + K b M <t i <t C 1 i < 1, c = µ 1 1 µ 2. Ci 2 <t i <t Let y :, b] X be a function defined by y = ϕ and yt = T tϕ on J. Clearly, y t K b MH + M b ϕ B, where K b = sup t b Kt, M b = sup t b Mt. Theorem 3.1. If the hypotheses Hf, Hg, He, HI and H are satisfied, the the initial value problem has at least one mild solution. Proof. Let Sb be the space Sb = {x :, b] X x =, x J PC} endowed with the supremum norm. b. Let Γ : Sb Sb be the map defined by, t [, ] Γxt = T tg, ϕ + gt, x t + y t + T t sfs, x s + y s, es, τ, x τ + y τ dτds + T t t i I i x ti + y ti, t J. <t i <t It is easy to see that x t + y t B K b MH + M b ϕ B + K b x t, where x t = sup s t xs. Thus Γ is well defined and with the values in Sb. In addition, from the axioms of phase space, the Lebesgue dominated convergence theorem and the condition Hf, Hg, He and HI, we can show that Γ is continuous. Step 1. For < λ < 1, set {x PC : x = λγx} is bounded. 3.1

6 T. Gunasekar, F. Paul Samuel, M. Mallika Arjunan, J. Nonlinear Sci. Appl , Let x λ be a solution of x = λγx for < λ < 1, we have x λt + y t B K b MH + M b ϕ B + K b x λ t. Let v λ t = K b MH + M b ϕ B + K b x λ t, for each t J. Then x λ t = λγx λ t Γxt M A β C1 ϕ B + C 2 + A β C2 + M + A β C 1 + M <t i <t C 1 i v λ s + M x λ t M A β C1 ϕ B + C 2 + A β C2 + M + A β C 1 + M <t i <t C 1 i v λ s + M <t i <t C 2 i αsω v λ s + <t i <t C 2 i αsω v λ s + which implies that v λ t K b MH + M b + M A β A β C1 K b ϕ B + K b C 2 M K b M A β +K b C 1 + M <t i <t C 1 i v λ s + M αsω v λ s + γmτφv λ τdτ ds γmτφv λ τdτ ds, γmτφv λ τdτ <t i <t C 2 i ds. Consequently, v λ t c + MK b 1 µ 2 αsω v λ s + γmτφv λ τdτ ds. Let us take the right-hand side of the above inequality as β λ t. Then β λ = c and and Since Ω is nondecreasing v λ t β λ t, t b β λ t MK b αtω v λ t + 1 µ 2 β λ t MK b αtω β λ t + 1 µ 2 γmsφv λ sds. γmsφβ λ sds. Let W λ t = β λ t + γmsφβ λsds. Then W λ = β λ and W λ t β λ t This implies that W λ t = β λ t + γmtφβ λt Wλ t W λ MK b αtωw λ t + γmtφw λ t 1 µ 2 ˆmt ΩW λ t + φw λ t ds b Ωs + φs ds ˆmsds c Ωs + φs

7 T. Gunasekar, F. Paul Samuel, M. Mallika Arjunan, J. Nonlinear Sci. Appl , which implies that the functions β λ t are bounded on J. Thus, the function v λ t are bounded on J, and x λ. are also bounded on J. Step 2. Next, we show that Γ is χ-contraction. To clarify this, we decompose Γ in the form Γ = Γ 1 + Γ 2, for t, where Γ 1 xt = T tg, ϕ + gt, x t + y t + T t t i I i x ti + y ti. Γ 2 xt = T t sfs, x s + y s, <t i <t es, τ, x τ + y τ dτds. Firstly, we show that Γ 1 is Lipschitz continuous. Take x 1, x 2 Sb arbitrary, on account of Definition 2.1 and hypotheses, we get Γ 1 x 1 t Γ 1 x 2 t Hence Γ 1 is Lipschitz continuous, and A β Lg x 1t x 2t B + M A β L g K b + MK b A β K b L g + M L A β = K b L g + M n L i x 1ti x 2ti B n L i x 1 x 2 b n L i x 1 x 2 b. n L i. Next, take bounded subset W Sb arbitrary. Since analytic semigroup is equicontinuous, T t sfs, w s + y s, es, τ, w τ + y τ dτ piecewise equicontinuous; and from Lemma 2.6 χ PC W = sup{χw t, t [, b]}, and from [2], Lemma 3.4.7, χωw τ K 1 χw τ, where K 1 is constant, we have χγ 2 W t χ T t sfs, w s + y s, ηs ηs sup <θ sup <θ ηs sup <τ s es, τ, w τ + y τ dτ ds χ[w s + θ + ys + θ] + χ[ es, τ, x τ + y τ dτ] ds χ[w s + θ + ys + θ] + L o χ[ωw s + θ + ys + θ] χw τ + L χωw τ ds ds Since χ PC W 1 + K 1 L ηsds for each bounded set W PC. χ PC ΓW = χ PC Γ 1 W + Γ 2 W χ PC Γ 1 W + χ PC Γ 2 W L + ηsds χ PC W, Γ is χ-contraction. In view of Lemma 2.3, i.e. Darbo fixed point theorem, we conclude that Γ has atleast one fixed point in W. Let x be a fixed of Γ on Sb, then z = x + y is a mild solution of So we deduce the existence of a mild solution of

8 T. Gunasekar, F. Paul Samuel, M. Mallika Arjunan, J. Nonlinear Sci. Appl , Theorem 3.2. Assume that Hf,He,HI,H are satisfied. Furthermore, we suppose [ H1 A β n b K b C 1 + M Ci 1 + M αsds lim sup Ωτ + L ] φτ < 1. τ τ Then the initial value problem has at least one mild solution. Proof. Proceeding in the proof of Theorem 3.1, we refer that the map Γ given by 3.1 is continuous from Sb into Sb. Furthermore, there exists k > such that ΓB k B k, where B k = {x Sb : x b k}. In fact, if we assume that the assertion is false, then k > there exists x k B k anf t k J such that k < Γx k t k. This yields that k < Γx k t k M g, ϕ + A β C1 x ktk + y tk B + C 2 k + M αsω x ks + y s B + es, τ, x kτ + y τ dτ ds + M Ci 1 x kti + y ti B + Ci 2 <t i <t M g, ϕ + A β C1 K b MH + M b ϕ B + C 1 K b k + C 2 b [ ] + M αsds Ω K b MH + M b ϕ B + K b k + L φk b MH + M b ϕ B + K b k + M n Ci 1 K b MH + M b ϕ B + Ci 1 K b k + Ci 2, which implies that [ A β 1 < K b C 1 + M b n C 1 i Ω + M αsds lim sup k [ A β n K b C 1 + M Ci 1 + M ] [ K b MH + M b ϕ B + K b k + L φk b MH + M b ϕ B + K b k b αsds k lim sup τ + L ] φτ τ τ a contradiction. By means of lemma 2.4, as the proof of Theorem 3.1, we conclude that has at least a mild solution. 4. Example In this section, we apply some of the results established in this paper. In the next applications, B will be the phase space C L 2 h, X see [19]. We study, the first-order neutral integrodifferential equation with unbounded delay d dt + ut, ξ π at, ξ, s tf us, ξ, ut, = ut, π, bt s, η, ξus, ηdηds = 2 x 2 ut, ξ t [, b], π < 1 bt s, η, ξus, ηdηds qs, τ, u τ dτds, t [, b], ξ [, π], 4.1 uτ, ξ = ϕτ, ξ, τ, ξ π, 4.3 ut i ξ = i ] 4.2 a i t i sus, ξds, 4.4 where, ϕ C L 2 h, X, < t 1 <... < t n < b and

9 T. Gunasekar, F. Paul Samuel, M. Mallika Arjunan, J. Nonlinear Sci. Appl , a The functions bs, η, ξ, bs,η,ξ ξ { π π L g := max 1 hs are measurable, bs, η, π = bs, η, = and i bs,η,ξ 2dηdsdξ 1/2 } ξ : i =, 1 < ; i b The function at, ξ, θ is continuous in J [, π], ] and at, ξ, θdθ = mt, ξ < c The function q. is continuous such that qt, s, ξ Ω ξ, where Ω. is positive, continuous and nondecreasing in [,. d The function F u 1, u 2 Ω 1 u 1 + u 2, where Ω 1. is positive, continuous and nondecreasing in [,. e The functions a i C[, ; R and L 1 i := a i s 2 hs ds 1/2 < for all i = 1, 2,..., n. Assuming that the conditions a-e are varified, the problem can be modeled as the abstract impulsive Cauchy problem by defining gt, ψξ := ft, φ, ψξ := I i ψξ := π bs, v, ξψs, vdvds, 4.5 at, ξ, τf τ φτ, ξ, ψτ, θ, u θ dθ dτ, 4.6 a i sψs, ξds. 4.7 Moreover, gt,., ft,.,. and I i, i = 1,..., n, are bounded linear operators, the range of g. is contained in X 1/2, A 1/2 gt,. L g, I i L 1 i, i = 1,..., n and ft, φ, ψ αtω φ B + ψ for every t [, b], where αt := µt,s 2 hs ds1/2. References [1] R. Agarwal, M. Meehan, D. O Regan, Fixed point theory and applications, in: Cambridge Tracts in Mathematics, Cambridge University Press, New York, 21, [2] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, Measure of noncompactness and condensing operators, Translated from the Russian by A. Iacob, Basel; Boston :Berlin : Birkhauser, [3] J. M. Ayerbe, T. D. Benavides and G.L. Acedo, Measure of noncmpactness in in metric fixed point theorem, Birkhauser, Basel, [4] S. Baghlt, M. Benchohra, Perturbed functional and neutral functional evolution equations with infinite delay in Frechet spaces, Electron. J. Differ. Equ [5] J. Banas, K. Goebel, Measure of Noncompactness in Banach Space, in: Lecture Notes in Pure and Applied Matyenath, Dekker, New York, , 1, 2.2, 2.4, 2.5 [6] J. Banas, W. G. El-Sayed, Measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variation, J. Math. Anal. and Appl., , [7] J. Banas and A. Martinon, Measure of noncmpactness in Banach sequnence spaces, Portugalie Math , [8] G. Emmanuele, Measures of weak noncompactness and fixed point theorems, Bull. Math. Soc. Sci. Math. R. S. Roumanie, , [9] Z. Fan, Q. Dong and G. Li, Semilinear differential equations with nonlocal conditions in Banach spaces, Int. J. Nonlinear Sci. 2 26, [1] M. Benchohra, J. Henderson, S. K. Ntouyas, Existence results for impulsive multivalued semilinear neutral functional inclusions in Banach spaces, J. Math. Anal. Appl [11] M. Benchohra, S. Djebali, T. Moussaoui, Boundary value problems for double perturbed first order ordinary differential systems, Electron. J. Qual.Theory Differ. Equ [12] F. S. De Blasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie, , [13] Y.K. Chang, A. Anguraj and M. Mallika Arjunan, Existence results for impulsive neutral functional differential equations with infinite delay, Nonlinear Anal. Hybrid Syst [14] Q. Dong, Z. Fan and G. Li, Existence of solutions to nonlocal neutral functional differential and integrodifferential equations, Int. J. Nonlinear Sci

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