International Journal of Analysis and Applications ISSN 2291-8639 Volume 7, Number 1 (2015), 22-37 http://www.etamaths.com EXISTENCE OF QUASILINEAR NEUTRAL IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS IN BANACH SPACE B. RADHAKRISHNAN Abstract. In this paper, we devoted to study the existence of mild solutions for quasilinear impulsive integrodifferential equation in Banach spaces. The results are established by using Hausdorff’s measure of noncompactness and the fixed point theorems. Application is provided to illustrate the theory. 1. Introduction In various fields of engineering and physics, many problems that are related to lin- ear viscoelasticity, nonlinear elasticity have mathematical models and are described by the problems of differential or integral equations or integrodifferential equations. Our work centers on the problems described by the integrodifferential models. It is important to note that when we describe the systems which are functions of space and time by partial differential equations, in some situations, such a formulation may not accurately model the physical system because, while describing the system as a function at a given time, it may fail to take into account the effect of past his- tory. Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention during the last few decades [1, 2, 3]. A good guide to the literature for neutral functional differential equations is the book by Hale and Verduyn Lunel [4] and the references therein. The existence of solution to evolution equations with nonlocal conditions in Ba- nach space was studied first by Byszewski [5, 6]. Byszewski and Lakshmikanthan [7] proved an existence and uniqueness of solutions of a nonlocal Cauchy problem in Banach spaces. Ntouyas and Tsamatos [8] studied the existence for semilinear evolution equations with nonlocal conditions. The problem of existence of solutions of evolution equations in Banach space has been studied by several authors [9, 10]. However, one may easily visualize that abrupt changes such as shock, harvesting and disasters may occur in nature. These phenomena are short time perturbations whose duration is negligible in comparison with the duration of the whole evolution process. Consequently, it is natural to assume, in modeling these problems, that these perturbations act instantaneously, that is in the form of impulses. The theory of impulsive differential equation [11, 12, 13] is much richer than the corresponding theory of differential equations without impulsive effects. The impulsive condition ∆u(ti) = u(t + i ) −u(t − i ) = Ii(u(t − i )), i = 1, 2, . . . ,m, 2010 Mathematics Subject Classification. 34A37, 34K05, 34K40, 47H10. Key words and phrases. Mild solution, neutral differential equation, impulsive condition, Haus- dorff measure of noncompactness, fixed point theorem. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 22 QUASILINEAR NEUTRAL IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS 23 is a combination of traditional initial value problems and short-term perturbations whose duration is negligible in comparison with the duration of the process. Liu [14] discussed the iterative methods for the solution of impulsive functional differential systems. Measures of noncompactness are a very useful tool in many branches of mathe- matics. They are used in the fixed point theory, linear operators theory, theory of differential and integral equations and others [15]. There are two measures which are the most important ones. The Kuratowski measure of noncompactness σ(X) of a bounded set X in a metric space is defined as infimum of numbers r > 0 such that X can be covered with a finite number of sets of diameter smaller than r. The Hausdorff measure of noncompactness χ(X) defined as infimum of numbers r > 0 such that X can be covered with a finite number of balls of radii smaller than r. There exist many formulae on χ(X) in various spaces [15, 18]. Let E be a Banach space and F be a subspace of E. Let χE(X), χF(X), σE(X), σF(X) denote Hausdorff and Kuratowski measures in spaces E,F, respectively. Then, for any bounded X ⊂ F we have χE(X) ≤ χF(X) ≤ σF(X) = σE(X) ≤ 2χE(X). The notion of a measure of weak compactness was introduced by De Blasi [16] and was subsequently used in numerous branches of functional analysis and the theory of differential and integral equations. Several authors have studied the mea- sures of noncompactness in Banach spaces [17, 18, 19]. Motivated by [9, 15, 20, 21], in this paper, we study the existence results for quasilinear equation represented by first-order neutral integrodifferential equations using the semigroup theory and the measure of noncompactness. 2. Preliminaries We consider the quasilinear integrodifferential equations with impulsive and non- local condition of the form d dt [ x(t) + e ( t,x(t), ∫ t 0 k(t,s,x(s))ds )] + A(t,x(t))x(t) = f(t,x(t)) + ∫ t 0 g(t,s,x(s))ds, t ∈ [0,b], t 6= tk,(1) x(0) + h(x) = x0,(2) ∆x(tk) = Ik(x(tk)), k = 1, 2, 3, . . . ,n,(3) where A : [0,b] × X → X is a continuous function in Banach space X, x0 ∈ X, f : [0,b] × X → X, g : Λ × X → X, h : PC([0,b],X) → X, e : [0,b] × X × X → X, k : Λ × X → X and ∆x(tk) = x(t+k ) − x(t − k ), for all k = 1, 2, . . . ,m; 0 = t0 < t1 < t2 < ... < tm < tm+1 = b; constitutes an impulsive condition. Here Λ = {(t,s) : 0 ≤ s ≤ t ≤ b}. Let X be a Banach space with norm || · ||. Let PC([0,b],X) consist of functions u from [0,b] into X, such that x(t) is continuous at t 6= ti and left continuous at t = ti and the right limit x(t + i ) exists, for i = 1, 2, 3, . . . ,n. Evidently PC([0,b],X) is a Banach space with the norm ‖x‖PC = sup t∈[0,b] ‖x(t)‖, 24 RADHAKRISHNAN and denoted L([0,b],X) by the space of X-valued Bochner integrable functions on [0,b] with the form ‖x‖L = ∫ b 0 ‖x(t)‖dt. The Hausdorff’s measure of noncompactness χY is defined by χ(B) = inf{r > 0, B can be covered by finite number of balls with radii r}, for bounded set B in a Banach space Y . Lemma 2.1 [15]. Let Y be a real Banach space and B,E ⊆ Y be bounded, with the following properties: (i) B is precompact if and only if χX (B) = 0. (ii) χY (B) = χY (B̄) = χY (conB), where B̄ and conB mean the closure and convex hull of B respectively. (iii) χY (B) ≤ χY (E), where B ⊆ E. (iv) χY (B + E) ≤ χY (B) + χY (E), where B + E = {x + y : x ∈ B, y ∈ E}. (v) χY (B∪E) ≤ max{χY (B), χY (E)}. (vi) χY (λB) ≤ |λ|χY (B), for any λ ∈ R. (vii) If the map F : D(F) ⊆ Y → Z is Lipschitz continuous with constant r, then χZ (FB) ≤ rχY (B), for any bounded subset B ⊆ D(F), where Z be a Banach space. (viii) χY (B) = inf{dY (B,E); E ⊆ Y is precompact} = inf{dY (B,E); E ⊆ Y is finite valued}, where dY (B,E) means the non- symmetric (or symmetric) Hausdorff distance between B and E in Y . (ix) If {Wn}+∞n=1 is decreasing sequence of bounded closed nonempty subsets of Y and lim n→∞ χY (Wn) = 0, then +∞⋂ n=1 Wn is nonempty and compact in Y. The map F : W ⊆ Y → Y is said to be a χY -contraction if there exists a positive constant r < 1 such that χY (F(B)) ≤ rχY (B) for any bounded closed subset B ⊆ W, where Y is a Banach space. Lemma 2.2 (Darbo-Sadovskii [15]). If W ⊆ Y is bounded closed and convex, the continuous map F : W → W is a χY -contraction, the map F has atleast one fixed point in W. We denote by χ the Hausdorff’s measure of noncompactness of X and also denote χc by the Hausdorff’s measure of noncompactness of PC([0,b],X). Before we prove the existence results, we need the following Lemmas. Lemma 2.3 [22] If W ⊆ PC([0,b],X) is bounded, then χ(W(t)) ≤ χc(W), for all t ∈ [0,b], where W(t) = {u(t); u ∈ W}⊆ X. Furthermore if W is equicontinuous on [0,b], then χ(W(t)) is continuous on [0,b] and χc(W) = sup{χ(W(t)), t ∈ [0,b]}. Lemma 2.4 [22, 23]. If {un}∞n=1 ⊂ L1([0,b],X) is uniformly integrable, then the function χ({un(t)}∞n=1) is measurable and χ ({∫ t 0 un(s)ds }∞ n=1 ) ≤ 2 ∫ t 0 χ({un(s)}∞n=1)ds.(4) QUASILINEAR NEUTRAL IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS 25 Lemma 2.5 If W ⊆PC([0,b],X) is bounded and equicontinuous, then χ(W(t)) is continuous and χ (∫ t 0 W(s)ds ) ≤ ∫ t 0 χ(W(s))ds, for all t ∈ [0,b],(5) where ∫ t 0 W(s)ds = {∫ t 0 u(s)ds : u ∈ W } . The C0 semigroup Uu(t,s) is said to be equicontinuous if (t,s) → {Uu(t,s)u(s) : u ∈ B} is equicontinuous for t > 0, for all bounded set B in X. The following lemma is obvious. Lemma 2.6 If the evolution family {Uu(t,s)}0≤s≤t≤b is equicontinuous and η ∈ L([0,b],R+), then the set {∫ t 0 Uu(t,s)u(s)ds, ||u(s)|| ≤ η(s), for a.e s ∈ [0,b] } , is equicontinuous for t ∈ [0,b]. We know that, for any fixed u ∈ PC([0,b],X) there exist a unique continuous function Uu : [0,b] × [0,b] → B(X) defined on [0,b] × [0,b] such that Uu(t,s) = I + ∫ t s Au(w)Uu(w,s)dw,(6) where B(X) denote the Banach space of bounded linear operators from X to X with the norm ||F|| = sup {||Fu|| : ||u|| = 1}, and I stands for the identity operator on X, Au(t) = A(t,u(t)), we have Uu(t,t) = I, Uu(t,s)Uu(s,r) = Uu(t,r), (t,s,r) ∈ [0,b] × [0,b] × [0,b], ∂Uu(t,s) ∂t = Au(t)Uu(t,s), for almost all t,s ∈ [0,b]. 3. The Existence of Mild Solution Definition 3.1 A function x ∈PC([0,b],X) is said to be a mild solution of (1)−(3) if it satisfies the integral equation x(t) = Ux(t, 0)x0 −Ux(t, 0)h(x) + Ux(t, 0)e(0,x(0), 0) −e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) + ∫ t 0 A(s,x(s))Ux(t,s)e ( s,x(s), ∫ s 0 k(s,τ,x(τ))dτ ) ds + ∫ t 0 Ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(τ))dτ ] ds + ∑ 0 0 such that ||h(x)|| ≤ N0, for all u ∈PC([0,b]; X). 26 RADHAKRISHNAN (H3) (i) The nonlinear function f : [0,b] ×X → X satisfies the Carathèodory- type conditions; that is, f(·,x) is measurable for all x ∈ X and f(t, ·) is continuous, for a.e t ∈ [a,b]. (ii) There exists a function α ∈L([0,b],R+) such that for every x ∈ X, we have ‖f(t,x)‖≤ α(t)(1 + ‖x‖), a.e t ∈ [0,b]. (iii) There exists a function mf ∈L([0,b],R+) such that, for every bounded K ⊂ X, we have χ(f(t,K)) ≤ mf (t)χ(K), a.e t ∈ [0,b]. (H4) (i) The nonlinear function g : [0,b]×[0,b]×X → X satisfies the Carathèodory- type conditions; i.e., g(·, ·,x) is measurable, for all x ∈ X and g(t,s, ·) is continuous for a.e t ∈ [a,b]. (ii) There exist two functions β1 ∈L([0,b],R+) and β2 ∈L([0,b],R+) such that for every x ∈ X, we have ‖g(t,s,x(s))‖≤ β1(t)β2(s)(1 + ‖x(s)‖), a.e t ∈ [0,b]. (iii) There exist functions mg,ng ∈L([0,b],R+) such that, for every bound- ed K ⊂ X, we have χ(g(t,s,K)) ≤ mg(t)ng(s)χ(K), a.e t ∈ [0,b]. Assume that the finite bound of ∫ t 0 mg(s)ds is G0. (H5) (i) The function e : [0,b] ×X ×X → X satisfies the Carathèodory-type conditions; that is, e(·,x,x1) is measurable, for all x,x1 ∈ X and e(t, ·, ·) is continuous, for a.e t ∈ [0,b]. (ii) There exists a function γ ∈L([0,b],R+) such that for every x,x1 ∈ X, we have ‖e(t,x,x1)‖≤ γ(t)(1 + ‖x‖) + ‖x1‖, a.e t ∈ [0,b]. (iii) The nonlinear function q : [0,b]×[0,b]×X → X satisfies the Caratheodory- type conditions; i.e. k(·, ·,x) is measurable, for all x ∈ X and k(t,s, ·) is continuous, for a.e t ∈ [0,b]. (iv) There exist two functions ω1 ∈ L([0,b],R+) and ω2 ∈ L([0,b],R+) such that for every x ∈ X, we have ‖k(t,s,x(s))‖≤ ω1(t)ω2(s)(1 + ‖x(s)‖), a.e t ∈ [0,b]. (v) There exists a function η ∈L([0,b],R+) such that for every x,x1 ∈ X, we have ‖A(t,x(t))e(t,x,x1)‖≤ η(t)‖e(t,x,x1)‖, a.e t ∈ [0,b]. (vi) There exists a function me ∈L([0,b],R+) such that, for every bounded K, K1 ⊂ X, we have χ(e(t,K,K1)) ≤ me(t)(χ(K) + ϕ(K1)), a.e t ∈ [0,b]. Assume that the finite bound of ∫ t 0 me(s)ds is G1. (vii) There exist functions mk,nk ∈L([0,b],R+) such that for every bound- ed K ⊂ X, we have χ(k(t,s,K)) ≤ mk(t)nk(s)χ(K), a.e t ∈ [0,b]. Assume that the finite bound of ∫ t 0 mk(s)ds is G2. QUASILINEAR NEUTRAL IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS 27 (H6) For every t ∈ [0,b] and there exist positive constants N1 and N2, the scalar equation m(t) = M0N0 + γ1(1 + m(s)) + M0γ0 + M0C1ω(t)(1 + m(s)) + γ(t)C1 ∫ t 0 η(t)ω1(s)ds +M0 ∫ t 0 [ α(s)(1 + m(s))ds + C0 ∫ t 0 β1(s)(1 + m(s))ds + n∑ k=1 dk ] , where C0 = ∫ s 0 β(t)dt. (H7) Ik : X → X is continuous. There exist constants dk > 0 k = 1, 2, 3, . . . ,n such that ‖Ik(x(tk))‖≤ n∑ k=1 dk, where, k = 1, 2, 3, . . . ,n. For any bounded subset K ⊂ X, and there is a constant lk > 0 such that χ(Ik(K)) ≤ n∑ k=1 liχ(K), k = 1, 2 . . . ,n. Theorem: 3.1. If assumptions (H1) − (H7) holds, then the quasilinear neutral impulsive problem (1) − (3) has at least one mild solution. Proof. Let m(t) be a solution of the scalar equation m(t) = M0N0 + γ1(1 + m(s)) + M0γ0 + M0C1ω(t)(1 + m(s)) + γ(t)C1 ∫ t 0 η(t)ω1(s)ds +M0 ∫ t 0 [ α(s)(1 + m(s))ds + C0 ∫ t 0 β1(s)(1 + m(s))ds + n∑ k=1 dk ] .(7) Let us assume that the finite bound of ∫ t 0 β2(s)ds is C0, for t ∈ [0,b]. Consider the map F : PC([0,b],X) →PC([0,b],X) defined by (Fx)(t) = Ux(t, 0)h(x) + Ux(t, 0)e(0,x(0), 0) −e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) + ∫ t 0 A(s,x(s))Ux(t,s)e ( s,x(s), ∫ s 0 k(s,τ,x(τ))dτ ) ds + ∫ t 0 Ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(τ))dτ ] ds + ∑ 0 0, there is a sequence {xk}∞k=1 ⊆ Wn, such that (see, e.g. [24], QUASILINEAR NEUTRAL IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS 29 pp.125). χ(Wn+1(t)) = χ(FWn(t)) ≤ 2χ ( e(t,{xk(s)}∞k=1, ∫ t 0 k(t,s,{xk(t)}∞k=1)ds) ) +2M0η(t) ∫ t 0 χ ( e(s,{xk(s)}∞k=1, ∫ s 0 k(s,τ,{xk(τ)}∞k=1)dτ) ) ds +2M0 ∫ t 0 χ ( f(s,{xk(s)}∞k=1) ) ds + 4M0 ∫ t 0 ∫ s 0 χ ( g(s,τ,{uk(τ)}∞k=1) ) dτds +2M0 n∑ i=1 χ ( Ik({uk(tk)}∞k=1) ) + � ≤ 2me(t)χ{xk(t)}∞k=1 + 2mk(t) ∫ t 0 mk(s)χ{xk(s)}∞k=1ds +2M0η(t) [∫ t 0 me(s)χ{xk(s)}∞k=1ds + 2 ∫ t 0 ∫ s 0 mk(s)mk(τ)χ{xk(τ)}∞k=1dτds ] +2M0 ∫ t 0 mf (s)χ ( {uk(s)}∞k=1 ) ds+4M0 ∫ t 0 ∫ s 0 mg(s)ng(τ)χ ( {uk(τ)}∞k=1 ) dτds +2M0 n∑ i=1 liχ ( {uk(tk)}∞k=1 ) + � ≤ 2 [ me(t) + mk(t)G2 + M0η(t)G1 ] χ(Wn(t)) + 2M0 [∫ t 0 {2G2mk(s) +mf (s)}χ(Wn(s))ds + 2G0 ∫ t 0 ng(s)χ(Wn(s))ds ] + 2M0 n∑ k=1 lkχ(Wn(tk))+�. Since � > 0 is arbitrary, it follows that from the above inequality that χ(Wn+1(t)) ≤ 2 [ me(t) + mk(t)G2 + M0η(t)G1 ] χ(Wn(t)) +2M0 [∫ t 0 [2G2mk(s) + mf (s) + 2G0ng(s)]χ(Wn(s)) ] ds +2M0 n∑ k=1 lkχ(Wn(tk), for all t ∈ [0,b].(9) Because Wn is decreasing for n, we have λ(t) = lim n→∞ χ(Wn(t)), for all t ∈ [0,b]. From (9), we have λ(t) ≤ 2 [ me(t) + mk(t)G2 + M0η(t)G1 ] λ(t) +2M0 [∫ t 0 [2G2mk(s) + mf (s) + 2G0ng(s)]λ(s)ds+ n∑ k=1 lkλ(tk) ] , for t ∈ [0,b], which implies that λ(t) = 0, for all ti ∈ [0,b]. By Lemma 2.3, we know that lim n→∞ χ(Wn(t)) = 0. Using Lemma 2.1 we know that W = ∞⋂ n=1 Wn is convex 30 RADHAKRISHNAN compact and nonempty in PC([0,b],X) and F(W) ⊂ W. By the Schauder fixed point theorem, there exist at least one mild solution u of the initial value problem (1) − (3), where x ∈ W is a fixed point of the continuous map F. � Remark 3.2. If the functions f, g and Ii are compact or Lipschitz continuous (see e.g [5, 7]), then (H3) − (H7) is automatically satisfied. In some of the early related results in references and above results, it is supposed that the map h is uniformly bounded. In fact, if h is compact, then it must be bounded on bounded set. Here we give an existence result under growth condition of f,g and Ii, when h is not uniformly bounded. Precisely, we replace the assumptions (H3) − (H6) by (H8) There exists a function p ∈L([0,b],R+) and a increasing function φ : R+ → R+ such that ‖f(t,x)‖≤ Lf (t)φ(‖x‖), for a.e t ∈ [0,b], for all x ∈PC([0,b],X). (H9) There exist two functions Lg ∈ L([0,b],R+)and L̂g ∈ L([0,b],R+) and a increasing function Ψ : R+ → R+ such that ‖g(t,s,x)‖≤ Lg(t)L̂g(s)Ψ(‖x‖), for a.e t ∈ [0,b] and for all Lg ∈PC([0,b],X). Assume that the finite bound of ∫ t 0 Lg(s)ds is G3. (H10) There exists a function Le ∈ L([0,b],R+) and a increasing function Γ : R+ → R+ such that ‖e(t,x,x1)‖≤ Le(t)Γ(‖x‖) + ‖x1‖ for a.e t ∈ [0,b] and for all Lg ∈PC([0,b],X). Assume that the finite bound of ∫ t 0 Le(s)ds is G5. (H11) There exist two functions Lk ∈ L([0,b],R+)and L̂k ∈ L([0,b],R+) and a increasing function Θ : R+ → R+ such that ‖k(t,s,x)‖≤ Lk(t)L̂k(s)Θ(‖x‖), for a.e t ∈ [0,b] and for all Lk ∈PC([0,b],X). Assume that the finite bound of ∫ t 0 Lk(s)ds is G4. Theorem: 3.2. Suppose that the assumptions (H1) − (H2) and (H8) − (H11) are satisfied, then the equation (1) − (3) has at least one mild solution if lim r→∞ sup 1 r { M0 [ ϕ(r) + Le(t) ] + Le(t)(Γ‖x‖) +G3Lk(t)Θ(r) + η(t)M0 [ G4Γ(r) + G3Θ(r) ∫ t 0 L̂k(s)ds ] +M0 [ φ(r) ∫ t 0 Lf (s)ds + G2Ψ(r) ∫ t 0 L̂g(s)ds + n∑ k=1 dk ]} < 1,(10) where ϕ(r) = sup{||h(x)||, ||x|| ≤ r}. QUASILINEAR NEUTRAL IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS 31 Proof. The inequality (10) implies that there exist a constant r > 0 such that M0 [ ϕ(r) + Le(t) ] + Le(t)(Γ‖x‖) + G3Lk(t)Θ(r) + η(t)M0 [ G4Γ(r) +G3Θ(r) ∫ t 0 L̂k(s)ds ] + M0 [ φ(r) ∫ t 0 p(s)ds + G2Ψ(r) ∫ t 0 L̂g(s)ds + n∑ k=1 dk ] < r, As in the proof of Theorem 3.1, let W0 = {x ∈PC([0,b],X), ||x(t)|| ≤ r} and W1 = con FW0. Then for any x ∈ W1, we have ‖x(t)‖ ≤ ‖Ux(t, 0)h(x)‖ + ‖Ux(t, 0)e(0,x(0), 0)‖ + ‖e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) ‖ + ∫ t 0 ‖A(t,x(t))Ux(t,s)e ( t,x(t), ∫ s 0 k(s,τ,x(τ))dτ ) ‖ds + ∫ t 0 ‖Ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(s))dτ ] ‖ds + ∑ 0 0 such that ‖h(x) −h(y)‖≤ L0‖x−y‖, x,y ∈PC([0,b],X). Theorem: 4.1. Suppose that the assumptions (H1)−(H12) are satisfied, then the equation (1) − (3) has at least one mild solution provided that M0[L0 + h4(t)]χc(B) + 2 [ me(t) + mk(t)G2 + M0η(t)G1 ] +2M0 [∫ t 0 {2G2mk(s) + mf (s) + 2G0ng(s)}ds + n∑ k=1 lk ] < 1.(11) Proof. Consider the map F : PC([0,b],X) → PC([0,b],X) is defined by F = F1 + F2, where (F1x)(t) = Ux(t, 0)h(u) + Ux(t, 0)e(0,x(0), 0), (F2u)(t) = ∫ t 0 A(t,x(t))Ux(t,s)e ( t,x(t), ∫ s 0 k(s,τ,x(τ))dτ ) ds −e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) + ∫ t 0 Ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(τ))dτ ] ds + ∑ 0 0, there is a sequence {xk}∞k=1 ⊂ B such that χ(F2(B(t)) ≤ 2χ({F2xi(t)}∞n=1 + �. QUASILINEAR NEUTRAL IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS 33 Note that B and F2B are equicontinuous, we can get from Lemma 2.1, Lemma 2.4, Lemma 2.5 and using the assumptions we get χ(F2B(t)) ≤ 2χ ( e(t,{xk(s)}∞k=1, ∫ t 0 k(t,s,{xk(t)}∞k=1)ds) ) 2M0η(t) ∫ t 0 χ ( e(s,{xk(s)}∞k=1, ∫ s 0 k(s,τ,{xk(τ)}∞k=1)dτ) ) ds +2M0 ∫ t 0 χ ( f(s,{xk(s)}∞k=1) ) ds + 4M0 ∫ t 0 ∫ s 0 χ ( g(s,τ,{uk(τ)}∞k=1) ) dτds +2M0 n∑ i=1 χ ( Ik({uk(tk)}∞k=1) ) + � ≤ 2me(t)χ{xk(t)}∞k=1 + 2mk(t) ∫ t 0 mk(s)χ{xk(s)}∞k=1ds +2M0η(t) [∫ t 0 me(s)χ{xk(s)}∞k=1ds + 2 ∫ t 0 ∫ s 0 mk(s)mk(τ)χ{xk(τ)}∞k=1dτds ] +2M0 ∫ t 0 mf (s)χ ( {uk(s)}∞k=1 ) ds+4M0 ∫ t 0 ∫ s 0 mg(s)ng(τ)χ ( {uk(τ)}∞k=1 ) dτds +2M0 n∑ i=1 liχ ( {uk(tk)}∞k=1 ) + � ≤ 2 [ me(t) + mk(t)G2 + M0η(t)G1 ] χ(B) +2M0 [∫ t 0 {2G2mk(s) + mf (s)}χ(B))ds + 2G0 ∫ t 0 ng(s)χ(B)ds ] +2M0 n∑ k=1 lkχ(B)+�. Since � > 0 is arbitrary, it follows that from the above inequality that χc(F2B(t)) ≤ 2 [ me(t) + mk(t)G2 + M0η(t)G1 ] χc(B) +2M0 [∫ t 0 {2G2mk(s) + mf (s) + 2G0ng(s)}ds + n∑ k=1 lk ] χc(B)(13) for any bounded B ⊂ W. Now, for any subset B ⊂ W, due to Lemma 2.1, (12) and (13) we have χc(FB) ≤ χc(F1B) + χc(F2B) ≤ M0[L0 + h4(t)]χc(B) + 2 [ me(t) + mk(t)G2 + M0η(t)G1 ] χc(B) +2M0 [∫ t 0 {2G2mk(s) + mf (s) + 2G0ng(s)}ds + n∑ k=1 lk ] χc(B)(14) By (14) we know that F is a χc-contraction on W. By Lemma 2.2, there is a fixed point x of F in W, which is a solution of (1) − (3). This completes the proof. Theorem: 4.2. Suppose that the assumptions (H1)−(H12) are satisfied, then the equation (1)−(3) has at least one mild solution if (15) and the following condition 34 RADHAKRISHNAN are satisfied. M0L0 + lim r→∞ sup 1 r { M0Le(t) + Le(t)Γ(r) + G3Lk(t)Θ(r) +η(t)M0 [ G4Γ(r) + G3Θ(r) ∫ t 0 L̂k(s)ds ] +M0 [ φ(r) ∫ t 0 Lf (s)ds + G2Ψ(r) ∫ t 0 L̂g(s)ds + n∑ k=1 dk ]} < 1.(15) Proof. From the equation (15) and fact that L0 < 1, there exists a constant r > 0 such that M0 ( rL0 + ‖h(0)‖ + Le(t) ) + Le(t)(Γ‖x‖) + G3Lk(t)Θ(r) +η(t)M0 [ G4Γ(r) + G3Θ(r) ∫ t 0 L̂k(s)ds ] +M0 [ φ(r) ∫ t 0 Lf (s)ds + G2Ψ(r) ∫ t 0 L̂g(s)ds + n∑ k=1 dk ] } < r. We define W0 = {x ∈ PC([0,b],X), ‖x(t)‖ ≤ r, for all t ∈ [0,b]}. Then for every x ∈ W0, we have ‖Fx(t)‖ ≤ ‖Ux(t, 0)h(u)‖ + ‖Ux(t, 0)e(0,x(0), 0)‖ + ‖e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) ‖ + ∫ t 0 ‖A(t,x(t))Ux(t,s)e ( t,x(t), ∫ s 0 k(s,τ,x(τ))dτ ) ‖ds + ∫ t 0 ‖Ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(s))dτ ] ‖ds + ∑ 0 0 such that ‖CW−1‖≤M1. Theorem: 5.1. If the assumptions (H1) − (H13) are satisfied, then the system (16) − (18) is controllable on J. 36 RADHAKRISHNAN Proof. Using the assumption (H13), for an arbitrary function u(·), define the control v(t) = W−1 [ u1 −Ux(b, 0)x0 + Ux(b, 0)h(x) −Ux(b, 0)e(0,x(0), 0) +e ( b,x(b), ∫ b 0 k(b,s,x(s))ds ) − ∫ b 0 A(b,x(b))Ux(b,s)e ( b,x(b), ∫ s 0 k(s,τ,x(τ))dτ ) ds − ∫ b 0 Ux(b,s) [ f(s,x(s)) − ∫ s 0 g(s,τ,x(τ))dτ ] ds − ∑ 0