J. Nonlinear Sci. Appl. 4 (2011), no. 2, 102–114 The Journal of Nonlinear Science and Applications http://www.tjnsa.com EXISTENCE OF GLOBAL SOLUTIONS FOR IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS S. SIVASANKARAN1 , M. MALLIKA ARJUNAN2 AND V. VIJAYAKUMAR3,∗ Dedicated to Professor Themistocles M. Rassias on the occasion of his sixtieth birthday Communicated by Choonkil Park Abstract. In this paper, we study the existence of global solutions for a class of impulsive abstract functional differential equation with nonlocal conditions. The results are obtained by using the Leray-Schauder alternative fixed point theorem. An example is provided to illustrate the theory. 1. Introduction In this work, we discuss the existence of global solutions for an impulsive abstract functional differential equation with nonlocal conditions in the form u0 (t) = Au(t) + f (t, ut , u(ρ(t))), t ∈ I = [0, ∞), u0 = g(u, ϕ) ∈ B, (1.1) (1.2) ∆u(ti ) = Ii (uti ), (1.3) where A is the infinitesimal generator of a C0 -semigroup of bounded linear operators (T (t))t≥0 defined on a Banach space (X, k · k); I is an interval of the form [0, ∞); 0 < t1 < t2 < · · · < ti < · · · are pre-fixed numbers; the history ut : (−∞, 0] → X, ut (θ) = u(t + θ), belongs to some abstract phase space B defined axiomatically; ρ : [0, ∞) → [0, ∞), f : I × B × X → X, Ii : B → X are Date: Received: September 19, 2010; Revised: November 24, 2010. ∗ Corresponding author c 2011 N.A.G. ° 2000 Mathematics Subject Classification. Primary 34A37, 34K30, 35R10, 35R12, 47D06, 49K25. Key words and phrases. Impulsive functional differential equations; mild solutions; global solutions; semigroup theory. 102 IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS 103 appropriate functions and the symbol ∆ξ(t) represents the jump of the function ξ at t, which is defined by ∆ξ(t) = ξ(t+ ) − ξ(t− ). Differential equations arise in many real world problems such as physics, population dynamics, ecology, biological systems, biotechnology, optimal control, and so forth. Much has been done the assumption that the state variables and systems parameters change continuously. 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 equations has become an active area of investigation due to their applications in the field such as mechanics, electrical engineering, medicine biology and so on; see for instance [1, 3, 4, 7, 8, 9, 23, 26, 30, 35, 37, 38]. For more details on this theory and on its applications we refer to the monographs of Benchohra [10], Lakshmikantham [29], Bainov and Simeonov [5] and Samoilenko and Perestyuk [39], and the papers of [2, 15, 16, 17, 18, 25, 31], where numerous properties of their solutions are studied and detailed bibliographies are given. The literature concerning differential equations with nonlocal conditions also include ordinary differential equations, partial differential equations, abstract partial functional differential equations. Related to this matter, we cite among others works, [6, 11, 12, 13, 14, 20, 22, 28, 33, 34]. Our main results can been seen as a generalization of the work in [21] and the above mentioned impulsive partial functional differential equations. 2. Preliminaries In this paper, A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (T (t))t≥0 defined on a Banach space (X, k · k) and M, δ are positive constants such that kT (t)k ≤ M e−δt for every t ≥ 0. For the theory of strongly continuous semigroup, refer the reader to Pazy [36]. To consider the impulsive condition (1.3), it is convenient to introduce some additional concepts and notations. 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 denote by PC([σ, τ ]; X) the space formed by all normalized piecewise continuous functions from [σ, τ ] into X. In particular, we introduce the space PC formed by all functions u : [0, a] → X such that u is continuous at t 6= ti , i = 1, ..., n. It is clear that, PC([σ, τ ]; X) endowed with the norm of uniform convergence, is a Banach space. On the other hand, the notation PC([0, ∞); X) stands for the space formed by all bounded normalized piecewise continuous functions u : [0, ∞) → X such that u|[0,a] ∈ PC([0, a]; X) for all a > 0, endowed with the uniform convergence topology. In what follows, for the case I = [0, a], we set t0 = 0, tn+1 = a, and for u ∈ PC([0, a]; X), we denote by u ei ∈ C([ti , ti+1 ]; X), i = 0, 1, ..., n, the function 104 S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR given by ( u ei (t) = u(t), u(t+ i ), for t ∈ (t, ti+1 ], for t = ti . ei , i = 0, 1, ..., n, for the sets Moreover, for B ⊂ PC, we employ the notation B ei = {e B ui : u ∈ B}. Lemma 2.1. A set B ⊂ PC is relatively compact in PC([0, a]; X) if, and only if, ei , is relatively compact in the space C([ti , ti+1 ]; X) for every i = 0, 1, ..., n. the set B In this paper, the phase space B is a linear space of functions mapping (−∞, 0] into X endowed with a seminorm k · kB and verifying the following axioms. (A) If x : (−∞, a) → X, a > 0, is continuous on [0, a) and x0 ∈ B, then for every t ∈ [0, a) the following conditions hold: (i) xt is in B. (ii) kx(t)k ≤ Hkxt kB . (iii) kxt kB ≤ K(t) sup{kx(s)k : 0 ≤ s ≤ t} + M (t)kx0 kB , where H > 0 is a constant; K, M : [0, ∞) → [1, ∞), K is continuous, M is locally bounded and H, K, M are independent of x(·). (B) The space B is complete. Remark 2.2. In this paper, K is a positive constant such that supt≥0 {K(t), M (t)} ≤ K. We know from that the functions M (·), K(·) are bounded if, for instance, B is a fading memory space, see [27, pp. 190] for additional details. Example 2.3. The phase space PC r × Lp (ρ, X) Let r ≥ 0, 1 ≤ p < ∞, and ρ : (−∞, −r] → R be a non-negative measurable function which satisfies the conditions (g-5)-(g-6) of [14]. Briefly, this means that ρ is locally integrable and there exists a non-negative locally bounded function γ on (−∞, 0] such that ρ(ξ + θ) ≤ γ(ξ)ρ(θ), for all ξ ≤ 0 and θ ∈ (−∞, −r) \ Nξ , where Nξ ⊆ (−∞, −r) is a set whose Lebesgue measure zero. The space B = PC r ×Lp (ρ, X) consists of all classes of functions ψ : (−∞, 0] → X such that ψ is continuous on [−r, 0], Lebesgue-measurable, and ρkψkp is Lebesgue integrable on (−∞, −r). The seminorm in PC r × Lp (ρ, X) is defined as follows: ³ Z −r ´ p1 p kψkB = sup{kψ(θ)k : −r ≤ θ ≤ 0} + ρ(θ)kψ(θ)k dθ . −∞ The space B = PC r × Lp (ρ, X) satisfies axioms (A) and (B). Moreover, when 1 r = 0 and p = 2, one can take H = 1, M (t) = γ(−t) 2 and K(t) = 1 + ¢ 12 ¡R0 ρ(θ)dθ for t ≥ 0 (see [27, Theorem 1.3.8]). We also note that if the −t conditions (g-5)-(g-7) of [27] hold, then B is a uniform fading memory. For additional details concerning phase space, we cite [27] In this paper, h : [0, ∞) → R be a positive, continuous and non-decreasing function such that h(0) = 1 and limt→∞ h(t) = ∞. In the sequel, PC([0, ∞); X) IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS 105 and (PC h )0 (X) are the spaces defined by x|[0,a] ∈ PC([0, a]; X), ∀a ∈ (0, +∞) \ {ti : i ∈ N}, PC([0, ∞); X) = x : [0, ∞) → X : x0 = 0, kxk∞ = supt≥0 kx(t)k < +∞, n o kx(t)k (PC)0h (X) = x ∈ PC([0, ∞); X) : lim =0 , t→∞ h(t) endowed with the norms kxk∞ = supt≥0 kx(t)k and kxkPC h = supt≥0 spectively. Additionally, we define the space n o BPC 0h (X) = u ∈ PC(R, X) : u0 ∈ B, u|[0,∞) ∈ PC 0h (X) kx(t)k , h(t) re- endowed with the norm kukBPC 0h = ku0 kB + ku|[0,∞) kh . We recall here the following compactness criterion. Lemma 2.4. A bounded set B ⊂ (PC)0h (X) is relatively compact in (PC)0h (X) if, and only if: (a) The set Ba = {u|[0,a] : u ∈ B} is relatively compact in PC([0, a]; X), for all a ∈ (0, ∞) \ {ti ; i ∈ N}. (b) limt→∞ kx(t)k = 0, uniformly for x ∈ B. h(t) Theorem 2.5. ([19, Theorem 6.5.4] Leray-Schauder Alternative Theorem). Let D be a closed convex subset of a Banach space (Z, k · kZ ) and assume that 0 ∈ D. Let F : D → D be a completely continuous map, then the set {x ∈ D : x = λF (x), 0 < λ < 1} is unbounded or the map F has a fixed point in D. Theorem 2.6. ([32, Theorem 4.3.2]). Let D be a convex, bounded and closed subset of a Banach space (Z, k · kZ ) and F : D → D be a condensing map. Then F has a fixed point in D. 3. Existence of mild solutions In this section, we study the existence of mild solutions for the nonlocal abstract Cauchy problem (1.1)-(1.3). To treat this system, we introduce the following conditions. H1 The function ρ : [0, ∞) → [0, ∞) is continuous and ρ(t) ≤ t for every t ≥ 0. H2 The function f : I ×B×X → X is continuous and there exist an integrable function m : [0, ∞) → [0, ∞) and a nondecreasing continuous function W : [0, ∞) → (0, ∞) such that kf (t, ψ, x)k ≤ m(t)W (kψkB + kxk), for every (t, ψ, x) ∈ [0, ∞) × B × X. H3 The function g : BPC 0h (X) × B → B is continuous and there exists Lg ≥ 0 such that kg(u, ϕ) − g(v, ϕ)kB ≤ Lg ku − vkBPC 0h , u, v ∈ BPC 0h (X). 106 S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR H4 The function g(·, ϕ) : BPC 0h (X) → B is completely continuous and uniformly bounded. In the sequel, N is a positive constant such that kg(ψ, ϕ)kB ≤ N for every ψ ∈ BPC 0h (X). H5 The maps Ii : B → X are completely continuous and uniformly bounded, i ∈ N. In what follows, we use the notation Ni = sup{kIi (φ)k : φ ∈ B}. H6 There are positive constants Li , such that kIi (ψ1 ) − Ii (ψ2 )k ≤ Li kψ1 − ψ2 kB , ψ1 , ψ2 ∈ B, i ∈ N. In this work, we adopt the following concept of mild solutions for (1.1)-(1.3). Definition 3.1. A function u ∈ PC(R, X) is called a mild solution of system (1.1)-(1.3) if u0 = g(u, ϕ) and Z t u(t) = T (t)g(u, ϕ)(0) + T (t − s)f (s, us , u(ρ(s)))ds 0 X + T (t − ti )Ii (uti ), t ∈ I = [0, ∞). 0<ti <t Now, we can establish the principal results of this section. Theorem 3.2. Assume H1 , H2 , H4 , H5 and H6 be hold and that the following properties are verified. © ª (a) The set T (t)f (s, ψ, x) : (s, ψ, x) ∈ [0, t]×Br (0, B)×Br (0, X) is relatively compact in X for every t ≥ 0 Rand all r > 0. t 1 (b) For every K > 0, limt→∞ h(t) m(s)(Kh(s))ds = 0. 0 R∞ R ∞ ds If M K(1 + H) 0 m(s)ds < c W (s) , where c = KN (1 + H)(1 + M H) P + MK ∞ i=1 Ni , then there exists a mild solution for the system (1.1)-(1.3). Proof. Let Γ : BPC 0h (I : X) → BPC 0h (X) be the operator defined by (Γu)0 = g(u, ϕ) and Z t Γu(t) = T (t)g(u, ϕ)(0) + T (t − s)f (s, us , u(ρ(s)))ds 0 X + T (t − ti )Ii (uti ), t ≥ 0. 0<ti <t Next we prove that Γ verifies the conditions in Theorem 2.5. To begin, we note that for u ∈ BPC 0h (X) and t ≥ 0, ku(t)k ≤ ku|I kh h(t). If K is the constant in Remark 2.2, then Z t ∞ MNH M M X kΓu(t)k ≤ + m(s)W (2KkukBPC 0h h(s))ds + Ni , h(t) h(t) h(t) 0 h(t) i=1 which from (b) permits us to conclude that Γ is a function from BPC 0h (X) into BPC 0h (X). Let (xn )n∈N be a sequence in BPC 0h (X) and x ∈ BPC 0h (X) such that xn → x in BPC 0h (X). Let ε > 0 and η = supn∈N kxn kBPC 0h . From condition (b), we can IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS select L > 0 such that Z t ∞ 2M N H M 2M X ε + m(s)W (2Kηh(s))ds + Ni ≤ , h(t) h(t) 0 h(t) i=1 2 107 t≥L (3.1) From the properties of the functions f , g, we can choose Nε ∈ N such that ε kg(xn , ϕ) − g(x, ϕ)kB ≤ , (3.2) 6(M H + 1) Z L ε kf (s, xns , xn (ρ(s))) − f (s, xs , x(ρ(s)))kds ≤ , (3.3) 6M 0 and X kIi (xnti ) − Ii (xti )k ≤ 0≤ti ≤L ε , 6M n ≥ Nε for every n ≥ Nε . Under these conditions, we see that n kΓxn (t) − Γx(t)k o ε sup : t ∈ [0, L], n ≥ Nε ≤ . h(t) 2 (3.4) (3.5) Moreover, for t ≥ L and n ≥ Nε , we find that ∞ kΓxn (t) − Γx(t)k kg(xn , ϕ)(0) − g(x, ϕ)(0)k 2M X ≤M + Ni h(t) h(t) h(t) i=1 Z t M kf (s, xns , xn (ρ(s))) − f (s, xs , x(ρ(s)))kds, + h(t) 0 Z t ∞ 2M N H 2M X M ≤ m(s)W (2Kηh(s))ds + + Ni , h(t) h(t) 0 h(t) i=1 and hence, sup n kΓxn (t) − Γx(t)k h(t) o ε : t ≥ L, n ≥ Nε ≤ . 2 (3.6) From the inequalities (3.2)-(3.6), it follows that kΓxn − ΓxkBPC 0h ≤ ε for every n ≥ Nε , which proves that Γ is continuous. To prove that Γ is completely continuous, we introduce the decomposition Γ = Γ1 + Γ2 + Γ3 , where (Γ1 u)0 = g(u, ϕ), (Γ2 u)0 = 0, (Γ3 u)0 = 0 and Γ1 u(t) =T (t)g(u, ϕ)(0), t ≥ 0 Z t Γ2 u(t) = T (t − s)f (s, us , u(ρ(s)))ds, t ≥ 0 0 X Γ3 u(t) = T (t − ti )Ii (uti ), t ≥ 0 (3.7) (3.8) (3.9) 0<ti <t From Lemma 2.4 and the properties of the semigroup (T (t))t≥0 , it is easy to see that Γ1 is completely continuous. Next, we prove that Γ2 is also completely con© 0 tinuous. From [24, Lemma 3.1], we infer that, Γ2 Br (0, BPC h )|[0,a] = Γ2 x|[0,a] : 108 S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR ª x ∈ Br (0, BPC 0h ) is relatively compact in PC([0, a]; X) for every a > 0. Considering this property, Lemma 2.4 and the fact that M kΓ2 x(t)k ≤ h(t) h(t) Z t m(s)W (2Krh(s))ds → 0, as t → ∞, 0 uniformly for x ∈ Br (0, BPC 0h ), we can conclude that the set Γ2 (Br (0, BPC 0h )) is relatively compact in BPC 0h (X) for every r > 0. Next, by using Lemma 2.1, we prove that Γ3 is also completely continuous. We observe that the continuity of Γ3 is obvious. On the other hand, for r > 0, t ∈ [ti , ti+1 ], i ≥ 1, and u ∈ Br = Br (0, PC([0, a]; X)), we have that there exists re > 0 such that Pi t ∈ (ti , ti+1 ), Pj=1 T (t − tj )Ij (Bre(0; B)), i g [Γ3 u]i (t) ∈ t = ti+1 , j=1 T (ti+1 − tj )Ij (Bre(0; B)), Pi−1 T (t − t )I (B (0; B)) + I (B (0; B)), t = t , i j j re i re i j=1 where Bre(0; B) is an open ball of radius re. From condition H5 it follows that [Γ^ 3 (Br )]i (t) is relatively compact in X, for all t ∈ [ti , ti+1 ], i ≥ 1. Moreover, using the compactness of the operators {Ii }i and the continuity of (T (t))t≥0 , we conclude that for ε > 0, there exists δ > 0 such that kT (t)z − T (s)zk ≤ ε, z∈ n [ Ii (Bre(0, B)), (3.10) i=1 for all ti , i = 1, · · · , n, t, s ∈ (ti , ti+1 ] with |t − s| < δ. Under these conditions, for u ∈ Br , t ∈ [ti , ti+1 ], i ≥ 0, and 0 < |h| < δ with t + h ∈ [ti , ti+1 ], we have that g g k[Γ 3 u]i (t + h) − [Γ3 u]i (t)k ≤ M nε, t ∈ I, (3.11) which proves that the set of functions [Γ^ 3 (Br )]i , i ≥ 0, is uniformly equicontinuous. Now, from Lemma 2.1, we conclude that Γ3 is completely continuous. Thus, Γ is completely continuous. To finish the proof, we obtain a priori estimates for the solutions of the integral equation {u = λΓu, λ ∈ (0, 1)}. Let λ ∈ (0, 1) and uλ ∈ BPC 0h (X) be a solution of λΓz = z. By using the notation αλ (t) = K(sups∈[0,t] kuλ (s)k+kuλ0 kB ), for t ∈ I we see that Z t λ ku (t)k ≤ M N H + M 0 Z 0 ∞ X i=1 t ≤ MNH + M m(s)W (kuλs kB + kuλ (ρ(s))k)ds + M m(s)W ((1 + H)αλ (s))ds + M ∞ X i=1 Ni , Ni , IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS 109 and hence, αλ (t) Z t ≤ Kkg(u, ϕ)kB + KM N H + KM λ m(s)(W (1 + H)α (s))ds + KM 0 Z Ni , i=1 t ≤ KN (1 + M H) + KM ∞ X λ m(s)W ((1 + H)α (s))ds + KM 0 ∞ X Ni . i=1 Denoting by βλ the right hand side of last inequality, we see that βλ0 (t) ≤ KM m(t)W ((1 + H)βλ (t)), which implies that Z (1+H)βλ (t) (1+H)βλ (0)=c ds ≤ M K(1 + H) W (s) Z Z ∞ ∞ m(s)ds < 0 c ds . W (s) This inequality permits us to conclude that the set of functions {βλ : λ ∈ (0, 1)} is bounded in C(R) and as consequence that the set {uλ ; λ ∈ (0, 1)} is bounded in BPC 0h (X). Now the assertion is a consequence of Theorem 2.5. The proof is complete. ¤ Theorem 3.3. Assume assumptions H1 , H2 , H3 , H5 and H6 be hold. If the conditions (a) and (b) of Theorem 3.2 are valid and Z +∞ Lg (1 + M H) + M lim inf r→∞ 0 ∞ X m(s)W ((1 + 2K)rh(s)) ds + M Li < 1, rh(s) i=1 (3.10) then there exists a mild solution of (1.1)-(1.3). Proof. Let Γ, Γ1 , Γ2 , Γ3 be the operators introduced in the proof of Theorem 3.2, we claim that there exists r > 0 such that Γ(Br ) ⊂ Br , where Br = Br (0, BPC 0h (X)). In fact, if this property is false, then for every r > 0, there 110 S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR r r exist xr ∈ Br and tr ≥ 0 such that r < k(Γxr )0 kB + k Γxh(t(tr ) ) k. Consequently kg(xr , ϕ)(0)k r ≤ kg(xr , ϕ)kB + M h(tr ) Z tr M + m(s)W (Kkxr0 kB + K sup kxr (θ)k + kxr (ρ(s))k)ds h(tr ) 0 θ∈[0,s] ∞ ∞ X Li kxt k M X i + kI (0)k + M i r r h(t ) i=1 h(t ) i=1 MH M ≤ Lg kxr kBPC oh + kg(0, ϕ)kB + Lg kxr kBPC 0h + kg(0, ϕ)kB r h(t ) h(tr ) Z tr M + m(s)W (Kr + Krh(s) + rh(s))ds h(tr ) 0 ∞ ∞ X M X kIi (0)k + Mr Li + h(tr ) i=1 i=1 Z tr ≤ Lg (1 + M H)r + (1 + M )kg(0, ϕ)kB + M 0 and then, Z +∞ 1 ≤ Lg (1 + M H) + M lim inf r→∞ 0 m(s)((2K + 1)rh(s)) ds h(s) ∞ X m(s)((2K + 1)rh(s)) ds + M Li , rh(s) i=1 which is contrary to (3.10). Let r > 0 such that Γ(Br ) ⊂ Br . From the proof of the Theorem 3.2, we know that Γ2 is a completely continuous on Br . Moreover, from the estimate, kΓ1 u − Γ1 vkBPC 0h ≤ Lg ku − vkBPC 0h + Hkg(u, ϕ) − g(v, ϕ)kB ≤ Lg (1 + H)ku − vkBPC 0h , we infer that Γ1 is a contraction on Br . It is easy to see that ∞ X kΓ3 u − Γ3 vkBPC 0h ≤ M Li ku − vkBPC 0h , u, v ∈ BPC 0h , i=1 which implies that Γ3 is a contraction in BPC 0h (X) from which we conclude that Γ is a condensing operator on Br . The assertion is now a consequence of Theorem 2.6. ¤ 4. An application To complete this work, we study the existence of global solutions for a concrete partial differential equation with nonlocal conditions. In the sequel, X = L2 [0, π] and A : D(A) ⊂ X → X is the operator Ax = x00 with domain D(A) = {x ∈ X : x00 ∈ X, x(0) = x(π) = 0}. It is well known that A is the infinitesimal generator of an analytic semigroup (T (t))t≥0 on X. Furthermore, A has discrete spectrum with eigenvalues of the form −n2 , n ∈ N, and corresponding normalized IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS 111 1 eigenfunctions given by zn (x) = ( π2 ) 2 sin(nx); the set of functions {zn : n ∈ N} P −n2 t is an orthonormal basis for X and T (t)x = ∞ hx, zn izn for every x ∈ X n=1 e and all t ≥ 0. It follows from this last expression that (T (t))t≥0 is a compact semigroup on X and that kT (t)k ≤ e−t for every t ≥ 0. As phase space, we choose the space B = PC 0 × L2 (ρ, X), see Example 2.3, and we assume that the conditions (g5) and (g7) in [27] are verified. Under these conditions, B is a fading memory space, and as consequence, there exists K > 0 such that supt≥0 {M (t), K(t)} ≤ K. Consider the delayed impulsive partial differential equation with nonlocal conditions Z t ∂ ∂2 a(t, s − t)w(s, ξ)ds + b(t, w(ρ(t), ξ)), (4.1) w(t, ξ) = 2 w(t, ξ) + ∂t ∂ξ −∞ w(t, 0) =w(t, π) = 0, t ≥ 0, (4.2) ∞ X w(s, ξ) = Li w(ti + s, ξ) + ϕ(s, ξ), s ≤ 0, ξ ∈ [0, π] (4.3) i=1 ∆w(ti , ·) =w(t+ i , ·) Z − w(·, t− i ) π = pi (ξ, w(ti , s))ds, (4.4) 0 for t ≥ 0 and ξ ∈ [0, π], and where (Li )i∈N , (ti )i∈N are sequences of real numbers and ϕ ∈ B. To treat this system in the abstract form (1.1)–(1.3), we assume the following conditions: P 2 (a) Assume ∞ i=1 |Li |h(ti ) < ∞, the functions ρ : [0, ∞) → [0, ∞), a : R → R 1 ³R ´ 2 0 a2 (t,s) are continuous and ρ(t) ≤ t and L1 (t) = ds < ∞, for every −∞ ρ(s) t ≥ 0. (b) Assume b : [0, ∞)×R → R is continuous and that there exists a continuous function L2 : [0, ∞) → [0, ∞) such that |b(t, x)| ≤ L2 (t)|x| for every t ≥ 0 and x ∈ R. (c) The P functions g : BPC 0h (X) → B, f : [0, ∞) × B × X → X by g(u, ϕ) = ϕ+ ∞ i=1 Li uti and Z 0 f (t, ψ, x)(ξ) = a(t, s)ψ(s, ξ)ds + b(t, x(ρ(t), ξ)). −∞ (d) The functions pi : [0, π] × R → R, i ∈ N, are continuous and there are positive constants Li such that |pi (ξ, s) − pi (ξ, s)| ≤ Li |s − s|, ξ ∈ [0, π], s, s ∈ R. Rπ By defining the operator Ii : X → X by Ii (x)(ξ) = 0 pi (ξ, x(0, s))ds, i ∈ N, ξ ∈ [0, π], we can transform system (4.1)-(4.4) into the abstract system (1.1)-(1.3). Moreover, it is easy to see that f, g are continuous, kf (t, ψ, x)k ≤ L1 (t)kψkB + L2 (t)kxk, for all (t, ψ, x) ∈ [0, ∞) P and g verifies conditions H3 with Lg : K ∞ i=1 Li h(ti ). The next result is a direct consequence from Theorem 3.3. We omit the proof. 112 S. SIVASANKARAN, M. MALLIKA ARJUNAN AND V. VIJAYAKUMAR R∞ P Proposition 4.1. If K ∞ i=1 Li h(ti ) + (1 + 2K) 0 (L1 (s) + L2 (s))ds P∞ + K i=1 Li < 1, then there exists a mild solution of (4.1)-(4.4). References [1] H. Akca, A. Boucherif and V. Covachev, Impulsive functional differential equations with nonlocal conditions, Inter. J. Math. Math. Sci., 29:5 (2002), 251-256. 1 [2] A. Anguraj and K. Karthikeyan, Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions, Nonlinear Anal., 70(7) (2009), 27172721. 1 [3] A. Anguraj and M. 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VIJAYAKUMAR 2 Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore - 641 114, Tamil Nadu, India E-mail address: arjunphd07@yahoo.co.in 3 Department of Mathematics, Info Institute of Engineering, Kovilpalayam, Coimbatore - 641 107, Tamil Nadu, India E-mail address: vijaysarovel@gmail.com