ARTICLE IN PRESS Nonlinear Analysis: Hybrid Systems ( ) – Contents lists available at ScienceDirect Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps Nadjet Abada a , Mouffak Benchohra b,∗ , Hadda Hammouche c a Département de Mathématiques, Université Mentouri de Constantine, Algérie b Laboratoire de Mathématiques, Université de Sidi Bel Abbès, BP 89, 22000 Sidi Bel Abbès, Algérie c Département de Mathématiques, Université Kasdi Merbah de Ouargla, Algérie article abstract info Article history: Received 29 January 2009 Accepted 19 May 2010 In this paper, we shall establish sufficient conditions for the existence of integral solutions for some nondensely defined impulsive semilinear functional differential inclusions with state-dependent delay in separable Banach spaces We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators © 2010 Elsevier Ltd All rights reserved Keywords: Partial differential inclusions Impulses Multivalued jumps State-dependent delay Integral solution Semigroup Introduction In this paper, we shall be concerned with the existence of integral solutions defined on a compact real interval for first order impulsive semilinear functional inclusions with state-dependent delay in a separable Banach space of the form: y (t ) ∈ Ay(t ) + F (t , yρ(t ,yt ) ), ∆y(ti ) ∈ Ik (ytk ), y(t ) = φ(t ), t ∈ J = [0, b], k = 1, 2, , m, t ∈ (−∞, 0], (1.1) (1.2) (1.3) where F : J × D → P (E ) is a given multivalued map with nonempty convex compact values, D is the phase space defined axiomatically (see Section 2) which contains the mappings from (−∞, 0] into E, φ ∈ D , = t0 < t1 < · · · < tm < tm+1 = b, Ik : D → P (E ), k = 1, 2, , m are bounded valued multivalued maps, P (E ) is the collection of all nonempty subsets of E , ρ : I × D → (−∞, b], A : D(A) ⊂ E → E is a nondensely defined closed linear operator on E, and E a real separable Banach space with norm |.| For any function y defined on (−∞, b] \ {t1 , t2 , , tm } and any t ∈ J, we denote by yt the element of D defined by yt (θ ) = y(t + θ ), θ ∈ (−∞, 0] In recent years, impulsive differential and partial differential equations have been the object of much investigation because they can describe various models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics We refer the reader to the monographs of Bainov and Simeonov [1], Benchohra ∗ Corresponding author Fax: +213 48 54 43 44 E-mail addresses: n65abada@yahoo.fr (N Abada), benchohra@univ-sba.dz (M Benchohra), h.hammouche@yahoo.fr (H Hammouche) 1751-570X/$ – see front matter © 2010 Elsevier Ltd All rights reserved doi:10.1016/j.nahs.2010.05.008 Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps, Nonlinear Analysis: Hybrid Systems (2010), doi:10.1016/j.nahs.2010.05.008 ARTICLE IN PRESS N Abada et al / Nonlinear Analysis: Hybrid Systems ( ) – et al [2], Lakshmikantham et al [3], and Samoilenko and Perestyuk [4] where numerous properties of their solutions are studied, and a detailed bibliography is given Semilinear functional differential equations and inclusions with or without impulses have been extensively studied where the operator A generates a C0 -semigroup Existence and uniqueness, among other things, are derived; see the books of Ahmed [5,6], Benchohra et al [7], Heikkila and Lakshmikantam [8], Kamenski et al [9] and the papers by Ahmed [10,11], Liu [12] and Rogovchenko [13,14] In [15] Abada et al have studied the controllability of a class of impulsive semilinear functional differential inclusions in Fréchet spaces by means of the extrapolation method [16,17], and in [18] the existence of mild and extremal mild solutions for first-order semilinear densely defined impulsive functional differential inclusions in separable Banach spaces with local and nonlocal conditions has been considered To the best of our knowledge, there are very few results for impulsive evolution inclusions with multivalued jump operators; see [18–20] The notion of the phase space D plays an important role in the study of both qualitative and quantitative theory A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato [21] (see also [22,23]) For a detailed discussion on this topic we refer the reader to the book by Hino et al [24] For the case where the impulses are absent (i.e Ik = 0, k = 1, , m), an extensive theory has been developed for the problem (1.1)–(1.3) We refer to Belmekki et al [25], Corduneanu and Lakshmikantham [26], Hale and Kato [21], Hino et al [24], Lakshmikantham et al [27] and Shin [28] The literature related to ordinary and partial functional differential equations with delay for which ρ(t , ψ) = t is very extensive; see for instance the books [29–32] and the papers therein On the other hand, functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equations has received great attention in the last year, see, for instance, [33,34] and the references therein The literature related to partial functional differential equations with state-dependent delay is limited (see [35–37]) This paper is organized as follows In Section 2, we will recall briefly some basic definitions and preliminary facts which will be used throughout the following sections In Section we give some examples of operators with nondense domain In Section 4, we prove existence of integral solutions for problem (1.1)–(1.3) Our approach will be based for the existence of integral solutions, on a fixed point theorem of Dhage [38] for the sum of a contraction map and a completely continuous map In Section we present some examples of phase spaces Finally in Section we give an example to illustrate the abstract theory The results of the present paper extend to a nondensely defined operator some ones considered in the previous literature Preliminaries In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper For ψ ∈ D the norm of ψ is defined by ψ D = sup{|ψ(θ )| : θ ∈ (−∞, 0]} Also B(E ) denotes the Banach space of bounded linear operators from E into E, with norm N B(E ) = sup{|N (y)| : |y| = 1} L1 (J , E ) denotes the Banach space of measurable functions y : J −→ E which are Bochner integrable normed by b y L1 |y(t )|dt = To consider the define the solution of problem (1.1)–(1.3), it is convenient to introduce some additional concepts and notations Consider the following space Bb = y : (−∞, b] → E , yk ∈ C (Jk , E ) and there exist y(tk− ), y(tk+ ) with y(tk ) = y(tk− ), y(t ) = φ(t ), t ≤ where yk is the restriction of y to Jk = (tk , tk+1 ], k = 0, , m Let y b = y0 D + sup{|y(s)| : ≤ s ≤ b}, · b be the semi-norm in Bb defined by y ∈ Bb In this work, we will employ an axiomatic definition for the phase space D which is similar to those introduced in [24] Specifically, D will be a linear space of functions mapping (−∞, 0] into E endowed with a semi norm D , and satisfies the following axioms introduced at first by Hale and Kato in [21]: (A1 ) There exist a positive constant H and functions K (·), M (·) : R+ → R+ with K continuous and M locally bounded, such that for any b > 0, if y : (−∞, b] → E, y ∈ D , and y(·) is continuous on [0, b], then for every t ∈ [0, b] the following conditions hold: (i) yt is in D ; (ii) |y(t )| ≤ H yt D ; (iii) yt D ≤ K (t ) sup{|y(s)| : ≤ s ≤ t } + M (t ) y0 (A2 ) The space D is complete D, and H , K and M are independent of y(·) Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps, Nonlinear Analysis: Hybrid Systems (2010), doi:10.1016/j.nahs.2010.05.008 ARTICLE IN PRESS N Abada et al / Nonlinear Analysis: Hybrid Systems ( ) – In what follows we use the following notations Kb = sup{K (t ) : t ∈ J } and Mb = sup{M (t ) : t ∈ J } Definition 2.1 ([39]) Let E be a Banach space An integrated semigroup is a family of operators (S (t ))t ≥0 of bounded linear operators S (t ) on E with the following properties: (i) S (0) = 0; (ii) t → S (t ) is strongly continuous; s (iii) S (s)S (t ) = (S (t + r ) − S (r ))dr , for all t , s ≥ Definition 2.2 An integrated semigroup (S (t ))t ≥0 is called exponential bounded, if there exists constant M ≥ and ω ∈ R such that |S (t )| ≤ Meωt , for t ≥ Moreover (S (t ))t ≥0 is called nondegenerate if S (t )y = 0, for all t ≥ 0, implies y = Definition 2.3 An operator A is called a generator of an integrated semigroup, if there exists ω ∈ R such that (ω, +∞) ⊂ ρ(A), and there exists a strongly continuous exponentially bounded family (S (t ))t ≥0 of linear bounded operators such that ∞ S (0) = and (λI − A)−1 = λ e−λt S (t )dt for all λ > ω If A is the generator of an integrated semigroup (S (t ))t ≥0 which is locally Lipschitz, then from [39], S (·)y is continuously d S (t )y defines a bounded operator on the set differentiable if and only if y ∈ D(A) In particular S (t )y := dt E1 := {y ∈ E : t → S (t )y is continuously differentiable on [0, ∞)} and (S (t ))t ≥0 is a C0 semigroup on D(A) Here and hereafter, we assume that A satisfies the Hille–Yosida condition, that is, there exists M > and ω ∈ R such that (ω, ∞) ⊂ ρ(A) and sup{(λI − ω)n |(λI − A)−n | : λ > ω, n ∈ N} ≤ M Note that, since A satisfies the Hille–Yosida condition, S (t ) B(E ) ≤ Meωt , t ≥ 0, where M and ω are the constants considered in the Hille–Yosida condition (see [40]) Let (S (t ))t ≥0 , be the integrated semigroup generated by A Consider the Cauchy Problem y (t ) = Ay(t ) + f (t ), t ∈ [0, b], y(0) = y0 ∈ E (2.1) Then we have the following Theorem 2.1 ([40]) Let f : [0, b] → E be a continuous function Then for y0 ∈ D(A), there exists a unique continuous function y : [0, b] → E of the Cauchy Problem (2.1) such that (i) t y(s)ds ∈ D(A) for t ∈ [0, b], (ii) y(t ) = y0 + A (iii) |y(t )| ≤ Me ωt t t y(s)ds + |y0 | + t e f (s)ds, t ∈ [0, b], −ωs |f (s)|ds , t ∈ [0, b] Moreover, from [39,40] y satisfies the variation of constants formula, y(t ) = S (t )y0 + t d dt S (t − s)f (s)ds, t ≥ (2.2) Let Bλ = λR(λ, A) := λ(λI − A)−1 Then [40] for all y ∈ D(A), Bλ y → y as λ → ∞ Also from the Hille–Yosida condition (with n = 1) it easy to see that limλ→∞ |Bλ y| ≤ M |y|, since |Bλ | = |λ(λI − A)−1 | ≤ Mλ λ−ω Thus limλ→∞ |Bλ | ≤ M Also if y is given by (2.2), then t y(t ) = S (t )y0 + lim λ→∞ S (t − s)Bλ f (s)ds, t ≥ (2.3) Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps, Nonlinear Analysis: Hybrid Systems (2010), doi:10.1016/j.nahs.2010.05.008 ARTICLE IN PRESS N Abada et al / Nonlinear Analysis: Hybrid Systems ( ) – We finish this section, with notations, definitions, and some results from multivalued analysis Let (X , d) be a metric space We use the notations: Pcl (X ) = {Y ∈ P (X ) : Y closed}, Pbd (X ) = {Y ∈ P (X ) : Y bounded} Pc v (X ) = {Y ∈ P (X ) : Y convex}, Consider Hd : P (X ) × P (X ) → R+ Pcp (X ) = {Y ∈ P (X ) : Y compact} {∞} given by Hd (A, B) = max{sup d(a, B), sup d(A, b)}, a∈A b∈B where d(A, b) = infa∈A d(a, b), d(a, B) = infb∈B d(a, b) Then (Pbd,cl (X ), Hd ) is a metric space and (Pcl (X ), Hd ) is a generalized metric space (see [41]) A multivalued map N : J → Pcl (X ) is said to be measurable if, for each x ∈ X , the function Y : J → R defined by Y (t ) = d(x, N (t )) = inf{d(x, z ) : z ∈ N (t )}, is measurable Definition 2.4 A measurable multivalued function F : J → Pbd,cl (X ) is said to be integrably bounded if there exists a function w ∈ L1 (J , R+ ) such that v ≤ w(t ) a.e t ∈ J for all v ∈ F (t ) A multivalued map F : X → P (X ) is convex (closed) valued if F (x) is convex (closed) for all x ∈ X F is bounded on bounded sets if F (B) = x∈B F (x) is bounded in X for all B ∈ Pb (X ) i.e supx∈B {sup{|y| : y ∈ F (x)}} < ∞ F is upper semi-continuous (u.s.c for short) on X if for each x0 ∈ X the set F (x0 ) is nonempty, closed subset of X , and for each open set U of X containing F (x0 ), there exists an open neighborhood V of x0 such that F (V ) ⊆ U.G is said to be completely continuous if F (B) is relatively compact for every B ∈ Pbd (X ) If the multivalued map F is completely continuous with nonempty compact valued, then G is u.s.c if and only if F has closed graph i.e xn → x∗ , yn → y∗ , yn ∈ G(x∗ ) imply y∗ ∈ G(x∗ ) Definition 2.5 A multivalued map F : J × D → P (E ) is said to be Carathéodory if (i) t −→ F (t , u) is measurable for each u ∈ D (ii) u −→ F (t , u) is u.s.c for almost each t ∈ J Definition 2.6 A multivalued operator N : J → Pcl (X ) is called (a) contraction if and only if there exists γ > such that Hd (N (x), N (y)) ≤ γ d(x, y), for each x, y ∈ X , with γ < 1, (b) N has a fixed point if there exists x ∈ X such that x ∈ N (x) For more details on multivalued maps and the proof of the known results cited in this section we refer interested reader to the books of Deimling [42], Gorniewicz [43] and Hu and Papageorgiou [44] Examples of operators with nondense domain In this section we shall present examples of linear operators with nondense domain satisfying the Hille–Yosida estimate More details can be found in the paper by Da Prato and Sinestrari [45] Example 3.1 Let E = C ([0, 1], R) and the operator A : D(A) → E defined by Ay = y , where D(A) = {y ∈ C ((0, 1), R) : y(0) = 0} Then D(A) = {y ∈ C ((0, 1), R) : y(0) = 0} = E Example 3.2 Let E = C ([0, 1], R) and the operator A : D(A) → E defined by Ay = y , where D(A) = {y ∈ C ((0, 1), R) : y(0) = y(1) = 0} Then D(A) = {y ∈ C ((0, 1), R) : y(0) = y(1) = 0} = E Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps, Nonlinear Analysis: Hybrid Systems (2010), doi:10.1016/j.nahs.2010.05.008 ARTICLE IN PRESS N Abada et al / Nonlinear Analysis: Hybrid Systems ( ) – Example 3.3 Let us set for some α ∈ (0, 1) E = C0α ([0, 1], R) = y : [0, 1] → R : y(0) = and sup 0≤t and A satisfied the Hille–Yosida condition Set γ > For the phase space, we choose D to be defined by D = PC γ = {φ ∈ PC ((−∞, 0], E ) : lim eγ θ φ(θ ) exists in E } θ→−∞ Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps, Nonlinear Analysis: Hybrid Systems (2010), doi:10.1016/j.nahs.2010.05.008 ARTICLE IN PRESS 12 N Abada et al / Nonlinear Analysis: Hybrid Systems ( ) – with norm φ = γ sup θ∈(−∞,0] eγ θ |φ(θ )|, φ ∈ PC γ Notice that the phase space D satisfies axioms (A1 ), (A2 ) (see [24] for more details) By making the following change of variables y(t )(ξ ) = v(t , ξ ), t ∈ [0, 1], ξ ∈ [0, π], φ(θ )(ξ ) = v0 (θ , ξ ), θ ≤ 0, ξ ∈ [0, π], F (t , ϕ)(ξ ) = m(t )a(t , ϕ(0, ξ )), t ∈ [0, 1], ξ ∈ [0, π], φ ∈ Cγ ρ(t , ϕ) = t − σ (ϕ(0, 0)) γk (−s)[−|v0 (s, ξ )|, |v0 (s, ξ )|]ds, Ik (ytk ) = −∞ the problem (6.1)–(6.4) takes the abstract form (1.1)–(1.3) Moreover, we have F (t , ϕ) P ≤ m(t )(b1 ϕ D + b2 ) for all (t , ϕ) ∈ I × D with ∞ ds ψ(s) ∞ = ds b1 s + b2 = +∞ The next results are consequence of Theorem 4.2 and Remark 4.1 Theorem 6.1 Let φ ∈ B be such that condition (Hφ ) holds, then problem (6.1)–(6.4) has a mild solution Corollary 6.1 Let φ ∈ B be continuous and bounded, then problem (6.1)–(6.4) has a mild solution Acknowledgement The authors are grateful to the referee for his/her remarks References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] D.D Bainov, P.S Simeonov, Systems with Impulsive Effect, Horwood, Chichister, 1989 M Benchohra, J Henderson, S.K Ntouyas, Impulsive Differential Equations and Inclusions, Vol 2, Hindawi Publishing Corporation, New York, 2006 V Lakshmikantham, D.D Bainov, P.S Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989 A.M Samoilenko, N.A Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995 N.U Ahmed, Semigroup Theory with Applications to Systems and Control, in: Pitman Research Notes in Mathematics Series, vol 246, Longman Scientific & Technical, Harlow, 1991, John Wiley & Sons, New York N.U Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co Pte Ltd, Hackensack, NJ, 2006 M Benchohra, L Gorniewicz, S.K Ntouyas, Controllability of Some Nonlinear Systems in Banach Spaces (The Fixed Point Theory Approach), Pawel Wlodkowic University College, Wydawnictwo Naukowe NOVUM, Plock, 2003 S Heikkila, V Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations, Marcel Dekker Inc., New York, 1994 M Kamenskii, V Obukhovskii, P Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, Berlin, 2001 N.U Ahmed, Systems governed by impulsive differential inclusions on Hilbert spaces, Nonlinear Anal 45 (2001) 693–706 N.U Ahmed, Optimal control for impulsive systems in Banach spaces, Int J Differ Equ Appl (1) (2000) 37–52 J.H Liu, Nonlinear impulsive evolution equations, Dynam Contin Discrete Impuls Systems (1999) 77–85 Yuri V Rogovchenko, Impulsive evolution systems: main results and new trends, Dyn Contin Discrete Impuls Syst (1) (1997) 57–88 Yuri V Rogovchenko, Nonlinear impulsive evolution systems and applications to population models, J Math Anal Appl 207 (2) (1997) 300–315 N Abada, M Benchohra, H Hammouche, A Ouahab, Controllability of impulsive semilinear functional differential inclusions with finite delay in Fréchet spaces, Discuss Math Differ Incl Control Optim 27 (2) (2007) 329–347 G Da Prato, E Grisvard, On extrapolation spaces, Rend Accad Naz Lincei 72 (1982) 330–332 K.J Engel, R Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000 N Abada, M Benchohra, H Hammouche, Existence and controllability results for impulsive partial functional differential inclusions, Nonlinear Anal 69 (2008) 2892–2909 I Benedetti, An existence result for impulsive functional differential inclusions in Banach spaces, Discuss Math Differ Incl Control Optim 24 (2004) 13–30 S Migorski, A Ochal, Nonlinear impulsive evolution inclusions of second order, Dynam Systems Appl 16 (2007) 155–173 J Hale, J Kato, Phase space for retarded equationswith infinite delay, Funkcial Ekvac 21 (1978) 11–41 F Kappel, W Schappacher, Some considerations to the fundamental theory of infinite delay equation, J Differential Equations 37 (1980) 141–183 K Schumacher, Existence and continuous dependencefor differential equations with unbounded delay, Arch Ration Mech Anal 64 (1978) 315–335 Y Hino, S Murakami, T Naito, Functional-Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, vol 1473, Springer-Verlag, Berlin, 1991 M Belmekki, M Benchohra, K Ezzinbi, S.K Ntouyas, Existence results for some partial functional differential equations with infinite delay, Nonlinear Stud 15 (4) (2008) 373–385 C Corduneanu, V Lakshmikantham, Equations with unbounded delay, Nonlinear Anal (1980) 831–877 V Lakshmikantham, L Wen, B Zhang, Theory of Differential Equations with Unbounded Delay, in: Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1994 Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps, Nonlinear Analysis: Hybrid Systems (2010), doi:10.1016/j.nahs.2010.05.008 ARTICLE IN PRESS N Abada et al / Nonlinear Analysis: Hybrid Systems ( ) – 13 [28] J.S Shin, An existence of functional differential equations, Arch Ration Mech Anal 30 (1987) 19–29 [29] J.K Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977 [30] J.K Hale, S Verduyn Lunel, Introduction to Functional -Differential Equations, in: Applied Mathematical Sciences, vol 99, Springer-Verlag, New York, 1993 [31] V Kolmanovskii, A Myshkis, Introduction to the Theory and Applications of Functional–Differential Equations, in: Mathematics and its Applications, vol 463, Kluwer Academic Publishers, Dordrecht, 1999 [32] J Wu, Theory and Applications of Partial Functional Differential Equations, in: Applied Mathematical Sciences, vol 119, Springer-Verlag, New York, 1996 [33] O Arino, K Boushaba, A Boussouar, A mathematical model of the dynamics of the phytoplankton-nutrient system Spatial heterogeneity in ecological models (Alcal de Henares, 1998), Nonlinear Anal RWA (1) (2000) 69–87 [34] D.R Will, C.T.H Baker, Stepsize control and continuity consistency for state-dependent delay-differential equations, J Comput Appl Math 53 (2) (1994) 163–170 [35] N Abada, R.P Agarwal, M Benchohra, H Hammouche, Existence results for nondensely impulsive semilinear functional differential equations with state-dependente delay, Asian–Eur J Math (4) (2008) 449–468 [36] E Hernández, A Prokopczyk, L Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal RWA (2006) 510–519 [37] A.V Rezounenko, J Wu, A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors, J Comput Appl Math 190 (1–2) (2006) 99–113 [38] B.C Dhage, Multivalued maping and fixed point, Nonlinear Funct Anal Appl 10 (2005) 359–378 [39] W Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J Math 59 (1987) 327–352 [40] H Kellermann, M Hieber, Integrated semigroup, J Funct Anal 84 (1989) 160–180 [41] M Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991 [42] K Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin, New York, 1992 [43] L Gòrniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, vol 495, Kluwer Academic Publishers, Dordrecht, 1999 [44] Sh Hu, N Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer Academic Publishers, Dordrecht, 1997 [45] G Da Prato, E Sinestrari, Differential operators with non-dense domains, Ann Sc Norm Super Pisa Sci 14 (1987) 285–344 [46] E Sinestrari, Continuous interpolation spaces and spatial regularity in nonlinear Volterra integrodifferential equations, J Integral Equations (1983) 287–308 [47] B.C Dhage, E Gastori, S.K Ntouyas, Existence theory for perturbed functional differential inclusions, Commun Appl Nonlinear Anal 13 (2006) 1–14 [48] K Yosida, Functional Analysis, 6th ed., Springer-Verlag, Berlin, 1980 [49] A Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983 Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps, Nonlinear Analysis: Hybrid Systems (2010), doi:10.1016/j.nahs.2010.05.008 [...]... Differential Equations with Unbounded Delay, in: Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1994 Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state- dependent delay and multivalued jumps, Nonlinear Analysis: Hybrid Systems (2010), doi:10.1016/j.nahs.2010.05.008 ARTICLE IN PRESS N Abada et al / Nonlinear Analysis:... Springer-Verlag, Berlin, 1980 [49] A Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983 Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state- dependent delay and multivalued jumps, Nonlinear Analysis: Hybrid Systems (2010), doi:10.1016/j.nahs.2010.05.008 ... Prokopczyk, L Ladeira, A note on partial functional differential equations with state- dependent delay, Nonlinear Anal RWA 7 (2006) 510–519 [37] A.V Rezounenko, J Wu, A non-local PDE model for population dynamics with state- selective delay: local theory and global attractors, J Comput Appl Math 190 (1–2) (2006) 99–113 [38] B.C Dhage, Multivalued maping and fixed point, Nonlinear Funct Anal Appl 10 (2005)... ≥0 and that S (t ) B(E ) ≤ e−µt for t ≥ 0 for some constant µ > 0 and A satisfied the Hille–Yosida condition Set γ > 0 For the phase space, we choose D to be defined by D = PC γ = {φ ∈ PC ((−∞, 0], E ) : lim eγ θ φ(θ ) exists in E } θ→−∞ Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state- dependent delay and multivalued jumps, ... Hammouche, Existence and controllability results for impulsive partial functional differential inclusions, Nonlinear Anal 69 (2008) 2892–2909 I Benedetti, An existence result for impulsive functional differential inclusions in Banach spaces, Discuss Math Differ Incl Control Optim 24 (2004) 13–30 S Migorski, A Ochal, Nonlinear impulsive evolution inclusions of second order, Dynam Systems Appl 16 (2007)... de Henares, 1998), Nonlinear Anal RWA 1 (1) (2000) 69–87 [34] D.R Will, C.T.H Baker, Stepsize control and continuity consistency for state- dependent delay -differential equations, J Comput Appl Math 53 (2) (1994) 163–170 [35] N Abada, R.P Agarwal, M Benchohra, H Hammouche, Existence results for nondensely impulsive semilinear functional differential equations with state- dependente delay, Asian–Eur J... Iterative Technique for Nonlinear Discontinuous Differential Equations, Marcel Dekker Inc., New York, 1994 M Kamenskii, V Obukhovskii, P Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, Berlin, 2001 N.U Ahmed, Systems governed by impulsive differential inclusions on Hilbert spaces, Nonlinear Anal 45 (2001)... equationswith infinite delay, Funkcial Ekvac 21 (1978) 11–41 F Kappel, W Schappacher, Some considerations to the fundamental theory of infinite delay equation, J Differential Equations 37 (1980) 141–183 K Schumacher, Existence and continuous dependencefor differential equations with unbounded delay, Arch Ration Mech Anal 64 (1978) 315–335 Y Hino, S Murakami, T Naito, Functional- Differential Equations with. .. Infinite Delay, in: Lecture Notes in Mathematics, vol 1473, Springer-Verlag, Berlin, 1991 M Belmekki, M Benchohra, K Ezzinbi, S.K Ntouyas, Existence results for some partial functional differential equations with infinite delay, Nonlinear Stud 15 (4) (2008) 373–385 C Corduneanu, V Lakshmikantham, Equations with unbounded delay, Nonlinear Anal 4 (1980) 831–877 V Lakshmikantham, L Wen, B Zhang, Theory of Differential. .. follows that D is a phase space which satisfies Axioms (A1 ) and (A2 ) 1 Moreover, for r = 0 and p = 2 this space coincides (see [24]) with C0 × L2 (g , E ), H = 1, M (t ) = G(−t ) 2 and 0 K (t ) = 1 + g (s)ds 1 2 , for t ≥ 0 −t 6 An example To apply our abstract results, we consider the partial functional differential equations with state dependent delay of the form ∂ ∂ v(t , ξ ) ∈ − v(t , ξ ) + m(t )a(t ... systems and applications to population models, J Math Anal Appl 207 (2) (1997) 300–315 N Abada, M Benchohra, H Hammouche, A Ouahab, Controllability of impulsive semilinear functional differential... One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000 N Abada, M Benchohra, H Hammouche, Existence and controllability results for impulsive partial functional differential... delay-differential equations, J Comput Appl Math 53 (2) (1994) 163–170 [35] N Abada, R.P Agarwal, M Benchohra, H Hammouche, Existence results for nondensely impulsive semilinear functional differential