mathematical and computer modelling

Applied Mathematics and Computation 186 (2007) 294–301 www.elsevier.com/locate/amc On state-dependent delay partial neutral functional–differential equations Eduardo Herna´ndez M a,* , Mark A McKibben b a b Departamento de Matema´tica, ICMC, Universidade de Sa˜o Paulo, Caixa Postal 668, 13560-970 Sa˜o Carlos SP, Brazil Department of Mathematics and Computer Science, Goucher College, 1021 Dulaney Valley Road, Baltimore, MD 21204, USA Abstract In this paper, we study the existence of mild solutions for a class of abstract partial neutral functional–differential equations with state-dependent delay Ó 2006 Elsevier Inc All rights reserved Keywords: Abstract Cauchy problem; Neutral equations; State-dependent delay; Semigroup of linear operators; Unbounded delay Introduction The purpose of this article is establish the existence of mild solutions for a class of abstract neutral functional–differential equations with state-dependent delay described by the form d Dðut Þ ¼ ADðut Þ þ F ðt; xqðt;xt Þ Þ; dt x0 ¼ u B; t I ¼ ½0; aŠ; ð1:1Þ ð1:2Þ where A is the infinitesimal generator of a compact C0-semigroup of bounded linear operators (T(t))tP0 on a Banach space X; the function xs : (À1, 0] ! X, xs(h) = x(s + h), belongs to some abstract phase space B described axiomatically; F, G are appropriate functions; and Dw = w (0) À G(t, w), where w is in B 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 equation has received a significant amount of attention in the last years, see for instance [1–11] and the references therein We also cite [12,9,13] for the case of neutral differential equations with dependent delay The literature related to partial functional–differential equations with state-dependent delay is limited, to our knowledge, to the recent works [14,15] Abstract neutral differential equations arise in many areas of applied mathematics For this reason, they have largely been studied during the last few decades The literature related to ordinary neutral differential * Corresponding author E-mail addresses: lalohm@icmc.usp.br (E Herna´ndez M.), mmckibben@goucher.edu (M.A McKibben) 0096-3003/$ - see front matter Ó 2006 Elsevier Inc All rights reserved doi:10.1016/j.amc.2006.07.103 E Herna´ndez M., M.A McKibben / Applied Mathematics and Computation 186 (2007) 294–301 295 equations is very extensive, thus, we refer the reader to [16] only, which contains a comprehensive description of such equations Similarly, for more on partial neutral functional–differential equations and related issues we refer to Adimy and Ezzinbi [17], Hale [18], Wu and Xia [19] and [20] for finite delay equations, and Herna´ndez and Henriquez [21,22] and Herna´ndez [23] for unbounded delays Preliminaries Throughout this paper, A : D(A) & X ! X is the infinitesimal generator of a compact C0-semigroup of line is a positive constant such that kT ðtÞk M e for every ear operators (T(t))tP0 on a Banach space X and M t I = [0, a] For background information related to semigroup theory, we refer the reader to Pazy [24] In this work we will employ an axiomatic definition for the phase space B which is similar to those introduced in [25] Specifically, B will be a linear space of functions mapping (À1, 0] into X endowed with a seminorm k Á kB , and satisfies the following axioms: (A) If x : (À1, r + b] ! X, b > 0, is such that xj[r, r+b] C([r, r + b] : X) and xr B, then for every t [r, r + b] the following conditions hold: (i) xt is in B, (ii) kxðtÞk H kxt kB , (iii) kxt kB Kðt À rÞ supfkxðsÞk : r s tg þ Mðt À rÞkxr kB , where H > is a constant; K, M : [0, 1) ! [1, 1), K is continuous, M is locally bounded, and H, K, M are independent of x(Æ) (A1) For the function x(Æ) in (A), the function t ! xt is continuous from [r, r + b] into B (B) The space B is complete Example 2.1 (The phase spaces Cg, C 0g ) Let g : (À1, 0] ! [1, 1) be a continuous, non-decreasing function with g(0) = 1, which satisfies conditions (g-1), (g-2) of [25] Briefly, this means that the function cðtÞ :¼ supÀ1 in Z and for a bounded function n : I ! Z and t [0, a] we employ the notation knkZ,t for knkZ;t ¼ supfknðsÞkZ : s ½0; tŠg: ð2:3Þ We will simply write knkt when no confusion arises The remainder of the paper is divided into two sections The existence of mild solutions for the abstract Cauchy problem (1.1)–(1.2) is studied in Section 3, and Section is devoted to a discussion of some applications To conclude the current section, we recall the following well-known result, referred to as the Leray Schauder Alternative, for convenience Theorem 2.1 [26, Theorem 6.5.4] Let D be a convex subset of a Banach space X and assume that D Let G : D ! D be a completely continuous map Then, either the set {x D : x = kG(x), < k < 1} is unbounded or the map G has a fixed point in D Existence results In this section we discuss the existence of mild solutions for the abstract system (1.1)–(1.2) We begin by introducing the following conditions: (Hu) Let RðqÀ Þ ¼ fqðs; wÞ : ðs; wÞ I  B; qðs; wÞ 0g The function t ! ut is well defined and continuous from RðqÀ Þ into B, and there exists a continuous and bounded function J u : RðqÀ Þ ! ð0; 1Þ such that kut kB J u ðtÞkukB for every t RðqÀ Þ (H1) The function F : I  B ! X satisfies the following properties: (a) The function F(Æ, w) : I ! X is strongly measurable, for every w B (b) The function F ðt; ÁÞ : B ! X is continuous, for each t I (c) There exist a continuous non decreasing function W : [0, 1) ! (0, 1) and an integrable function m : I ! [0, 1) such that kF ðt; wÞk mðtÞW ðkwkB Þ; ðt; wÞ I  B: (H2) The function G : R  B ! X is continuous and there exists LG > such that kGðt; w1 Þ À Gðt; w2 Þk LG kw1 À w2 kB ; ðt; wi Þ I  B: (H3) Let S(a) = {x : (À1, a] ! X : x0 = 0; x C([0, a] : X)} endowed with the norm of uniform convergence on [0, a] and y : (À1, a] ! X be the function defined by y0 = u on (À1, 0] and y(t) = T(t)u(0) on [0, a] Then, for every bounded set Q such that Q & S (a), the set of functions {t ! G(t, xt + yt) : x Q} is equicontinuous on [0, a] Remark 3.2 We remark that condition Hu is frequently satisfied by functions that are continuous and bounded In fact, if the space B satisfies axiom C2 in [25], then there exists a constant L > such that kukB Lsuph60 kuðhÞk for every u B that is continuous and bounded, see [25, Proposition 7.1.1] for details Consequently, kut kB L suph60 kuðhÞk kukB kukB for every u B n f0g continuous and bounded and every t We also observe that Cr · Lp(g; X) satisfies axiom C2 if g(Æ) is integrable on (À1, r], see [25, p 10] E Herna´ndez M., M.A McKibben / Applied Mathematics and Computation 186 (2007) 294–301 297 Motivated by general semigroup theory, we adopt the following concept of mild solution Definition 3.1 A function x : (À1, a] ! X is a mild solution of the abstract Cauchy problem (1.1)–(1.2) if x0 = u, xqðs;xs Þ B, for every s I, and Z t T ðt À sÞF ðs; xqðs;xs Þ Þ ds; t I: xðtÞ ¼ T ðtÞðuð0Þ À Gð0; uÞÞ þ Gðt; xt Þ þ The proof of Lemma 3.1 is routine; the details are left to the reader Lemma 3.1 Let x : (À1, a] ! X be a function such that x0 = u and xj½0;aŠ Cð½0; aŠ : X Þ Then kxs kB ðM a þ J u0 ÞkukB þ K a supfkxðhÞk; h ½0; maxf0; sgŠg; s RðqÀ Þ [ ½0; aŠ; where J u0 ¼ supt2RðqÀ Þ J u ðtÞ and the notation in (2.3) has been used Now, we can prove our first existence result Theorem 3.2 Let u B and assume that conditions H1, H2, Hu hold If   Z W ðnÞ a e K a LG þ M lim inf mðsÞ ds < 1; n!1þ n then there exists a mild solution of (1.1)–(1.2) Proof Consider the space Y = {u C(I : X) : u(0) = u(0)} endowed with the uniform convergence topology, and define the operator C : Y ! Y by Z t CxðtÞ ¼ T ðtÞðuð0Þ À Gð0; uÞÞ þ Gðt; xt Þ þ T ðt À sÞF ðs; xqðs;xs Þ Þ ds; t I; where x : ðÀ1; aŠ ! X is the extension of x to (À1, a] such that x0 ¼ u From our assumptions it is easy to see that Cx Y  : ðÀ1; aŠ ! X be the extension of u to (À1, a] such that u  ðhÞ ¼ uð0Þ on I We claim that there exists Let u r > such that CðBr ð ujI ; Y ÞÞ & Br ð ujI ; Y Þ Indeed, suppose to the contrary that this assertion is false Then, for every r > there exist xr Br ð ujI ; Y Þ and tr I such that r < kCxr(tr) À u(0)k Then, from Lemma 3.1 we find that r < kCxr ðtr Þ À uð0Þk kT ðtr Þuð0Þ À uð0Þk þ kT ðtr ÞGð0; uÞ À Gð0; uÞk þ kGðtr ; ðxr Þtr Þ À Gð0; uÞk Z tr þ kT ðtr À sÞkkF ðs; ðxr Þqðs;ðxr Þs Þ Þk ds e þ 1ÞH kukB þ kT ðtr ÞGð0; uÞ À Gð0; uÞk þ LG ðK a kxr À uð0Þktr þ ðM a þ HK a þ 1ÞkukÞ ðM Z tr e þM mðsÞW ððM a þ J u0 ÞkukB þ K a ðkxr À uð0Þka þ kuð0ÞkÞÞ ds e þ 1ÞH Þkuk þ kT ðtr ÞGð0; uÞ À Gð0; uÞk þ LG ðK a r þ ðM a þ HK a þ 1ÞkukÞ ðð M B Z tr e þM mðsÞW ððM a þ J u0 þ H ÞkukB þ K a rÞ ds and hence   Z a e lim inf W ðnÞ K a LG þ M mðsÞ ds ; n!1 n which contradicts our assumption ujI ; Y ÞÞ & Br ð ujI ; Y Þ and consider the decomposition C = C1 + C2 where Let r > be such that CðBr ð C1 xðtÞ ¼ T ðtÞðuð0Þ À Gð0; uÞÞ þ Gðt; xt Þ; t I; Z t C2 xðtÞ ¼ T ðt À sÞF ðs; xqðs;xs Þ Þ ds; t I: E Herna´ndez M., M.A McKibben / Applied Mathematics and Computation 186 (2007) 294–301 298 From the proof of [14, Theorem 2.2], we know that C2 is completely continuous Moreover, using the phase space axioms we find that kC1 uðtÞ À C1 vðtÞk LG K a ku À vka ; t I; which proves that C1 is a contraction on Br ð ujI ; Y Þ, so that C is a condensing operator on Br ð ujI ; Y ÞÞ The existence of a mild solution for (1.1)–(1.2) is now a consequence of [27, Theorem 4.3.2] This completes the proof h Theorem 3.3 Let conditions Hu, H1, H3 be satisfied Assume that q(t, w) t for every ðt; wÞ I  B, that G is completely continuous, and that there exist positive constants c1, c2 such that kGðt; wÞk c1 kwkB þ c2 for every ðt; wÞ I  B If l = À Kac1 > and Z e Ka Z a M ds ; mðsÞ ds < W ðsÞ l C where   h i e kGð0; uÞk þ c1 ðM a þ M e HK a kuk þ K a M e HK a ÞkukB þ c2 ; C ¼ M a þ J u0 þ M l then there exists a mild solution of (1.1)–(1.2) Proof On the space BC ¼ fu : ðÀ1; aŠ ! X ; u0 ¼ 0; ujI CðI; X Þg kuka = sups2[0, a]ku(s)k, we define the operator C : BC ! BC by  0; t ðÀ1; 0Š; Rt CxðtÞ ¼ T ðtÞGð0; uÞ À Gðt; xt Þ þ T ðt À sÞF ðs; xqðs;xs Þ Þ ds; t I; endowed with the norm where x : ðÀ1; aŠ ! X is defined by the relation x ¼ y þ x on (À1, a] In preparation for using Theorem 2.1, we establish a priori estimates for the solutions of the integral equation z = kCz, k (0, 1) Let xk be a solution of z = kCz, k (0, 1) If ak(s) = suph2[0, s]kxk(h)k, then from Lemma 3.1 and the fact that qðs; ðxk Þs Þ s, we find that Z e kxk ðtÞk kT ðtÞGð0; uÞk þ c1 kxt kB þ c2 þ M t e HK a ÞkukB þ K a ak ðsÞÞ ds mðsÞW ððM a þ J u þ M e HK a ÞkukB þ K a aðtÞÞ þ c2 e kGð0; uÞk þ c1 ððM a þ M 6M Z t e e HK a ÞkukB þ K a ak ðsÞÞ ds: þM mðsÞW ððM a þ J u þ M Consequently, e kGð0; uÞk þ c1 ððM a þ M e HK a Þkuk þ K a aðtÞÞ þ c2 kak ðtÞk M B Z t e e HK a ÞkukB þ K a ak ðsÞÞ ds þM mðsÞW ððM a þ J u þ M and so e e HK a ÞkukB þ c2 Š ½ M kGð0; uÞk þ c1 ðM a þ M l e Z t M e HK a ÞkukB þ K a ak ðsÞÞds: mðsÞW ððM a þ J u þ M þ l e HK a Þkuk þ K a aðtÞ, we obtain after a rearrangement of terms that By denoting nk ðtÞ ¼ ðM a þ J u0 þ M B kak ðtÞk e kGð0; uÞk þ c1 ðM a þ M e HK a Þkuk þ K a ½ M e HK a ÞkukB þ c2 Š nk ðtÞ ðM a þ J u0 þ M l e Ka Z t M mðsÞW ðnk ðsÞÞ ds: þ l ð3:4Þ E Herna´ndez M., M.A McKibben / Applied Mathematics and Computation 186 (2007) 294–301 299 Denoting by bk(t) the right-hand side of (3.4), it follows that b0k ðtÞ and hence Z bk ðtÞ bk ð0Þ¼C e Ka M mðtÞW ðbk ðtÞÞ l Z e Ka Z a M ds ds ; mðsÞ ds < W ðsÞ W ðsÞ l C which implies that the set of functions {bk(Æ) : k (0, 1)} is bounded in CðI : RÞ Thus, the set {xk(Æ) : k (0, 1)} is bounded on I To prove that C is completely continuous, we consider the decomposition C = C1 + C2 introduced in the proof of Theorem 3.2 From the proof of [14, Theorem 2.2] we know that C2 is completely continuous and from the assumptions on G we infer that C1 is a compact map It remains to show that C1 is continuous Let ðun Þn2N be a sequence in BC and u BC such that un ! u From the phase space axioms we infer that ðun Þs ! us uniformly on [0, a] as n ! and that U ¼ ½0; aŠ  fðun Þs ; us : s ½0; aŠ; n Ng is relatively compact in ½0; aŠ  B Thus, G is uniformly continuous on U, so that Gðs; ðun Þs Þ ! Gðs; us Þ uniformly on [0, a] as n ! 1, which shows that C1 is continuous These remarks, in conjunction with Theorem 2.1, enable us to conclude that there exists a mild solution for (1.1) and (1.2) The proof is complete h Examples We conclude this work with two applications of our previous abstract results In the sequel, X = L2([0, p]) and A : D(A) & X ! X is the operator Af = f00 with domain D(A) :¼ {f X : f00 X, f(0) = f(p) = 0} It is well known that A is the infinitesimal generator of a compact C0-semigroup of bounded linear operators (T(t))tP0 on X Moreover, A has discrete spectrum, the eigenvalues are Àn2, n N, with corresponding nor1=2 malized P eigenvectors zn ðnÞ :¼ ðp2 Þ sinðnnÞ, the set fzn : n Ng is an orthonormal basis of X, and Àn2 t T ðtÞx ¼ n¼1 e < x, zn > zn for x X Consider the differential system   Z t d uðt; nÞ þ a1 ðs À tÞuðs; nÞ ds dt À1  Z t  o ¼ uðt; nÞ þ a1 ðs À tÞuðs; nÞ ds on À1 Z t þ a2 ðs À tÞuðs À q1 ðtÞq2 ðkuðtÞkÞ; nÞ ds; t I ¼ ½0; aŠ; n ½0; pŠ; ð4:5Þ À1 together with the initial conditions uðt; 0Þ ¼ uðt; pÞ ¼ 0; t P 0; uðs; nÞ ¼ uðs; nÞ; s 0; n p: ð4:6Þ ð4:7Þ In the sequel, B ¼ C  L2 ðg; X Þ is the space introduced in Example 2.2; u B with the identification u(s)(s) = u(s, s); the functions : R ! R, qi : [0, 1) ! [0, 1), i = 1, 2, are continuous; and !1=2 Z ðai ðsÞÞ ds Li ¼ < 1; i ¼ 1; 2: gðsÞ À1 By defining the operators D; G; F : I  B ! X and q : I  B ! R by DðwÞ ¼ wð0; nÞ À GðwÞðnÞ Z GðwÞðnÞ ¼ À a1 ðsÞwðs; nÞ ds; À1 E Herna´ndez M., M.A McKibben / Applied Mathematics and Computation 186 (2007) 294–301 300 F ðt; wÞðnÞ ¼ Z a2 ðsÞwðs; nÞ ds; À1 qðs; wÞ ¼ s À q1 ðsÞq2 ðkwð0ÞkÞ; we can transform system (4.5)–(4.7) into the abstract system (1.1)–(1.2) Moreover, G, F are bounded linear operators, kGkLðB;X Þ L1 and kF kLðB;X Þ L2 The next result is an immediate consequence of Theorem 3.2 Theorem 4.4 Let u B be such that condition Hu holds and assume that Z 1=2 ! 1þ gðhÞ dh ðL1 þ aL2 Þ < 1: ð4:8Þ Àa Then there exists a mild solution of (4.5)–(4.7) The proof of Corollary 4.1 follows directly from Theorem 4.4 and Remark 3.2 Corollary 4.1 Let u B continuous and bounded, and assume that (4.8) holds Then, there exists a mild solution of (4.5)–(4.7) on I To conclude this section, we briefly consider the differential system d o2 ½uðt; nÞ þ uðt À r; nފ ¼ ½uðt; nÞ þ uðt À r; nފ þ a1 ðtÞb1 ðuðt À rðkuðtÞkÞ; nÞÞ; dt on t I ¼ ½0; aŠ; n ½0; pŠ; ð4:9Þ uðt; 0Þ ¼ uðt; pÞ ¼ 0; uðs; nÞ ¼ uðs; nÞ; s 0; n ½0; pŠ: ð4:10Þ ð4:11Þ For this system, we take u B ¼ C 0g ðX Þ and assume that the functions a1 : I ! R, b1 : R  J ! R, r : R ! Rþ are continuous and that there exist positive constants d1, d2 such that jb1(t)j d1jtj + d2 for every t R Let D, G : B ! X , F : ½0; aŠ  B ! X and q : ½0; aŠ  B ! R be the operators defined by D(w)(n) = w(0, n) À G(w)(n), G(w)(n) = Àw(Àr, n), F(t, w)(n) = a1(t)b1(w(0, n)) and q(t, w) = t À r(kw(0)k) Using these definitions, we can represent the system (4.8)–(4.10) in the abstract form (1.1) and (1.2) Moreover, G is a bounded linear operator on B with kGðwÞkLðB;X Þ gðÀrÞ; F is continuous and kF ðt; wÞk a1 ðtÞ½d kwkB þ d pŠ for all ðt; wÞ I  B As such, the following results follow from Theorem 3.2 and Remark 3.2 Ra Theorem 4.5 If u B satisfies condition Hu and gðÀrÞ þ d a1 ðsÞ ds < 1, then there exists a mild solution of (4.9)–(4.11) Corollary 4.2 If u is continuous and bounded on (À1, 0] and gðÀrÞ þ d solution of (4.9)–(4.11) Ra a1 ðsÞ ds < 1, then there exists a mild References [1] Ovide Arino, Khalid Boushaba, Ahmed Boussouar, A mathematical model of the dynamics of the phytoplankton-nutrient system: spatial heterogeneity in ecological models (Alcala´ de Henares, 1998), Nonlinear Anal RWA (1) (2000) 69–87 [2] Walter Aiello, H.I Freedman, J Wu, 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Q & S (a), the set of functions {t ! G(t, xt + yt) : x Q} is equicontinuous on [0, a] Remark 3.2 We remark that condition Hu is frequently satisfied by functions that are continuous and bounded