Hernandez, E.

TanakaAki, SueliM.

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Electronic Journal of Qualitative Theory of Differential Equations2009, No. 50, 1-8; http://www.math.u-szeged.hu/ejqtde/GLOBAL SOLUTIONS FOR ABSTRACT FUNCTIONALDIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONSEDUARDO HERNA´NDEZ M∗ AND SUELI M. TANAKA AKIAbstract. In this paper we study the existence of global solutions for a class ofabstract functional differential equation with nonlocal conditions. An applicationis considered.1. IntroductionIn this paper, we discuss the existence of global solutions for an abstract functionaldifferential equation with nonlocal conditions in the form{u′(t) = Au(t) + f(t, ut, u(ρ(t))), t ∈ I = [0,∞),u0 = g(u, ϕ) ∈ B,(1.1)where A is the infinitesimal generator of a C0-semigroup of bounded linear operators(T (t))t≥0 defined on a Banach space (X, ‖ · ‖); the history ut : (−∞, 0]→ X, definedby ut(θ) := u(t+ θ), belongs to some abstract phase space B defined axiomatically,ϕ ∈ B and ρ, f, g are appropriated functions.The study of initial value problems with nonlocal conditions arises to deal spe-cially with some situations in physics. For the importance of nonlocal conditionsin different fields, we refer the reader to [1, 2, 3, 4, 5] and the references containedtherein.Differential systems with nonlocal conditions are considered in different works. InByszewski [1] is studied the existence of mild, strong and classical solutions for theabstract Cauchy problemx′(t) = Ax(t) + f(t, x(t)), t ∈ [0, a],x(0) = x0 + q(t1, t2, t3, ..., tn, x(·)) ∈ X,where A is the generator of a C0-semigroup of linear operators on a Banach spaceX; f, q are appropriated functions; 0 < t1 < .... < tn ≤ a are prefixed numbers andthe symbol q(t1, t2, t3, ..., tn, x(·)) is used in the sense that the function x(·) can beevaluated only in the points ti, for instance q(t1, t2, t3, ..., tn, x(·)) =∑ni=1 αix(ti).The literature concerning differential equations with nonlocal conditions also in-clude ordinary differential equations, partial differential equations, abstract partialfunctional differential, integro-differential equations. Related to this matter, we cite* Corresponding author.2000 Mathematics Subject Classification. 35R10, 34K30, 49K25, 47D06.Key words and phrases. Abstract Cauchy problem, semigroup of linear operators.EJQTDE, 2009 No. 50, p. 1among others works, [1, 2, 3, 4, 5, 6, 7, 8, 9]. To the best of our knowledge, thestudy of the existence of “ global ” solutions for equations described in the abstractform (1.1), is a untreated topic, and this fact, is the main motivation of this work.In this paper, A is the infinitesimal generator of a strongly continuous semigroupof bounded linear operators (T (t))t≥0 defined on a Banach space (X, ‖ · ‖) and M, δare positive constants such that ‖ T (t) ‖≤Me−δt, for every t ≥ 0. In this work, thephase space B is a linear space of functions mapping (−∞, 0] into X endowed witha seminorm ‖ · ‖B and verifying the following axioms.(A) If x : (−∞, a)→ X, a > 0, is continuous on [0, a) and x0 ∈ B, then for everyt ∈ [0, a) the following conditions hold:(i) xt is in B.(ii) ‖ x(t) ‖≤ H ‖ xt ‖B.(iii) ‖ xt ‖B≤ K(t) sup{‖ x(s) ‖: 0 ≤ s ≤ t}+M(t) ‖ x0 ‖B,where H > 0 is a constant; K,M : [0,∞) → [1,∞), K is continuous, M islocally bounded and H,K,M are independent of x(·).(A1) For the function x(·) in (A), the function t → xt is continuous from [0, a)into B.(B) The space B is complete.Remark 1. 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 afading memory space, see [10, p. 190] for additional details.Example 1.1. The phase space Cr × Lp(̺,X)Let r ≥ 0, 1 ≤ p < ∞, and let ̺ : (−∞,−r] 7→ R be a nonnegative measurablefunction which satisfies the conditions (g-5)-(g-6) of [10]. Briefly, this means that ̺is locally integrable and there exists a non-negative locally bounded function γ on(−∞, 0] such that ̺(ξ + θ) ≤ γ(ξ)̺(θ), for all ξ ≤ 0 and θ ∈ (−∞,−r) \Nξ, whereNξ ⊆ (−∞,−r) is a set whose Lebesgue measure zero.The space B = Cr×Lp(̺,X) consists of all classes of functions ψ : (−∞, 0] 7→ Xsuch that ψ is continuous on [−r, 0], Lebesgue-measurable, and ̺‖ψ‖p is Lebesgueintegrable on (−∞,−r). The seminorm in Cr × Lp(̺,X) is defined as follows:‖ ψ ‖B:= sup{‖ψ(θ)‖ : −r ≤ θ ≤ 0} +(∫ −r−∞̺(θ)‖ψ(θ)‖pdθ)1/p.The space B = Cr × Lp(̺,X) satisfies axioms (A), (A1), and (B). Moreover,when r = 0 and p = 2, one can then take H = 1, M(t) = γ(−t)1/2 and K(t) =1 +(∫ 0−t ̺(θ) dθ)1/2for t ≥ 0 (see [10, Theorem 1.3.8]). We also note that if theconditions (g-5)-(g-7) of [10] hold, then B is a uniform fading memory.For additional details concerning phase space, we cite [10].In this paper, h : [0,∞) → R is a positive, continuous and non-decreasing functionsuch that h(0) = 1 and limt→∞ h(t) =∞. In the sequel, C0(X) and C0h(X) are thespaces defined byC0(X) = {x ∈ C([0,∞) : X) : limt→∞‖ x(t) ‖= 0},C0h(X) = {x ∈ C([0,∞) : X) : limt→∞‖ x(t) ‖h(t)= 0},endowed with the norms ‖ x ‖0= supt≥0 ‖ x(t) ‖ and ‖ x ‖h= supt≥0‖x(t)‖h(t) respec-tively. Additionally, we define the spaceBC0h(X) = {u ∈ C(R,X) ;u0 ∈ B, u|[0,∞) ∈ C0h(X)}endowed with the norm ‖ u ‖BC0h=‖ u0 ‖B + ‖ u|[0,∞) ‖h.We recall here the following compactness criterion.Lemma 1.1. A set B ⊂ C0h(X) is relatively compact in C0h(X) if, and only if, B isequicontinuous, B(t) = {x(t) : x ∈ B} is relatively compact in X for all t ≥ 0 andlimt→∞‖x(t)‖h(t) = 0 uniformly for x ∈ B.2. Existence of mild solutionsIn this section, we study the existence of mild solutions for the nonlocal abstractCauchy problem (1.1). 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 integrablefunction m : [0,∞) → [0,∞) and a nondecreasing continuous function W :[0,∞) → (0,∞) such that ‖ f(t, ψ, x) ‖≤ m(t)W (‖ ψ ‖B + ‖ x ‖), for every(t, ψ, x) ∈ [0,∞) × B ×X.H3 The function g : BC0h(X) × B → B is continuous and there exists Lg ≥ 0such that‖ g(u, ϕ) − g(v, ϕ) ‖B≤ Lg ‖ u− v ‖BC0h, u, v ∈ BC0h(X).H4 The function g(·, ϕ) : BC0h(X) → B is completely continuous and uniformlybounded. In the sequel, N is a positive constant such that ‖ g(ψ,ϕ) ‖B≤ Nfor every ψ ∈ BC0h(X).In this work, we adopt the following concept of mild solutions for (1.1).Definition 2.1. A function u ∈ C(R,X) is called a mild solution of system (1.1) ifu0 = g(u, ϕ) andu(t) = T (t)g(u, ϕ)(0) +∫ t0T (t− s)f(s, us, u(ρ(s)))ds, t ∈ I = [0,∞).Now, we can establish the principal results of this section.Theorem 2.1. Assume H1,H2 and H4 be hold and that the followings propertiesare verified.EJQTDE, 2009 No. 50, p. 3(a) The set {T (t)f(s, ψ, x) : (s, ψ, x) ∈ [0, t] ×Br(0,B) ×Br(0,X)} is relativelycompact in X for every t ≥ 0 and all r > 0.(b) For every K > 0, limt→∞1h(t)∫ t0 m(s)W (Kh(s))ds = 0.If MK(1+H)∫∞0 m(s)ds <∫∞cdsW (s) , where c = KN(1 +H)(1 +MH), then thereexists a mild solution for system (1.1).Proof. Let Γ : BC0h(I : X) → BC0h(X) be the operator defined by (Γu)0 = g(u, ϕ)andΓu(t) = T (t)g(u, ϕ)(0) +∫ t0T (t− s)f(s, us, u(ρ(s)))ds, t ≥ 0.Next we prove that Γ verifies the conditions in [11, Theorem 6.5.4]. To begin, wenote that for u ∈ BC0h(X) and t ≥ 0, ‖ u(t) ‖≤ ‖ u|I ‖h h(t). If K is the constant inRemark 1, then‖ Γu(t) ‖h(t)≤MNHh(t)+Mh(t)∫ t0m(s)W (2K ‖ u ‖BC0hh(s))ds,which from (b) permits us to conclude that Γ is a function from BC0h(X) intoBC0h(X).Let (xn)n∈N be a sequence in BC0h(X) and x ∈ BC0h(X) such that xn → x inBC0h(X). Let ε > 0 and N = supn∈N ‖ xn ‖BC0h. From condition (b), we can selectL > 0 such that2MNHh(t)+Mh(t)∫ t0m(s)W (2KNh(s))ds ≤ε2, t ≥ L.(2.1)From the properties of the functions f, g, we can choose Nε ∈ N such that‖ g(xn, ϕ) − g(x, ϕ) ‖B ≤ε4(MH + 1),(2.2)∫ L0‖ f(s, xns , xn(ρ(s)))− f(s, xs, x(ρ(s))) ‖ ds ≤ε4M,(2.3)for every n ≥ Nε. Under these conditions, we see thatsup{‖ Γxn(t)− Γx(t) ‖h(t): t ∈ [0, L], n ≥ Nε} ≤ε2.(2.4)Moreover, for t ≥ L and n ≥ Nε, we find that‖ Γxn(t)− Γx(t) ‖h(t)≤ M‖ g(xn, ϕ)(0) − g(x, ϕ)(0) ‖h(t)+Mh(t)∫ t0‖ f(s, xns , xn(ρ(s))) − f(s, xs, x(ρ(s))) ‖ ds≤2MNHh(t)+Mh(t)∫ t0m(s)W (2KNh(s))ds,EJQTDE, 2009 No. 50, p. 4and hence,sup{‖ Γxn(t)− Γx(t) ‖h(t): t ≥ L, n ≥ Nε} ≤ε2.(2.5)From the inequalities (2.2), (2.4) and (2.5), it follows that ‖ Γxn − Γx ‖BC0h≤ 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, where (Γ1u)0 = g(u, ϕ), (Γ2u)0 = 0 andΓ1u(t) = T (t)g(u, ϕ)(0), t ≥ 0,(2.6)Γ2u(t) =∫ t0T (t− s)f(s, us, u(ρ(s)))ds, t ≥ 0.(2.7)From Lemma 1.1 and the properties of the semigroup (T (t))t≥0, it is easy to see thatΓ1 is completely continuous. Next, we prove that Γ2 is also completely continuous.From [12, Lemma 3.1], we infer that Γ2Br(0,BC0h) |[0,a]= {Γ2x|[0,a]; x ∈ Br(0,BC0h)}is relatively compact in C([0, a],X) for every a > 0. Considering this property,Lemma 1.1 and the fact that‖ Γ2x(t) ‖h(t)≤Mh(t)∫ t0m(s)W (2Krh(s))ds→ 0, as t→∞,uniformly for x ∈ Br(0,BC0h), we can conclude that the set Γ2(Br(0,BC0h)) is rela-tively compact in BC0h(X) for every r > 0. Thus, Γ is completely continuous.To finish the proof, we obtain a priori estimates for the solutions of the integralequation u = λΓu, λ ∈ (0, 1)}. Let λ ∈ (0, 1) and uλ ∈ BC0h(X) be a solution ofλΓz = z. By using the notation αλ(t) = K(sups∈[0,t] ‖ uλ(s) ‖ + ‖ uλ0 ‖B), for t ∈ Iwe see that‖ uλ(t) ‖ ≤ MNH +M∫ t0m(s)W (‖ uλs ‖B + ‖ uλ(ρ(s)) ‖)ds,≤ MNH +M∫ t0m(s)W ((1 +H)αλ(s))ds,and hence,αλ(t) ≤ K ‖ g(u, ϕ) ‖B +KMNH + KM∫ t0m(s)W (1 +H)αλ(s))ds,≤ KN(1 +MH) + KM∫ t0m(s)W ((1 +H)αλ(s))ds.Defining by βλ the right hand side of the last inequality, we see that β′λ(t) ≤KMm(t)W ((1 +H)βλ(t)), which implies that∫ (1+H)βλ(t)(1+H)βλ(0)=cdsW (s)≤MK(1 +H)∫ ∞0m(s)ds <∫ ∞cdsW (s).EJQTDE, 2009 No. 50, p. 5This 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 inBC0h(X).Now the assertion is a consequence of [11, Theorem 6.5.4]. The proof is complete.Theorem 2.2. Assume assumptions H1, H2 and H3 be hold. If the conditions (a)and (b) of Theorem 2.1 are valid and(2.8) Lg(1 +MH) +M lim infr→∞∫ +∞0m(s)W ((1 + 2K)rh(s))rh(s)ds < 1,then there exists a mild solution of (1.1).Proof. Let Γ,Γ1,Γ2 be the operators introduced in the proof of Theorem 2.1. Weclaim that there exists r > 0 such that Γ(Br) ⊂ Br, where Br = Br(0,BC0h(X)). Infact, if this property is false, then for every r > 0, there exist xr ∈ Br and tr ≥ 0such that r <‖ (Γxr)0 ‖B + ‖Γxr(tr)h(tr) ‖. Consequently,r ≤ ‖ g(xr, ϕ) ‖B +M‖ g(xr, ϕ)(0) ‖h(tr)+Mh(tr)∫ t0m(s)W (K ‖ xr0 ‖B +K supθ∈[0,s]‖ xr(θ) ‖ + ‖ xr(ρ(s)) ‖)ds≤ Lg ‖ xr ‖BC0h+ ‖ g(0, ϕ) ‖B +MHh(tr)Lg ‖ xr ‖BC0h+Mh(tr)‖ g(0, ϕ) ‖B +Mh(tr)∫ tr0m(s)W (Kr + Krh(s) + rh(s))ds≤ Lg(1 +MH)r + (1 +M) ‖ g(0, ϕ) ‖B +M∫ tr0m(s)((2K + 1)rh(s))h(s)dsand then,1 ≤ Lg(1 +MH) +M lim infr→∞∫ +∞0m(s)W ((2K + 1)rh(s))rh(s)ds,which is contrary to (2.8).Let r > 0 such that Γ(Br) ⊂ Br. From the proof of Theorem 2.1, we know thatΓ2 is a completely continuous on Br. Moreover, from the estimate,‖ Γ1u− Γ1v ‖BC0h≤ Lg ‖ u− v ‖BC0h+H ‖ g(u, ϕ) − g(v, ϕ) ‖B≤ Lg(1 +H) ‖ u− v ‖BC0h,we infer that Γ1 is a contraction on Br from which we conclude that Γ is a condensingoperator on Br.The assertion is now a consequence of [13, Theorem 4.3.2].EJQTDE, 2009 No. 50, p. 63. An applicationTo complete this work, we study the existence of global solutions for a concretepartial differential equation with nonlocal conditions. In the sequel, X = L2[0, π]and A : D(A) ⊂ X → X is the operator Ax = x′′ with domain D(A) := {x ∈ X :x′′ ∈ X, x(0) = x(π) = 0}. It is well known that A is the infinitesimal generator ofan analytic semigroup (T (t))t≥0 on X. Furthermore, A has discrete spectrum witheigenvalues of the form −n2, n ∈ N, and corresponding normalized eigenfunctionsgiven by zn(x) = (2π )12 sin(nx); the set of functions {zn : n ∈ N} is an orthonormalbasis for X and T (t)x =∑∞n=1 e−n2t〈x, zn〉zn for every x ∈ X and all t ≥ 0. Itfollows from this last expression that (T (t))t≥0 is a compact semigroup on X andthat ‖ T (t) ‖≤ e−t for every t ≥ 0.As phase space, we choose the space B = C0×L2(̺,X), see Example 1.1, and weassume that the conditions (g5) and (g7) in [10] are verified. Under these conditions,B is a fading memory space, and as consequence, there exists K > 0 such thatsupt≥0{M(t),K(t)} ≤ K.Consider the delayed partial differential equation with nonlocal conditions∂∂tw(t, ξ) =∂2∂ξ2w(t, ξ) +∫ t−∞a(t, s− t)w(s, ξ) ds + b(t, w(ρ(t), ξ)),(3.1)w(t, 0) = w(t, π) = 0, t ≥ 0,(3.2)w(s, ξ) =∞∑i=1Liw(ti + s, ξ) + ϕ(s, ξ), s ≤ 0, ξ ∈ [0, π],(3.3)for t ≥ 0 and ξ ∈ [0, π], and where (Li)i∈N, (ti)i∈N are sequences of real numbers andϕ ∈ B.To treat this system in the abstract form (1.1), we assume∑∞i=1 | Li | h(ti) <∞,the functions ρ : [0,∞) → [0,∞), a : R2 → R are continuous, ρ(t) ≤ t and L1(t) =(∫ 0−∞a2(t,s)̺(s) ds)1/2 < ∞, for every t ≥ 0. We also assume that b : [0,∞) × R → Ris continuous and that there exists a continuous function L2 : [0,∞) → [0,∞) suchthat |b(t, x)| ≤ L2(t)|x|, for every t ≥ 0 and x ∈ R.By defining the functions g : BC0h(X) → B, f : [0,∞)×B ×X → X by g(u, ϕ) =ϕ+∑∞i=1 Liuti andf(t, ψ, x)(ξ) =∫ 0−∞a(t, s)ψ(s, ξ)ds + b(t, x(ρ(t), ξ)),we can transform system (3.1)-(3.3) into the abstract system (1.1). Moreover, it iseasy to see that f, g are continuous, ‖ f(t, ψ, x) ‖≤ L1(t) ‖ ψ ‖B +L2(t) ‖ x ‖, forall (t, ψ, x) ∈ [0,∞) and g verifies conditions H3 with Lg = K∑∞i=1 Lih(ti).The next result is a direct consequence of Theorem 2.2. We omit the proof.Proposition 3.1. If K∑∞i=1 Lih(ti)+ (1+2K)∫∞0 (L1(s)+L2(s))ds < 1, then thereexists a mild solution of (3.1)-(3.3).EJQTDE, 2009 No. 50, p. 7References[1] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinearevolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (2) 494-505 (1991).[2] L. 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Lee, Nonlinear nonlocal Cauchy problems in Banach spaces, Appl.Math. Lett. 18 (4) 401-407 (2005).[15] D. Bahuguna, Existence, uniqueness and regularity of solutions to semilinear nonlocalfunctional differential problems, Nonlinear Anal. 57 (7-8) 1021-1028 (2004).[16] L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of asolution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 40 (1)11-19 (1991).[17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equa-tions, Applied Mathematical Sciences, 44, Springer-Verlag, New York-Berlin (1983).(Received May 5, 2009)Eduardo Herna´ndez M & Sueli M. Tanaka AkiDepartamento de Matema´tica, Instituto de Cieˆncias Matema´ticas de Sa˜o Carlos, Uni-versidade de Sa˜o Paulo, Caixa Postal 668, 13560-970 Sa˜o Carlos, SP, BrazilE-mail address: lalohm@icmc.usp.brE-mail address: smtanaka@icmc.usp.brEJQTDE, 2009 No. 50, p. 8

Global solutions for abstract functional differential equations with nonlocal conditions

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