In this paper, the Leray-Schauder Alternative is used to investigate the existence of mild solutions to first-order nonlinear functional integrodifferential evolution equations with nonlocal conditions in Banach spaces

Zuomao Yan

Abstract. In this paper, the Leray-Schauder Alternative is used to investigate the ex-istence of mild solutions to first-order nonlinear functional integrodifferential evolution equations with nonlocal conditions in Banach spaces. AMS subject classifications: 34K30, 34A60, 34G20 Key words: nonlinear functional integrodifferential evolution equations, fixed point, non

CiteSeerX

Nonlinear functional integrodifferential evolution equations with nonlocal conditions in Banach spaces

Hrčak - Portal of scientific journals of Croatia

MATHEMATICAL COMMUNICATIONS 35Math. Commun., Vol. 14, No. 1, pp. 35-45 (2009)Nonlinear functional integrodifferential evolutionequations with nonlocal conditions in Banach spacesZuomao Yan1,∗1 Department of Mathematics, Hexi University, Zhangye, Gansu 734 000, PR ChinaReceived September 9, 2008; accepted January 15, 2009Abstract. In this paper, the Leray-Schauder Alternative is used to investigate the ex-istence of mild solutions to first-order nonlinear functional integrodifferential evolutionequations with nonlocal conditions in Banach spaces.AMS subject classifications: 34K30, 34A60, 34G20Key words: nonlinear functional integrodifferential evolution equations, fixed point, non-local conditions1. IntroductionThis paper is concerned with the existence of mild solutions for nonlinear functionalintegrodifferential evolution equations with nonlocal conditions. More precisely, weconsider the following nonlocal Cauchy problem on a general Banach space X :x′(t) = A(t)x(t) + F(t, x(σ1(t)), ..., x(σn(t)),∫ t0h(t, s, x(σn+1(t)))ds), t ∈ J,x(0) + g(x) = x0, (1)where J = [0, b], the family {A(t) : 0 ≤ t ≤ b} of an unbounded linear operatorgenerates linear evolution systems. Nonlinear operators F : J × Xn+1 → X, h :J × J ×X, g : C(J,X)→ X, σi : J → J, i = 1, ..., n+ 1, are given functions.The study of abstract nonlocal Cauchy problem was motivated by the paperof Byszewski and Lakshmikantham [4, 5]. In [5], the author has considered theexistence and uniqueness of mild, strong, and classical solutions of the nonlocalCauchy problem, where the operator A(t) = A generates a strongly continuoussemigroup. Subsequently, many papers have been interested in the nonlocal Cauchyproblem and that stems mainly from the observation that nonlocal conditions aremore realistic than the usual ones in treating physical problems, e.g. Byszewski andAkca [6], Ntouyas and Tsamatos [15], Lin and Liu [14], Ezzinbi and Fu [9], Liang etal. [13], Benchohra et al. [3], Ezzinbi and Liu [8], Aizicovici and Mckibben [1].The fundamental tools used in the existence proofs of all above-mentioned worksare essentially fixed-point arguments; Schauder,s fixed point theorem [6], Leray-Schauder Alternative [15], the Sadovskii fixed point theorem [12], Schaefer, fixedpoint theorem [2].∗Corresponding author. Email address: yanzuomao@163.com (Z.Yan)http://www.mathos.hr/mc c©2009 Department of Mathematics, University of Osijek36 Z.YanIn this paper, we shall investigate the existence of mild solutions of a nonlocalCauchy problem (1) in Banach spaces, by means of a different method, that is, byusing the theory of evolution families, Banach,s contraction principle and Leray-Schauder Alternative.This paper will be organized as follows. In Section 2, we will briefly recall somepreliminary facts which will be used in the paper. Section 3 is devoted to theexistence of mild solutions of problem (1). Finally, a concrete example is presentedin Section 4 to show the application of our main results.2. PreliminariesThroughout this section, for the family {A(t) : 0 ≤ t ≤ b} of linear operators, weneed the following assumptions (see [11]).(I) The domain D(A) of {A(t) : 0 ≤ t ≤ b} is dense in the Banach space X andindependent of t, A(t) is a closed linear operator.(II) For each t ∈ J, the resolvent R(λ,A(t)) exists for all λ with Reλ ≥ 0 andthere exists K > 0 such that‖ R(λ,A(t)) ‖≤ K(|λ|+ 1) .(III) For any t, s, τ ∈ J, there exists a 0 < δ < 1 and K > 0 such that‖ (A(t)−A(τ))A−1(s) ‖≤ K|t− τ |δ,and for each t ∈ J and some λ ∈ ρ(A(t)), the resolvent R(λ,A(t)) set of A(t) is acompact operator.Definition 1 (see [16]). A family of linear operators {U(t, s) : 0 ≤ s ≤ t ≤ b} on Xis called an evolution system if the following two conditions are satisfied:(a) U(t, s) ∈ L(X) the space of bounded linear transformations on X whenever0 ≤ s ≤ t ≤ b and for each x ∈ X the mapping (t, s)→ U(t, s)x is continuous;(b) U(t, s)U(s, τ) = U(t, τ) whenever 0 ≤ τ ≤ s ≤ t ≤ b.Definition 2. A continuous function x(·) : J → X is said to be a mild solution toproblem (1) if for all x0 ∈ X, it satisfies the following integral equationx(t) = U(t, 0)[x0 − g(x)]+∫ t0U(t, s)F(s, x(σ1(s)), ..., x(σn(s)),∫ s0h(s, τ, x(σn+1(τ)))dτ)ds. (2)Lemma 1 (see [10]). If conditions (I)-(III) are satisfied, then the family {A(t) :0 ≤ t ≤ b} generates a unique linear evolution system {U(t, s) : 0 ≤ s ≤ t ≤ b} is acompact linear operator on X whenever t− s > 0(0 ≤ s < t ≤ b).Lemma 2 (Leray-Schauder Nonlinear Alternative [7] ). Let X be a Banach spacewith Ω ⊂ X convex. Assume V is a relatively open subset of Ω with 0 ∈ V andG : V → Ω is a compact map. Then eitherNonlinear functional integrodifferential evolution equations 37(I) G has a fixed point in V, or(II) there exists a point v ∈ ∂V such that v ∈ λG(v) for some λ ∈ (0, 1).Further we assume the following hypotheses:(H1) U(t, s) is a compact linear operator on X whenever t − s > 0 and thereexists a constant M > 0, such that ‖ U(t, s) ‖≤M, 0 ≤ s < t ≤ b.(H2) The function F : J × Xn+1 → X is continuous and there exist constantsL > 0, L1 ≥ 0, such that for all xi, yi ∈ X, i = 1, ...n+ 1, we have‖ F (t, x1, x2, ..., xn+1)− F (t, y1, y2, ..., yn+1) ‖≤ L[ n+1∑i=1‖ xi − yi ‖],andL1 = maxt∈J‖ F (t, 0, ..., 0) ‖ .(H3) The function h : J × J × X → X is continuous and there exist constantsN > 0, N1 ≥ 0, such that for all x, y ∈ X,‖ h(t, s, x)− h(t, s, y) ‖≤ N ‖ x− y ‖,andN1 = max0≤s≤t≤b‖ h(t, s, 0) ‖ .(H4) σi : J → J, i = 1, ..., n+ 1, are continuous functions such that σi(t) ≤ t, i =1, ..., n+ 1.(H5) (i)The function g(·) : C(J,X) → X is continuous and there exists a δ ∈(0, b) such that g(φ) = g(ψ) for any φ, ψ ∈ C := C(J,X) with φ = ψ on [δ, b].(ii) There is a continuous nondecreasing function Λ : [0,∞)→ (0,∞) such that‖ g(φ) ‖≤ Λ(‖ φ ‖), φ ∈ C.(H6) There exists a constant M∗ > 0 such thatM∗[M(‖ x0 ‖ +Λ(M∗)) +Mb(LbN1 + L1)]K0 > 1,where K0 = eML(n+Nb)b.3. Main resultTheorem 1. Suppose that assumptions (H1)-(H6) are satisfied. Then the nonlocalCauchy problem (1) has at least one mild solution on J .Proof. Consider the space C := C(J,X) the Banach space of all continuous func-tions from J to X endowed with sup norm.Let L0 := 2ML(n + Nb) and we introduce in the space C the equivalent normdefined as‖ φ ‖V := supt∈Je−L0t ‖ φ(t) ‖ .38 Z.YanThen, it is easy to see that V := (C(J,X), ‖ · ‖V ) is a Banach space. Fix v ∈ C andfor t ∈ J, φ ∈ V, we now define an operator(Qvφ)(t) = U(t, 0)[x0 − g(v)]+∫ t0U(t, s)F(s, φ(σ1(s)), ..., φ(σn(s)),∫ s0h(s, τ, φ(σn+1(τ)))dτ)ds. (3)Since U(·, 0)(x0 − g(v)) ∈ C(J,X), so it follows from (H1)-(H4) that (Qvφ)(t) ∈ Vfor all φ ∈ V. Let φ, ψ ∈ V, we havee−L0t ‖ (Qvφ)(t)− (Qvψ)(t) ‖≤ e−L0t∫ t0wwwwU(t, s)[F(s, φ(σ1(s)), ..., φ(σn(s)),∫ s0h(s, τ, φ(σn+1(τ)))dτ)−F(s, ψ(σ1(s)), ..., ψ(σn(s)),∫ s0h(s, τ, ψ(σn+1(τ)))dτ)]wwwwds≤ML∫ t0e−L0t[‖ φ(σ1(s))− ψ(σ1(s)) ‖ + · · ·+ ‖ φ(σn(s))− ψ(σn(s)) ‖+wwww∫ s0h(s, τ, φ(σn+1(τ)))dτ −∫ s0h(s, τ, ψ(σn+1(τ)))dτwwww]ds≤ML∫ t0e−L0t[‖ φ(s)− ψ(s) ‖ + · · ·+ ‖ φ(s)− ψ(s) ‖+N∫ s0‖ φ(σn+1(τ))− ψ(σn+1(τ)) ‖ dτ]ds≤ML∫ t0e−L0t[n ‖ φ(s)− ψ(s) ‖ +Nb ‖ φ(s)− ψ(s) ‖]ds≤ML∫ t0eL0((s−t)[ne−L0s ‖ φ(s)− ψ(s) ‖ +Nb sups∈Je−L0s ‖ φ(s)− ψ(s) ‖]ds≤ML(n+Nb)∫ t0eL0(s−t)ds ‖ φ− ψ ‖V≤ ML(n+Nb)L0‖ φ− ψ ‖V , t ∈ J,which implies thate−L0t ‖ (Qvφ)(t)− (Qvψ)(t) ‖≤ 12 ‖ φ− ψ ‖V , t ∈ J.Thus‖ Qvφ−Qvψ ‖V≤ 12 ‖ φ− ψ ‖V , φ, ψ ∈ V.Therefore, Qv is a strict contraction. By Banach,s contraction principle we con-clude that Qv has a unique fixed point φv ∈ V and equation (3) has a unique mildsolution on [0, b]. Setv˜(t) :={v(t) if t ∈ (δ, b],v(δ) if t ∈ [0, δ].Nonlinear functional integrodifferential evolution equations 39From (3), we haveφv˜(t) = U(t, 0)[x0 − g(v˜)] (4)+∫ t0U(t, s)F(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ))dτ)ds.Consider the map, P : Cδ = C([δ, b], X)→ Cδ defined by(Pv)(t) = φv˜(t), t ∈ [δ, b]. (5)We shall show that P satisfy all conditions of Lemma 2. The proof will be given inseveral steps.Step 1. P maps bounded sets into bounded sets in Cδ.Indeed, it is enough to show that there exists a positive constant L such that foreach v ∈ Cr(δ) := {φ ∈ Cδ; supδ≤t≤b‖ φ(t) ‖≤ r} one has ‖ Pv ‖≤ L.Let v ∈ Cr(δ), then for t ∈ (0, b], we have‖ φv˜(t) ‖ ≤‖ U(t, 0)[x0 − g(v˜)] ‖+∫ t0wwwwU(t, s)F(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ))dτ)wwwwds≤M [‖ x0 + g(v˜) ‖]+M∫ t0[wwwwF(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ))dτ)−F (s, 0, ..., 0)wwww+ ‖ F (s, 0, ..., 0) ‖]ds≤M [‖ x0 ‖ + ‖ g(v˜) ‖] +M∫ t0{L[‖ φv˜(s) ‖ + · · ·+ ‖ φv˜(s) ‖+∫ s0[‖ h(s, τ, φv˜(σn+1(τ))− h(s, τ, 0) ‖ + ‖ h(s, τ, 0) ‖]dτ]+ L1}ds≤M [‖ x0 ‖+Λ(r)] +M∫ t0{L[n ‖ φv˜(s) ‖ +b(N ‖ φv˜(s) ‖ +N1)]+L1}ds≤M [‖ x0 ‖+Λ(r)] +Mb(LbN1 + L1)+ML(n+Nb)∫ t0sups∈(0,b]‖ φv˜(s) ‖ ds.Thus‖ Pv ‖≤M [‖ x0 ‖ +Λ(r)] +Mb(LbN1 + L1) +ML(n+Nb)rb = L.Step 2. P maps bounded sets into equicontinuous sets of Cδ.Let δ ≤ t1 < t2 ≤ b, and Cr(δ) a bounded set as in Step 1. Let v ∈ Cr(δ), we40 Z.Yanhave‖ Pv(t2)− Pv(t1) ‖≤‖ [U(t2, 0)− U(t1, 0)][x0 − g(v˜)] ‖+∫ t20wwww[U(t2, s)F(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ)))dτ)wwwwds−∫ t10wwww[U(t1, s)F(s, φv˜σ1(s)), ..., φv˜σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ)))dτ)wwwwds≤‖ U(t2, 0)− U(t1, 0) ‖ [‖ x0 ‖ +Λ(r)] +∫ t10‖ U(t2, s)− U(t1, s) ‖wwwwF(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ)))dτ)wwwwds+M∫ t2t1wwwwF(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜2(σn+1(τ)))dτ)wwwwds.Noting thatwwwwF(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ)))dτ)wwww≤wwwwF(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ)))dτ)−F (s, 0, ..., 0)wwww+ ‖ F (s, 0, ..., 0) ‖≤ L[‖ φv˜(σ1(s)) ‖ + · · ·+ ‖ φv˜(σn(s)) ‖ +wwww∫ s0h(s, τ, φv˜(σn+1(τ)))dτwwww]+ L1≤ L[‖ φv˜(s) ‖ + · · ·+ ‖ φv˜(s) ‖ +∫ s0[‖ h(s, τ, φv˜(τ))− h(s, τ, 0) ‖+ ‖ h(s, τ, 0) ‖]dτ]+ L1≤ L[n ‖ φv˜(s) ‖ +b[N sups∈[δ,b]‖ φv˜(s) ‖ +N1]]+ L1≤ L[(n+Nb) sups∈[δ,b]‖ φv˜(s) ‖ +bN1]+ L1 ≤ L[(n+Nb)r + bN1] + L1.We see that ‖ Pv(t2)−Pv(t1) ‖ tend to zero independently of v ∈ Cr(δ) as t2−t1 → 0,since the compactness of U(t, s) for t− s > 0 implies the continuity in the uniformoperator topology. Thus the family of functions {(Pv) : v ∈ Cr(δ)} is equicontinuouson [δ, b].Step 3. The set {P (v)(t) : v ∈ Cr(δ)} is relatively compact in Cδ.Let δ < t ≤ s ≤ b be fixed and ε a real number satisfying 0 < ε < t, for v ∈ Cr(δ),Nonlinear functional integrodifferential evolution equations 41we define(Pεv)(t) = U(t, 0)[x0 − g(v˜)]+∫ t−ε0U(t, s)F(s, φv˜σ1(s)), ..., φv˜σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ)))dτ)ds= U(t, 0)[x0 − g(v˜)] + U(t, t− ε)×∫ t−ε0U(t− ε, s)F(s, φv˜σ1(s)), ..., φv˜σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ)))dτ)ds.Using the compactness of U(t, s) for t−s > 0, we obtain the set {(Pεv)(t) : v ∈ Cr(δ)}is precompact v ∈ Cr(δ) for every ε, 0 < ε < t. Moreover, for every v ∈ Cr(δ) wehave‖ (Pv)(t)− (Pεv)(t) ‖≤∫ tt−εwwwwU(t, s)F(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ)))dτ)wwwwds≤M∫ tt−ε[wwwwF(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ)))dτ)wwww]ds≤M∫ tt−ε[L(n+Nb) sups∈[δ,b]‖ φv˜(s) ‖ +LbN1 + L1]ds≤M∫ tt−ε[L(n+Nb)r + LbN1 + L1]ds.Therefore, there are precompact sets arbitrarily close to the set {(Pv) : v ∈ Cr(δ)}.Hence the set {(Pv) : v ∈ Cr(δ)} is a precompact in Cδ.Step 4. P : Cδ → Cδ is continuous.From (4) and (H1)-(H5), we deduce that for v1, v2 ∈ Cr(δ), t ∈ (0, b],‖ φv˜1(t)− φv˜2(t) ‖≤‖ U(t, 0)[g(v˜1)− g(v˜2)] ‖+∫ t0wwwwU(t, s)[F(s, φv˜1(σ1(s)), ..., φv˜1(σn(s)),∫ s0h(s, τ, φv˜1(σn+1(τ)))dτ)−F(s, φv˜2(σ1(s)), ..., φv˜2(σn(s)),∫ s0h(s, τ, φv˜2(σn+1(τ))))]wwwwds≤M ‖ g(v˜1)− g(v˜2) ‖+M∫ t0L[‖ φv˜1(σ1(s))− φv˜2(σ1(s)) ‖ + · · ·+ ‖ φv˜1(σn(s))− φv˜2(σn(s)) ‖+wwww∫ s0h(s, τ, φv˜1(σn+1(τ)))dτ))−∫ s0h(s, τ, φv˜2(σn+1(τ)))dτwwww]ds42 Z.Yan≤M ‖ g(v˜1)− g(v˜2) ‖+ML∫ t0[‖ φv˜1(s)− φv˜2(s) ‖ + · · ·+ ‖ φv˜1(s)− φv˜2(s) ‖+N∫ s0[‖ φv˜1(σn+1(s))− φv˜2(σn+1(s)) ‖]ds≤M ‖ g(v˜1)− g(v˜2) ‖ +ML∫ t0[n ‖ φv˜1(s)− φv˜2(s) ‖+Nb ‖ φv˜1(s)− φv˜2(s) ‖]ds≤M ‖ g(v˜1)− g(v˜2) ‖ +ML(n+Nb)∫ t0sups∈(0,b]‖ φv˜1(s)− φv˜2(s) ‖ ds.Using again the Gronwall,s inequality, that for t, v1, v2 as abovesupt∈(0,b]‖ φv˜1(t)− φv˜2(t) ‖≤MeML(n+Nb)b ‖ g(v˜1)− g(v˜2) ‖,for all t ∈ [0, b], which implies that‖ Pv1 − Pv2 ‖≤Me[ML(n+Nb)+β]b ‖ g(v˜1)− g(v˜2) ‖ .for all t ∈ [δ, b], v1, v2 ∈ Cr(δ). Therefore, P is continuous.Step 5. We now show that there exists an open set V ∗ ⊆ Cδ with v /∈ λPv forλ ∈ (0, 1) and v ∈ ∂V ∗. Let λ ∈ (0, 1) and let v ∈ Cδ be a possible solution ofv = λP (v) for some 0 < λ < 1. Thus, for each t ∈ (0, b],v(t) = λφv˜(t) = λU(t, 0)[x0 − g(v˜)] (6)+λ∫ t0U(t, s)F(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ))dτ)ds.This implies by (H1)-(H5) that for each t ∈ J we have ‖ v(t) ‖≤‖ φv˜(t) ‖ and‖ φv˜(t) ‖ ≤M [‖ x0 ‖ +g(v˜) ‖]+M∫ t0[wwwwF(s, φv˜(σ1(s)), ..., φv˜(σn(s)),∫ s0h(s, τ, φv˜(σn+1(τ))dτ)wwww]ds≤M [‖ x0 ‖ +Λ(‖ v˜ ‖)] +Mb(LbN1 + L1)+ML(n+Nb)∫ t0sups∈(0,b]‖ φv˜(s) ‖ ds.Making use of the Gronwall,s inequality, such thatsupt∈(0,b]‖ φv˜(t) ‖ ≤[M(‖ x0 ‖ +Λ(‖ v˜ ‖)) +Mb(LbN1 + L1)]eML(n+Nb)b,and the previous inequality holds. Consequently,‖ v ‖ ≤[M(‖ x0 ‖ +Λ(‖ v˜ ‖)) +Mb(LbN1 + L1)]eML(n+Nb)b,Nonlinear functional integrodifferential evolution equations 43and therefore‖ v ‖[M(‖ x0 ‖ +Λ(‖ v˜ ‖)) +Mb(LbN1 + L1)]K0 ≤ 1.Then, by (H6), there exists M∗ such that ‖ v ‖6= M∗. SetV ∗ = {v ∈ C([δ, b], X); supδ≤t≤b‖ v(t) ‖< M∗}.As a consequence of Steps 1-4, together with the Arzela-Ascoli theorem it sufficesto show that P : V ∗ → Cδ is a compact map.From the choice of V ∗, there is no x ∈ ∂V ∗ such that v ∈ λPv for λ ∈ (0, 1). Asa consequence of Lemma 2, we deduce that P has a fixed point V ∗. Let x = φv˜∗ .Then, we havex(t) = U(t, 0)[x0 − g(v˜∗)] (7)+∫ t0U(t, s)F(s, x(σ1(s)), ..., x(σn(s)),∫ s0k(s, τ)h(τ, x(σn+1(τ)))dτ)ds.Noting that x = φv˜∗ = (P v˜∗)(t) = v˜∗, t ∈ [δ, b]. By (H5)(i), we obtaing(x) = g(v˜∗).This implies that x is Q has a fixed point in V ∗ ⊂ C(J,X). Hence, problem (1) hasa mild solution and completes the proof of Theorem 1.Remark 1. In [6], Byszewski and Akca discussed a related semilinear nonlocal prob-lem when g is convex and compact on a given ball. In this paper, we consider thecase when g is continuous but without imposing severe compactness conditions andconvexity.Remark 2. Condition (H5) on g in the above theorem is an extension of the cor-responding conditions in paper [15], [13].4. ApplicationTo illustrate the application of the obtained results of this paper, we study thefollowing example in this section:zt(t, x) =∂2∂x2a0(t, x)z(t, x)+a1(t)z(sin t, x) + sin z(t, x) +11 + t2∫ t0a2(s)z(sin s, x)ds,z(t, 0) = z(t, pi) = 0, (8)z(0, x) +∫ 1δ[z(s, x) + log(1 + |z(s, x)|)]ds = z0(x), 0 ≤ t ≤ 1, 0 ≤ x ≤ pi,where δ > 0, z0(x) ∈ X = L2([0, pi]) and z0(0) = z0(pi) = 0. Here, the functiona0(t, x) is continuous and uniformly Ho¨lder continuous in t.44 Z.YanLet X = L2([0, pi]) and the operators A(t) be defined byA(t)w = a0(t, x)w′′with the domain D(A) = {w ∈ X : w,w′′ are absolutely continuous, w′′ ∈ X,w(0) =w(1) = 0}, then A(t) generates an evolution system U(t, s) satisfying assumptions(I)-(III)(see [11]).We assume that the function ai(·) is continuous on [0, 1], and li = sup0≤s≤1|ai(s)| <1, i = 1, 2.Define F : [0, 1]×X×X → X,h : [0, 1]× [0, 1]×X → X and g : C([0, 1], X)→ XbyF(t, z,∫ t0h(t, s, z(σ(s)))ds)(x) = a1(t)z(sin t, x) + sin z(t, x)+11 + t2∫ t0a2(s)z(sin s, x)ds,∫ t0h(t, s, z(σ(s)))(x)ds =11 + t2∫ t0a2(s)z(sin s, x)ds,andg(z)(x) =∫ 1δ[z(s, x) + log(1 + |z(s, x)|)]ds, x ∈ C([0, 1], X).It is easy to see that with these choices, the assumptions (H1)-(H5) of Theorem 1are satisfied. In particular, the constants are L = 1 + l1 + l2, N = l2 and L1 =N1 = 0. If we assume that MeM(1+l1+l2)(1+l2)(1 − δ) < 12 and choose the constantM∗ = max{8M ‖ x0 ‖ eM(1+l1+l2)(1+l2), 1}, then1 >M [‖ x0 ‖ +(M∗ + log(1 +M∗))(1− δ)]eM(1+l1+l2)(1+l2)M∗.Now condition (H6) in Section 2 holds and hence by Theorem 1, we deduce thatnonlocal Cauchy problem (8) has a mild solution on [0, 1].References[1] S.Aizicovici, M.Mckibben, Existence results for a class of abstract nonlocal Cauchyproblems, Nonlinear Anal. 39(2000), 649-668.[2] K.Balachandran, J.Y. Park, M.Chandrasekaran, Nonlocal Cauchy problemsfor delay integrodifferential equations of sobolev type in Banach spaces, Appl. Math.Lett. 15(2002), 845-854.[3] N.Benchohra, E.Gatsori, J. Henderson, S.K. Ntouyas, Nondensely definedevolution impulsive differential inclusions with nonlocal conditions, J. Math. Anal.Appl. 286(2003), 307-325.[4] L. Byszewski., V. Lakshmikantham, Theorem about existence and uniqueness ofa solutions of a nonlocal Cauchy problem in a Banach space, Appl. Anal. 40(1990),11-19.[5] L.Byszewski, Theorems about the existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162(1991), 494-505.Nonlinear functional integrodifferential evolution equations 45[6] L.Byszewski, H.Akca, Existence of solutions of a semilinear functional-differentialevolution nonlocal problem, Nonlinear Anal. 34(1998), 65-72.[7] J.Dugundji, A.Granas, Fixed point theory, Monografie Mat. PWN, Warsaw, 1982.[8] K.Ezzinbi, J. H. Liu, Nondensely defined evolution equations with nonlocal condi-tions, Math. Comput. Model. 36(2002), 1027-1038.[9] K.Ezzinbi, X. Fu, Existence and regularity of solutions for some neutral partial dif-ferential equations with nonlocal conditions, Nonlinear Anal. 57(2004), 1029-1041.[10] W.E.Fitzgibbon, Semilinear functional differential equations in Banach spaces, J.Differential Equations, 29(1978), 1-14.[11] A.Freidman, Partial Differential Equations, Holt, Rinehat and Winston, New York,1969.[12] X.Fu, K.Ezzinibi, Existence of solutions for neutral functional evolution equationswith nonlocal conditions, Nonlinear Anal. Theory Methods Appl. 54(2003), 215-227.[13] J. Liang, J.H. Liu, T. J. Xiao, Nonlocal Cauchy problems governed by compact op-erator families, Nonlinear Anal. 57(2004), 183-189.[14] Y.Lin, J. H. Liu, Semilinear integrodifferential equations with nonlocal Cauchy prob-lem, Nonlinear Anal. 26(1996), 1023-1033.[15] S.K. Ntouyas, P.Ch.Tsamatos, Global existence for semilinear evolution equationswith nonlocal conditions, J. Math. Anal. Appl. 210(1997), 679-687.[16] A.Pazy, Semigroup of linear operators and applications to partial differential equa-tions, Springer-Verlag, New York, 1983.

HRČAK - Portal of Croatian Scientific and Professional Journals

MATHEMATICAL COMMUNICATIONS 35Math. Commun., Vol. 14, No. 1, pp. 35-45 (2009)Nonlinear functional integrodifferential evolutionequations with nonlocal conditions in Banach spacesZuomao Yan1,∗1 Department of Mathematics, Hexi University, Zhangye, Gansu 734 000, PR ChinaReceived September 9, 2008; accepted January 15, 2009Abstract. In this paper, the Leray-Schauder Alternative is used to investigate the ex-istence of mild solutions to first-order nonlinear functional integrodifferential evolutionequations with nonlocal conditions in Banach spaces.AMS subject classifications: 34K30, 34A60, 34G20Key words: nonlinear functional integrodifferential evolution equations, fixed point, non-local conditions1. IntroductionThis paper is concerned with the existence of mild solutions for nonlinear functionalintegrodifferential evolution equations with nonlocal conditions. More precisely, weconsider the following nonlocal Cauchy problem on a general Banach space X :x′(t) = A(t)x(t) + F(t, x(σ1(t)), ..., x(σn(t)),∫ t0h(t, s, x(σn+1(t)))ds), t ∈ J,x(0) + g(x) = x0, (1)where J = [0, b], the family {A(t) : 0 ≤ t ≤ b} of an unbounded linear operatorgenerates linear evolution systems. Nonlinear operators F : J × Xn+1 → X, h :J × J ×X, g : C(J,X)→ X, σi : J → J, i = 1, ..., n+ 1, are given functions.The study of abstract nonlocal Cauchy problem was motivated by the paperof Byszewski and Lakshmikantham [4, 5]. In [5], the author has considered theexistence and uniqueness of mild, strong, and classical solutions of the nonlocalCauchy problem, where the operator A(t) = A generates a strongly continuoussemigroup. Subsequently, many papers have been interested in the nonlocal Cauchyproblem and that stems mainly from the observation that nonlocal conditions aremore realistic than the usual ones in treating physical problems, e.g. Byszewski andAkca [6], Ntouyas and Tsamatos [15], Lin and Liu [14], Ezzinbi and Fu [9], Liang etal. [13], Benchohra et al. [3], Ezzinbi and Liu [8], Aizicovici and Mckibben [1].The fundamental tools used in the existence proofs of all above-mentioned worksare essentially fixed-point arguments; Schauder,s fixed point theorem [6], Leray-Schauder Alternative [15], the Sadovskii fixed point theorem [12], Schaefer, fixedpoint theorem [2].∗Corresponding author. Email address: yanzuomao@163.com (Z.Yan)http://www.mathos.hr/mc c©2009 Department of Mathematics, University of Osijek36 Z.YanIn this paper, we shall investigate the existence of mild solutions of a nonlocalCauchy problem (1) in Banach spaces, by means of a different method, that is, byusing the theory of evolution families, Banach,s contraction principle and Leray-Schauder Alternative.This paper will be organized as follows. In Section 2, we will briefly recall somepreliminary facts which will be used in the paper. Section 3 is devoted to theexistence of mild solutions of problem (1). Finally, a concrete example is presentedin Section 4 to show the application of our main results.2. PreliminariesThroughout this section, for the family {A(t) : 0 ≤ t ≤ b} of linear operators, weneed the following assumptions (see [11]).(I) The domain D(A) of {A(t) : 0 ≤ t ≤ b} is dense in the Banach space X andindependent of t, A(t) is a closed linear operator.(II) For each t ∈ J, the resolvent R(λ,A(t)) exists for all λ with Reλ ≥ 0 andthere exists K > 0 such that‖ R(λ,A(t)) ‖≤ K(|λ|+ 1).(III) For any t, s, τ ∈ J, there exists a 0 < δ < 1 and K > 0 such that‖ (A(t)−A(τ))A−1(s) ‖≤ K|t− τ |δ,and for each t ∈ J and some λ ∈ ρ(A(t)), the resolvent R(λ,A(t)) set of A(t) is acompact operator.Definition 1 (see [16]). A family of linear operators {U(t, s) : 0 ≤ s ≤ t ≤ b} on Xis called an evolution system if the following two conditions are satisfied:(a) U(t, s) ∈ L(X) the space of bounded linear transformations on X whenever0 ≤ s ≤ t ≤ b and for each x ∈ X the mapping (t, s)→ U(t, s)x is continuous;(b) U(t, s)U(s, τ) = U(t, τ) whenever 0 ≤ τ ≤ s ≤ t ≤ b.Definition 2. A continuous function x(·) : J → X is said to be a mild solution toproblem (1) if for all x0 ∈ X, it satisfies the following integral equationx(t) = U(t, 0)[x0 − g(x)]+∫ t0U(t, s)F(s, x(σ1(s)), ..., x(σn(s)),∫ s0h(s, τ, x(σn+1(τ)))dτ)ds. (2)Lemma 1 (see [10]). If conditions (I)-(III) are satisfied, then the family {A(t) :0 ≤ t ≤ b} generates a unique linear evolution system {U(t, s) : 0 ≤ s ≤ t ≤ b} is acompact linear operator on X whenever t− s > 0(0 ≤ s < t ≤ b).Lemma 2 (Leray-Schauder Nonlinear Alternative [7] ). Let X be a Banach spacewith Ω ⊂ X convex. Assume V is a relatively open subset of Ω with 0 ∈ V andG : V → Ω is a compact map. Then eitherNonlinear functional integrodifferential evolution equations 37(I) G has a fixed point in V, or(II) there exists a point v ∈ ∂V such that v ∈ λG(v) for some λ ∈ (0, 1).Further we assume the following hypotheses:(H1) U(t, s) is a compact linear operator on X whenever t − s > 0 and thereexists a constant M > 0, such that ‖ U(t, s) ‖≤M, 0 ≤ s < t ≤ b.(H2) The function F : J × Xn+1 → X is continuous and there exist constantsL > 0, L1 ≥ 0, such that for all xi, yi ∈ X, i = 1, ...n+ 1, we have‖ F (t, x1, x2, ..., xn+1)− F (t, y1, y2, ..., yn+1) ‖≤ L[ n+1∑i=1‖ xi − yi ‖],andL1 = maxt∈J‖ F (t, 0, ..., 0) ‖ .(H3) The function h : J × J × X → X is continuous and there exist constantsN > 0, N1 ≥ 0, such that for all x, y ∈ X,‖ h(t, s, x)− h(t, s, y) ‖≤ N ‖ x− y ‖,andN1 = max0≤s≤t≤b‖ h(t, s, 0) ‖ .(H4) σi : J → J, i = 1, ..., n+ 1, are continuous functions such that σi(t) ≤ t, i =1, ..., n+ 1.(H5) (i)The function g(·) : C(J,X) → X is continuous and there exists a δ ∈(0, b) such that g(φ) = g(ψ) for any φ, ψ ∈ C := C(J,X) with φ = ψ on [δ, b].(ii) There is a continuous nondecreasing function Λ : [0,∞)→ (0,∞) such that‖ g(φ) ‖≤ Λ(‖ φ ‖), φ ∈ C.(H6) There exists a constant M∗ > 0 such thatM∗[M(‖ x0 ‖ +Λ(M∗)) +Mb(LbN1 + L1)]K0> 1,where K0 = eML(n+Nb)b.3. Main resultTheorem 1. Suppose that assumptions (H1)-(H6) are satisfied. Then the nonlocalCauchy problem (1) has at least one mild solution on J .Proof. Consider the space C := C(J,X) the Banach space of all continuous func-tions from J to X endowed with sup norm.Let L0 := 2ML(n + Nb) and we introduce in the space C the equivalent normdefined as‖ φ ‖V := supt∈Je−L0t ‖ φ(t) ‖ .38 Z.YanThen, it is easy to see that V := (C(J,X), ‖ · ‖V ) is a Banach space. Fix v ∈ C andfor t ∈ J, φ ∈ V, we now define an operator(Qvφ)(t) = U(t, 0)[x0 − g(v)]+∫ t0U(t, s)F(s, φ(σ1(s)), ..., φ(σn(s)),∫ s0h(s, τ, φ(σn+1(τ)))dτ)ds. (3)Since U(·, 0)(x0 − g(v)) ∈ C(J,X), so it follows from (H1)-(H4) that (Qvφ)(t) ∈ Vfor all φ ∈ V. Let φ, ψ ∈ V, we havee−L0t ‖ (Qvφ)(t)− (Qvψ)(t) ‖≤ e−L0t∫ t0wwwwU(t, s)[F(s, φ(σ1(s)), ..., φ(σn(s)),∫ s0h(s, τ, φ(σn+1(τ)))dτ)−F(s, ψ(σ1(s)), ..., ψ(σn(s)),∫ s0h(s, τ, ψ(σn+1(τ)))dτ)]wwwwds≤ML∫ t0e−L0t[‖ φ(σ1(s))− ψ(σ1(s)) ‖ + · · ·+ ‖ φ(σn(s))− ψ(σn(s)) ‖+wwww∫ s0h(s, τ, φ(σn+1(τ)))dτ −∫ s0h(s, τ, ψ(σn+1(τ)))dτwwww]ds≤ML∫ t0e−L0t[‖ φ(s)− ψ(s) ‖ + · · ·+ ‖ φ(s)− ψ(s) ‖+N∫ s0‖ φ(σn+1(τ))− ψ(σn+1(τ)) ‖ dτ]ds≤ML∫ t0e−L0t[n ‖ φ(s)− ψ(s) ‖ +Nb ‖ φ(s)− ψ(s) ‖]ds≤ML∫ t0eL0((s−t)[ne−L0s ‖ φ(s)− ψ(s) ‖ +Nb sups∈Je−L0s ‖ φ(s)− ψ(s) ‖]ds≤ML(n+Nb)∫ t0eL0(s−t)ds ‖ φ− ψ ‖V≤ ML(n+Nb)L0‖ φ− ψ ‖V , t ∈ J,which implies thate−L0t ‖ (Qvφ)(t)− (Qvψ)(t) ‖≤12‖ φ− ψ ‖V , t ∈ J.Thus‖ Qvφ−Qvψ ‖V≤12‖ φ− ψ ‖V , φ, ψ ∈ V.Therefore, Qv is a strict contraction. By Banach,s contraction principle we con-clude that Qv has a unique fixed point φv ∈ V and equation (3) has a unique mildsolution on [0, b]. Setṽ(t) :={v(t) if t ∈ (δ, b],v(δ) if t ∈ [0, δ].Nonlinear functional integrodifferential evolution equations 39From (3), we haveφṽ(t) = U(t, 0)[x0 − g(ṽ)] (4)+∫ t0U(t, s)F(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ))dτ)ds.Consider the map, P : Cδ = C([δ, b], X)→ Cδ defined by(Pv)(t) = φṽ(t), t ∈ [δ, b]. (5)We shall show that P satisfy all conditions of Lemma 2. The proof will be given inseveral steps.Step 1. P maps bounded sets into bounded sets in Cδ.Indeed, it is enough to show that there exists a positive constant L such that foreach v ∈ Cr(δ) := {φ ∈ Cδ; supδ≤t≤b‖ φ(t) ‖≤ r} one has ‖ Pv ‖≤ L.Let v ∈ Cr(δ), then for t ∈ (0, b], we have‖ φṽ(t) ‖ ≤‖ U(t, 0)[x0 − g(ṽ)] ‖+∫ t0wwwwU(t, s)F(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ))dτ)wwwwds≤M [‖ x0 + g(ṽ) ‖]+M∫ t0[wwwwF(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ))dτ)−F (s, 0, ..., 0)wwww+ ‖ F (s, 0, ..., 0) ‖]ds≤M [‖ x0 ‖ + ‖ g(ṽ) ‖] +M∫ t0{L[‖ φṽ(s) ‖ + · · ·+ ‖ φṽ(s) ‖+∫ s0[‖ h(s, τ, φṽ(σn+1(τ))− h(s, τ, 0) ‖ + ‖ h(s, τ, 0) ‖]dτ]+ L1}ds≤M [‖ x0 ‖+Λ(r)] +M∫ t0{L[n ‖ φṽ(s) ‖ +b(N ‖ φṽ(s) ‖ +N1)]+L1}ds≤M [‖ x0 ‖+Λ(r)] +Mb(LbN1 + L1)+ML(n+Nb)∫ t0sups∈(0,b]‖ φṽ(s) ‖ ds.Thus‖ Pv ‖≤M [‖ x0 ‖ +Λ(r)] +Mb(LbN1 + L1) +ML(n+Nb)rb = L.Step 2. P maps bounded sets into equicontinuous sets of Cδ.Let δ ≤ t1 < t2 ≤ b, and Cr(δ) a bounded set as in Step 1. Let v ∈ Cr(δ), we40 Z.Yanhave‖ Pv(t2)− Pv(t1) ‖≤‖ [U(t2, 0)− U(t1, 0)][x0 − g(ṽ)] ‖+∫ t20wwww[U(t2, s)F(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ)))dτ)wwwwds−∫ t10wwww[U(t1, s)F(s, φṽσ1(s)), ..., φṽσn(s)),∫ s0h(s, τ, φṽ(σn+1(τ)))dτ)wwwwds≤‖ U(t2, 0)− U(t1, 0) ‖ [‖ x0 ‖ +Λ(r)] +∫ t10‖ U(t2, s)− U(t1, s) ‖wwwwF(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ)))dτ)wwwwds+M∫ t2t1wwwwF(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ2(σn+1(τ)))dτ)wwwwds.Noting thatwwwwF(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ)))dτ)wwww≤wwwwF(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ)))dτ)−F (s, 0, ..., 0)wwww+ ‖ F (s, 0, ..., 0) ‖≤ L[‖ φṽ(σ1(s)) ‖ + · · ·+ ‖ φṽ(σn(s)) ‖ +wwww∫ s0h(s, τ, φṽ(σn+1(τ)))dτwwww]+ L1≤ L[‖ φṽ(s) ‖ + · · ·+ ‖ φṽ(s) ‖ +∫ s0[‖ h(s, τ, φṽ(τ))− h(s, τ, 0) ‖+ ‖ h(s, τ, 0) ‖]dτ]+ L1≤ L[n ‖ φṽ(s) ‖ +b[N sups∈[δ,b]‖ φṽ(s) ‖ +N1]]+ L1≤ L[(n+Nb) sups∈[δ,b]‖ φṽ(s) ‖ +bN1]+ L1 ≤ L[(n+Nb)r + bN1] + L1.We see that ‖ Pv(t2)−Pv(t1) ‖ tend to zero independently of v ∈ Cr(δ) as t2−t1 → 0,since the compactness of U(t, s) for t− s > 0 implies the continuity in the uniformoperator topology. Thus the family of functions {(Pv) : v ∈ Cr(δ)} is equicontinuouson [δ, b].Step 3. The set {P (v)(t) : v ∈ Cr(δ)} is relatively compact in Cδ.Let δ < t ≤ s ≤ b be fixed and ε a real number satisfying 0 < ε < t, for v ∈ Cr(δ),Nonlinear functional integrodifferential evolution equations 41we define(Pεv)(t) = U(t, 0)[x0 − g(ṽ)]+∫ t−ε0U(t, s)F(s, φṽσ1(s)), ..., φṽσn(s)),∫ s0h(s, τ, φṽ(σn+1(τ)))dτ)ds= U(t, 0)[x0 − g(ṽ)] + U(t, t− ε)×∫ t−ε0U(t− ε, s)F(s, φṽσ1(s)), ..., φṽσn(s)),∫ s0h(s, τ, φṽ(σn+1(τ)))dτ)ds.Using the compactness of U(t, s) for t−s > 0, we obtain the set {(Pεv)(t) : v ∈ Cr(δ)}is precompact v ∈ Cr(δ) for every ε, 0 < ε < t. Moreover, for every v ∈ Cr(δ) wehave‖ (Pv)(t)− (Pεv)(t) ‖≤∫ tt−εwwwwU(t, s)F(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ)))dτ)wwwwds≤M∫ tt−ε[wwwwF(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ)))dτ)wwww]ds≤M∫ tt−ε[L(n+Nb) sups∈[δ,b]‖ φṽ(s) ‖ +LbN1 + L1]ds≤M∫ tt−ε[L(n+Nb)r + LbN1 + L1]ds.Therefore, there are precompact sets arbitrarily close to the set {(Pv) : v ∈ Cr(δ)}.Hence the set {(Pv) : v ∈ Cr(δ)} is a precompact in Cδ.Step 4. P : Cδ → Cδ is continuous.From (4) and (H1)-(H5), we deduce that for v1, v2 ∈ Cr(δ), t ∈ (0, b],‖ φṽ1(t)− φṽ2(t) ‖≤‖ U(t, 0)[g(ṽ1)− g(ṽ2)] ‖+∫ t0wwwwU(t, s)[F(s, φṽ1(σ1(s)), ..., φṽ1(σn(s)),∫ s0h(s, τ, φṽ1(σn+1(τ)))dτ)−F(s, φṽ2(σ1(s)), ..., φṽ2(σn(s)),∫ s0h(s, τ, φṽ2(σn+1(τ))))]wwwwds≤M ‖ g(ṽ1)− g(ṽ2) ‖+M∫ t0L[‖ φṽ1(σ1(s))− φṽ2(σ1(s)) ‖ + · · ·+ ‖ φṽ1(σn(s))− φṽ2(σn(s)) ‖+wwww∫ s0h(s, τ, φṽ1(σn+1(τ)))dτ))−∫ s0h(s, τ, φṽ2(σn+1(τ)))dτwwww]ds42 Z.Yan≤M ‖ g(ṽ1)− g(ṽ2) ‖+ML∫ t0[‖ φṽ1(s)− φṽ2(s) ‖ + · · ·+ ‖ φṽ1(s)− φṽ2(s) ‖+N∫ s0[‖ φṽ1(σn+1(s))− φṽ2(σn+1(s)) ‖]ds≤M ‖ g(ṽ1)− g(ṽ2) ‖ +ML∫ t0[n ‖ φṽ1(s)− φṽ2(s) ‖+Nb ‖ φṽ1(s)− φṽ2(s) ‖]ds≤M ‖ g(ṽ1)− g(ṽ2) ‖ +ML(n+Nb)∫ t0sups∈(0,b]‖ φṽ1(s)− φṽ2(s) ‖ ds.Using again the Gronwall,s inequality, that for t, v1, v2 as abovesupt∈(0,b]‖ φṽ1(t)− φṽ2(t) ‖≤MeML(n+Nb)b ‖ g(ṽ1)− g(ṽ2) ‖,for all t ∈ [0, b], which implies that‖ Pv1 − Pv2 ‖≤Me[ML(n+Nb)+β]b ‖ g(ṽ1)− g(ṽ2) ‖ .for all t ∈ [δ, b], v1, v2 ∈ Cr(δ). Therefore, P is continuous.Step 5. We now show that there exists an open set V ∗ ⊆ Cδ with v /∈ λPv forλ ∈ (0, 1) and v ∈ ∂V ∗. Let λ ∈ (0, 1) and let v ∈ Cδ be a possible solution ofv = λP (v) for some 0 < λ < 1. Thus, for each t ∈ (0, b],v(t) = λφṽ(t) = λU(t, 0)[x0 − g(ṽ)] (6)+λ∫ t0U(t, s)F(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ))dτ)ds.This implies by (H1)-(H5) that for each t ∈ J we have ‖ v(t) ‖≤‖ φṽ(t) ‖ and‖ φṽ(t) ‖ ≤M [‖ x0 ‖ +g(ṽ) ‖]+M∫ t0[wwwwF(s, φṽ(σ1(s)), ..., φṽ(σn(s)),∫ s0h(s, τ, φṽ(σn+1(τ))dτ)wwww]ds≤M [‖ x0 ‖ +Λ(‖ ṽ ‖)] +Mb(LbN1 + L1)+ML(n+Nb)∫ t0sups∈(0,b]‖ φṽ(s) ‖ ds.Making use of the Gronwall,s inequality, such thatsupt∈(0,b]‖ φṽ(t) ‖ ≤[M(‖ x0 ‖ +Λ(‖ ṽ ‖)) +Mb(LbN1 + L1)]eML(n+Nb)b,and the previous inequality holds. Consequently,‖ v ‖ ≤[M(‖ x0 ‖ +Λ(‖ ṽ ‖)) +Mb(LbN1 + L1)]eML(n+Nb)b,Nonlinear functional integrodifferential evolution equations 43and therefore‖ v ‖[M(‖ x0 ‖ +Λ(‖ ṽ ‖)) +Mb(LbN1 + L1)]K0≤ 1.Then, by (H6), there exists M∗ such that ‖ v ‖6= M∗. SetV ∗ = {v ∈ C([δ, b], X); supδ≤t≤b‖ v(t) ‖< M∗}.As a consequence of Steps 1-4, together with the Arzela-Ascoli theorem it sufficesto show that P : V ∗ → Cδ is a compact map.From the choice of V ∗, there is no x ∈ ∂V ∗ such that v ∈ λPv for λ ∈ (0, 1). Asa consequence of Lemma 2, we deduce that P has a fixed point V ∗. Let x = φṽ∗ .Then, we havex(t) = U(t, 0)[x0 − g(ṽ∗)] (7)+∫ t0U(t, s)F(s, x(σ1(s)), ..., x(σn(s)),∫ s0k(s, τ)h(τ, x(σn+1(τ)))dτ)ds.Noting that x = φṽ∗ = (P ṽ∗)(t) = ṽ∗, t ∈ [δ, b]. By (H5)(i), we obtaing(x) = g(ṽ∗).This implies that x is Q has a fixed point in V ∗ ⊂ C(J,X). Hence, problem (1) hasa mild solution and completes the proof of Theorem 1.Remark 1. In [6], Byszewski and Akca discussed a related semilinear nonlocal prob-lem when g is convex and compact on a given ball. In this paper, we consider thecase when g is continuous but without imposing severe compactness conditions andconvexity.Remark 2. Condition (H5) on g in the above theorem is an extension of the cor-responding conditions in paper [15], [13].4. ApplicationTo illustrate the application of the obtained results of this paper, we study thefollowing example in this section:zt(t, x) =∂2∂x2a0(t, x)z(t, x)+a1(t)z(sin t, x) + sin z(t, x) +11 + t2∫ t0a2(s)z(sin s, x)ds,z(t, 0) = z(t, π) = 0, (8)z(0, x) +∫ 1δ[z(s, x) + log(1 + |z(s, x)|)]ds = z0(x), 0 ≤ t ≤ 1, 0 ≤ x ≤ π,where δ > 0, z0(x) ∈ X = L2([0, π]) and z0(0) = z0(π) = 0. Here, the functiona0(t, x) is continuous and uniformly Hölder continuous in t.44 Z.YanLet X = L2([0, π]) and the operators A(t) be defined byA(t)w = a0(t, x)w′′with the domain D(A) = {w ∈ X : w,w′′ are absolutely continuous, w′′ ∈ X,w(0) =w(1) = 0}, then A(t) generates an evolution system U(t, s) satisfying assumptions(I)-(III)(see [11]).We assume that the function ai(·) is continuous on [0, 1], and li = sup0≤s≤1|ai(s)| <1, i = 1, 2.Define F : [0, 1]×X×X → X,h : [0, 1]× [0, 1]×X → X and g : C([0, 1], X)→ XbyF(t, z,∫ t0h(t, s, z(σ(s)))ds)(x) = a1(t)z(sin t, x) + sin z(t, x)+11 + t2∫ t0a2(s)z(sin s, x)ds,∫ t0h(t, s, z(σ(s)))(x)ds =11 + t2∫ t0a2(s)z(sin s, x)ds,andg(z)(x) =∫ 1δ[z(s, x) + log(1 + |z(s, x)|)]ds, x ∈ C([0, 1], X).It is easy to see that with these choices, the assumptions (H1)-(H5) of Theorem 1are satisfied. In particular, the constants are L = 1 + l1 + l2, N = l2 and L1 =N1 = 0. If we assume that MeM(1+l1+l2)(1+l2)(1 − δ) < 12 and choose the constantM∗ = max{8M ‖ x0 ‖ eM(1+l1+l2)(1+l2), 1}, then1 >M [‖ x0 ‖ +(M∗ + log(1 +M∗))(1− δ)]eM(1+l1+l2)(1+l2)M∗.Now condition (H6) in Section 2 holds and hence by Theorem 1, we deduce thatnonlocal Cauchy problem (8) has a mild solution on [0, 1].References[1] S.Aizicovici, M.Mckibben, Existence results for a class of abstract nonlocal Cauchyproblems, Nonlinear Anal. 39(2000), 649-668.[2] K.Balachandran, J.Y. Park, M.Chandrasekaran, Nonlocal Cauchy problemsfor delay integrodifferential equations of sobolev type in Banach spaces, Appl. Math.Lett. 15(2002), 845-854.[3] N.Benchohra, E.Gatsori, J. Henderson, S.K. Ntouyas, Nondensely definedevolution impulsive differential inclusions with nonlocal conditions, J. Math. Anal.Appl. 286(2003), 307-325.[4] L. Byszewski., V. Lakshmikantham, Theorem about existence and uniqueness ofa solutions of a nonlocal Cauchy problem in a Banach space, Appl. Anal. 40(1990),11-19.[5] L.Byszewski, Theorems about the existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162(1991), 494-505.Nonlinear functional integrodifferential evolution equations 45[6] L.Byszewski, H.Akca, Existence of solutions of a semilinear functional-differentialevolution nonlocal problem, Nonlinear Anal. 34(1998), 65-72.[7] J.Dugundji, A.Granas, Fixed point theory, Monografie Mat. PWN, Warsaw, 1982.[8] K.Ezzinbi, J. H. Liu, Nondensely defined evolution equations with nonlocal condi-tions, Math. Comput. Model. 36(2002), 1027-1038.[9] K.Ezzinbi, X. Fu, Existence and regularity of solutions for some neutral partial dif-ferential equations with nonlocal conditions, Nonlinear Anal. 57(2004), 1029-1041.[10] W.E.Fitzgibbon, Semilinear functional differential equations in Banach spaces, J.Differential Equations, 29(1978), 1-14.[11] A.Freidman, Partial Differential Equations, Holt, Rinehat and Winston, New York,1969.[12] X.Fu, K.Ezzinibi, Existence of solutions for neutral functional evolution equationswith nonlocal conditions, Nonlinear Anal. Theory Methods Appl. 54(2003), 215-227.[13] J. Liang, J.H. Liu, T. J. Xiao, Nonlocal Cauchy problems governed by compact op-erator families, Nonlinear Anal. 57(2004), 183-189.[14] Y.Lin, J. H. Liu, Semilinear integrodifferential equations with nonlocal Cauchy prob-lem, Nonlinear Anal. 26(1996), 1023-1033.[15] S.K. Ntouyas, P.Ch.Tsamatos, Global existence for semilinear evolution equationswith nonlocal conditions, J. Math. Anal. Appl. 210(1997), 679-687.[16] A.Pazy, Semigroup of linear operators and applications to partial differential equa-tions, Springer-Verlag, New York, 1983.

Nonlinear functional integrodifferential evolution equations with nonlocal conditions in Banach spaces

Abstract

Similar works

Full text

Available Versions

CiteSeerX

Hrčak - Portal of scientific journals of Croatia

HRČAK - Portal of Croatian Scientific and Professional Journals