Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 27-41 Existence Results for Semilinear Fractional Functional Differential Equations with State-Dependent Delay∗ Yong-Kui Chang1, M.Mallika Arjunan2, G. M. N’Guér ékata3 and V. Kavitha4 1Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 7300070, P.R. China 2Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore- 641 114, Tamil Nadu, India 3 Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, M.D. 21251, USA 4 Department of Mathematics, Karunya University, Karunya Nagar, Coimbatore- 641 114, Tamil Nadu, India Email: lzchangyk@163.com, arjunphd07@yahoo.co.in, Gaston.N’Guerekata@morgan.edu, kavi velubagyam@yahoo.co.in Abstract According to theories for α-resolvent family (Sα(t))t≥0 and fixed point methods, this pa- per is mainly concerned with existence of mild solutions to a semilinear fractional functional differential equation with state-dependent delay in a complex Banach spaceX. Some sufficient conditions are estab- lished without the compactness of (Sα(t))t≥0. Keywords Fractional differential equations α-resolvent family State-dependent delay Fixed point 1. Introduction In this paper, we establish the existence of mild solutions to the following fractional func- ∗This research is supported by NNSF of China (10901075), the Key Project of Chinese Ministry of Education (210226), the Scientific Research Fund of Gansu Provincial Education Department (0804-08) and “Qing Lan” Talent Engineering Funds (QL-05-16A) by Lanzhou Jiaotong University. ISSN 1078-6236 International Institute for General Systems Studies, Inc. 28 Chang:Existence Results for Semilinear Fractional Functional . . . . . . tional differential equation with state-dependent delay Dαx(t) = Ax(t) + f(t, xρ(t,xt)), t ∈ J = [0, b], (1) x0 = ϕ ∈ B, (2) where b > 0, 0 < α < 1 and A : D(A) ⊂ X → X is the infinitesimal generator of an α-resolvent family (Sα(t))t≥0 defined on a complex Banach space X. The function xs : (−∞, 0] → X, xs(θ) = x(s + θ), belongs to some phase space B that will be defined later ( see Section 2), f : J ×B → X, ρ : J ×B → (−∞, b] are appropriate functions and ϕ belongs to the phase space B with ϕ(0) = 0. The fractional derivative Dα is understood here in the Riemann-Liouville sense. The theory of functional differential equations has emerged as an important branch in non- linear analysis. It is worth mentioning that several important practical problems have lead to investigations of functional differential equations of various types ( see the books of Hale et al. [11], Wu [31], and the references 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 equation has gained great attention in the last decades, we refer to [4, 5, 9, 14, 15, 22] and the references therein. Differential equations of fractional order play a very important role in describing some real world problems. For example some problems in physics, mechanics and other fields can be described with the help of fractional differential equations, see [2, 7, 17, 24, 25] and references therein. The theory of differential equations of fractional order has recently received much attention and now constitutes a significant branch in differential equations. Lots of research papers and monographs have appeared devoted to fractional differential equations, for example see [1, 3, 6, 8, 18, 19, 20, 21, 26, 27, 28, 30, 33, 34] and the references therein. Motivated by the above mentioned works, the purpose of this paper is to investigate the existence results of mild solutions to a semilinear fractional functional differential equation with state-dependent delay described in the general abstract form (1)-(2). The main technique is based upon the α-resolvent family (Sα(t))t≥0 combined with suitable fixed point theorems. This paper is organized as follows. In Section 2, we introduce notations, definitions and some lemmas which are used in the sequel. In section 3, we prove the existence of mild solutions Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 29 for the problem (1)-(2). 2. Preliminaries From now on, we set J = [0, b]. We denote by X a complex Banach space with norm ‖ · ‖,C(J,X) the space of all X-valued continuous functions on J , endowed with the topology of uniform convergence with norm ‖x‖∞ := sup t∈J ‖x(t)‖. And L(X) the Banach space of all linear and bounded operators on X. Moreover, Br(z0,Z) denotes the closed ball with center at z0 and radius r > 0 in Z. Definition 2.1 [25] Assume that f ∈ Cm(R+,X). If α ∈ (m− 1,m), where m ∈ N, then the Riemann-Liouville fractional derivative of order α ∈ (m− 1,m) is the expression Dα t f(t) = dm dtm ∫ t 0 gm−α(t− s)f(s)ds, where for β > 0 gβ(t) =  tβ−1 Γ(β) for t > 0, 0 for t ≥ 0. Definition 2.2 [25] Let α > 0 and f : R+ → R be in L1(R+,X). Then the Riemann-Liouville integral is given by: Iαf(t) = 1 Γ(α) ∫ t 0 (t− s)α−1f(s)ds. Recall that the Laplace transform of a function f ∈ L1(R+,X) is defined by: f̂(λ) = ∫ ∞ 0 e−λtf(t)dt, Re(λ) > ω, if the integral is absolutely convergent for Re(λ) > ω. 30 Chang:Existence Results for Semilinear Fractional Functional . . . . . . Definition 2.3 [26] Let A be a closed and linear operator with domain D(A) defined on a Banach space X and α > 0. Let ρ(A) be the resolvent set of A. We call A the generator of an α-resolvent family if there exists ω ≥ 0 and a strongly continuous function Sα : R+ → L(X) satisfying Sα(0) = I such that {λα : Re(λ) > ω} ⊂ ρ(A) and (λα −A)−1x = ∫ ∞ 0 e−λtSα(t)xdt, Re(λ) > ω, x ∈ X. In this case, Sα(t) is called the α-resolvent family generated by A. For construction of solution by using α-resolvent family, we refer to [26] and the references therein. We also refer to [23, 29] for more information about resolvent or solution operator. Remark 2.1 [26] Note that if A is the generator of an α-resolvent family (Sα(t))t≥0 then the Laplace transform of Sα(t) is Ŝα(λ) = (λα −A)−1. In this paper, we will employ the axiomatic definition of the phase space B introduced by Hale and Kato in [12] and follow the terminology used in [16]. Thus, (B, ‖ · ‖B) will be a seminormed linear space of functions mapping (−∞, 0] to X, and satisfying the following axioms: (A1) If x : (−∞, b) → X with b > 0, is continuous on [0, b] and x0 ∈ B, then for every t ∈ [0, b) the following conditions hold: (i) xt ∈ B; (ii) There exists a positive constant H such that ||x(t)|| ≤ H‖xt‖B ; (iii) There exist two functions K(·),M(·) : R+ → [1,+∞) independent of x(t) with K continuous and M locally bounded such that ‖xt‖B ≤ K(t) sup{||x(s)|| : 0 ≤ s ≤ t}+M(t)‖x0‖B. Denote Kb = sup{K(t) : t ∈ [0, b]} and Mb = sup{M(t) : t ∈ [0, b]}. (A2) For the function x(·) in (A1), xt is a B-valued continuous function on [0, b]. (A3) The space B is complete. Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 31 An example of phase space B satisfying (A1) − (A3) is the following space C0 g (see [31], pp.44), where g : [−∞, 0]→ [0,∞) is a given continuous nondecreasing function such that: (i) g(0) = 1 and g(−∞) =∞. (ii) The function G(t) = sup { g(t+s) g(t) : −∞ < s ≤ −t } is locally bounded for t ≥ 0. Let C0 g = {φ : (−∞, 0]→ X;φ is continuous and lim s→−∞ |φ(s)| g(s) = 0} Then C0 g , together with the following norm: ‖φ‖C0 g = sup |φ(s)| g(s) , satisfies axioms (A1)− (A3). The next lemma is a consequence of the phase space axioms and is proved in [14]. Lemma 2.1 ([14]) Let ϕ ∈ B and I = (γ, 0] be such that ϕt ∈ B for every t ∈ I . Assume that there exists a locally bounded function Jϕ : I → [0,∞) such that ‖ϕt‖B ≤ Jϕ(t)‖ϕ‖B for every t ∈ I . If x : (−∞, b]→ R is continuous on J and x0 = ϕ, then ‖xs‖B ≤ (Mb + Jϕ(max{γ,−|s|})‖ϕ‖B +Kb‖x‖max{0,s}, for s ∈ (γ, b], where we denotedKb = sup t∈J K(t) andMb = sup t∈J M(t), ‖x‖max{0,s} = sup {‖x(θ)‖, θ ∈ [0,max{0, s}]}. To conclude the current section, we recall the following well-known results. Theorem 2.2 [10, Theorem 6.5.4]. Let D be a closed convex subset of a Banach space X and assume that 0 ∈ D. Let Γ : D → D be a completely continuous map. Then, either the set {x ∈ D : x = λΓ(x), 0 < λ < 1} is unbounded or the map Γ has a fixed point in D. Lemma 2.3 ([13, 32]) Suppose b ≥ 0, α > 0 and a(t) is a nonnegative function locally in- tegrable on 0 ≤ t < T ( for some T ≤ +∞), and suppose u(t) is nonnegative and locally integrable on 0 ≤ t < T with u(t) ≤ a(t) + b ∫ t 0 (t− s)α−1u(s)ds 32 Chang:Existence Results for Semilinear Fractional Functional . . . . . . on this interval; then u(t) ≤ a(t) + ∫ t 0 [ ∞∑ n=1 (bΓ(α))n Γ(nα) (t− s)nα−1a(s) ] ds. 3. Existence Results In this section, we present and prove the existence results for the fractional differential problem (1)-(2). First, we present its mild solution. Definition 3.1 A function x : (−∞, b] → X is called a mild solution of (1)-(2) if x0 = φ, xρ(s,xs) ∈ B for each s ∈ J and x(t) = ∫ t 0 Sα(t− s)f(s, xρ(s,xs))ds, for each t ∈ J. (3) We are now in a position to state and prove our existence result for the problem (1)-(2). For the study of this, we first list the following hypotheses: (H1) There exist M > 0 and δ > 0 such that ‖Sα(t)‖L(X) ≤Meδt, t ∈ J . (H2) The function f : J → B → X is completely continuous and there exists a continuous function µ : J → (0,+∞) such that ‖f(t, ψ)‖ ≤ µ(t)‖ψ‖B, (t, ψ) ∈ J × B. (H3) The function t → ϕt is well defined and continuous from the set R(ρ−) = {ρ(s, ψ) : (s, ψ) ∈ J × B, ρ(s, ψ) ≤ 0} into B. Moreover, there exists a continuous and bounded function Jϕ : R(ρ−)→ (0,∞) such that ‖ϕt‖B ≤ Jϕ(t)‖ϕ‖B for every t ∈ R(ρ−). Remark 3.1 For more details on the hypothesis (H3), we refer to [14]. Theorem 3.1 Assume that the hypotheses (H1)-(H3) hold, then the problem (1)-(2) has at least one mild solution on (−∞, b]. Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 33 Proof. Let Y = {u ∈ C(J,X) : u(0) = ϕ(0) = 0} endowed with the uniform convergence topology and N : Y → Y be the operator defined by Nx(t) = ∫ t 0 Sα(t− s)f(s, xρ(s,xs))ds, for each t ∈ J. (4) where x : (−∞, b] → X is such that x0 = ϕ and x = x on J . From axiom (A1) and our assumption on ϕ, we infer that Nx(·) is well defined and continuous. Let ϕ : (−∞, b] → X be the extension of ϕ to (−∞, b] such that ϕ(0) = ϕ(0) = 0 on J and Jϕ = sup{Jϕ : s ∈ R(ρ−)}. We will prove that N(·) is completely continuous from Br(0, Y ) to Br(0, Y ). Step 1: N is continuous on Br(0, Y ). Let (xn)n∈N be a sequence in Br(0, Y ) and x ∈ Br(0, Y ) such that xn → x in Y . From the axiom (A1), it is easy to see that (xn)s → xs uniformly for s ∈ (−∞, b] as n → ∞. By (H2) we have ‖f(s, xn ρ(s,(xn)s) )− f(s, x ρ(s,(x)s) )‖ ≤ ‖f(s, xn ρ(s,(xn)s) )− f(s, x ρ(s,(xn)s) )‖+ ‖f(s, x ρ(s,(xn)s) )− f(s, x ρ(s,(x)s) )‖, which implies that f(s, xn ρ(s,(xn)s) ) → f(s, x ρ(s,(x)s) ) as n → ∞ for each x ∈ J . By axiom (A1), Lemma 2.1 and dominated convergence theorem, we obtain ‖N(xn)−N(x)‖ = sup t∈J ∥∥∥∥∫ t 0 Sα(t− s) [ f(s, xn ρ(s,(xn)s) )− f(s, x ρ(s,(x)s) ) ] ds ∥∥∥∥ → 0 as n→∞. Thus, N(·) is continuous. Step 2: N maps bounded sets into bounded sets. Let µ∗ = sup 0≤τ≤b µ(τ). If x ∈ Br(0, Y ), from Lemma 2.1, follows that ‖xρ(t,xt)‖B ≤ r ∗ = (Mb + J ϕ )‖ϕ‖B +Kbr. (5) and so ‖N(x)(t)‖ = ∥∥∥∥∫ t 0 Sα(t− s)f(s, xρ(s,xs))ds ∥∥∥∥ 34 Chang:Existence Results for Semilinear Fractional Functional . . . . . . ≤ ∫ t 0 ‖Sα(t− s)‖L(X)‖f(s, xρ(s,xs))‖ds ≤M ∫ t 0 eδ(t−s)µ(s)‖xρ(s,xs)‖Bds ≤Mµ∗r∗ ∫ t 0 eδ(t−s)ds ≤Mµ∗r∗ eδb δ . This implies that ‖N(x)‖∞ ≤Mµ∗r∗ eδb δ = l. Step 3: N maps bounded sets into equicontinuous sets. Let t1, t2 ∈ J with t1 > t2 and x ∈ Br(0, Y ). Then ‖N(x)(t1)−N(x)(t2)‖ = ∥∥∥∥∫ t1 t2 Sα(t1 − s)f(s, xρ(s,xs))ds + ∫ t2 0 [ Sα(t1 − s)− Sα(t2 − s) ] f(s, xρ(s,xs))ds ∥∥∥∥ ≤ I1 + I2, where I1 = ∫ t1 t2 ‖Sα(t1 − s)f(s, xρ(s,xs))‖ds, I2 = ∫ t2 0 ∥∥∥[Sα(t1 − s)− Sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ ds. Here I1 and I2 tend to 0 as t1 → t2 independently of x ∈ Br(0, Y ). In fact, I1 = ∫ t1 t2 ‖Sα(t1 − s)f(s, xρ(s,xs))‖ds ≤ ∫ t1 t2 ‖Sα(t1 − s)‖L(X)‖f(s, xρ(s,xs))‖ds ≤M ∫ t1 t2 eδ(t1−s)µ(s)‖xρ(s,xs)‖Bds ≤Mµ∗r∗ ∫ t1 t2 eδ(t1−s)ds = Mµ∗r∗ [eδ(t1−t2) − 1 δ ] . Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 35 Hence lim t1→t2 I1 = 0. And I2 = ∫ t2 0 ∥∥∥[Sα(t1 − s)− Sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ ds → 0, since f is compact and Sα is strongly continuous, ∥∥∥[Sα(t1 − s)− Sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ → 0 as t1 → t2 uniformly for x ∈ Br(0, Y ). We conclude that lim t1→t2 I2 = 0. Step 4: The operator N maps Br(0, Y ) into a relatively compact set in X. Indeed from the strong continuity of Sα(·) and (H2), the set { Sα(t− s)f(s, xρ(s,xs)), t, s ∈ [0, b], x ∈ Br(0, Y )} is relatively compact in X. Moreover, for x ∈ Br(0, Y ), using the mean value theorem for the Bochner integral, we obtain Nx(t) ∈ tconv{Sα(t− s)f(s, xρ(s,xs)) : s ∈ [0, b], x ∈ Br(0, Y )}, for all t ∈ [0, b]. Consequently the set {Nx(t) : x ∈ Br(ϕ|J , Y )} is relatively compact in X, for every t ∈ [0, b]. Step 5: A priori bounds. Set Λ = {x ∈ Y such that x = λN(x) for some 0 < λ < 1}. Let x ∈ Λ. Then for each t ∈ [0, b], we have ‖x(t)‖ ≤ λ ∫ t 0 ‖Sα(t− s)‖L(X)‖f(s, xρ(s,xs))‖ds ≤M ∫ t 0 eδ(t−s)µ(s)‖xρ(s,xs)‖Bds ≤Mµ∗ ∫ t 0 eδ(t−s) [ (Mb + J ϕ )‖ϕ‖B +Kb‖x‖max{0,s} ] ds ≤Mµ∗ ∫ t 0 eδ(t−s) [ (Mb + J ϕ )‖ϕ‖B +Kb‖x‖s ] ds ≤Mµ∗(Mb + J ϕ )‖ϕ‖B eδb δ +Mµ∗Kb ∫ t 0 eδ(t−s)(t− s)α−1(t− s)1−α‖x‖sds ≤ θ1 + θ2 ∫ t 0 (t− s)α−1‖x‖sds, where θ1 = Mµ∗(Mb + J ϕ )‖ϕ‖B eδb δ θ2 = Mµ∗Kbe δbb1−α. 36 Chang:Existence Results for Semilinear Fractional Functional . . . . . . In view of Lemma 2.3, we have for all t ∈ J , ‖x(t)‖ ≤ θ1 [ 1 + ∫ t 0 ∞∑ n=1 (θ2Γ(α))n Γ(nα) (t− s)nα−1 ] ds ≤ θ1 [ 1 + ∞∑ n=1 (θ2Γ(α))nbnα nαΓ(nα) ] = θ1 [ 1 + ∞∑ n=1 [θ2Γ(α)bα]n Γ(nα+ 1) ] ≤ θ1Λα[θ2Γ(α)bα], where Λα[θ2Γ(α)bα] = ∞∑ n=0 [θ2Γ(α)bα]n Γ(nα+ 1) is the Mittag-Leffer function. This implies that ‖x‖∞ ≤ θ1Λα[θ2Γ(α)bα]. Hence combining Step 1–Step 5 and using the Theorem 2.2, we obtain that N has a fixed point which is a mild solution of (1)-(2) on (−∞, b]. Next, we give an existence result when the nonlinearity f has a sublinear growth with its state variable. Let us list the following condition: (H2∗) The function f : J → B → X is completely continuous such that there exist a continuous function µ : J → (0,+∞) and a continuous nondecreasing function W : [0,+∞) → (0,+∞) satisfying ‖f(t, ψ)‖ ≤ µ(t)W (‖ψ‖B), (t, ψ) ∈ J × B, lim inf ξ→+∞ W (ξ) ξ = γ < +∞. Theorem 3.2 Assume that the hypotheses (H1), (H2∗) and (H3) are satisfied. Then the problem ( 1)-(2) admits at least one mild solution on (−∞, b] provided that Meδbµ∗Kb δ γ < 1,where µ∗ = sup 0≤τ≤b µ(τ). (6) Proof. Let N be the operator defined by (4). We shall complete the proof by Schauder’s fixed point theorem. Let r∗ be defined as (5). We claim that there exists a positive number r such that NBr(0, Y ) ⊆ Br(0, Y ). If it is not true, then for each r > 0, there exists xr(·) ∈ Br(0, Y ), but Nxr /∈ Br(0, Y ), that is, Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 37 ‖N(xr)(t)‖ > r for some t(r) ∈ J , where t(r) denotes t depending on r. However, on the other hand, we have from (H1), (H2∗) that r < ‖N(xr)(t)‖ = ∥∥∥∥∫ t 0 Sα (t− s) f ( s, xρ(s,xrs) ) ds ∥∥∥∥ ≤ M ∫ t 0 eδ(t−s)µ (s)W (r∗) ds ≤ M eδb δ µ∗W (r∗) . Dividing both sides by r and taking the lower limit, we get Meδbµ∗Kb δ γ ≥ 1, where contradicts (6). Hence for some positive r, NBr(0, Y ) ⊆ Br(0, Y ). Just the same as the proof in Theorem 3.1, we can show that N is continuous on Br(0, Y ) and N maps Br(0, Y ) into a relatively compact set in X. Next we prove that the family {Nx : x ∈ Br(0, Y )} is an equicontinuous family of functions. Let t1, t2 ∈ J with t1 > t2 and x ∈ Br(0, Y ). Then ‖N(x)(t1)−N(x)(t2)‖ = ∥∥∥∥∫ t1 t2 Sα(t1 − s)f(s, xρ(s,xs))ds + ∫ t2 0 [ Sα(t1 − s)− Sα(t2 − s) ] f(s, xρ(s,xs))ds ∥∥∥∥ ≤ I1 + I2. where I1 = ∫ t1 t2 ‖Sα(t1 − s)f(s, xρ(s,xs))‖ds, I2 = ∫ t2 0 ∥∥∥[Sα(t1 − s)− Sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ ds. Here I1 and I2 tend to 0 as t1 → t2 independently of x ∈ Br(0, Y ). Indeed, I1 = ∫ t1 t2 ‖Sα(t1 − s)f(s, xρ(s,xs))‖ds 38 Chang:Existence Results for Semilinear Fractional Functional . . . . . . ≤ ∫ t1 t2 ‖Sα(t1 − s)‖L(X)‖f(s, xρ(s,xs))‖ds ≤M ∫ t1 t2 eδ(t1−s)µ(s)W (‖xρ(s,xs)‖B)ds ≤Mµ∗W (r∗) ∫ t1 t2 eδ(t1−s)ds = Mµ∗W (r∗) [ eδ(t1−t2) − 1 δ ] . Hence lim t1→t2 I1 = 0. And I2 = ∫ t2 0 ∥∥∥[Sα(t1 − s)− Sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ ds → 0, since f is compact and Sα is strongly continuous, ∥∥∥[Sα(t1 − s)− Sα(t2 − s) ] f(s, xρ(s,xs)) ∥∥∥ → 0 as t1 → t2 uniformly for x ∈ Br(0, Y ). We deduce that lim t1→t2 I2 = 0. Thus, by the Arzela-Ascoli theorem N is a completely continuous operator. 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