IMPULSIVE NEUTRAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH STATE DEPENDENT DELAYS AND INTEGRAL CONDITION

Transcription

1 Electronic Journal of Differential Equations, Vol. 213 (213), No. 273, pp ISSN: URL: or ftp ejde.math.txstate.edu IMPULSIVE NEUTRAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH STATE DEPENDENT DELAYS AND INTEGRAL CONDITION JAYDEV DABAS, GANGA RAM GAUTAM Abstract. In this article, we establish the existence of a solution for an impulsive neutral fractional integro-differential state dependent delay equation subject to an integral boundary condition. The existence results are proved by applying the classical fixed point theorems. An example is presented to demonstrate the application of the results established. 1. Introduction Let X be a Banach space and P C t := P C([ d, t]; X), d >, t T <, be a Banach space of all such functions φ : [ d, t] X, which are continuous everywhere except for a finite number of points t i, i = 1, 2,..., m, at which φ(t i ) and φ(t i ) exists and φ(t i) = φ(t i ), endowed with the norm φ t = sup φ(s) X, φ P C t, dst where X is the norm in X. In this article we study an impulsive neutral fractional integro-differential equation of the form [ Dt α ] (1.1) = f(t, x ρ(t,xt), B(x)(t)), t J = [, T ], T <, t t k, x(t k ) = I k (x(t k )), x (t k ) = Q k (x(t k )), k = 1, 2,..., m, (1.2) x(t) = φ(t), t [ d, ], (1.3) ax () bx (T ) = q(x(s))ds, a b, b, (1.4) where x denotes the derivative of x with respect to t and D α t, α (1, 2) is Caputo s derivative. The functions f : J P C X X, g : J P C X, and q : X X are given continuous functions where P C = P C([ d, ], X) and for any x P C T = P C([ d, T ], X), t J, we denote by x t the element of P C defined by 2 Mathematics Subject Classification. 26A33, 34K5, 34A12, 34A37, 26A33. Key words and phrases. Fractional order differential equation; nonlocal condition; contraction; impulsive condition. c 213 Texas State University - San Marcos. Submitted May 3, 213. Published December 17,

2 2 J. DABAS, G. R. GAUTAM EJDE-213/273 x t (θ) = x(t θ), θ [ d, ]. In the impulsive conditions for = t < t 1 < < t m < t m1 = T, Q k, I k C(X, X), (k = 1, 2,..., m), are continuous and bounded functions. We have x(t k ) = x(t k ) x(t k ) and x (t k ) = x (t k ) x (t k ). The term Bx(t) is given by Bx(t) = K(t, s)x(s)ds, (1.5) where K C(D, R ), is the set of all positive functions which are continuous on D = {(t, s) R 2 : s t < T and B t = sup t [,t] K(t, s)ds <. The study of fractional differential equations has been gaining importance in recent years due to the fact that fractional order derivatives provide a tool for the description of memory and hereditary properties of various phenomena. Due to this fact, the fractional order models are capable to describe more realistic situation than the integer order models. Fractional differential equations have been used in many field like fractals, chaos, electrical engineering, medical science, etc. In recent years, we have seen considerable development on the topics of fractional differential equations, for instance, we refer to the articles [8, 1, 26, 27]. The theory of impulsive differential equations of integer order is well developed and has applications in mathematical modelling, especially in dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, disease, and so forth. For general theory and applications of fractional order differential equations with impulsive conditions, we refer the reader to the references [1, 7, 11, 17, 21, 22, 28, 29, 3]. Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, etc. For a detailed description of the integral boundary conditions, we refer the reader to some recent papers [4, 5, 6, 13, 16, 17] and the references therein. On the other hand, we know that the delay arises naturally in systems due to the transmission of signal or the mechanical transmission. Moreover, the study of fractional order problems involving various types of delay (finite, infinite and state dependant) considered in Banach spaces has been receiving attention, see [2, 3, 8, 12, 14, 15, 18, 19, 2, 23, 24, 25] and references cited in these articles. In [17] authors have established the existence and uniqueness of a solution for the following system D α t x(t) = f(t, x t, Bx(t)), t J = [, T ], t t k, x(t k ) = Q k (x(t k )), k = 1, 2,..., m, x (t k ) = I k (x(t k )), k = 1, 2,..., m, x(t) = φ(t), t (, ], ax () bx (T ) = q(x(s))ds, (1.6) the results are proved by using the contraction and Krasnoselkii s fixed point theorems. This paper is motivated from some recent papers treating the boundary value problems for impulsive fractional differential equations [4, 5, 13, 17, 3]. To the best of our knowledge, there is no work available in literature on impulsive neutral fractional integro-differential equation with state dependent delay and with an integral boundary condition. In this article, we first establish a general

3 EJDE-213/273 FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS 3 framework to find a solution to system (1.1) (1.4) and then by using classical fixed point theorems we proved the existence and uniqueness results. 2. Preliminaries In this section, we shall introduce notations, definitions, preliminary results which are required to establish our main results. We continue to use the function spaces introduced in the earlier section. For the following definitions we refer to the reader to see the monograph of Podlunby [27]. Definition 2.1. Caputo s derivative of order α for a function f : [, ) R is defined as D α t f(t) = 1 Γ(n α) for n 1 α < n, n N. If α < 1, then D α t f(t) = (t s) n α 1 f (n) (s)ds = J n α f (n) (t), (2.1) 1 Γ(1 α) (t s) α f (1) (s)ds. (2.2) Definition 2.2. The Riemann-Liouville fractional integral operator for order α >, of a function f : R R and f L 1 (R, X) is defined by J t f(t) = f(t), J α t f(t) = 1 where Γ( ) is the Euler gamma function. f(s)ds, α >, t >, (2.3) Lemma 2.3 ( [3]). For α >, the general solution of fractional differential equations D α t x(t) = is given by x(t) = c c 1 t c 2 t 2 c 3 t 3 c n 1 t n 1 where c i R, i =, 1,..., n 1, n = [α] 1 and [α] represent the integral part of the real number α. Lemma 2.4 ([21, Lemma 2.6]). Let α (1, 2), c R and h : J R be continuous function. A function x C(J, R) is a solution of the following fractional integral equation x(t) = w h(s)ds (w s) α 1 h(s)ds x c(t w), (2.4) if and only if x is a solution of the following fractional Cauchy problem D α t x(t) = h(t), t J, x(w) = x, w. (2.5) As a consequence of Lemma 2.3 and Lemma 2.4 we have the following result. Lemma 2.5. Let α (1, 2) and f : J P C X X be continuously differentiable function. A piecewise continuous differential function x(t) : ( d, T ] X is a

4 4 J. DABAS, G. R. GAUTAM EJDE-213/273 solution of system (1.1) (1.4) if and only if x satisfied the integral equation φ(t), t [ d, ], φ() { t (t s) α 1 bt 1 T ab b q(x(s))ds x(t) = k Q i(x(t i )) g(s, x ρ(s,x s))ds f(s, x ρ(s,x s), B(x)(s))ds (t s) α 1 f(s, x ρ(s,xs), B(x)(s))ds, t [, t 1 ],... φ() k I i(x(t i )) k (t { t i)q i (x(t i )) bt q(x(s))ds (t s) α 1 k Q i(x(t i )) Proof. If t [, t 1 ], then D α t [ ab 1 b g(s, x ρ(s,x s))ds f(s, x ρ(s,x s), B(x)(s))ds (t s) α 1 f(s, x ρ(s,xs), B(x)(s))ds, t (t k, t k1 ]. ] = f(t, x ρ(t,xt), B(x)(t)), x(t) = φ(t), t [ d, ]. (2.6) (2.7) Taking the Riemann-Liouville fractional integral of (2.7) and using the Lemma 2.4, we have (2.8) = a b t f(s, x ρ(s,xs), B(x)(s))ds, using the initial condition, we get a = φ(), then (2.8) becomes = φ() b t Similarly, if t (t 1, t 2 ], then D α t [ (2.9) ] = f(t, x ρ(t,xt), B(x)(t)) (2.1) x(t 1 ) = x(t 1 ) I 1(x(t 1 )), (2.11) x (t 1 ) = x (t 1 ) Q 1(x(t 1 )). (2.12) Again apply the Riemann-Liouville fractional integral operator on (2.1) and using the lemma 2.4, we obtain = a 1 b 1 t f(s, x ρ(s,xs), B(x)(s))ds, (2.13)

5 EJDE-213/273 FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS 5 rewrite (2.13) as 1 x(t 1 ) (t 1 s) α 1 = a 1 b 1 t 1 1 (t 1 s) α 1 f(s, x ρ(s,xs), B(x)(s))ds, (2.14) due to impulsive condition (2.11) and the fact that x(t 1 ) = x(t 1 ), we may write (2.14) as 1 x(t 1 ) I 1 (x(t 1 )) (t 1 s) α 1 1 (2.15) (t 1 s) α 1 = a 1 b 1 t 1 Now from (2.9), we have x(t 1 ) 1 = φ() b t 1 (t 1 s) α 1 1 (t 1 s) α 1 (2.16) From (2.15) and (2.16), we get a 1 = φ() b t 1 b 1 t 1 I 1 (x(t 1 )), hence (2.14) can be written as = φ() b t 1 b 1 (t t 1 ) I 1 (x(t 1 )) (2.17) On differentiating (2.13) with respect to t at t = t 1, and incorporate second impulsive condition (2.12), we obtain x (t 1 ) Q 1(x(t 1 )) 1 1 = b 1 (t s) α 2 Γ(α 1) g(s, x ρ(s,x s))ds (t 1 s) α 2 Γ(α 1) f(s, x ρ(s,x s), B(x)(s))ds, Now differentiating (2.9), with respect to t at t = t 1, we get x (t 1 ) = b 1 1 (t 1 s) α 2 Γ(α 1) g(s, x ρ(s,x s))ds (t 1 s) α 2 Γ(α 1) f(s, x ρ(s,x s), B(x)(s))ds. From (2.18) and (2.19), we obtain b 1 = b Q 1 (x(t 1 )). Thus, (2.17) become = φ() b t I 1 (x(t 1 )) (t t 1)Q 1 (x(t 1 )) (2.18) (2.19) (2.2)

6 6 J. DABAS, G. R. GAUTAM EJDE-213/273 Similarly, for t (t 2, t 3 ], we can write the solution of the problem as = φ() b t I 1 (x(t 1 )) I 2(x(t 2 )) (t t 1)Q 1 (x(t 1 )) (t t 2)Q 2 (x(t 2 )) In general, if t (t k, t k1 ], then we have the result = φ() b t I i (x(t i )) (t t i )Q i (x(t i )) (2.21) Finally, we use the integral boundary condition ax () bx (T ) = q(x(s))ds, where x () calculated from (2.9) and x (T ) from (2.2). On simplifying, we get the following value of the constant b, b = b 1 a b{ b q(x(s))ds Q i (x(t i )) Γ(α 1) f(s, x ρ(s,x s), B(x)(s))ds. Γ(α 1) g(s, x ρ(s,x s))ds On summarizing, we obtain the desired integral equation (2.6). Conversely, assuming that x satisfies (2.6), by a direct computation, it follows that the solution given in (2.6) satisfies system (1.1) (1.4). This completes the proof of the lemma. 3. Existence result The function ρ : J P C [ d, T ] is continuous and φ() P C. Let the function t ϕ t be well defined and continuous from the set R(ρ ) = {ρ(s, ψ) : (s, ψ) [, T ] P C into P C. Further, we introduce the following assumptions to establish our results. (H1) There exist positive constants L f1, L f2, L q and L g, such that f(t, ψ, x) f(t, χ, y) X L f1 ψ χ P C L f2 x y X, g(t, ψ) g(t, χ) X L g ψ χ P C, t J, ψ, χ P C, x, y X, q(x) q(y) X L q x y X, x, y X. (H2) There exist positive constants L Q, L I, L q, such that Q k (x) Q k (y) X L Q x y X, I k (x) I k (y) X L I x y X. (H3) The functions Q k, I k, q are bounded continuous and there exist positive constants C 1, C 2, C 3, such that Q k (x) X C 1, I k (x) X C 2, q(x) X C 3, x X. Our first result is based on the Banach contraction theorem.

7 EJDE-213/273 FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS 7 Theorem 3.1. Let the assumptions (H1) (H2) are satisfied with { = m(l I T L Q ) L g Γ(α 1) bt ( T Lq ml Q a b b 1 L g 1 (L f1 L f2 B ) ) (L f1 L f2 B ) < 1. Γ(α 1) Then (1.1) (1.4) has a unique solution. Proof. We transform problem (1.1) (1.4) into a fixed point problem. Consider the operator P : P C T P C T defined by φ(t), t [ d, ], φ() { t (t s) α 1 bt 1 T ab b q(x(s))ds P x(t) = m Q i(x(t i )) g(s, x ρ(s,x s))ds f(s, x ρ(s,x s), B(x)(s))ds (t s) α 1 f(s, x ρ(s,xs), B(x)(s))ds, t [, t 1 ]... φ() k I i(x(t i )) k (t { t i)q i (x(t i )) bt 1 T ab b q(x(s))ds m Q i(x(t i )) g(s, x ρ(s,x s))ds T f(s, x ρ(s,x s), B(x)(s))ds (t s) α 1 (t s) α 1 f(s, x ρ(s,xs), B(x)(s))ds, t (t k, t k1 ]. Let x, x P C T and t [, t 1 ]. Then P (x) P (x ) X g(s, x ρ(s,xs)) g(s, x ρ(s,x )) Xds s bt 1 T q(x(s)) q(x a b{ (s)) X ds Q i (x(t i b )) Q i(x (t i )) X Γ(α 1) g(s, x ρ(s,x s)) g(s, x ρ(s,x )) Xds s { Γ(α 1) f(s, x ρ(s,x s), B(x)(s)) f(s, x ρ(s,x s ), B(x )(s)) X ds f(s, x ρ(s,xs), B(x)(s)) f(s, x ρ(s,x ), B(x )(s)) s X ds bt ( T a b b L q ml Q 1 L g ) Γ(α 1) L g 1 (L f1 L f2 B ) In a similar way for t (t k, t k1 ], we have P (x) P (x ) X Γ(α 1) (L f1 L f2 B ) x x P CT.

8 8 J. DABAS, G. R. GAUTAM EJDE-213/273 I i (x(t i )) I i(x (t i )) X (t t i ) Q i (x(t i )) Q i(x (t i )) X g(s, x ρ(s,xs)) g(s, x ρ(s,x )) Xds s bt 1 T q(x(s)) q(x a b{ (s)) X ds Q i (x(t i b )) Q i(x (t i )) X Γ(α 1) g(s, x ρ(s,x s)) g(s, x ρ(s,x s )) Xds Γ(α 1) f(s, x ρ(s,x s), B(x)(s)) f(s, x ρ(s,x s ), B(x )(s)) X ds f(s, x ρ(s,xs), B(x)(s)) f(s, x ρ(s,x ), B(x )(s)) s X ds bt ( T a b b L q ml Q 1 L g { ml I mt L Q Γ(α 1) L g 1 ) (L f1 L f2 B ) x x P CT. Γ(α 1) (L f1 L f2 B ) x x P CT Since < 1, implies that the map P is a contraction map and therefore has a unique fixed point x P C T, hence system (1.1) (1.4) has a unique solution on the interval [ d, T ]. This completes the proof of the theorem. Our second result is based on Krasnoselkii s fixed point theorem. Theorem 3.2. Let B be a closed convex and nonempty subset of a Banach space X. Let P and Q be two operators such that (i) P x Qy B, whenever x, y B. (ii) P is compact and continuous. (iii) Q is a contraction mapping. Then there exists z B such that z = P zqz. Theorem 3.3. Let the function f, g be continuous for every t [, T ], and satisfy the assumptions (H1) (H3) with { = Γ(α 1) L g bt ( T a b b L q 1 L g 1 ) (L f1 L f2 B ) Γ(α 1) (L f1 L f2 B ) < 1. Then system (1.1) (1.4) has at least one solution on [ d, T ]. Proof. Choose r [ φ() ml I r mt L Q r Γ(α 1) L gr bt a b (T b L qr ml Q r 1 L gr 1 (L f1r L f2 B r)) Γ(α 1) (L f1r L f2 B r) Define P CT r = {x P C T : x P CT r, then P CT r is a bounded, closed convex subset in P C T. Consider the operators N : P CT r P Cr T and P : P Cr T P Cr T for ].

9 EJDE-213/273 FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS 9 t J k = (t k, t k1 ], defined by N(x) = φ() I i (x(t i )) (t t i )Q i (x(t i )) bt a b P (x) = bt 1 (T s) q(x(s))ds a b{ α 2 b Γ(α 1) g(s, x ρ(s,x s))ds Γ(α 1) f(s, x ρ(s,x s), B(x)(s))ds We complete the proof in the following steps: Step 1. Let x, x P CT r then, N(x) P (x ) X φ() X I i (x(t i )) X bt a b Q i (x(t i )) (3.1) (3.2) (t t i ) Q i (x(t i )) X Q i (x(t i )) X bt 1 a b{ b Γ(α 1) g(s, x ρ(s,x s )) Xds Γ(α 1) f(s, x ρ(s,x s ), B(x )(s)) X ds g(s, x ρ(s,x )) Xds s f(s, x ρ(s,x ), B(x )(s)) s X ds [ φ() mc 2 mt C 1 Γ(α 1) L gr mc 1 1 L gr 1 (L f1r L f2 B r)) ] Γ(α 1) (L f1r L f2 B r) r. q(x (s)) X ds bt a b (T b C 3 Which shows that P CT r is closed with respect to both the maps. Step 2. N is continuous. Let x n x be sequence in P CT r, then for each t J k N(x n ) N(x) X I i (x n (t i )) I i(x(t i )) X bt a b (t t i ) Q i (x n (t i )) Q i(x(t i )) X Q i (x n (t i )) Q i(x(t i )) X. Since the functions Q k and I k, k = 1,..., m, are continuous, hence N(x n ) N(x), as n. Which implies that the mapping N is continuous on P CT r.

10 1 J. DABAS, G. R. GAUTAM EJDE-213/273 Step 3. The fact that the mapping N is uniformly bounded is a consequence of the following inequality. For each t J k, k =, 1,..., m and for each x P C r T, we have N(x) X φ() X bt a b I i (x(t i )) X Q i (x(t i )) X φ() mc 2 mt C 1 (t t i ) Q i (x(t i )) X bt a b mc 1. Step 4. Now, to show that N is equi-continuous, let l 1, l 2 J k, t k l 1 < l 2 t k1, k = 1,..., m, x P CT r, we have N(x)(l 2 ) N(x)(l 1 ) X (l 2 l 1 ) Q i (x(t i )) X b(l 2 l 1 ) a b Q i (x(t i )) X. As l 2 l 1, then N(x)(l 2 ) N(x)(l 1 ) implies that N is an equi-continuous map. Combining the Steps 2 to 4, together with the Arzela Ascoli s theorem, we conclude that the operator N is compact. Step 5. Now, we show that P is a contraction mapping. Let x, x P CT r and t J k, k = 1,..., m, we have P (x) P (x ) X g(s, x ρ(s,xs)) g(s, x ρ(s,x )) Xds s bt 1 a b{ b. { q(x(s)) q(x (s)) X ds Γ(α 1) g(s, x ρ(s,x s)) g(s, x ρ(s,x s )) Xds Γ(α 1) f(s, x ρ(s,x s), B(x)(s)) f(s, x ρ(s,x s ), B(x )(s)) X ds f(s, x ρ(s,xs), B(x)(s)) f(s, x ρ(s,x ), B(x )(s)) s X ds bt ( T a b b L q 1 L g ) Γ(α 1) L g 1 (L f1 L f2 B ) x x P C r T Γ(α 1) (L f1 L f2 B ) x x P C r T As < 1, it implies that P is a contraction map. Thus all the assumptions of the Krasnoselkii s theorem are satisfied. Hence we have that the set P CT r has a fixed point which is the solution of system (1.1) (1.4) on ( d, T ]. This completes the proof of the theorem.

11 EJDE-213/273 FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS Example Consider the following example to demonstrate the application of the results established. D α t [ x(t i ) = 1 47 x(t σ(x)ds] = et x(t σ(x(t))) 25 x 2 (t σ(x(t))) i d t [, T ], t t i, γ i (t i s)x(s) ds, x (t i ) = 25 x(t) = φ(t), t ( d, ], x () x (T ) = i d cos(t s) xes 4 x ds, γ i (t i s)x(s) ds, 9 sin( 1 4 x(s))ds, where γ i C([, ), X), σ C(X, [, )), < t 1 < t 2 < < t n < T. Set γ >, and choose P C γ as P C γ = {φ P C((, ], X) : lim t d eγt φ(t) exist with the norm φ γ = sup t (, ] e γt φ(t), φ P C γ. We set B(x)(t) = ρ(t, ϕ) = t σ(ϕ()), (t, ϕ) J P C γ, f(t, ϕ) = et (ϕ) 25 (ϕ) 2, (t, ϕ) J P Cγ, g(t, ϕ) = ϕ 47 ds, ϕ P Cγ, xe s cos(t s) (4 x) ds, (t, x) I P Cγ, Q k (x(t k )) = I k (x(t k )) = i d i d γ i (t i s)x(s) ds, 25 γ i (t i s)x(s) ds. 9 We can see that all the assumptions of Theorem 3.1 are satisfied with Further, we observe that { ml I mt L Q t ϕ χ f(t, ϕ) f(t, χ) e 25 t J, ϕ, χ P C γ, t x y B(x) B(y) e 4 t J, x, y P C γ, g(t, ϕ) g(t, χ) 1 47 ϕ χ, t J, ϕ, χ P Cγ, Q k (x(t k )) Q k (y(t k )) γ 1 x y, x, y X, 25 I k (x(t k )) I k (y(t k )) γ 1 x y, x, y X, 9 q(x) q(y) 1 x y, x, y X. 4 Γ(α 1) L g bt ( T a b b L q ml Q 1 L g

12 12 J. DABAS, G. R. GAUTAM EJDE-213/273 1 ) (L f1 L f2 B ) Γ(α 1) (L f1 L f2 B ).513γ.534 < 1. We fix γ = d γ i(t i s)ds <, < t 1 < t 2 < t 3 < 1, α = 3/2, T = 1. This implies that there exists a unique solution of the considered problem in this section. Acknowledgements. The research work of J. Dabas has been partially supported by the Sponsored Research & Industrial Consultancy, Indian Institute of Technology Roorkee, project No.IITR/SRIC/247/F.I.G(Scheme-A). References [1] S. Abbas, M. Benchohra; Impulsive partial hyperbolic Functional Differential Equations of fractional order with State-Dependent delay, Fractional calculus and Applied analysis, 13 (3): ISSN , (21). [2] S. Abbas, M. Benchohra, A. Cabada; Partial neutral functional integro-differential,equations of fractional order with delay, Boundary Value Problems, (212). [3] R. P. Agarwal, B. d. Andrade; On fractional integro-differential equations with statedependent delay, Computers and Mathematics with Applications, 62, , (211). [4] B. Ahmad, A. Alsaedi, B. Alghamdi; Analytic approximation of solutions of the forced duffing equation with integral boundary conditions, Nonlinear Anal, RWA 9, (28). [5] B. Ahmad, J. J. Nieto; Existence results for nonlinear boundary value problems of fractional integro-differential equations with integral boundary conditions, Bound. Value Probl, doi:1.1155/29/ pages. Article ID (29). [6] B. Ahmad, S. Sivasundaram; Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal, Hybrid System. 4, , (21). [7] M. M. Arjunan, V. Kavitha; Existence results for impulsive neutral functional differential equations with state-dependent delay, Electronic Journal of Qualitative Theory of Differential Equations, No. 26, 1 13, (29). [8] A. Babakhani, E. Enteghami; Existence of positive solutions for multiterm fractional differential equations of finite delay with polynomial coefficients, Abstract and Applied Analysis, Article ID 76892, 12 pages, (29). [9] M. Belmekki, M. Benchohra; Existence results for fractional order semilinear functional differential equations with nondense domain, Nonlinear Analysis,72, , (21). [1] M. Belmekki, K. Mekhalfi, S. K. Ntouyas; Semilinear functional differential equations with fractional order and finite delay, Malaya Journal of Matematika, 1(1) pp.73 81, (212). [11] M. Benchohra, F. Berhoun; Impulsive fractional differential equations with state dependent delay, Communications of Applied Analysis, no.2, 14, , (21). [12] M. Benchohra, S. Litimein, G. N Guerekata; On fractional integro-differential inclusions with state-dependent delay in Banach spaces, Applicable Analysis, 1 16, (211). [13] A. Boucherif; Second-order boundary value problems with integral boundary conditions, Nonlinear Anal, TMA 7, , (29). [14] J. P. Carvalho d. Santos, C. Cuevas, B. d. Andrade; Existence results for a fractional equation with state-dependent delay, Advances in Difference Equations, Article ID 64213, (211). [15] J. P. Carvalho d. Santos, M. M. Arjunan; Existence results for fractional neutral integrodifferential equations with state-dependent delay, Computers and Mathematics with Applications, 62, , (211). [16] A. Chauhan, J. Dabas; Existence of mild solutions for impulsive fractional order semilinear evoluation equations with nonlocal conditions, Electronic Journal of Differential Equations. 211 no.17 pp.1 1, (211). [17] A. Chauhan, J. Dabas, M. Kumar; Integral boundary-value problem for impulsive fractional functional integro-differential equation with infinite delay, Electronic Journal of Differential Equations. 212 no.229 pp. 1 13, (212). [18] M. A. Darwish, S. K. Ntouyas; Semilinear functional differential equations of fractional order with state-dependent delay, Electronic Journal of Differential Equations, Vol. 29, ISSN: , (29).

13 EJDE-213/273 FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS 13 [19] M. A. Darwish, S. K. Ntouyas; Functional differential equations of fractional order with state-dependent delay, Dynamic Systems and Applications, 18, , (29). [2] F. Li; Existence of the mild solutions for delay fractional integro-differential equations with almost sectorial operators, Abstract and Applied Analysis, Article ID , 22 pages, (212). [21] M. Feckan, Y. Zhou, J.Wang, On the concept and existence of solution for impulsive fractional differential equations, Comm. Nonl. Sci. Num. Sum. 17, 35 36, (212). [22] T. L. Guo, W. Jiang; Impulsive fractional functional differential equations, Computers and Mathematics with Applications, 64, , (212). [23] J. K. Hale, J. Kato; Phase space for retarded equations with infinite delay, Funkcialaj Ekvacioj, 21, 11 41, (1978) [24] V. Kavitha, P. Z. Wang, R. Murugesu; Existence results for neutral functional fractional differential equations with state dependent-delay, Malaya Journal of Matematik, 561, 1(1), 5 51, (212). [25] L. Kexue, J. Junxiong; Existence and uniqueness of mild solutions for abstract delay fractional differential equations, Computers and Mathematics with Applications, 62, , (211). [26] V. Lakshmikantham; Theory of fractional functional differential equations, Nonlinear Analysis, 69, , (28). [27] I. Podlubny; Fractional differential equations, Acadmic press, New York (1993). [28] C. Ravicharndran, M. M. Arjunan; Existence result for impulsive fractional semilinear functional integro-differential equations in Banach spaces, Journal of Fractional Calculus and Applications, ISSN: , No. 8, Vol. 3, pp. 1 11, July (212). [29] X. Shu, F. Xu; The existence of solutions for impulsive fractional partial neutral differential equations, Preprint submitted to Journal of Mathematics, (212). [3] J. Wang, Y. Zhou, M. Feckan; On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comp. Math. Appl. doi:1.116/ j.camwa, 64, 38 32, (212). Jaydev Dabas Department of Applied Science and Engineering, IIT Roorkee, Saharanpur Campus, Saharanpur-2471, India address: jay.dabas@gmail.com Ganga Ram Gautam Department of Applied Science and Engineering, IIT Roorkee, Saharanpur Campus, Saharanpur-2471, India address: gangaiitr11@gmail.com