ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.21(216) No.3, pp.151-16 Mild Solution for Nonlocal Fractional Functional Differential Equation with not Instantaneous Impulse Ganga Ram Gautam, Jaydev Dabas Department of Applied Science and Engineering, IIT Roorkee, Saharanpur Campus, Saharanpur-2471, India. (Received 16 May 214, accepted 28 February 216) Abstract: This paper is concerned with the problem of the existence results of the mild solution for nonlocal fractional differential equations with state dependent delay subject to not instantaneoumpulse. The existence results are proved with the help of fixed point theorems. One example involving partial differential equations presented to illustrate the existence result. Keywords: Fractional order differential equation; Functional differential equations; Impulsive conditions; Fixed point theorem; 1 Introduction In the last few decades fractional differential equations are more interested in developing the numerical methods and theoretical analysis, because it is recently proved that it has memory and hereditary property which is valuable in serval field of engineering and sciences. For some recent development theory and applications of fractional differential equations reader can see the monographs [1 5] and with various conditions such anitial, impulse and nonlocal, one can see the papers [6 12, 14, 21, 22, 26, 28 3] and reference therein. It has been seen that the theory of existence result for fractional differential equations not yet sufficiently elaborated, compared to that of theory of ordinary differential equations. Due to this fact, it important and necessary to study the existence result for semi-linear functional fractional differential equations. The spacial type of functional differential equations delay differential equationn which delay may state-dependent or constant with different type of conditions. See for more details of relevant update theory of state dependent delay in the cited papers [13, 15 2, 23 25, 27]. Zhou et al. [11] study the following Cauchy problem c D q t [x(t) h(t, x t )] + Ax(t) = f(t, x t ), t (, a] (1) x (ϑ) + (g(x t1,, x tp ))(ϑ) = φ(ϑ), ϑ [ r, ], (2) and prove the existence and uniqueness result of mild solutions by using Krasnoselskii s fixed point theorem. Chauhan et al. [6] find the definition of mild solution with the help of Laplace transformation and established the existence and uniqueness results of a mild solution applying the Banach and Krasnoselskii s Fixed Point Theorems for the following model equation d α dt x(t) + Ax(t) = f(t, x, x(a 1(t)),, x(a m (t))), t J = [, T ], t t k (3) x t=tk = I k (x(t k )), k = 1, 2,... m, (4) x() + g(x) = x. (5) Bahuguna [7] establish the existence, uniqueness and continuation of a mild solution of problem (3) with out impulsive condition and taking the order α = 1 and condition (5). They proved some regularity results under different conditions. Corresponding author. E-mail address: gangaiitr11@gmail.com Copyright c World Academic Press, World Academic Union IJNS.216.6.15/911
152 International Journal of Nonlinear Science, Vol.21(216), No.3, pp. 151-16 Author [8] also study the problem (3) with out impulsive condition and taking the nonlocal condition (5) and established existence results. Hernandez et al. [9] use first time not instantaneoumpulsive condition for semi-linear abstract differential equation of the form u (t) = Au(t) + f(t, u(t)), t (, t i+1 ], i =, 1,, N, (6) u(t) = g i (t, u(t)), t (t i, ], i = 1, 2,, N, u() = u, (7) and introduced the concepts of mild and classical solution. They established the existence results by using fixed point theorems. Further, Pierri et al. [1] extend the results of [9] in the study of the problem (6)-(7) using the theory of analytic semigroup and fractional power of closed operators and established the existence results of solutions. Kumar et al. [28] have studied the the following fractional order problem with not instantaneoumpulse C D β t u(t) + Au(t) = f(t, u(t), g(u(t))), t (, t i+1 ], i =, 1,, N, (8) u(t) = g i (t, u(t)), t (t i, ], i = 1, 2,, N, u() = u H, (9) by using the Banach fixed point theorem with condensing map established the existence and uniqueness results. Motivated by the above said work, we consider the following fractional functional differential equations of the form C D α t u(t) = Au(t) + J 1 α f(t, u ρ(t,ut ), u(a 1 (t)),, u(a m (t))), t (, t i+1 ] J, i =, 1,, N, (1) u(t) = g i (t, u(t)), t (t i, ], i = 1, 2,, N, (11) u(t) + h(u t1,, u tp )(t) = ϕ(t), t (, ], (12) where C Dt α is Caputo s fractional derivative of order < α 1, J 1 α is Riemann-Liouville fractional integral operator and J = [, T ] is operational interval. The map A : D(A) X X is a closed linear sectorial operator defined on a Banach space (X, X ), = t = s < t 1 s 1 t 2 < < t N s N t N+1 = T, are pre-fixed numbers, g i C((t i, ] X; X) for all i = 1, 2,, N. f : J B h X m X, for each of j = 1, 2,, m, the map a j is defined on [, T ] into (, T ] satisfying some properties, h : B p h X such that < t 1 < t 2 < < t p < T and ρ : J B h (, T ] are appropriate functions. The history function u t : (, ] X is element of B h and defined by u t (θ) = u(t + θ), θ (, ]. Here impulses are not instantaneous means these impulses start abruptly at the points t i and their action continues on the interval [t i, ] and B h is a phase space defined in next section. To the best of our knowledge, system (1)-(12) is an untreated topic yet in the literature and this fact is the motivation of the present work. Further, this paper has four sections, second section provides some basic definitions, theorems, notations and lemma. Third section is equipped with existence results of the mild solution of the considered problem in this paper and fourth section is concerned with an example. 2 Preliminaries Let (X, X ) be a complex Banach space of functions with the norm u X = sup t J { u(t) : u X} and L(X) denotes the Banach space of all bounded linear operators from X into X equipped with its natural topology. Due to infinite delay we use abstract phase space B h as defined in [12] details are as follow: Assume that h : (, ] (, ) is a continuous functions with l = h(s)ds <, s (, ]. For any a >, we define B = {ψ : [ a, ] X such that ψ(t) is bounded and measurable}, and equipped the space B with the norm ψ [ a,] = sup s [ a,] ψ(s) X, ψ B. Let us define B h = {ψ : (, ] X, s.t. for any a c >, ψ [ c,] B & h(s) ψ [s,] ds < }. If B h is endowed with the norm ψ Bh = h(s) ψ [s,]ds, ψ B h, then it is clear that (B h, Bh ) is a complete Banach space. We consider the another space B h := P C((, T ]; X), T <, IJNS email for contribution: editor@nonlinearscience.org.uk
G. Gautam and J. Dabas: Mild Solution for Nonlocal Fractional Functional Differential Equation 153 be a Banach space of all such functions u : (, T ] X, which are continuous every where except for a finite number of points t i (, T ), i = 1, 2,..., N, at which u(t + i ) and u(t i ) exists and endowed with the norm u B = sup{ u(s) X : s [, T ]} + ϕ Bh, u B h, h where B to be a semi-norm in B h For a function u B h and i {, 1,..., N}, we introduce the function ū i C([t i, t i+1 ]; X) given by { u(t), for t (ti, t ū i (t) = i+1 ], u(t + i ), for t = t i. If u : (, T ] X is a function s.t. u B h then for all t J, the following conditions hold: (C 1 ) u t B (C 2 ) u(t) X H u t B (C 3 ) u t Bh K(t) sup{ u(s) X : s t} + M(t) ϕ Bh, where H > is constant; K, M : [, ) [, ), K( ) is continuous, M( ) is locally bounded and K, M are independent of u(t). (C 4ϕ ) The function t ϕ t is well defined and continuous from the set R(ρ ) = {ρ(s, ψ) : (s, ψ) [, T ] B h } into B h and there exists a continuous and bounded function J ϕ : R(ρ ) (, ) such that ϕ t Bh J ϕ (t) ϕ Bh for every t R(ρ ). Lemma 1 ([13])Let u : (, T ] X be function such that u = ϕ, u Jk C(J k, X) and if (C 4ϕ ) hold, then u s Bh (M b + J ϕ ) ϕ Bh + K b sup{ u(θ) X ; θ [, max{, s}]}, s R(ρ ) J, where J ϕ = sup t R(ρ ) J ϕ (t), M b = sup s [,T ] M(s) and K b = sup s [,T ] K(s). Definition 1 Caputo s derivative of order α > with lower limit a, for a function f : [a, ) R is defined as C a Dt α 1 f(t) = Γ(n α) a (t s) n α 1 f (n) (s)ds = a J n α t f (n) (t), where a, n N. The Laplace transform of the Caputo derivative of order α > is given as n 1 L{ C Dt α f(t); λ} = λ α ˆf(λ) λ α k 1 f k (); n 1 < α n. Definition 2 A two parameter function of the Mittag-Lefller type is defined by the series expansion E α,β (y) = k= k= y k Γ(αk + β) = 1 2πi c µ α β e µ µ α dµ, α, β >, y C, y where C is a contour which starts and ends at and encircles the disc µ y 1 α integral of this function given by counter clockwise. The Laplace e λt t β 1 E α,β (ωt α )dt = λα β λ α ω, Reλ > ω 1 α, ω >. For more details on the above definitions one can see the monographs of I. Podlubny [5]. Definition 3 [14] A closed and linear operator A is said to be sectorial if there are constants ω R, θ [ π 2, π], M >, such that the following two conditions are satisfied: IJNS homepage: http://www.nonlinearscience.org.uk/
154 International Journal of Nonlinear Science, Vol.21(216), No.3, pp. 151-16 (1) (θ,ω) = {λ C : λ ω, arg(λ ω) < θ} ρ(a) (2) R(λ, A) L(X) M λ ω, λ (θ,ω), where X is the complex Banach space with norm denoted. X. Definition 4 [15] Let A be a closed and linear operator with domain D(A) defined on a Banach space X and α >. We say that A is the generator of a solution operator if there exists ω and a strongly continuous function S α : R + L(X) such that {λ α : Reλ > ω} ρ(a) and λ α 1 (λ α I A) 1 u = In this case, S α (t) is called the solution operator generated by A. e λt S α (t)udt, Reλ > ω, u X. Lemma 2 Let f satisfies the uniform Holder condition with exponent β (, 1] and A is a sectorial operator. Consider the fractional equations of order < α < 1 C a D α t u(t) = Au(t) + J 1 α f(t), t J = [a, T ], a, u(a) = u. (13) Then a function u(t) C([a, T ], X) is the solution of the equation (13) if it satisfies the following integral equation u(t) = S α (t a)u + where S α (t) is solution operator generated by A defined as 1 S α (t) = e λt λ α 1 (λ α I A) 1 dλ, 2πi Γ is a suitable path lying on θ,ω. Proof. Let t = w + a, then the problem (13) translated into the form Γ a D α wũ(w) = Aũ(w) + J 1 α f(w), S α (t s)f(s)ds, (14) Now, applying the Laplace transform, we have ũ() = u. λ α L{ũ(w)} λ α 1 ũ() = AL{ũ(w)} + L{J 1 α f(w)}. We get λ α L{ũ(w)} λ α 1 ũ() = AL{ũ(w)} + 1 λ 1 α L{ f(w)}. (15) Since (λ α I A) 1 exists, that is λ α ρ(a), from (15), we obtain L{ũ(w)} = λ α 1 (λ α I A) 1 ũ() + (λ α I A) 1 1 L{ f(w)}. λ1 α Therefore, by taking the inverse Laplace transformation, we have Let w = t a, we obtain ũ(w) = E α,1 (Aw α )ũ() + w E α,1 (A(w τ) α ) f(τ)dτ. u(t) = E α,1 (A(t a) α )u + a E α,1 (A(t a τ) α )f(τ)dτ. IJNS email for contribution: editor@nonlinearscience.org.uk
G. Gautam and J. Dabas: Mild Solution for Nonlocal Fractional Functional Differential Equation 155 This the same as Let S α (t) = E α,1 (At α ), then we have u(t) = E α,1 (A(t a) α )u + u(t) = S α (t a)u + This completes the proof of the Lemma. Following definition of mild solution is based on definition 2.1 in [9]. a E α,1 (A(t s) α )f(s)ds. a S α (t s)f(s)ds. Definition 5 A function u : (, T ] X s.t. u B h is called a mild solution of the problem (1)-(12) if u() = ϕ(), u(t) = g j (t, u(t)) for t (t j, s j ] and each j = 1, 2,, N, satisfies the following integral equation S α (t)(ϕ() h(u t1,, u tp )()) + u(t) = S α(t s)f(s, u ρ(s,us), u(a 1 (s)),, u(a m (s)))ds, for all t [, t 1 ], S α (t )g i (, u( )) + S α (t s)f(s, u ρ(s,us ), u(a 1 (s)),, u(a m (s)))ds, for all t [, t i+1 ], and every i = 1, 2,, N. In fact, from the lemma 2 it is easy to see that definition 5 holds, so the proof is omitted. Theorem 3 (Theorem 3.4 [6]) 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 u + Qw B, whenever u, w B. (ii) P is compact and continuous. (iii) Q is a contraction mapping. Then there exists z B such that z = P z + Qz. 3 Main Result This section is equipped with existence results of mild solutions for the nonlocal impulsive system (1)-(12). If A A α (θ, ω ), then strongly continuous S α (t) Me ωt. Let M S := sup t T S α (t) L(X). So we have S α (t) L(X) M S. To prove our results we assume the function ρ : [, T ] B h (, T ] is continuous and ϕ B If u B h we defined ū : (, T ) X as the extension of u to (, T ] such that ū(t) = ϕ. We defined ũ : (, T ) X such that ũ = u + x where x : (, T ) X is the extension of ϕ B h such that x(t) = S α (t)ϕ() for t J. Now, we introduce the following axioms: (H 1 ) There exist positive constant L f such that f(t, φ, u 1, u m ) f(t, ξ, w 1, w m ) X L f [ φ ξ Bh + for t J, φ, ξ B h, (u 1, u m ), (w 1, w m ) X m. (H 2 ) There exist positive constant L h such that for all t i J and ϕ, φ B (H 3 ) There exist positive constant L gi such that for all t J and y, z X. h(ϕ(t 1 ),, ϕ(t p )) h(φ(t 1 ),, φ(t p )) X L h ϕ φ Bh, g i (t, y) g i (t, z) X L gi y z X, m u i w i X ], i=1 IJNS homepage: http://www.nonlinearscience.org.uk/
156 International Journal of Nonlinear Science, Vol.21(216), No.3, pp. 151-16 (H 4 ) f is continuous function and there exist a positive constant M such that f(t, φ, u 1, u m ) X M 1, for (u 1, u m ) X m, φ B h and t J. (H 5 ) There exists positive constants M 2, M 3 such that h((ϕ(t 1 ),, ϕ(t p )) X M 2 ; g i (t, y) X M 3, for y X, (ϕ(t 1 ),, ϕ(t p )) B p Theorem 4 Let the assumptions (H 1 ) (H 3 ) hold and the constant δ = max{ M S ( K L h + T L f (K b + m)), M S (L gi + T L f (K b + m))} < 1, for i = 1, 2,, N. Then there exists a unique mild solution u(t) on J of the system (1)-(12). Proof. Let ϕ : (, T ) X be the extension of ϕ to (, T ] such that ϕ(t) = ϕ() on J. Consider the space B h = {u B h : u() = ϕ()} and u(t) = ϕ(t), for t (, ] endowed with the uniform convergence topology. Let us consider a operator P : B h B h defined as P u() = ϕ(), P u(t) = g i(t, ū(t)) for t (t i, ] and S α (t)(ϕ() h(ū t1,, ū tp )()) + P u(t) = S α(t s)f(s, ū ρ(s,ūs ), ū(a 1 (s)),, ū(a m (s)))ds, t [, t 1 ], S α (t )g i (, ū( )) + S α (t s)f(s, ū ρ(s,ūs), ū(a 1 (s)),, ū(a m (s)))ds, t [, t i+1 ], where ū : (, T ] X is such that ū() = ϕ and ū = u on J. It is obvious that P is well defined. Let u(t), u (t) B h for t [, t 1 ], we have P u(t) P u (t) X S α (t) L(X) ( h(ū t1,, ū tp )()) h(ū t 1,, ū t p )()) X For t [, t i+1 ], we estimate as For t (t j, s j ], we get + S α (t s) L(X) f(s, ū ρ(s,ūs ), ū(a 1 (s)),, ū(a m (s))) f(s, ū ρ(s,ū s ), ū (a 1 (s)),, ū (a m (s))) X ds M S ( K L h + T L f (K b + m)) u u B P u(t) P u (t) X S α (t ) L(X) g i (, ū( )) g i (, ū ( )) X Gathering above results, we obtain + S α (t s) L(X) f(s, ū ρ(s,ūs ), ū(a 1 (s)),, ū(a m (s))) f(s, ū ρ(s,ū s ), ū (a 1 (s)),, ū (a m (s))) X ds M S (L gi + T L f (K b + m)) u u B P u(t) P u (t) X L gj u u B, j = 1, 2,, N. h P u(t) P u (t) X δ u u B Since δ < 1, which implies that P is a contraction map and there exists a unique fixed point which is the mild solution of system (1)-(12). This completes the proof of the theorem. Theorem 5 Let the assumptions (H 2 ) (H 5 ) hold and max{ M S K L h, M S L gi } < 1. Then system (1)-(12) has atleast one mild solution u(t) on J. IJNS email for contribution: editor@nonlinearscience.org.uk
G. Gautam and J. Dabas: Mild Solution for Nonlocal Fractional Functional Differential Equation 157 Proof. Let ϕ : (, T ) X be the extension of ϕ to (, T ] such that ϕ(t) = ϕ() on J. Consider the space B h = {u B h : u() = ϕ()} and u(t) = ϕ(t), for t (, ] endowed with the uniform convergence topology. Choose r max{ M S ( ϕ() Bh +M 2 )+ M S T M 1, M S M 3 + M S T M 1 } and consider space B r = {u B h : u r}, then it is clear that B r is a bounded, closed and convex subset in B Let us define the operator P and Q on B r by P u() = ϕ(), P u(t) = g i (t, ū(t)) for t (t i, ] and { Sα (t)(ϕ() h(ū P u(t) = t1,, ū tp )()), t [, t 1 ], S α (t )g i (, ū( )), t [, t i+1 ], { Qu(t) = S α(t s)f(s, ū ρ(s,ūs ), ū(a 1 (s)),, ū(a m (s)))ds, t [, t 1 ], S α (t s)f(s, ū ρ(s,ūs), ū(a 1 (s)),, ū(a m (s)))ds, t [, t i+1 ], where ū : (, T ] X is such that u() = ϕ and ū = u on J. Let u, w B r, then for t [, t 1 ] we have P u(t) + Qw(t) X S α (t) L(X) ( ϕ() X + h(ū t1,, ū tp )()) X For t [, t i+1 ], we estimate as For all t [, T ] + S α (t s) L(X) f(s, w ρ(s, ws), w(a 1 (s)),, w(a m (s))) X ds M S ( ϕ() X + M 2 ) + M S T M 1. P u(t) + Qw(t) X S α (t ) L(X) g i (, ū( )) X + S α (t s) L(X) f(s, w ρ(s, ws), w(a 1 (s)),, w(a m (s))) X ds M S M 3 + M S T M 1. P u(t) + Qw(t) X max{ M S ( ϕ() X + M 2 ) + M S T M 1, M S M 3 + M S T M 1 } r, which implies that P u(t) + Qw(t) B r. Now, we shall show that P is contraction map. Let u(t), u (t) B r for t [, t 1 ], we have P u(t) P u (t) X S α (t) L(X) ( h(ū t1,, ū tp )()) h(ū t 1,, ū t p )()) X For t [, t i+1 ], we estimate as M S K L h u u B For t (t j, s j ], we get By above results, we obtain P u(t) P u (t) X S α (t ) L(X) g i (, ū( )) g i (, ū ( )) X M S L gi u u B P u(t) P u (t) X L gj u u B, j = 1, 2,, N. h P u(t) P u (t) X max{ M S K L h, M S L gi } u u B Since max{ M S K L h, M S L gi } < 1, which implies that P is a contraction map. Next, we shall show that Q is completely continuous on B r. So consider a sequence u n u in B r, for t [, t i+1 ], we have Qu n (t) Qu(t) X S α (t s) L(X) f(s, ū n ρ(s,ū n s ), ūn (a 1 (s)),, ū n (a m (s))) f(s, ū ρ(s,ūs), ū(a 1 (s)),, ū(a m (s))) X ds. IJNS homepage: http://www.nonlinearscience.org.uk/
158 International Journal of Nonlinear Science, Vol.21(216), No.3, pp. 151-16 Since function f is continuous, so Qu n (t) Qu(t) X as n. Which implies that Q is continuous. It is easy to prove that Q maps bounded set into bounded set in B r. To do this we have for all t [, T ] Qu(t) X T M S M 1 = C. Finally, we show Q is a family of equi-continuous functionn B r. Let l 1, l 2 [, t i+1 ], such that l 1 < l 2 t i+1. We have Q(u)(l 2 ) Q(u)(l 1 ) X l1 S α (l 2 s) S α (l 1 s) L(X) f(s, ū ρ(s,ūs ), ū(a 1 (s)),, ū(a m (s))) X ds + l2 l 1 S α (l 2 s) L(X) f(s, ū ρ(s,ūs ), ū(a 1 (s)),, ū(a m (s))) X ds l1 M 1 S α (l 2 s) S α (l 1 s) L(X) ds + M S M 1 (l 2 l 1 ). Since S α (t) is strongly continuous, so lim l2 l 1 S α (l 2 s) S α (l 1 s) L(X) =, which implies that Q(u)(l 2 ) Q(u)(l 1 ) X as l 2 l 1. This proves that Q is a family of equi-continuous functions. So, we conclude that the operator Q is a completely continuous or compact operator by Arzela-Ascoli s theorem. Therefor by the theorem 3 there exist atleast one fixed point in B r. Hence we conclude that the system (1)-(12) has a mild solution u(t) on J. This completes the proof of the theorem. 4 Example Consider the following nonlocal impulsive fractional partial differential equation of the form α 2 u(t, x) = tα y 2 u(t, x) + 1 Γ(1 α) 1 16 s (t s) α e 2(ν s) u(ν ρ 1 (ν)ρ 2 ( u )), u(a 1 (ν)),, u(a m (ν)), x)dνds, (t, x) N i=1[, t i+1 ] [, π], (16) u(t, ) = u(t, π) =, t, (17) p π u(t, x) + K(x, ξ)u(t i, ξ)dξ = ϕ(t, x), t (, ], x [, π], (18) i=1 u(t, x) = G i (t, u(t, x)), x [, π], t (t i, ], (19) where α t is Caputo s fractional derivative of order α (, 1], = t α = s < t 1 s 1 < < t N s N < t N+1 = 1 are fixed real numbers, ϕ B h, and p is a positive integer, < t < t 1,, < t p < 1. Let X = L 2 [, π] and define the operator A : D(A) X X by Aw = w with the domain D(A) := {w X : w, w are absolutely continuous, w X, w() = = w(π)}. Then Aw = n 2 (w, w n )w n, w D(A), n=1 2 where w n (x) = π sin(nx), n N is the orthogonal set of eigenvectors of A. It is well known that A is the infinitesimal generator of an analytic semigroup (T (t)) t in X and is given by T (t)ω = e n2t (ω, ω n )ω n, for all ω X, and every t >. n=1 IJNS email for contribution: editor@nonlinearscience.org.uk
G. Gautam and J. Dabas: Mild Solution for Nonlocal Fractional Functional Differential Equation 159 The subordination principle of solution operator implies that A is the infinitesimal generator of a solution operator {S α (t)} t, such that S α (t) L(X) M S for t [, 1]. Let h(s) = e 2s, s < then l = h(s)ds = 1 2 <, for t (, ] and define ϕ Bh = h(s) sup ϕ(θ) L 2ds. θ [s,] Hence for (t, ϕ) [, 1] B h, where ϕ(θ)(x) = ϕ(θ, x), (θ, x) (, ] [, π]. Set u(t)(x) = u(t, x), and ρ(t, ϕ) = ρ 1 (t)ρ 2 ( ϕ() ) we have f(t, u(t ρ 1 (t)ρ 2 ( u )), u(a 1 (t)),, u(a m (t))) = 1 16 e 2(ν) ϕ, u(a 1 (ν)),, u(a m (ν)))(x)dν, g i (t, u)(x) = G i (t, u(t, x)) = G i (t, u)(x), n h(u t1,, u tp )(t) = K q u ti (t) i= where K q z(x) = π k(x, ξ)z(ξ)dξ for z L2 [, π], x [, π] then with these settings the equations (16)-(19) can be written in the abstract form of equation(1)-(12). We assume that ρ i : [, ) [, ), i = 1, 2, are continuous functions and taking M S = K = K b = 1. The functions f, g i and h are Lipschitz with Lipschitz constants L f, Lg i and L h respectively with max{l h + L f (1 + m), L gi + L f (1 + m) : i = 1,, N} < 1, then there exists a unique mild solution u on [, 1] by the Theorem 4. References [1] S. G. Samko et al. Fractional integrals and derivatives theory and applications. Gordon and Breach, Yverdon. (1993). [2] K. S. Miller and B. Ross. An introduction to the fractional falculus and differential equations. John Wiley, New York. (1993). [3] A. A. Kilbas et al. Theory and applications of fractional differential equations. North-Holland Mathematics Studies. (26). [4] V. Lakshmikantham et al. Theory of fractional dynamic systems. Cambridge Scientific Publishers. (29). [5] I. Podlubny. Fractional Differential Equations. Acadmic Press New York. (1999). [6] A. Chauhan and J. Dabas. Existence of mild solutionmpulsive fractional order semilinear evolution equations with non local conditions. Electronic. J. Diff. Equ. 17(211): 1 1. [7] D. Bahuguna. Existence, uniqueness and regularity of solutions to semilinear nonlocal functional differential problems. Nonlinear Anal. 57(24): 121 128. [8] L. Byszewski and H. Acka. Existence of solutions of a semilinear functional differential evoluation nonlocal problems. Nonlinear Anal. 34(1998): 65 72. [9] E. Hernandez and D. O Regan, On a new class of abstract impulsive differential equations. Pro. American Math. Society. 141(213): 1641 1649. [1] M. Pierri et al. Existence of solutions for semi-linear abstract differential equations with not instantaneoumpulses. Appl. Math. Comput. 219(213): 6743 6749. [11] Y. Zhou and F. Jiao. Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59(21): 163 177 [12] J. Dabas et al. Existence of mild solution for impulsive fractional equation with infinity delay. Int. J. Diff. Equ. (211). [13] M. Benchohra and F. Berhoun. Impulsive fractional differential equations with state dependent delay. Commun. Appl. Anal. 14(21): 213 224. [14] M. Haase. The functional calculus for sectorial operators. Operator theory Advan. (26). IJNS homepage: http://www.nonlinearscience.org.uk/
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