In recent papers [7, 8], we used Schauder’s ﬁxed-point theorem to obtain local exis- tence, and Tychonov’s ﬁxed-point theorem to obtain global existence of solution of the **fractional** integrodiﬀerential **equations** (1.1) and (1.2). The existence of extremal (maxi- mal and minimal) solutions of the **fractional** integrodiﬀrential **equations** (1.1) and (1.2) using comparison principle and Ascoli lemma has been investigated in [9].

We study the solvability of the **fractional** integrodiﬀerential **equations** of neutral type with infinite delay in a Banach space X. An existence result of mild solutions to such problems is obtained under the conditions in respect of Kuratowski’s measure of noncompactness. As an application of the abstract result, we show the existence of solutions for an integrodiﬀerential equation.

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Therefore, the main result of this paper also includes the existence as well as approximation results for the solutions of above mentioned initial value problems of **fractional** differential **equations** as special cases. Again our approach here in this paper is different than that employed in the related paper of Dhage [3].

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**integrodifferential** **equations** of neutral type with finite delay and nonlocal conditions in a Banach space X . The existence of mild solutions is proved by means of measure of noncompactness. As an application, the existence of mild solutions for some **integrodifferential** equation is obtained.

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noncompactness, we discuss the existence of solutions for a boundary value problem of impulsive integrodiﬀerential **equations** of **fractional** order α ∈ (1, 2]. Our results improve and generalize some known results in (Zhou and Chu in Commun. Nonlinear Sci. Numer. Simul. 17:1142-1148, 2012; Bai et al. in Bound. Value Probl. 2016:63, 2016). Finally, an example is given to illustrate that our result is valuable.

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The aim of this paper is to discuss the existence of weak solutions for a nonlinear two-point boundary value problem of integrodiﬀerential **equations** of **fractional** order α ∈ (1, 2]. Our analysis relies on the Krasnoselskii ﬁxed point theorem combined with the technique of measure of weak noncompactness.

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In this article, we study a neutral **fractional** **integrodifferential** equation supplemented with nonlocal flux type integral boundary conditions. The existence and uniqueness results are obtained by using Banach fixed point theorem and Leray-Schauder nonlinear alternative theorem. The obtained results are illustrated by examples at the end.

that the large number of **fractional** derivatives does not constitute a disadvantage, since they can be used in diﬀerent models which provide the best reﬂection of the behavior of the system. In many simultaneously occurring processes in modeling of the real world phenomena to obtain data, the ﬁeld observations are needed. The modeling of a dynamical system based on the ﬁeld observations becomes uncertain and vagueness or fuzziness, which is inherent in the systems behavior rather than being purely random or deterministic. The study of interval and fuzzy diﬀerential **equations** is an area of mathematics that has recently received a lot of attention (see e.g. [4, 11, 17, 18, 19, 20]). Recently, there are some papers dealing with the existence of solution for nonlinear set valued and fuzzy **fractional** diﬀerential **equations** whose methods are based on the monotone method, the method of upper and lower solutions and ﬁxed point theorems [1, 2, 5, 6, 7, 10, 17]. Among of them, we can ﬁnd results on existence of solution for fuzzy diﬀerential **equations** in presence of

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This paper deals with some existence results for a boundary value problem involving a nonlinear integrodi ﬀ erential equation of **fractional** order q ∈ 1, 2 with integral boundary conditions. Our results are based on contraction mapping principle and Krasnosel’ski˘ ı’s fixed point theorem. Copyright q 2009 B. Ahmad and J. J. Nieto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[26] J. Liang and T-J. Xiao, “Solvability of the Cauchy problems for infinite delay **equations**”, Nonlinear Analysis: Theory, Methods & Applications, vol.58, no.3-4, 2004, pp.271-297. [27] A.Pazy, Semigroups of Linear Operators and Applications to

β < 1 is the Riemann-Liouville **fractional** integral of order β . The theory of **fractional** calculus has been available and applicable to various fields of study. The investigation of the theory of **fractional** differential and integral **equations** has started quite recently. One can see the monographs of Kil- bas et.al. [11], Podlubny [15]. **Integrodifferential** **equations** arise in many engineering and scientific disciplines, often as approximation to partial differential **equations**, which repre- sent much of the continuum phenomena. **Integrodifferential** equation is an equation that the unknown function appears un- der the sign of integration and it also contains the derivatives of the unknown function. It can be classified into Fredholm **equations** and Volterra **equations**. The upper bound of the region for integral part of Volterra type is variable, while it is a fixed number for that of Fredholm type. However, in this paper, we focus on Fredholm **integrodifferential** **equations**.

Recently some interesting results on Caputo **fractional** resolvents have been given in [11, 12, 26]. We note that the properties of resolvent operators for Caputo derivative and Riemann–Liouville derivative are diﬀerent in essence, though neither of them has the semigroup property. For Caputo **fractional** resolvents T α (t), T α (0)x = x for every x ∈ X,

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We study the existence and uniqueness of mild solution of a class of nonlinear nonautonomous **fractional** integrodiﬀerential **equations** with infinite delay in a Banach space X. The existence of mild solution is obtained by using the theory of the measure of noncompactness and Sadovskii’s fixed point theorem. An application of the abstract results is also given.

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Eq.(1.1) is the mixed type of Eq.(1.3) and Eq.(1.4). It well enable us to study the nonlinear Volterra **integrodifferential** equation with delay. On the basis of the results in Eq.(1.4) we gen- eralize the method used in [33] to derive global existence and regularity of Eq.(1.1). The result obtained is a generalization and a continuation of [33]. The method used treats the equation in the domain of A with the graph norm employing results from linear semigroup theory concern- ing abstract inhomogeneous linear differential **equations**.

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with impulsive initial conditions and derived some suﬃcient conditions ensuring the exponential stability of solutions for the singular perturbed impulsive delay diﬀerential **equations** SPIDDEs. In this paper, we will improve the inequality established in 14 such that it is eﬀective for SPIDIDEs. By establishing an IDIDI, some suﬃcient conditions ensuring the exponential stability of any solution of SPIDIDEs for suﬃciently small ε > 0 are obtained. The results extend and improve the earlier publications, and which will be shown by the Remarks 3.2 and 3.5 provided later. An example is given to illustrate the theory.

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In this section we consider some control systems gov- erned by partial differential **equations**, **integrodifferential** **equations** and difference **equations** that can study using these results. Particularly, we work in details the con- trolled damped wave equation. Finally, we propose fu- ture investigations an open problem.

Let E and U be a pair of real Banach spaces with norms · and | · |, respectively. Let σ be a linear closed and densely defined operator with Dσ ⊆ E and let τ ⊆ X be a linear operator with Dσ and Rτ ⊆ X, a Banach space. In this paper we study the boundary controllability of nonlinear **fractional** integrodiﬀerential systems in the form

[18] G. W. Desch, R. Grimmer, and W. Schappacher, “Propagation of singularities by solutions of second order integrodiﬀerential **equations**,” in Volterra Integrodiﬀerential **Equations** in Banach Spaces and Applications (Trento, 1987), G. Da Prato and M. Iannelli, Eds., vol. 190 of Pitman Research Notes in Mathematics Series, pp. 101–110, Longman Scientific & Technical, Harlow, UK, 1989.

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real world applications, wherever (in physics, chemistry, biology, medecine, economy etc, see e.g, [16]) the evolution of a process depends on its history in an essentiel way. In recent year, the theory of **integrodifferential** **equations** with delay has been studied deeply in the literature. For more details, we refer to [2], [3], [5], [6], [7], [8], [9], [10], [17] and the references therein. In this paper, we are interested in the existence and regularity of solutions for the following neutral partial functional **integrodifferential** equation with finite delay

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(2.1) By a local mild solution of (1.1) on we mean that there exist a and a function defined from into such that is a mild solution of (1.1) refer to [8,22,25]. We define the **fractional** power by