Journal of Function Spaces

Volume 2016 (2016), Article ID 6165804, 9 pages

http://dx.doi.org/10.1155/2016/6165804

## About the Existence Results of Fractional Neutral Integrodifferential Inclusions with State-Dependent Delay in Fréchet Spaces

^{1}Department of Mathematics, C. B. M. College, Kovaipudur, Coimbatore, Tamil Nadu 641 042, India^{2}Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, 06530 Ankara, Turkey^{3}Institute of Space Sciences, Magurele, Bucharest, Romania

Received 21 October 2015; Accepted 16 December 2015

Academic Editor: Hugo Leiva

Copyright © 2016 Selvaraj Suganya et al. 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.

#### Abstract

A recent nonlinear alternative for multivalued contractions in Fréchet spaces thanks to Frigon fixed point theorem consolidated with semigroup theory is utilized to examine the existence results for fractional neutral integrodifferential inclusions (FNIDI) with state-dependent delay (SDD). An example is described to represent the hypothesis.

#### 1. Introduction

We are dealing in this paper with the existence of mild solutions for FNIDI with SDD in Fréchet spaces by making use of the fixed point theorem of Frigon [1, Corollary 3.5]. In Section 3 of this paper, we deliberate the neutral integrodifferential inclusions of fractional-order of the modelwherein and is the generator of an integral resolvent family characterized on a complex Banach space , the convolution integral within the equation is understood because of the Riemann-Liouville fractional integral (see [2]), is a multivalued map, ( is the family of nonempty subsets of ), , , and are apposite functions, and is theoretical phase space axioms characterized in Section 2.

For almost any continuous function characterized on and any , we designate by the part of characterized by for . Now, speaks to the historical backdrop of the state from every likely the current time .

The notion of a fractional derivative plays an important role in numerous technological innovation and scientific disciplines as the statistical modeling of frameworks and procedures in numerous fields, case in point, physical science, chemical industry, aerodynamics, electrodynamics of complex medium, and so forth. For information, we recommend the readers to refer to the treatise of Abbas et al. [3], Baleanu et al. [4], Podlubny [5], Diethelm [6], Kilbas et al. [7], and Zhou [8] and the papers of fractional differential and integrodifferential systems [9–12] and impulsive fractional differential systems [13–15] and the references cited therein.

We recall that the fractional differential inclusions (FDI) occur in the mathematical modeling of specific models in financial aspects, optimal control, and so forth and are usually investigated by numerous writers; see, for instance, [16–18] and the references therein. Fractional equation with delay properties arises in several fields such as biological and physical ones with state-dependent delay or nonconstant delay. Nowadays, the existence results of mild solutions for such problems became very attractive and several researchers are working on it. Recently, several number of papers have been written on the fractional-order problems with state-dependent delay [19–23] and the sources therein. In particular, in [20, 21], the authors analyzed the existence results for neutral differential systems with SDD in Banach spaces, whereas in [19, 22] the authors investigate the same type of problems with SDD and impulsive conditions by utilizing appropriate fixed point theorem. Also, the integrodifferential systems are experienced in numerous ranges of science, the place where it is imperative to deal with aftereffect or delay (e.g., control theory, biology, ecology, and medicine). Particularly, one dependably depicts a model which has inherited qualities by integrodifferential systems in implementation; see, for instance, [23–25].

The beginning stage of this work is reflected in [26–30]. Particularly, in [26], Agarwal et al. acquired the existence of mild solutions for FIDE of the structure in which and is a linear densely described operator of sectorial kind on a complex Banach space , whereas, in [30], the authors establish the existence results for FIDI of the model where and are the same as those mentioned in model (1).

The existence of mild solutions for the division of FNIDI with SDD in Fréchet spaces of the structure (1) is by all accounts an unread point. By utilizing a few speculations as a part of [27, 30], our desire here is to yield the existence results for the above model (1) utilizing a nonlinear alternative for multivalued contractions as of late created by Frigon [1, Corollary 3.5].

#### 2. Preliminary Notions

Below, we briefly present the mathematical tools required in this paper.

Let , , be the Banach space of all continuous functions from into making use of the standard norm

Allow to be the space of all bounded linear operators , having the common supremum norm

A measurable function is Bochner integrable if and only if is Lebesgue integrable (to get extra insights about Bochner integral, see the treatise of Yosida [31]).

Let signify the Banach space of all measurable functions which are Bochner integrable making use of the norm Recognize the space where is the restraint of to .

We expect that the phase space is a seminormed linear space of functions mapping into and fulfilling the subsequent elementary adages as a result of Hale and Kato (find illustration in [32–34]).()If is continuous on and , then for every the going hand in hand circumstances hold the following:(i) is in ;(ii);(iii), where is a constant and is continuous, is locally bounded, and , are independent of .()For function in , is a -valued continuous function on .()The space is complete.

Designate and .

The next step is to review some known results from the fractional calculus.

The Laplace transformation of a function is determined by if the integral is definitely convergent for . With a specific end goal to give an operator hypothetical methodology, we review the subsequent definition [2].

*Definition 1. *Let be a closed and linear operator on a Banach space . One addresses as the generator of an integral resolvent if one can find and a strongly continuous function to ensure that

For this situation, is known as the integral resolvent family produced by . For extra points of interest regarding this, we refer the reader to [35, Proposition 3.1 and Lemma 2.2].

*Remark 2. *The uniqueness and uniform continuity of the resolvent are long-familiar (see Benchohra and Litimein [30], Pruss [36]).

Before we complete this section, we display some long-familiar outcomes from multivalued research.

Indicate the following:

Proposition 3 (see [37, Proposition III.4]). *If and are compact valued measurable multifunctions, then the multifunction is measurable. If is a sequence of compact valued measurable multifunctions, then is measurable, and if is compact, then is measurable.*

*Remark 4. *The definitions of measurable, admissible contraction, metric space, and nonlinear alternative fixed point theorem [1, Corollary 3.5] are classical in multivalued analysis; hence, we keep off it.

*Remark 5. *For primary and surplus points on Fréchet spaces, we refer the reader to [30].

For each , specify the set of selections for by

For surplus points of benefit on multivalued maps, think about the treatise of Castaing and Valadier [37] and Graef et al. [38].

#### 3. The Main Results

In this part, we prove the existence outcomes for the structure (1). To commence, we delineate the mild solution for the structure (1).

*Definition 6. *One affirms that the function is a mild solution of the model (1) if for all , the constraint of to the period is continuous and one can find , in a way that a.e. , and fulfills the consecutive integral equation:

Set

We generally expect that is continuous. Moreover, we make the subsequent assumption:()function is continuous from into and we can find a continuous and bounded function in a way that

Lemma 7 (see [21, Lemma 3.1]). *If is a function to ensure that , then where .*

The successive hypotheses will be required in whatever is left of this paper.(H1)The solution operator is compact for , and we can find in a way that (H2) (i)The multivalued map is Carathéodory and there is certainly function and a continuous nondecreasing function in a way that (ii)For all , we can find to ensure that in which , in addition to, for all in conjunction with , joined with (H3) (i)There is a function and a continuous nondecreasing function to ensure that (ii)There is a constant in a way that (H4) (i)Function is continuous on , and there are certainly positive constants , in a way that (ii)For every , there is a function in a way that For each , we delineate, in , the family of seminorms byin which , , , and and accept that and , are a function from (H2)(ii) and (H4)(ii) appropriately.

Theorem 8. *Expect that (H1)–(H4) and () hold, and believe that and in which . At that point, model (1) has a mild solution on .*

*Proof. *We will transmute the structure (1) into a fixed point problem. Recognize the multivalued operator specified by with where . For , we express function by and then . For every function with , we designate by the function clear by If fulfills (12), we are able to decompose it as , , which suggests that , for each , and also the function fulfills where .

Let . For any , we sustain Along these lines, is a Banach space with the norm . We delimit the operator by with where .

It is vindicated that the operator has a fixed point if and only if has a fixed point. As a result, let us demonstrate that has a fixed point .*Remark 9*. (i) By condition and Lemma 7 in the above discussion, we have the subsequent estimates: where .

(ii) (iii) Presenting , is ought to be a solution of the inclusion for many and there may be in a way that, for any , we maintainFrom Remark 9(i), we have Thus, We conceive function characterized by Permit in a way that . Because of the aforementioned inequality, we maintain, for , Allow us to occupy the right-hand part of the overhead inequality as . Then, we sustain for all . Through the significance of , we get This leads us to the accompanying inequality for , where Next, we weigh the function Then we bring forth and for all . Applying the nondecreasing character of , we receive We characterize the function , , which suggests that From condition (25), we acquire Subsequently, for every , we have a constant in a way that and, consequently, . Due to the fact that , we certainly have . Fix Evidently, is a closed subset of . We should demonstrate that is a contraction and an admissible operator. Initially, we evaluate that is a contraction. In fact, consider and . Then, there may be such that From the hypotheses (H2)(ii) and (H3)(ii), we sustainAs a result, there is so that Recognize specified by Considering the fact that the multivalued operator is measurable (see Proposition 3), there is function , which is a measurable choice for . So, , and, from Remark 9(ii), we specifyFor every , give us a chance to characterize Again, from Remark 9(iii), we now have As a result Being practically equivalent to the connection gotten by exchanging the parts of and , it takes after that demonstrating that is a contraction for all and, from the second aspect of [30, Theorem 3.4], we realize that is likewise admissible contraction operator. With the decision of , there is no in a way that for many . From the nonlinear alternative fixed point theorem thanks to Frigon [1, Corollary 3.5], we realize that the operator has a fixed point . This intimates that , , is a fixed point of the operator , which is a mild solution of the structure (1).

#### 4. Illustrative Example

To exemplify our theoretical results, we treat the FNIDI with SDD of the structure where , , , stands for the operator with respect to the special variable are measurable in and continuous in , , and are appropriate functions. We expect that, for each , is lower semicontinuous (i.e., the set is open for all and accept that, for each , is upper semicontinuous (i.e., the set is open for each .

Recognize and the operator with domain Clearly, is densely defined in and is sectorial. Hence, is a generator of a solution operator on . For the phase space, we pick exists in invested with the norm Here, we note the phase space satisfying conditions , , and . Set The multivalued map is u.s.c. with compact convex values. Thus, (H1) and (H2) are fulfilled.

Presently, the existence of mild solutions can be reasoned from an immediate utilization of Theorem 8. In the perspective of phase space with the above examination, we have the accompanying results.

Corollary 10. *Let be continuous and bounded. Then, there is certainly at least one mild solution of the model (57) on .*

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The fourth author would like to dedicate this paper, with love and respect, to his doctoral supervisor Professor A. Anguraj, P.S.G. College of Arts & Science, Coimbatore 641014, India, on the occasion of his 54th birthday.

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