Caputo-Fabrizio fractional differential equations in Frechet spaces

This paper deals with some existence and uniqueness of solutions for a class of functional Caputo-Fabrizio fractional differential equations. Some applications are made of a generalization of the classical Darbo fixed point theorem for Frechet spaces associate with the concept of measure of noncompactness. The last section illustrates our results with some examples.

Existence solution of a system of differential equations using generalized Darbo's fixed point theorem

<abstract><p>In this paper, we proposed a generalized of Darbo's fixed point theorem via the concept of operators $ S(\bullet; .) $ associated with the measure of noncompactness. Using this generalized Darbo fixed point theorem, we have given the existence of solution of a system of differential equations. At the end, we have given an example which supports our findings.</p></abstract>

Existence results for some integro-differential equations with state-dependent nonlocal conditions in Fréchet Spaces

In this work, we present existence of mild solutions for partial integro-differential equations with state-dependent nonlocal local conditions. We assume that the linear part has a resolvent operator in the sense given by Grimmer. The existence of mild solutions is proved by means of Kuratowski’s measure of non-compactness and a generalized Darbo fixed point theorem in Fréchet space. Finally, an example is given for demonstration.

On the solvability of a nonlinear functional integral equations via measure of noncompactness in

Using the technique of a suitable measure of non-compactness and the Darbo fixed point theorem, we investigate the existence of a nonlinear functional integral equation of Urysohn type in the space of Lebesgue integrable functions Lp(RN). In this space, we show that our functional-integral equation has at least one solution. Finally, an example is also discussed to indicate the natural realizations of our abstract result.

Applications of measure of non-compactness and modified simulation function for solvability of nonlinear functional integral equations

In this work we introduce a modified version of simulation function and define a simulation type contraction mappings involving measure of non-compactness in the frame work of Banach space and derive some basic Darbo type fixed point results. Also, our theorem generalizes the Theorem 4 of [R. Arab, Some generalizations of Darbo fixed point theorem and its application, Miskolc Mathematical Notes, 18(2)(2017),595-610.] and extend some recent results. Further we show the applicability of obtained results to the theory of integral equations followed by two concrete examples.

On existence of solution of a class of quadratic-integral equations using contraction defined by simulation functions and measure of noncompactness

In this paper we have introduced a new type of contraction condition using a class of simulation functions, in the sequel using the new contraction definition, involving measure of noncompactness; we establish few results on existence of fixed points of continuous functions defined on a subset of Banach space. This result also generalizes other related results obtained in Arab [Arab, R., Some generalizations of Darbo fixed point theorem and its application, Miskolc Math. Notes, 18 (2017), No. 2, 595–610], Banas [Bana ´ s, J. and Goebel, K., ´ Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 60 (1980)]. The obtained results are used in establishing existence theorems for a class of nonlinear quadratic equation (which generalizes several types of fractional-quadratic integral equations such as Abel’s integral equation) defined on a closed and bounded subset of R. The existence of solution is established with the aid of a measure of noncompactness defined on function space C(I) introduced in [Banas, J. and Olszowy, L., ´ Measures of Noncompactness related to monotonicity, Comment. Math., 41 (2001), 13–23].

The controllability of nonlinear implicit fractional delay dynamical systems

This paper is concerned with the controllability of nonlinear fractional delay dynamical systems with implicit fractional derivatives for multiple delays and distributed delays in control variables. Sufficient conditions are obtained by using the Darbo fixed point theorem. Further, examples are given to illustrate the theory.

Application of measure of noncompactness for the system of functional integral equations

In this paper we introduce the notion of the generalized Darbo fixed point theorem and prove some fixed and coupled fixed point theorems in Banch space via the measure of non-compactness, which generalize the result of Aghajani et al.[6]. Our results generalize, extend, and unify several well-known comparable results in the literature. As an application, we study the existence of solutions for the system of integral equations.

On the well posed solutions for nonlinear second order neutral difference equations

In this work we investigate the existence of solutions, their uniqueness and finally dependence on parameters for solutions of second order neutral nonlinear difference equations. The main tool which we apply is Darbo fixed point theorem.