On the Application of Measure of Noncompactness to the Existence of Solutions for Fractional Differential Equations

Abstract

In this paper, we prove the existence of solutions for a boundary value problem of fractional differential equations. The technique relies on the concept of measures of noncompactness and Mönch's fixed point theorem.

Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces

“…This result relies on the set-valued analog of Mönch's fixed point theorem combined with the technique of measure of noncompactness. Recently, this has proved to be a valuable tool in studying fractional differential equations and inclusions in Banach spaces; for additional details, see the papers of Laosta et al [20], Agarwal et al [1], and Benchohra et al [7,8,9]. Our results here extend to the multivalued case some previous results in the literature and constitutes what we hope is an interesting contribution to this emerging field.…”

Solvability for a couple system of nonlinear fractional differential equations in a Banach space

“…Therefore, we extend some previous results in many respects, as see for example [4]- [11], [22], [23].…”

Some generalizations of Darbo fixed point theorem and its application

“…On the other hand, measures of noncompactness are very useful tools in the wide area of functional analysis such as the metric fixed point theory and the theory of operator equations in Banach spaces. They are also used in the study of functional equations, ordinary and partial differential equations, fractional partial differential equations, integral and integro-differential equations, optimal control theory, etc., see [1][2][3][4][5][6][12][13][14][15][16]. In our investigations, we apply the method associated with the technique of measures of noncompactness to generalize the Darbo fixed point theorem [9] and to extend some recent results of Aghajani et al [5].…”