Kuratowski MNC method on a generalized fractional Caputo Sturm–Liouville–Langevin q-difference problem with generalized Ulam–Hyers stability

In this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch’s fixed point theorem along with a theoretical method by terms of Kuratowski measure of noncompactness (MNC) and the Banach contraction principle (BCP). Furthermore, two dynamical notions of Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability are addressed for solutions of the supposed Sturm–Liouville–Langevin quantum boundary value problem (q-FBVP). Two examples are presented to show the validity and also the effectiveness of theoretical results. In the last part of the paper, we conclude our exposition with some final remarks and observations.

Existence of Mild Solutions for Multi-Term Time-Fractional Random Integro-Differential Equations with Random Carathéodory Conditions

In this paper, we investigate the existence of mild solutions to a multi-term fractional integro-differential equation with random effects. Our results are mainly relied upon stochastic analysis, Mönch’s fixed point theorem combined with a random fixed point theorem with stochastic domain, measure of noncompactness and resolvent family theory. Under the condition that the nonlinear term is of Carathéodory type and satisfies some weakly compactness condition, we establish the existence of random mild solutions. A nontrivial example illustrating our main result is also given.

Measure of noncompactness for nonlinear Hilfer fractional differential equation with nonlocal Riemann–Liouville integral boundary conditions in Banach spaces

This paper investigates the existence results and uniqueness of solutions for a class of boundary value problems for fractional differential equations with the Hilfer fractional derivative. The reasoning is mainly based upon Mönch’s fixed point theorem associated with the technique of measure of noncompactness. We illustrate our main findings, with a particular case example, included to show the applicability of our outcomes. The boundary conditions introduced in this work are of quite general nature and reduce to many special cases by fixing the parameters involved in the conditions.

Caputo-Hadamard fractional differential equations in banach spaces

This article deals with some existence results for a class of Caputo–Hadamard fractional differential equations. The results are based on the Mönch’s fixed point theorem associated with the technique of measure of noncompactness. Two illustrative examples are presented.

Fractional differential inclusions of Hilfer type under weak topologies in Banach spaces

In this paper, we present some results concerning the existence of weak solutions for some functional Hilfer and Hadamard fractional differential inclusions. The Mönch’s fixed point theorem and the concept of measure of weak noncompactness are the main tools used to carry out our results.

Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

In this paper, we present the existence result for Carathéodory type solutions for the nonlinear Sturm- Liouville boundary value problem (SLBVP) in Banach spaces on an arbitrary time scale. For this purpose, we introduce an equivalent integral operator to the SLBVP by means of Green’s function on an appropriate set. By imposing the regularity conditions expressed in terms of Kuratowski measure of noncompactness, we prove the existence of the fixed points of the equivalent integral operator. Mönch’s fixed point theorem is used to prove the main result. Finally, we also remark that it is straightforward to guarantee the existence of Carathéodory solutions for the SLBVP if Kuratowski measure of noncompactness is replaced by any axiomatic measure of noncompactness.

Weak Solutions for Nonlinear Fractional Differential Equations in Banach Spaces

We discuss the existence of weak solutions for a nonlinear boundary value problem of fractional differential equations in Banach space. Our analysis relies on the Mönch's fixed point theorem combined with the technique of measures of weak noncompactness.