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.

Boundary value problem for nonlinear fractional differential equations of variable order via Kuratowski MNC technique

In the present research study, for a given multiterm boundary value problem (BVP) involving the Riemann-Liouville fractional differential equation of variable order, the existence properties are analyzed. To achieve this aim, we firstly investigate some specifications of this kind of variable-order operators, and then we derive the required criteria to confirm the existence of solution and study the stability of the obtained solution in the sense of Ulam-Hyers-Rassias (UHR). All results in this study are established with the help of the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of our observed results.

Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order

A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads to the necessity of obtaining existence criteria for a boundary value problem for Hadamard fractional differential equations of variable order. Also, the stability in the sense of Ulam–Hyers–Rassias is investigated. The results are obtained based on the Kuratowski measure of noncompactness. An example illustrates the validity of the observed results.

On the Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order via Kuratowski MNC Technique

In this manuscript, we examine both the existence and the stability of solutions of the boundary value problems of Hadamard-type fractional differential equations of variable order. New outcomes are obtained in this paper based on the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of the observed results.

Investigation of the neutral fractional differential inclusions of Katugampola-type involving both retarded and advanced arguments via Kuratowski MNC technique

A class of the boundary value problem is investigated in this research work to prove the existence of solutions for the neutral fractional differential inclusions of Katugampola fractional derivative which involves retarded and advanced arguments. New results are obtained in this paper based on the Kuratowski measure of noncompactness for the suggested inclusion neutral system for the first time. On the one hand, this research concerns the set-valued analogue of Mönch fixed point theorem combined with the measure of noncompactness technique in which the right-hand side is convex valued. On the other hand, the nonconvex case is discussed via Covitz and Nadler fixed point theorem. An illustrative example is provided to apply and validate our obtained results.

Successive approximations for a differential equation in a Banach space via Constantin condition

We give some sufficient conditions for the convergence of the sequence of successive approximations to the unique solution of first order Cauchy problem in a Banach space. Our approach is based on a generalized Nagumo condition due to A. Constantin and the properties of the Kuratowski measure of noncompactness.

Local attractivity for integro-differential equations with noncompact semigroups

In this paper, we are devoted to study the existence and local attractivity of solutions for a class of integro-differential equations.Under the situation that the nonlinear term satisfy Carathéodory conditions and a noncompactness measure condition, we establish some existence and local attractivity of mild solutions by utilizing Mönch fixed point theorem, Kuratowski measure of noncompactness and resolvent operator theory in the sense of Grimmer.Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the right-hand real half axis 𝕉+. Moreover an example is given to illustrate our outcomes.

Spectral radius of S-essential spectra

In this paper, we study the spectral radius of some S-essential spectra of a bounded linear operator defined on a Banach space. More precisely, via the concept of measure of noncompactness,we show that for any two bounded linear operators $T$ and $S$ with $S$ non zero and non compact operator the spectral radius of the S-Gustafson, S-Weidmann, S-Kato and S-Wolf essential spectra are given by the following inequalities\begin{equation}\dfrac{\beta(T)}{\alpha(S)}\leq r_{e, S}(T)\leq \dfrac{\alpha(T)}{\beta(S)},\end{equation}where $\alpha(.)$ stands for the Kuratowski measure of noncompactness and $\beta(.)$ is defined in [11].In the particular case when the index of the operator $S$ is equal to zero, we prove the last inequalities for the spectral radius of the S-Schechter essential spectrum. Also, we prove that the spectral radius of the S-Jeribi essential spectrum satisfies inequalities 2) when the Banach space $X$ has no reflexive infinite dimensional subspace and the index of the operator $S$ is equal to zero (the S-Jeribi essential spectrum, introduced in [7]as a generalisation of the Jeribi essential spectrum).