On Local Generalized Ulam–Hyers Stability for Nonlinear Fractional Functional Differential Equation

We discuss the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Well-known Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. The simulation of an example is also given to show the applicability of our results.

Quasi-compactness of linear operators on Banach spaces: New properties and application to Markov chains

The purpose of this paper is to study the notion of quasi-compact linear operators acting in a Banach space. This class of operators contains the set of compact, polynomially compact, quasi-nilpotent and that of all Riesz operators. We show the equivalence between different definitions of quasi-compactness known in the mathematical literature and we present several general theorems about quasi-compact endomorphisms: stability under algebraic operations, extension of Schauder theorem and the Fredholm alternative. We also study the question of existence of invariant subspaces and we examine the class of semigroups for quasi-compact operators. The obtained results are used to describe Markov chains.

Infinitely many singular radial solutions for quasilinear elliptic systems

We prove the existence of infinitely many singular radial positive solutions for a quasilinear elliptic system with no variational structure [Formula: see text] where [Formula: see text] is the unit ball of [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text] are non-negative functions. We separate two fundamental classes (the sublinear and superlinear class), and we use respectively the Leray–Schauder Theorem and a method of monotone iterations to obtain the existence of many solutions with a property of singularity around the origin. Finally, we give a sufficient condition for the non-existence.

On the existence of classical solutions for differential-functional IBVP

We consider the initial-boundary value problem for second order differential-functional equations of parabolic type. Functional dependence in the equation is of the Hale type. By using Leray-Schauder theorem we prove the existence of classical solutions. Our formulation and results cover a large class of parabolic problems both with a deviated argument and integro-differential equations.