Measures of noncompactness in the study of solutions of nonlinear differential and integral equations

Abstract

The aim of this paper is to make an overview of some existence results for nonlinear differential and integral equations. Those results were obtained by the author and his co-workers during last years with some help of the technique of measures of noncompactness and a fixed point theorem of Darbo type. MSC: 47H08Keywords

Asymptotically Stable Solutions of a Generalized Fractional Quadratic Functional-Integral Equation of Erdélyi-Kober Type

“…Next, we give the construction of the measure of noncompactness in (R + ) which will be used as main tool of the proof of our main result; see [37,38] and references therein.…”

Some fixed point results via measure of noncompactness

“…In both theorems the notion of the compactness plays an essential role. On the other hand, the notion of a measure of noncompactness appears in various contexts and played an important role in several branch of mathematics, in particular, nonlinear functional analysis (see, for instance, [3,5,11,13,14]). Consequently, a natural question was appeared: "is it possible to get a fixed point if the compactness is dropped in well-known Schauder fixed point theorem".…”

Periodic solutions of Weyl fractional order integral systems†

“…Fractional integral equations are widely used to describe the medium with non-integer mass dimension in physics, viscoelasticity, electrochemistry and porous media. There are a lot of papers dealing with existence about solutions of Riemann-Liouville type or Erdélyi-Kober type fractional integral equations, fractional differential equations, and fractional optimal control problems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].…”