Some Variants of Krasnoselskii-Type Fixed Point Results for Equiexpansive Mappings with Applications

In this paper, we investigate the Krasnoselskii-type fixed point results for the operator F of two variables by assuming that the family F x , . : x is equiexpansive. The results may be considered as variants of the Krasnoselskii fixed point theorem in a general setting. We use our main results to obtain the existence of solutions of a fractional evolution differential equation. An example of a controlled system is given to illustrate the application.

The homoclinic breather wave solution, rational wave and n-soliton solution to a nonlinear differential equation

According to the homoclinic breather limit method, we obtain the homoclinic breather wave and rational wave of a nonlinear evolution differential equation. The n-soliton wave solutions are derived by utilizing the Hirota method. In addition, the graphs of these solutions are shown by selecting the appropriate parameters.

Existence of the Mild Solutions for Impulsive Fractional Equations with Infinite Delay

This paper is concerned with the existence and uniqueness of a mild solution of a semilinear fractional-order functional evolution differential equation with the infinite delay and impulsive effects. The existence and uniqueness of a mild solution is established using a solution operator and the classical fixed-point theorems.

A Parallel High-accuracy Method for the First-order Evolution Equation in Hilbert and Banach Spaces

We have developed a discretization method of an arbitrarily given order of accuracy with respect to the time discretization parameter for the first-order evolution differential equation in Banach space. The method includes two levels of parallelism: the operator exponential needed for the calculation of the evolution operator can be computed in parallel (inner parallelism) and then we can compute in parallel the evolution operator at various points of the time mesh (outer parallelism).