Operator Equation and Application of Variation Iterative Method

Abstract

In this paper, we study some semi-closed 1-set-contractive operators A and investigate the boundary conditions under which the topological degrees of 1-set contractive fields, deg  are equal to 1. Correspondingly, we can obtain some new fixed point theorems for 1-set-contractive operators which extend and improve many famous theorems such as the Leray-Schauder theorem, and operator equation, etc. Lemma 2.1 generalizes the famous theorem. The calculation of topological degrees and index are important things, which combine the existence of solution of for integration and differential equation and or approximation by iteration technique. So, we apply the effective modification of He's variation iteration method to solve some nonlinear and linear equations are proceed to examine some a class of integral-differential equations, to illustrate the effectiveness and convenience of this method.

Some Approximation in Cone Metric Space and Variational Iterative Method

“…In this section, we apply the variation iteration method (simple writing VIM) to Integral-differential equations below (see [6,7]). To illustrate the basic idea of the method, we consider:…”

“…In this section, we apply the effective modification method of He's VIM to solve some integral-differential equations. In [6][7][8] .…”

“…The variational iteration method (VIM) proposed by Ji-Huan He (see [6][7][8] recently has been intensively studied by scientists and engineers. the references cited therein) is one of the methods which have received much concern.…”

“…Fixed point theory has fascinated many researchers since 1922 with the celebrated Banach's fixed point theorem. There exist a fast literature on the topic and this is a very active field of research at present (See [1][2][3][4][5][6][7]).…”

“…First, we state following some extend conclusion ( [3,4]). Next, authors consider the variation iterative method to some integral and differential equations, and effective method ( [5][6][7][8] and T then we obtain that theorem 2.1 in [2].…”