Author(s)

Ning Chen, Jiqian Chen

School of Science, Southwest University of Science and Technology, Mianyang, P.R. China.

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 (I-A, Ω, p) 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.

Topology Degrees and Index; 1-Set-Contract Operators; Modified Variation Iteration Method; Integral-Differential Equation

N. Chen and J. Chen, "Operator Equation and Application of Variation Iterative Method," Applied Mathematics, Vol. 3 No. 8, 2012, pp. 857-863. doi: 10.4236/am.2012.38127.