Operator Equation and Application of Variation Iterative Method

Ning Chen, Jiqian Chen

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

DOI: 10.4236/am.2012.38127                

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 (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.

Keywords

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	D. Guo and V. Lashmikantham, “Nonlinear Problems in abstract Cones,” Academic Press, Inc., Boston, New York, 1988.
[2]	Y. J. Cui, F. Wang and Y. M. Zou, “Computation for the Fixed Index and Its Applications,” Nonlinear Analysis, Vol. 71, No. 1-2, 2009, pp. 219-226. doi:10.1016/j.na.2008.10.041
[3]	S. Y. Xu, “New Fixed Point Theorems for 1-Set-Contractive Operators in Banach Spaces,” Nonlinear Analysis, Vol. 67, No. 3, 2007, pp. 938-944. doi:10.1016/j.na.2006.06.051
[4]	N. Van Luong and N. X. Thuan, “Coupled Fixed Points in Partial Ordered Metric Spaces and Application,” Nonlinear Analysis, Vol. 74, No. 3, 2011, pp. 983-992. doi:10.1016/j.na.2010.09.055
[5]	N. Chen, and J. Q. Chen, “New Fixed Point Theorems for 1-Set-Contractive Operators in Banach Spaces,” Nonlinear Analysis, Vol. 6, No. 3, 2011, pp. 147-162.
[6]	G. Z. Li, “The Fixed Point Index and the Fixed Point Theorems of 1-Set-Contrac-Tive Mappings,” Proceedings of the American Mathematical Society, Vol. 104, No. 4, 1988, pp. 1163-1170. doi:10.1090/S0002-9939-1988-0969052-9
[7]	C. X. Zhu and Z. B. Xu, “Inequality and Solution of an Operator Equation,” Applied Mathematics Letters, Vol. 21, No. 6, 2008, pp. 607-611. doi:10.1016/j.aml.2007.07.013
[8]	R. Saadati, M. Dehghan, S. M. Vaezpour and M. Saravi, “The Convergence of He’s Variational Iteration for Solving Integral Equations,” Computers & Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2167-2171. doi:10.1016/j.camwa.2009.03.008
[9]	Y. F. Xu, “The Variational Iteration Method for Autonomous Ordinary Differential Equations with Fractional Order,” Journal of Hubei University Nationalities (Nature Science Edition), Vol. 29, No. 3, 2011, pp. 245-249.
[10]	G. B. Asghar and S. N. Jafar, “An Effective Modification of He’s Variational Iteration Method,” Nonlinear Analysis: Real World Application, Vol. 10, No. 5, 2009, pp. 2828-2833. doi:10.1016/j.nonrwa.2008.08.008
[11]	J. H. He, “Variational Iteration Approach to Schrodinger Equation,” Acta Mathematica Scienca, Vol. 21A, 2001, pp. 577-583.
[12]	S .Q. Wang and J. H. He, “Variational Iterative Method for Solving Integro-Differential Equations,” Physics Letters A, Vol. 367, No. 3, 2007, pp. 188-191. doi:10.1016/j.physleta.2007.02.049
[13]	Z. Z. Zhang and S. R. Lu, “Numerical Solution of Schrodinger Equation,” Journal of Shanxi Daton Universeity, Vol. 26, No. 2, 2010, pp. 22-24.
[14]	Y. F. Wang and L. B. Tang, “Direct Solution of One-Dimensional Schrodinger Equation through Finite Difference and MATLAB Matrix Computation,” INFRARED (MONTHLY), Vol. 31, No. 3, 2010, pp. 42-46.