Ishikawa Iterative Convergence And Mild Solutions Of Differential Equations The Existence Of,

Iterative methods for finding fixed points of nonlinear operator T are an important and active research area. And they have received vast investigation since these methods find applications in a variety of applied areas of inverse problems, the approximate solutions of equations and minimization of a function. So the research on convergence of iterative sequence is very meaningful.The research of semilinear evolution equations with infinite delay began in the 1970s of last century. The pioneering work on semilinear evolution equations with infinite delay is due to Travis and Webb and it has been studied extensively. Equations with delay are often more realistic to describe natural phenomena than those without delay. Therefore it is significance to study the existence of a class of semilinear evolution equations with unbounded delay.The purpose of this paper is to discuss the strong convergence of the modified Ishikawa iteration and establish the existence of mild solutions of the semilinear evolution equations with infinite delay. The results obtained are presented as follows:In the first chapter, we mainly discuss the strong convergence of the modified Ishikawa iteration.In this chapter, we employ the projection PK in Xu H ong -Kun to improve the Ishikawa iteration, and then extend the result to the real smooth and uniformly convex Banach space, we obtain the corresponding conclusions of Ishikawa iteration with the error sequences. As an application, we obtain the conclusions of approximating to the sets of zeros of m -accretive operator A . Also, we extend result for nonexpansive mappings in real smooth and uniformly convex Banach space to the case of asymptotically nonexpansive mappings.In the second chapter, we mainly discuss the existence of mild solutions of the semilinear evolution equations with infinite delay.In this chapter, we prove the existence of the mild solutions of the equations with infinite delay by using the Hausdorff′measure of noncompactness in infinite- dimensional Banach space, when the semigroup {T (t ) : t > 0} generated by the densely defined operator A loses the compactness. The compactness of {T (t ) : t > 0}or f and the Lipachitz condition of f are the special case of our conditions, we can extend and improve some related results.