Fixed point theorems for a sum of two mappings in locally convex spaces

International Journal of Mathematics and Mathematical Sciences 01/1994; DOI: 10.1155/S0161171294000967
Source: DOAJ


Cain and Nashed generalized to locally convex spaces a well known fixed point theorem of Krasnoselskii for a sum of contraction and compact mappings in Banach spaces. The class of asymptotically nonexpansive mappings includes properly the class of nonexpansive mappings as well as the class of contraction mappings. In this paper, we prove by using the same method some results concerning the existence of fixed points for a sum of nonexpansive and continuous mappings and also a sum of asymptotically nonexpansive and continuous mappings in locally convex spaces. These results extend a result of Cain and Nashed.

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    • "The freedom of choosing a more general notion of topology might remedy this difficulty. We should mention that others authors have already studied equation (1.1) in locally convex spaces [7] [23]. "
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    ABSTRACT: In this paper, we establish a fixed point result of Krasnoselskii type for the sum A+B, where A and B are continuous maps acting on locally convex spaces. Our results extend previous ones. We apply such results to obtain strong solutions for some quasi-linear elliptic equations with lack of compactness. We also provide an application to the existence and regularity theory of solutions to a nonlinear integral equation modeled in a Banach space. In the last section we develop a sequentially weak continuity result for a class of operators acting on vector-valued Lebesgue spaces. Such a result is used together with a geometric condition as the main tool to provide an existence theory for nonlinear integral equations in Lp(E).
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    ABSTRACT: In this article, we prove some results concerning the Krasnoselskii theorem on fixed points for the sum A + B of a weakly-strongly continuous mapping and an asymptotically nonexpansive mapping in Banach spaces. Our results encompass a number of previously known generalizations of the theorem.
    Journal of Inequalities and Applications 07/2011; 2011(1). DOI:10.1186/1029-242X-2011-28 · 0.77 Impact Factor
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