Fixed point theorems of convex-power 1-set-contraction operators in Banach spaces

Lvhuizi Zhao, Jingxian, Jingxian Sun

Description

In this article, we give the definition of a class of new operators, namely, convex-power 1-set-contraction operators in Banach spaces, and study the existence of fixed points of this class of operators. By using methods of approximation by operators, we obtain fixed point theorems of convex-power 1-set-contraction operators, which generalize fixed point theorems of 1-set-contraction operators in Banach spaces. By using the fixed point theorem, the existence of solutions of nonlinear Sturm-Liouville problems in Banach spaces is investigated under more general conditions than those used in former literatures. Mathematics Subject Classification 2010: 47H10 .

Sujets

• Fixed point theorem
• Banach space
• Sturm?Liouville theory

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 11
• Langue: English

Extrait

Lvhuizi and JingxianFixed Point Theory and Applications2012,2012:56 http://www.fixedpointtheoryandapplications.com/content/2012/1/56

R E S E A R C HOpen Access Fixed point theorems of convexpower 1set contraction operators in Banach spaces * Zhao Lvhuizi and Sun Jingxian

* Correspondence: jxsun7083@163. com Department of Mathematics, Xuzhou Normal University, Xuzhou, China

Abstract In this article, we give the definition of a class of new operators, namely, convex power 1setcontraction operators in Banach spaces, and study the existence of fixed points of this class of operators. By using methods of approximation by operators, we obtain fixed point theorems of convexpower 1setcontraction operators, which generalize fixed point theorems of 1setcontraction operators in Banach spaces. By using the fixed point theorem, the existence of solutions of nonlinear SturmLiouville problems in Banach spaces is investigated under more general conditions than those used in former literatures. Mathematics Subject Classification 2010: 47H10. Keywords:convexpower 1setcontraction, fixed point theorem, Banach spaces, SturmLiouville problems

0 Introduction For the need of studying differential equations and integral equations, Sun and Zhang [1] gave the definition of convexpower condensing operators and obtained the fixed point theorem of this class of operators. Li [2] gave the fixed point theorem of semi closed 1setcontraction operators. In this article, by combinating the definitions of convexpower condensing operators and 1setcontraction operators, we give the definition of convexpower 1setcontrac tion operators in Banach spaces and study the existence of fixed points of this class of new operators. The results in this article generalize the ones in [13]. By using the fixed point theorem, the existence of solutions of nonlinear SturmLiouville problems in Banach spaces is investigated under more general conditions than those used in for mer literatures.

1 Preliminaries Before providing the main results, we introduce some basic definitions and results (see [16]). In this article, we always assume thatEis a Banach space,D⊂E, anda(S) denotes the Kuratowski measure of noncompactness of a bounded setS⊂E. LetA:D®Ebe continuous. If there exists a constantk≥0, such that for any bounded subsetS⊂D, α (A ( S )) ≤kα (S ) .

© 2012 Lvhuizi and Jingxian; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.