Nonlinear integral equations, krasnoselskii fixed point theorem, ascoliarzela theorem. In the literature, there appears a huge number of papers studying the fixed point problems for the sum of two operators. The following krasoselskii type fixed point theorem is proved in dhage. Corrigendum to krasnosel skii type hybrid fixed point theorems and their applications to fractional integral equations h. Every contraction mapping on a complete metric space has a unique xed point. Some hybrid fixed point theorems of krasnoselskii type, which involve product of two operators, are proved in partially ordered normed linear spaces. First we show that t can have at most one xed point. As an application, the existence of solutions for perturbed integral equation is considered in p normed spaces.
Some previous and related results are extended and developed. Krasnoselskiitype fixed point theorems with applications to volterra integral. Sep 20, 2014 our fixed point results encompass the well known sadovskiis fixed point theorem and a number of its generalizations. Severalcorrelationsbetweentherelativecompactnessofthesetfe,k. Some extensions of the krasnoselskii fixed point theorems. The statement of the krasnosel skii twin fixed point theorem appears below and a proof can be found in 11. In particular, one new theorem complements the wellknown krasnoselskii fixedpoint theorem in the sense that the mapping t is expansive rather than contractive. There are diverse extensions of krasnoselskii fixed point theorem in the literature, and the operator b is often required to be contractive or expansive, or of any typical form involving control. As an application, the existence of solutions for perturbed integral equation is considered in pnormed spaces. This is also called the contraction mapping theorem. To show the usefulness and the applicability of our fixed point results we investigate the existence of mild solutions to a. The above refined formulations of the krasnoselskii fixed point theorems allow us to derive new classes of krasnoselskii type theorems.
We refer the reader who is interested in the fixed point theorems involving weakly continuous mapping to oregan 18,19 and the references therein. Corollary to the five functionals fixed point theorem. Pdf a class of expansivetype krasnoselskii fixed point theorem. A fixed point theorem of krasnoselskiischaefer type. Here, as an illustration, we apply corollary 2 of 16 together with theorem 2. Browderkrasnoselskiitype fixed point theorems in banach spaces. We delete the proof of theorem 14 and corollary 15 and rewrite the statements as follows. A generalization of krasnoselskii fixed point theorem for sums of compact and contractible maps with application.
Supposethecontrary,thenthereexistsaneighborhooduofit. In this paper, we investigate the fixed point problem of the sum of an expansive mapping and a compact mapping. Pdf a generalization of krasnoselskii fixed point theorem. Vedak no part of this book may be reproduced in any form by print, micro. This result is applied to prove the existence of asymptotically stable. Schaefers fixed point theorem will yield a tperiodic solution of 0. Nov 11, 2006 we prove a fixed point theorem which is a combination of the contraction mapping theorem and schaefers theorem which yields a t. These hybrid fixed point theorems are then applied to fractional integral equations for proving the. On the krasnoselskiitype fixed point theorems for the sum of continuous and asymptotically nonexpansive mappings in banach spaces.
Kis continuous, then there exists some c2ksuch that fc c. Research article krasnoselskiitype fixedset results. A note on krasnoselskii fixed point theorem springerlink. A generalization of krasnoselskii fixed point theorem for. The idea of the paper is to combine the notion of meirkeeler mapping and expansive krasnoselskii fixed point theorem. On the krasnoselskiitype fixed point theorems for the sum of. There are diverse extensions of krasnoselskii fixed point theorem in the literature, and the operator b is often required to be contractive or expansive, or. Pdf krasnoselskii type hybrid fixed point theorems and. A note on krasnoselskii fixed point theorem a note on krasnoselskii fixed point theorem. In 2005, hajji and hanebaly 7 presented a modular version of krasnoselskii fixed point theorem, for a i i.
This necessitates to search for a new fixed point theorem of krasnoselskii s type to deal with this problem. We first state a theorem of ambrosetti type see 17 for a proof. We now state the basic hybrid fixed point results by bedre et al. On krasnoselskiis cone fixed point theorem man kam kwong1,2 1 department of applied mathematics, the hong kong polytechnic university, hunghom, hong kong 2 department of mathematics, statistics, and computer science, university of illinois, chicago, il 606077045, usa. Existence of multiple positive solutions for nth order two. Some krasnoselskiitype fixed point theorems for meirkeeler. Lectures on some fixed point theorems of functional analysis by f.
An application of krasnoselskii fixed point theorem to. We also prove the sadovskii theorem for convex sets in normed spaces, where, and from it we obtain some fixed point theorems for the sum of two. Pdf critical type of krasnoselskii fixed point theorem. We prove a fixed point theorem which is a combination of the contraction mapping theorem and schaefers theorem which yields a t. Krasnoselskii type hybrid fixed point theorems and their. The analysis presented here reveals the essential characteristics of the krasnosel skii type fixed point theorem in strong topology setups. Krasnosel skii also presented many new general principles on solvability of a large variety of nonlinear equations, including onesided estimates, cone stretching and contractions, fixed point theorems for monotone operators and a combination of the schauder fixed point and contraction mapping theorems that was the genesis of condensing. Further, the results are used to prove the existence of periodic solutions of a nonlinear neutral differential equation with delay in the critical case. There we first relaxed the compactness assumption of s and derived a generalized noncompacttype krasnoselskii fixed point theorem. Some krasnoselskiitype fixed point theorems for meir. A fixed point theorem of krasnoselskiischaefer type burton. Krasnoselskii fixed point theorem for weakly continuous maps. Accepted august 2006 this paper presents a remark on a. Existence of positive solutions for m point boundary value problem for nonlinear fractional differential equation elshahed, moustafa and shammakh, wafa m.
Corrigendum to krasnoselskii type hybrid fixed point. We prove krasnosel skii type fixed point theorems in situations where the domain is not necessarily convex. Krasnoselskiitype fixed point theorems with applications to volterra. In this paper we have obtained some fixed point theorems on b metric space which is an extension of a fixed point theorem by hardy and reich 20. The theorem of krasnoselskii has been extended by many authors, for example, we refer to 14, 6, 7 and references therein. Xiangandyuanfixedpointtheoryandapplications20152015. Our results extend and complement the classical krasnoselskii fixed point theorem. A class of expansivetype krasnoselskii fixed point. Desale 5 department of mathematics and statistics, university of. We give the greens functions for the system with boundary conditions, and then obtain some useful properties of the greens functions.
In recent years, the krasnoselskii fixed point theorem for cone maps and its. In the last section, as an application of a krasnoselskii type theorem. The theory of krasnoselskii type fixed point theorem is initiated by dhage. Krasnoselskii s fixed point theorem for weakly continuous maps. The krasnoselskii fixed point theorem appears as a prototype for solving equations of the form and it initiates vast investigations in the direction of such mixed type problems.
Corrigendum corrigendum to krasnosel skii type hybrid fixed point theorems and their applications to fractional integral equations h. A note on krasnoselskii fixed point theorem fixed point. Krasnoselskii fixed point theorem for weakly continuous. An extension of leggettwilliams normtype theorem for coincidences and its application yang, aijun, topological methods in nonlinear analysis, 2011. Corrigendum corrigendum to krasnosel skii type hybrid. Krasnoselskii, proved the following fixed point theorem which is an important supplement to both the schauder fixed point theorem and the banach contraction principle. Krasnosel skii type hybrid fixed point theorems and their applications to fractional integral equations h.
We establish a criterion for the existence of at least one positive solution by utilizing krasnoselskii fixed point theorem. This paper presents a remark on a fixed point theorem of krasnoselskii type. Krasnoselskii type fixed point theorem for nonlinear expansion fuli wang and feng wang school of mathematics and physics changzhou university changzhou 2164, china email. The slight generalization of theorem 4 and dhage 8 using contraction is stated as follows.
Krasnoselskii type fixed point theorems for nonlinear expansion the main result of this section is the following krasnoselskii type. As is well known, krasnoselskiis fixed point theorem has a wide range. In the first part of this paper, we revisit the krasnoselskii theorem. The obtained results unify and significantly extend a number of previously known generalizations of the krasnosel skii fixed point theorem. For purposes of reference we will state without proof the relevant theorems. Generalized krasnoselskii fixed point theorem involving auxiliary functions in bimetric spaces and application to two point boundary value problem. I will only give proof for smooth g, although the milnor book explains how to extend this case to continuous g. In this paper, we prove several new krasnoselskii type. Krasnoselskiitype fixed point theorems with applications to. Krasnoselskii fixed point theorem for dissipative operators article pdf available in electronic journal of differential equations 2011147 november 2011.
In this paper, we establish some new fixed point theorems for the sum of two operators. In this paper, inspired by the idea of meirkeeler contractive mappings, we introduce meirkeeler expansive mappings, say mke, in order to obtain krasnoselskii type fixed point theorems in banach spaces. Research article krasnosel skii type hybrid fixed point. Multivalued versions of a krasnoselskiitype fixed point. In this paper we focus on three fixed point theorems and an integral equation. Pdf in this paper, we establish some new fixed point theorems for the sum of two operators. Fixed point theorems concern maps f of a set x into itself that, under certain conditions, admit a. We also prove the sadovskii theorem for sconvex sets in pnormed spaces, where 0 point theorems for the sum of two mappings. In this paper, we establish a single and multivalued version of a perov type fixed point theorem.
Most nice results are based on some clever selection. In particular, one new theorem complements the wellknown krasnoselskii fixed point theorem in the sense that the mapping t is expansive rather than contractive. An extension of krasnoselskiis fixed point theorem for contractions and compact mappings george l. We also prove the sadovskii theorem for sconvex sets in pnormed spaces, where 0 krasnoselskii type theorem. Krasnoselskiis fixed point theorem appeared as a prototyped for. Our fixed point results encompass the well known sadovskiis fixed point theorem and a number of its generalizations. Corollary of the five unctionalsf fixed point theorem with the su ciency conditions attained applying the krasnosel skii twin fixed point theorem at the end of the paper. Ams proceedings of the american mathematical society. On a fixed point theorem of krasnoselskii type and application to integral equations le thi phuong ngoc and nguyen thanh long received 15 april 2006. Positive solutions for a system of fourthorder differential. Corrigendum corrigendum to krasnosel skii type hybrid fixed. These hybrid fixed point theorems are then applied to fractional integral equations for proving the existence of solutions under certain monotonicity conditions blending with the existence of the upper or lower solution. Fixed point theorem, separate contraction mapping, periodic solution. Desale 5 department of mathematics and statistics, university of victoria, victoria, bc, canada vw r.
In particular, our results encompass the analogues of krasnoselskii s and sadovskiis fixed point theorems for sequentially weakly continuous mappings and a number of their generalizations. Sadovskiikrasnoselskii type fixed point theorems in. In recent years, the krasnoselskii fixed point theorem for cone maps and its many generalizations have been successfully applied to establish the existence of multiple solutions in the study of boundary value problems of various types. The krasnosel skii fixed point theorem appears as a prototype for solving equations of the form 1.
We prove krasnoselskii type fixed point theorems in situations where the domain is not necessarily convex. Research article krasnosel skii type hybrid fixed point theorems and their applications to fractional integral equations h. To show the usefulness and the applicability of our fixed point results we investigate the existence of mild solutions to a broad class of neutral differential equations. Some variants of a fixed point theorem of krasnoselskii and. In contrast, the contraction mapping theorem section3 imposes a strong continuity condition on f but only very weak conditions on x. By using the guokrasnoselskii fixedpoint theorem and the greens functions, some sufficient conditions for the existence of positive solutions are presented. A class of expansivetype krasnoselskii fixed point theorems.
Krasnoselskiitype fixed point theorems with applications. Lerayschaudertype fixed point theorems in banach algebras and application. The analysis presented here reveals the essential characteristics of the krasnoselskii type fixed point theorem in strong topology setups. Let be a nonempty, closed, convex, and bounded subset of the banach algebra. Finally we will give an example to illustrate our result. Sadovskiikrasnoselskii type fixed point theorems in banach. Krasnoselskii type fixed point theorems and applications yicheng liu and zhixiang li communicated by david s. Lectures on some fixed point theorems of functional analysis. Our results encompass a number of previously known generalizations of the theorem. A fixedpoint theorem of krasnoselskii sciencedirect. Dec 01, 2012 read a generalization of krasnoselskii fixed point theorem for sums of compact and contractible maps with application, open mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Vedak no part of this book may be reproduced in any.