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Fixed point results in b-metric spaces approach to the existence of a solution for nonlinear integral equations

  • Wutiphol SintunavaratEmail author
Original Paper

Abstract

The purpose of this work is to introduce new nonlinear mappings in setup of b-metric spaces and prove fixed point theorems for such mappings. Examples are provided in order to distinguish these results from the known ones. At the end of paper, we apply our fixed point result to prove the existence of a solution for the following nonlinear integral equation:
$$\begin{aligned} x(c) = \Omega (\phi (c),c ) + K(c,c,\phi (c))+\int ^b_a K(c,r,x(r))dr, \end{aligned}$$
(0.1)
where \(a,b\in \mathbb {R}\) with \(a<b\), \(x \in C[a,b]\) (the set of all continuous real functions defined on [ab]), \(\phi :[a,b]\rightarrow \mathbb {R}\), \(\Omega :\mathbb {R} \times [a,b]\rightarrow \mathbb {R}\) and \(K : [a,b] \times [a,b] \times \mathbb {R} \rightarrow \mathbb {R}\) are given mappings.

Keywords

\(\alpha \)-Admissible mappings \(\alpha \)-Regularity b-metric spaces Hölder inequality Nonlinear integral equations 

Mathematics Subject Classification

47H09 47H10 

Notes

Acknowledgments

The author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript.

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Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of Science and TechnologyThammasat University Rangsit CenterPathumthaniThailand

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