Fixed point of nonlinear contractions in modular spaces

Rudong Chen,Xue Wang

doi:10.1186/1029-242x-2013-399

Rudong Chen, Xue Wang

Open Access

https://doi.org/10.1186/1029-242x-2013-399

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Abstract

Razani et al. studied the fixed points of nonlinear and asymptotic contractions in the modular space in 2007. In this paper, we generalize the kind of nonlinear contraction that is the result of Razani et al. (Abstr. Appl. Anal. 2007:40575, 2007) and prove the existence and uniqueness of fixed points for the generalized nonlinear contractions in modular spaces.

Highlights

1 Introduction The notion of modular spaces, as a generalization of metric spaces, was introduced by Nakano [ ] in in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz [ ] in. These spaces were developed following the successful theory of Orlicz spaces, which replaces the particular integral form of the nonlinear functional, which controls the growth of members of the space, by an abstractly given functional with some good properties
For a current review of the theory of Musielak-Orlicz spaces and modular spaces, the reader is referred to the books of Musielak [ ] and Kozlowski [ ]
Fixed point theorems in modular spaces, generalizing the classical Banach fixed point theorem in metric spaces, have been studied extensively

Summary

Introduction

The notion of modular spaces, as a generalization of metric spaces, was introduced by Nakano [ ] in in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz [ ] in. In , Razani et al [ ] studied some fixed points of nonlinear and asymptotic contractions in the modular spaces. In , Kuaket and Kumam [ ] proved the existence of fixed points of asymptotic pointwise contractions in modular spaces.

Results

Conclusion