# A New Version of Schauder and Petryshyn Type Fixed Point Theorems in S-Modular Function Spaces

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1**

## 2. Preliminaries

- (a)
- $\rho \left(x\right)=0$ if and only if $x=0$;
- (b)
- $\rho \left(\alpha x\right)=\rho \left(x\right)$ for all scalar $\alpha $ with $\left|\alpha \right|=1$;
- (c)
- for all $x,y\in X$, $\rho (\alpha x+\beta y)\le \rho \left(x\right)+\rho \left(y\right)$ if $\alpha +\beta =1$ for any $\alpha ,\beta \ge 0$;

- (c’)
- $\rho (\alpha x+\beta y)\le {\alpha}^{s}\rho \left(x\right)+{\beta}^{s}\rho \left(y\right)$ if ${\alpha}^{s}+{\beta}^{s}=1$ for any $\alpha ,\beta \ge 0,$

**Definition**

**1.**

## 3. Main Results

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**2.**

**Proof.**

**Case-1**: Let $s=1$. By putting $c={{t}_{n}}^{\frac{-1}{2}}$ and $k={{t}_{n}}^{\frac{1}{2}}$, we have

**Case-2**: Let $0<s<1$. Set $c={t}_{n}^{-1}$ and $k={t}_{n}^{\frac{1-s}{s}}$, we obtain

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Application

**Lemma**

**1**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Nakano, H. Modulared Semi-Ordered Linear Spaces; Tokyo Mathematical Book Series; Maruzen Co. Ltd.: Tokyo, Japan, 1950. [Google Scholar]
- Musielak, J.; Orlicz, W. On modular spaces. Stud. Math.
**1959**, 18, 49–65. [Google Scholar] [CrossRef] - Krasnoselskii, M.A.; Rutickii, Y.B. Convex Functions and Orlicz Spaces; Noordhoff Ltd.: Groningen, The Netherlands, 1961. [Google Scholar]
- Kozlowski, W.M. Modular Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics; Marcel Dekker: New York, NY, USA, 1988; Volume 122. [Google Scholar]
- Khamsi, M.A.; Kozlowski, W.M. Fixed Point Theory in Modular Function Spaces; Springer: New York, NY, USA, 2015. [Google Scholar]
- Khamsi, M.A.; Kozlowski, W.M.; Reich, S. Fixed point theory in modular function spaces. Nonlinear Anal.
**1990**, 14, 935–953. [Google Scholar] [CrossRef] [Green Version] - Taleb, A.; Hanebaly, E. A fixed point theorem and its application to integral equations in modular function spaces. Proc. Am. Math. Soc.
**1999**, 128, 419–426. [Google Scholar] [CrossRef] - Ding, G.G. New Theory in Functional Analysis; Academic Press: Beijing, China, 2007. [Google Scholar]

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## Share and Cite

**MDPI and ACS Style**

Ramezani, M.; Baghani, H.; Ege, O.; De la Sen, M.
A New Version of Schauder and Petryshyn Type Fixed Point Theorems in *S*-Modular Function Spaces. *Symmetry* **2020**, *12*, 15.
https://doi.org/10.3390/sym12010015

**AMA Style**

Ramezani M, Baghani H, Ege O, De la Sen M.
A New Version of Schauder and Petryshyn Type Fixed Point Theorems in *S*-Modular Function Spaces. *Symmetry*. 2020; 12(1):15.
https://doi.org/10.3390/sym12010015

**Chicago/Turabian Style**

Ramezani, Maryam, Hamid Baghani, Ozgur Ege, and Manuel De la Sen.
2020. "A New Version of Schauder and Petryshyn Type Fixed Point Theorems in *S*-Modular Function Spaces" *Symmetry* 12, no. 1: 15.
https://doi.org/10.3390/sym12010015