Fixed point theorem for non-self mappings and its applications in the modular ‎space

Subject Areas : International Journal of Industrial Mathematics

R. Moradi ¹ , A. ‎Razani ²

1 - Department of Mathematics, Faculty of Science, Imam Khomeini International University, Postal code: 34149-16818, Qazvin, Iran.
2 - Department of Mathematics, Faculty of Science, Imam Khomeini International University, Postal code: 34149-16818, Qazvin, Iran.

Received: 2016-02-12 Accepted : 2016-02-12 Published : 2016-04-01

Keywords: Modular space, Non-self mappings, ‎Q‎uasi contraction, Krasnoseleskii's fixed point t, Integral&#x0D, &#x0D, ‎equation.‎,

Abstract :

‎In this paper, based on [A. Razani, V. Rako$\check{c}$evi$\acute{c}$ and Z. Goodarzi, Nonself mappings in modular spaces and common fixed point theorems, Cent. Eur. J. Math. 2 (2010) 357-366.] a fixed point theorem for non-self contraction mapping $T$ in the modular space $X_\rho$ is presented. Moreover, we study a new version of Krasnoseleskii's fixed point theorem for $S+T$, where $T$ is a continuous non-self contraction mapping and $S$ is continuous mapping such that $S(C)$ resides in a compact subset of $X_\rho$, where $C$ is a nonempty and complete subset of $X_\rho$, also $C$ is not bounded. Our result extends and improves the result announced by Hajji and Hanebally [A. Hajji and E. Hanebaly, Fixed point theorem and its application to perturbed integral equations in modular function spaces, Electron. J. Differ. Equ. 2005 (2005) 1-11]. As an application, the existence of a solution of a nonlinear integral equation on $C(I, L^\varphi) $ is presented, where $C(I, L^\varphi)$ denotes the space of all continuous function from $I$ to $L^\varphi$, $L^\varphi$ is the Musielak-Orlicz space and $I=[0,b] \subset \mathbb{R}$. In addition, the concept of quasi contraction non-self mapping in modular space is introduced. Then the existence of a fixed point of these kinds of mapping without $\Delta_2$-condition is proved. Finally, a three step iterative sequence for non-self mapping is introduced and the strong convergence of this iterative sequence is studied. Our theorem improves and generalized recent know results in the ‎literature.‎