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

‎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.‎