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.‎