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

In this paper, based on [a. razani, v. rakocevic 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 xp 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 xp, where c is a nonempty and complete subset of xp, 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;lφ) is presented, where c(i;lφ) denotes the space of all continuous function from i to lφ, lφ is the musielak-orlicz space and i = [0; b] 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 δ2-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