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. ۲ (۲۰۱۰) ۳۵۷-۳۶۶.] 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. ۲۰۰۵ (۲۰۰۵) ۱-۱۱]. 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=[۰,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_۲-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.‎

کلیدواژه ها:

Modular space ، Non-self mappings ، ‎Q‎uasi contraction ، Krasnoseleskii's fixed point theorem ، Integral ‎equation.‎