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

در اين مقاله، بر اساس مقاله [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.] قضيه نقطه ثابت اي براي غيرخود نگاشت Tدر فضاي مدلار X?ارايه شده است. بعلاوه، شكل جديدي از قضيه نقطه ثابت Krasnoseleskiiiبراي نگاشت S+Tكه در آن Tيكغيرخود نگاشت انقباضي پيوسته و Sنگاشت پيوسته به قسمي است كه ) S(Cدر يك زير مجموعه فشرده X?قرار دارد، كه در آن Cزير مجموعه غيرتھي كامل و بي كران از X?مي باشد. نتايج حاضر، نتايج ارايه شده توسط Hajji and Haneballyدر مقاله [A.Hajji and E. Hanebally, Fixed point theorem and its application to perturbed integral equations in modular function spaces, Electron. J. Differ. Equ. 2005 (2005) 1-11] را بھبود و تعمبم مي دھد. بعنوان يك كاربرد، وجود جواب يك معادله انتگرال غير خطي روي ) C(I,L?ارايه شده است، كه در آن ) C(I,L?نشان دھنده فضاي ھمه توابع پيوسته از Iبه L? ،L?فضاي Musielak-Orliczو .I=[0,b] ?R بعلاوه، مفھوم غيرخود نگاشت شبه انقباضي در فضاي مدولار معرفي مي شود. سپس وجود يك نقطه ثابت براي اين گونه نگاشت بدون شرط ?2ثابت مي شود. در پايان، يك دنباله سه مرحله اي تكرار شونده براي يك غيرخود نگاشت معرفي شده و ھمگرايي قوي اين دنباله تكرار شونده مطالعه مي شود. قضيه ارايه شده، نتايج موجود را، بھبود و تعميم مي دھد

‎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 Krasnoseleskiiʹs 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.‎