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The purpose of this paper is to define new concepts, such as T-orbitally w-completeness, orbitally wcontinuity and almost weakly w-contractive mapping in the modular metric spaces. We prove some fixed point theorems for these related concepts and mappings in this space. Further, we give an application using the technique in [Lj. B.Ćirić, B. Samet, H. Aydi, C. Vetro, Appl. Math. Comput., 218 (2011), 2398-2406 and show that our results can be applied to homotopy.doi:10.22436/jnsa.008.06.01 fatcat:xd4zng6kf5bptmijlvpixjk7t4