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The main objective of this work is to introduce a new type of non-linear contraction via C-class functions by using ?-admissible mapping. Our new results extend and generalize the very recent results of Karapinar and Sadarangani (2015. RACSAM. [37]). Illustrative examples are given to support our new findings. We have shown that our results satisfy the periodic fixed point results after modifying the contraction. Next, we extend our main findings from a self-mapping T to two self-mappings T; S. Also, an example is provided to justify the effectiveness of our new result on two self mappings, where the partially ordered structure fails. Finally, we apply our new findings to solve ordinary differential and non-linear integral equations.