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We establish existence results related to approximate fixed point property of special types of set-valued contraction mappings, in the setting of b-metric spaces. As consequences of the main theorem, we give some fixed point results which generalize and extend various fixed point theorems in the existing literature. A simple example illustrates the new theory. Finally, we apply our results to establishing the existence of solution for some differential and integral problems.

Approximate fixed points of set-valued mapping in b-metric space