Abstract:
We obtain a result by combining three prevalent trends of the fixed point theory, namely (i) replacement of the Lipschitz constants in contraction inequality by functions, (ii) considerations of functions without continuity assumption and (iii) use of binary relations in the space. Specifically, we define a Mizoguchi–Takahashi–Kannan type contraction, which is shown to have fixed points in a metric space with an appro- priate binary relation. The issue of the uniqueness of fixed point is separately considered. There are two illustrative examples, in one of which the discontinuity of the function occurs at a fixed point. We discuss Hyers–Ulam–Rassias stability of the fixed point problem and also establish a data dependence result.
Keywords:fixed point, MT -function, Kannan type contraction, binary relation, Hyers– Ulam–Rassias stability, data dependence
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