Abstract
In this paper, we introduce the notion of quadratic quasicontractive mapping and prove two generalizations of some classical fixed point theorems. Furthermore, we present some examples to support our main results.
Highlights
In 1962, Edelstein [1] proved the following fixed point theorem.Theorem 1
We have introduced the class of quadratic quasicontractive mapping and prove two generalizations of some classical fixed point theorems: Edelstein’s theorem, Hardy-Rogers’s theorem and Gregus’s theorem
We have presented some examples to support our main results
Summary
In 1962, Edelstein [1] proved the following fixed point theorem.Theorem 1. Let ( X, d) be a compact metric space and let T : X → X be a mapping satisfying inequality d ( Tx, Ty) < A · d ( x, Tx ) + B · d (y, Ty) + C · d ( x, y) for all x, y ∈ X and x 6= y, where A, B, C are positive and A + B + C = 1. A mapping there exists a ∈ 0, 2 such that d2 ( Tx, Ty) ≤ a · d2 ( x, Tx ) + a · d2 (y, Ty) + (1 − 2a) · d2 ( x, y) for all x, y ∈ X and a strict quadratic quasicontraction if in Relation (3) we have the strict inequality for all x, y ∈ X with x 6= y.
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