A Note on the Górnicki-Proinov Type Contraction

Abstract

In this paper, we propose a notion of the Górnicki-Proinov type contraction. Then, we prove the uniqueness and existence of the fixed point for such mappings in the framework of the complete metric spaces. Some illustrative examples are also expressed to strengthen the observed results.

Fixed point theorems in preordered sets

“…Assume d(z, T z) = ε > 0. It follows from T n x 0 → as n → ∞ that d(T n x 0 , T z) → d(z, T z), which implies that d(T n x 0 , T z) > 0 holds for infinitely many values of n. Then applying (10), we have…”

“…Attempting to obtain in preordered set results inspired by the recent work of Karapınar et al [10] encounters serious difficulties and opens the way for further research. Now we present a partial result in this direction.…”

On Jaggi-Suzuki-Type Hybrid Contraction Mappings

“…Among all, the authors observed a very general result in a general setting by using a simulation function. For more interesting results, see, e.g., [3,4].…”

Fixed Points of Proinov E-Contractions

“…In 2012, Karapinar-Samet proposed [6] generalized α − ψ-contraction. In this consideration, the role of α admissible mapping is to combine the problem of fixed point of "cyclic contractions" with the same problem in the framework of metric spaces endowed with a"partially ordered set", see, e.g., [7][8][9][10][11][12]. Later, this notion is refined by Popescu [13].…”