ON SOME F – CONTRACTION OF PIRI– KUMAM–DUNG–TYPE MAPPINGS IN METRIC SPACES

Introduction/purpose: This paper establishes some new results of Piri– Kumam–Dung-type mappings in a complete metric space.Тhe goal was to improve the already published results. Methods: Using the property of a strictly increasing function as well as the known Lemma formulated in (Radenović et al, 2017), the authors have proved that a Picard sequence is a Cauchy sequence. Results: New results were obtained concerning the F – contraction mappings of S in a complete metric space. To prove it, the authors used only property (W1). Conclusion:The authors believe that the obtained results represent a significant improvement of many known results in the existing literature.


Introduction and preliminaries
It is well known that the Banach fixed point theorem is the most celebrated result in Nonlinear analysis, Functional analysis, Mathematical Analysis, Topology and other mathematical disciplines. It shows that in a complete metric space, each contractive mapping has a unique fixed point. This result is used as a main tool for the existence of solutions of many non-linear problems. A great number of generalizations of this famous results appear in the literature. On the one hand, the usual contractive condition is replaced by a weakly contractive condition, while, on the other hand, the action space is replaced by some generalization of a standard metric space (as b  metric space, partial metric space, partial b  metric space, b  metric like space, G  metric space, b G  metric space, etc).
A fundamental role in the foundations of the constructions is played by the fixed point theorems in metric spaces. They have been intensively studied for quite some time. The Banach fixed point theorem (proved in 1922(proved in (Banach, 1922), provides a technique for solving a variety of problems in mathematical science and engineering.
In the further work we need the following notation.
In 2013, Secelean (Secelean, 2013) proved that condition (W2) in Definition 1.1 can be replaced by an equivalent condition (A1) inf F   , or Piri and Kumam (Piri & Kumam, 2014) introduced a new type of F  contraction, a so-called F  Suzuki contraction. With this notion, they generalized and extended the well-known fixed point results of Wardowski (Wardowski, 2012) and Secelean (Secelean, 2013 Then, S has a unique fixed point u   and for every u  the sequences   , n S u n  converges to u  .
be a complete metric space and : S    be an F  Suzuki contraction. Then S has a unique fixed point u   and for every u  the sequence   , n S u n  converges to u  .
It is well known (see, for instance, (Cosentino & Vetro, 2014)) that the contraction conditions for the mappings : . This fact inspired Dung and Hang (Dung & Hang, 2015) to introduce a new concept, a generalized F  contraction, and to prove several fixed point theorems for such mapping. Definition 1.3 (Dung & Hang, 2015) A mapping S of the metric space Theorem 1.4 (Dung & Hang, 2015) Let   ,   be a complete metric space and let : S    be a generalized F  contraction. If S or F is continuous, then S has a unique fixed point u   and for every u  the sequence   , n S u n  converges to u  . In this article, we will prove Theorem 1.2, Theorem 1.3 and Theorem 1.4 in the easier way: using only condition (W1) and the following Lemma.
Then S has a unique fixed point, say, u  and for all u  the sequence (2.2) Further, from the assumption that S is an  Then in each of these cases, S has a unique fixed point u   and for all u  the sequence   , n S u n  converges to u  . After all, we give the proof of Theorem 1.4 in an easier way: using only condition (W1) and Lemma 1.1.
be a complete metric space and : S    is a generalized F  contraction which satisfies condition (W1), that is, there

Conclusion
In this article we get some new results concerning the F  contraction mappings of S in a complete metric space. To prove it, we used only property (W1). We believe that this is a significant improvement of the known results in the existing literature.