Abstract
The present paper aims to introduce the concept of weak-fuzzy contraction mappings in the graph structure within the context of fuzzy cone metric spaces. We prove some fixed point results endowed with a graph using weak-fuzzy contractions. By relaxing the continuity condition of mappings involved, our results enrich and generalize some well-known results in fixed point theory. With the help of new lemmas, our proofs are straight forward. We furnish the validity of our findings with appropriate examples. This approach is completely new and will be beneficial for the future aspects of the related study. We provide an application of integral equations to illustrate the usability of our theory.
Highlights
IntroductionWe discuss a new concept that overlaps between metric fixed point theory and graph theory
The theory of fixed points centers on the process of solving the equation of the form T(μ) = μ
We introduce the weak-fuzzy contractions conditions from fuzzy cone metric spaces and prove some fixed point results for such mappings in the sense of Grabiec [24]
Summary
We discuss a new concept that overlaps between metric fixed point theory and graph theory. This new area yields interesting generalizations of the Banach contraction principle [1] in metric spaces endowed with a graph. Kramosil and Michálek [3] introduced the notion of fuzzy metric space. George and Veeramani [4] modified the description of fuzzy metric spaces due to Kramosil and Michálek. To find a solution to some special matrix equations was one of the great charms of the fixed point theorists. To this end, the work of EL-Sayed and Andre’ [7] was a pioneer one. Nieto and Rodriguez Lopez [8] extended the work of [6] and applied their results to solve some differential equations
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