Abstract
In this paper, we introduce the new concept of coupled fixed-point (FP) results depending on another function in fuzzy cone metric spaces (FCM-spaces) and prove some unique coupled FP theorems under the modified contractive type conditions by using “the triangular property of fuzzy cone metric.” Another function is self-mapping continuous, one-one, and subsequently convergent in FCM-spaces. In support of our results, we present illustrative examples. Moreover, as an application, we ensure the existence of a common solution of the two Volterra integral equations to uplift our work.
Highlights
Fixed-point theory is one of the most interesting areas of research
After the publication of this principle, many researchers have contributed their ideas to the problems on fixed points in the context of metric spaces for single-valued and multivalued mappings with different types of applications
We present an application of the two Volterra integral equations for a common solution to support our results. is new concept will play an important role in the theory of fixed point to prove more coupled FP and strongly coupled FP results in complete FCM-spaces with the application of different types of differential equations. is paper is organized as follows: Section 2 gives preliminary concepts
Summary
Fixed-point theory is one of the most interesting areas of research. In 1922, Banach [1] proved a “Banach contraction principle” stated as follows: “a single-valued contractive type mapping in a complete metric space has a unique FP.” After the publication of this principle, many researchers have contributed their ideas to the problems on fixed points in the context of metric spaces for single-valued and multivalued mappings with different types of applications. In 2015, Oner et al [25] introduced the concept of fuzzy cone metric spaces (FCM-spaces) and proved some basic properties and “a single-valued Banach contraction theorem for FP with the assumption that all the sequences are Cauchy.”. Is new concept will play an important role in the theory of fixed point to prove more coupled FP and strongly coupled FP results in complete FCM-spaces with the application of different types of differential equations. In the last section (Section 5), we present the conclusion of our work
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