Abstract
In this manuscript, we aim to provide a new hybrid type contraction that is a combination of a Jaggi type contraction and interpolative type contraction in the framework of complete metric spaces. We investigate the existence and uniqueness of such a hybrid contraction in separate theorems. We consider a solution to certain fractional differential equations as an application of the given results. In addition, we provide an example to indicate the genuineness of the given results.
Highlights
Introduction and PreliminariesFixed point theory is based on the solution of a simple equation, f ( x ) = x, for a self-mapping f on a non-empty set X
In 1837, Liouville [1] solved such differential equations by employing the method of successive approximations which implicitly brings a solution for thefixed point equation
Banach [3] was able to derive the abstraction of the method of successive approximation so that he proved a celebrated fixed point theorem in the framework of a complete metric space
Summary
Banach [3] was able to derive the abstraction of the method of successive approximation so that he proved a celebrated fixed point theorem in the framework of a complete metric space. After that, this outstanding result from Banach was characterized by Caccioppoli [4] in the context of complete metric spaces: Every contraction in a complete metric space possesses a unique fixed point. Fractional calculus and fractional differential equations have been studied intensely By taking this observation into account, as a second aim, we propose a solution to certain fractional differential equations by employing our main results.
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