Abstract
This paper involves extended b − metric versions of a fractional differential equation, a system of fractional differential equations and two-dimensional (2D) linear Fredholm integral equations. By various given hypotheses, exciting results are established in the setting of an extended b − metric space. Thereafter, by making consequent use of the fixed point technique, short and simple proofs are obtained for solutions of a fractional differential equation, a system of fractional differential equations and a two-dimensional linear Fredholm integral equation.
Highlights
Introduction and PreliminariesIn the last years, the fractional calculus branch [1, 2] has attracted great interest. ere exist many kinds of proposed fractional operators, for instance, we have the well-known Caputo, Riemann–Liouville, Grunwald–Letnikov derivative etc
Among all the papers dealing with fractional derivatives, fractional differential equations as an important research field have attained great deal of attention from many researchers
The standard definition for the Atangana–Baleanu fractional derivative involves an integral transform with a Mittag–Leffler function, where the kernel can be rewritten as a complex contour integral, which can be used to provide an analytic continuation of the definition to complex orders of differentiation [10]. ese lines are very important due to their applications in the field of natural science or engineering
Summary
The fractional calculus branch [1, 2] has attracted great interest. ere exist many kinds of proposed fractional operators, for instance, we have the well-known Caputo, Riemann–Liouville, Grunwald–Letnikov derivative etc. Is paper is concerned with this fact when considering the class of extended b-metric spaces. In 2017, the concept of extended b-metric spaces has been initiated by Kamran et al [11], by considering a control function at the right-hand side of the triangular inequality. An extended b-metric is a function πθ: R × R ⟶ [0, ∞) such that, for all η, ξ, σ ∈ R, we have the following:. Is (generalized) metric space has attracted many researchers where many real applications have been resolved. Definition 2 Let (R, πθ) be an extended b−metric space. Let (R, πθ) be an extended b − metric space.
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