Abstract
This paper deals with the fixed point theorems for mappings satisfying a contractive condition involving a gauge function φ when the underlying set is endowed with a b-metric. Our results generalize/extend the main results of Proinov and thus we obtain as special cases some results of Mysovskih, Rheinboldt, Gel’man, and Huang. We also furnish an example to substantiate the validity of our results. Subsequently, an existence theorem for the solution of initial value problem has also been established.
Highlights
Introduction and preliminaries TheBanach contraction principle has been extensively used to study the existence of solutions for the nonlinear Volterra integral equations and nonlinear integro-differential equations and to prove the convergence of algorithms in computational mathematics
These applications elicit the significance of fixed point theory
Mathematicians have been propelled to contribute enormously in the field of fixed point theory by finding the fixed point(s) of self-mappings or nonself-mappings defined on several ambient spaces and satisfying a variety of conditions
Summary
] Let (X, d) be a complete metric space and f : D ⊂ X → X be an operator satisfying d fx, f x ≤ φ d(x, fx) for all x ∈ D and fx ∈ D with d(x, fx) ∈ J, where φ is a Bianchini-Grandolfi gauge function on an interval J. Assume that f satisfies d(fx, fy) ≤ φ d(x, y) for all x, y ∈ X with d(x, y) ∈ J, where φ is a b-Bianchini-Grandolfi gauge function of order r ≥ on an interval J and with coefficient s ≥. Conclusions (i) and (iii) follow immediately from Theorem Let ξ be another fixed point of f in S; d(ξ , ξ ) ∈ J.
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