Fixed Point Theory in Bi-Operational Metric Spaces with Independent Contraction Operations

Abstract

The Banach contraction principle forms an integral part of nonlinear analysis and is of extensive applicability in fixed point theory and differential equations. Recently, operational metric spaces have been proposed, where in the triangle inequality, the additive structure was replaced by a binary operation ⊕. But in previous literature, the choice of the binary operation (⊕) is highly restricted (for example, either addition or max), with the implicit assumption of a zero element identity. In this paper, we introduce a completely novel concept of a bi-operational metric space with two unrelated binary operations, namely, ⊕, which controls the triangle inequality-type property, and ⊙, controlling the process of contraction. In case the conditions of monotonicity and continuity are satisfied in the usual sense, we prove a Banach-like fixed point theorem in the present context. The model does not involve any notion of scalar multiplication whatsoever. Further, we establish the general applicability of our method in proving that the classical metric space, multiplicative metric space, and ultrametric space are special cases. An example is illustrated on [2,3] with the operations a⊕b=(a−1)(b−1)+1 and a⊙b=(a−1)ln(b−1)+1, for which the inequality B(Tξ,T♭) ≤k⊙B(ξ,♭) leads to a unique fixed point.