A fixed point approach to growth modeling in biological systems using g-interpolative contractions in \(S_b\)-metric spaces

Abstract

In this paper, a new fixed point theoretical approach for modelling the growth dynamics in biological systems is presented by using \(g\)-interpolative contractions in \(S_b\)-metric spaces. We develop extensive existence and uniqueness results for fixed points under these generalized contractive conditions going beyond the classical metric space results, while allowing for the asymmetric and non-triangular nature found in biology's growth processes. Theoretical results are backed up by rigorous proofs, and are demonstrated using examples from population dynamics, modelling tumour growth and microbial systems. The key contributions are: (i) construction of \(g\)-interpolative-contracting type of Ćirić-Reich-Rus on \(S_b\)-metric spaces, (ii) proving the existence of fixed points under weaker continuity conditions, (iii) using the results to solve the nonlinear integral equations that model biological growth, and (iv) performing a numerical validation by comparing with empirical data. The framework serves as a unified understanding of the equilibrium states of biological systems in which the traditional models based on metrics are not enough.