A point of coincidence and common fixed point theorem for expansive type mappings in B-metric spaces
Fixed Point Methods and Optimization, Volume 2, Issue 3, December 2025, Pages 220–228
MULATU WOLDEGIORGIS
Department of Mathematics, Wachemo University, Hossana, SNNPRS, Ethiopia
MEAZA F. BOGALE
Department of Mathematics, Hampton University, Hampton, Virginia 23668, USA
ALEMAYEHU G. NEGASH
Department of Mathematics, Hampton University, Hampton, Virginia 23668, USA
SEMIRA HUSSEIN
Department of Mathematics, Jimma University, Jimma, Oromia State, Ethiopia
Abstract
This paper establishes new criteria for the existence and uniqueness of points of coincidence and common fixed points for pairs of self-mappings in \(b\)-metric spaces. We introduce a generalized expansive condition of the form $$d(fx,fy) + \frac{\beta}{s}\left[d(gx,fy) + d(gy,fx)\right] \geq \alpha_1 d(gx,gy) + \alpha_2 d(fx,gx) + \alpha_3 d(fy,gy),$$ where \(f\) and \(g\) are self-mappings on a \(b\)-metric space \((X,d)\) with coefficient \(s \geq 1\), and \(\alpha_1, \alpha_2, \alpha_3 \geq 0\), \(\beta \geq 0\) are parameters satisfying \(\alpha_1 + \alpha_2 + \alpha_3 > (1 + 2\beta)s\) and \(\beta < (1 + \alpha_3)^{-1}\). Under the assumptions that \(g(X) \subseteq f(X)\) and either \(f(X)\) or \(g(X)\) is complete, we prove: (1) Existence of points of coincidence for \(f\) and \(g\), (2) Uniqueness when \(\alpha_1 > 1 + 2\beta s^{-1}\), and (3) Existence of unique common fixed points when \(f\) and \(g\) are weakly compatible. These results generalize prior work on expansive mappings. The theory is validated through non-trivial examples where classical theorems fail, particularly in discontinuous \(b\)-metric spaces. Our approach provides a unified framework for analyzing expansive-type mappings in generalized metric spaces, with potential applications in functional analysis and nonlinear operator theory.
Cite this Article as
Mulatu Woldegiorgis, Meaza F. Bogale, Alemayehu G. Negash and Semira Hussein, A point of coincidence and common fixed point theorem for expansive type mappings in B-metric spaces, Fixed Point Methods and Optimization, 2(3), 220–228, 2025