Common fixed point and best simultaneous approximations theorems under releaxed conditions (Q2869585)

Let \(X\) be a set and \(f, g\) be self-mappings of \(X\). A~point \(x\) in \(X\) is called a coincidence point of \(f\) and \(g\) if \(f(x)= g(x)\). We say that \(f\) and \(g\) are occasionally weakly compatible (owc) if there is a point \(x\) in \(X\) which is a coincidence point of \(f\) and \(g\) at which \(f\) and \(g\) commute i.e., \((fg)(x)= (gf)(x)\). If \(C(g,f)\) denotes the set of coincidence points of the pair \((g,f)\), then the pair \((g,f)\) is called \(P\)-operator pair if \(d(u,gu)\leq\text{diam}\,C(g,f)\) for some \(u\in C(g,f)\).NEWLINENEWLINE In this paper, the authors obtain a common fixed point theorem for a \(P\)-operator pair and another common fixed point theorem for an owc pair on a set \(X\) together with the function \(d:X\times X\to[0,\infty)\) without using the triangle inequality, and assuming symmetry only on the set of points of coincidence. As an application, they establish a best simultaneous approximation result.