Generalized distance and new fixed point results (Q2805955)

Let \((X,d)\) be a cone metric space over a normal cone, and let \(q\) be a \(c\)-distance on \(X\), in the sense of \textit{H. Lakzian} and \textit{F. Arabyani} [Int. J. Math. Anal., Ruse 3, No. 21--24, 1081--1086 (2009; Zbl 1197.54065)]. Let \(f,g:X\to X\) be two mappings with \(f(X)\subseteq g(X)\) and \(g(X)\) being a complete subspace of~\(X\). Suppose that there exist mappings \(\alpha_i:X\to[0,1)\) for \(i\in\{1,2,3,4,5\}\) such that (i)~\(\alpha_i(fx)\leq \alpha_i(gx)\) for all \(i\) and \(x\in X\); (ii)~\((\alpha_1+\alpha_2+\alpha_3+2\alpha_4+2\alpha_5)(x)<1\) for all \(x\in X\); (iii)~\(q(fx,fy)+q(fy,fx)\preceq\alpha_1(gx)[q(gx,gy)+q(gy,gx)]+\alpha_2(gx)[q(gx,fx)+q(gx,fx)]+ \alpha_3(gx)[q(gy,fy)+q(fy,gy)]+\alpha_4(gx)[q(gx,fy)+q(fy,gx)]+\alpha_5(gx)[q(gy,fx)+q(fx,gy)]\) for all \(x\in X\). If \(f\) and \(g\) satisfy \(\inf\{\|q(fx,y)\|+\|q(gx,y)\|+\|q(gx,fx)\|:x\in X\}>0\) for all \(y\in X\) with \(y\neq fy\) or \(y\neq gy\), the authors prove that, under these conditions, \(f\) and \(g\) have a common fixed point in~\(X\). Some corollaries and examples are presented.