The existence of fixed and periodic point theorems in cone metric type spaces (Q2926399)

Let \((X,d,K)\) be a complete cone metric type space over a solid cone in the sense of \textit{A. S. Cvetković} et al. [Fixed Point Theory Appl. 2011, Article ID 589725, 15 p. (2011; Zbl 1221.54054)]. Suppose that \(f,g:X\to X\) satisfy \(d(fx,gy)\leq ad(x,y)+b[d(x,fx)+d(y,gy)]+c[d(x,gy)+d(y,fx)]\) for all \(x,y\in X\), where \(a,b,c\geq0\) and \(Ka+(K+1)b+(K^2+K)c<1\). The authors prove that \(f\) and \(g\) have a unique common fixed point in \(X\). Several corollaries are stated, including an application to the existence of periodic points.