Fixed point theorems for generalized contraction (Q1059889)

The authors prove a common fixed point theorem for self-mappings A, S, T of a complete metric space (X,d) satisfying \[ d(Sx,Ty)\leq a_ 1d(Sx,Ax)+a_ 2d(Ty,Ay)+a_ 3d(Sx,Ay)+a_ 4d(Ty,Ax)+a_ 5d(Ax,Ay) \] for all s,y\(\in X\), where \(a_ i\geq 0\) and \(\max \{(a_ 2+a_ 4),(a_ 3+a_ 4+a_ 5)\}<1\). Further conditions on the mappings involve continuity of A, a weak commutativity relationship between \(\{\) A,S\(\}\) and \(\{\) A,T\(\}\), and a certain joint asymptotic regularity assumption.