Suzuki type common fixed point theorems and applications (Q2863565)

The authors prove two results. The first one is: Let \(S\) and \(T\) be selfmaps of a complete metric space \(X\). Suppose that there exist nonnegative real numbers \(a_1, a_2, a_3, a_4, a_5\) which satisfy the conditions (i) \(a_1 + a_2 + a_3 + a_4 + a_5 < 1\) and (ii) \(a_2 = a_3\) or \(a_4 = a_5\). Assume for each \(x, y \in X\) that \(\beta\min\{d(x, ASx), d(y, Ty)\} < d(x, y)\) implies that (iii) \(d(Sx, Ty) \leq a_1d(x, y) + a_2d(x, Sx) + a_3d(y, Ty) + a_4d(x, Ty) + a_5d(y, Tx).\)). Then \(S\) and \(T\) have a unique common fixed point. NEWLINENEWLINEReviewer's remark. \textit{C. S. Wong} [Pac. J. Math. 48, 299--312 (1973; Zbl 0269.54028)] already proved this result without using the condition on \(\beta\). The second result of the paper is also a special case of the theorem of Wong.