Greguš type common fixed point theorems for compatible mappings of type (T) and variational inequalities
H. K. Pathak, S.M. S.M. Kang, Y. J., J. S. Jung · Publicationes Mathematicae Debrecen · 1995
Let T and I be two compatible mappings of type (T ) from a normed space X into itself and let C be a closed convex and bounded subset of X such that I(C) ⊇ (1 -k) • I(C) + k • T (C), where k ∈ (0, 1) is fixed andfor all x, y ∈ C, where 0 0. If, for some x 0 ∈ C, the sequence {x n } defined by Ix n+1 = (1 -k) • Ix n + k • T x n for all n ≥ 0 converges to a point z in C and if I is continuous at z, then T and I have a unique common fixed point.Further, if I is continuous at T z, then T and I have a unique common fixed point at which T is continuous.We have also applied this result to obtain the iterative solution of certain variational inequalities. Mathematics Subject Classification: Primary 54H25