On general hybrid contractions

Abstract

Khan et al. have studied the existence of solutions of functional equations f,x e Sx f) Tx and x = F,x e Sx n Tx under certain nonlinear hybrid contraction and asymptotic regularity conditions, generalizing many known results on coincidences and fixed points. However, most of their main theorems admit counter examples. In this paper, we rectify these results and obtain many coincidence and fixed point theorems in a more general setting.1991 Mathematics subject classification (Amer. Math. Soc): primary 47H10; secondary 54H25.

Random Coincidence and Fixed Point Theorem for Hybrid Maps in Separable Metric Spaces

“…In addition, (IT)-commutativity of a a hybrid pair T and f at a coincidence point x ∈ X is more general than its compatibility and weak compatibility at the same point. The following example shows that and (see also, Example 1 in [23]).…”

Some common fixed point theorems for hybrid pairs of maps without the completeness assumption

“…Hybrid fixed point theory for nonlinear single-valued and multivalued mappings is a development within the multivalued fixed point theory which has already made a considerable progress. For the work of this kind, one can be referred to George et al [4], Imdad et al [5], Imdad and Ali [6], Imdad et al [8], Kamran [10], Liu et al [13], Popa [19], Singh and Kumar [22], Singh and Mishra [23], Pathak [15], Pathak et al [16], Pathak and Khan [17] and references cited therein. Towards the utility side, the results of this kind offer techniques for solving functional equations and optimization problems (e,g.…”