Fixed point theorems for hybrid multi-valued mappings without using the Pompeiu–Hausdorff distance

In the present paper, we redefine the notion of $$T_f$$ -orbitally lower semi-continuous mapping, and obtain a hybrid coincidence point theorem satisfying new inequalities in the framework of $$\delta $$ -distance. We also obtain some sufficient conditions for the existence of common fixed points of a pair of asymptotically regular self-mappings on a complete metric space. Moreover, we show that the orbital continuity of a pair of non-commuting mappings of a complete metric space is equivalent to fixed point property. Several examples are given to illustrate the results. Our results extend several well-known fixed point theorems for both single valued and multi-valued mappings existing in the literature.