Orbital $\varphi$-regularity in coincidence and fixed point problems in metric spaces

The purpose of the present paper is to establish some (approximate) fixed point or coincidence theorems for set-valued mappings defined on metric spaces under the so-called orbital \varphi-regularity of the considered mappings. This is a type of (\varphi,\gamma)-regularity of set-valued mappings which is weaker than orbital regularity. In turn, it is used in the previous work of the author [12] and before by Ioffe in the work [7] to ensure the existence of (approximate) fixed points or (approximate) coincidence points.