The expansion and open question answering on convex orbital $\beta$-Lipschitz mappings

Abstract

Based on the impressive feature of convex orbital $\beta$-Lipschitz mappings, we are interested in presenting two ways to expand this idea to more generalizations. Firstly, convex orbital $\beta$-Lipschitz mappings are extended to quadratic weak orbital $\beta$-Lipschitz mappings. Moreover, the new type of decreasing mapping in inner product spaces is presented, and fixed point results for quadratic weak orbital $\beta$-Lipschitz mappings in Hilbert spaces are proved with the help of the proposed decreasingness. In a second way, the third open question on convex orbital $\beta$-Lipschitz mappings is answered in Banach spaces.