Let C be a nonempty, bounded, closed, and convex subset of a Banach space X and T : C → C be a monotone asymptotic nonexpansive mapping. In this paper, we investigate the existence of fixed points of T . In particular, we establish an analogue to the original Goebel and Kirk's fixed point theorem for asymptotic nonexpansive mappings.

We discuss Caristi’s fixed point theorem for mappings defined on a metric space endowed with a graph. This work should be seen as a generalization of the classical Caristi’s fixed point theorem. It extends some recent works on the extension of Banach contraction principle to metric spaces with graph.

A generic distance-regular graph is primitive of diameter at least two and valency at least three. We give a version of Derek Smith's famous theorem for reducing the classification of distance-regular graphs to that of primitive graphs. There are twelve cases-the generic case, four canonical imprimitive cases that reduce to the generic case, and seven exceptional cases. All distance-transitive graphs were previously known in six of the seven exceptional cases. We prove that the 6-cube is the only distance-transitive graph coming under the remaining exceptional case.

In this work, we define the concept of G-monotone nonexpansive multivalued mappings defined on a metric space with a graph G. Then we obtain sufficient conditions for the existence of fixed points for such mappings in hyperbolic metric spaces. This is the first kind of such results in this direction.

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