A Modified Proximal Point Algorithm and Some Convergence Results

In this paper, the convergence to minimizers of a convex function of a modified proximal point algorithm involving a single-valued nonexpansive mapping and a multivalued nonexpansive mapping in CAT(0) spaces is studied and a numerical example is given to support our main results.

On solving variational inequalities defined on fixed point sets of multivalued mappings in Banach spaces

We are concerned with the problem of solving variational inequalities which are defined on the set of fixed points of a multivalued nonexpansive mapping in a reflexive Banach space. Both implicit and explicit approaches are studied. Strong convergence of the implicit method is proved if the space satisfies Opial’s condition and has a duality map weakly continuous at zero, and the strong convergence of the explicit method is proved if the space has a weakly continuous duality map. An essential assumption on the multivalued nonexpansive mapping is that the mapping be single valued on its nonempty set of fixed points.

Generalized implicit viscosity approximation method for multivalued mappings in CAT(0) spaces

We prove strong convergence of the sequence generated by implicit viscosity approximation method involving a multivalued nonexpansive mapping in framework of CAT(0) space. Under certain appropriate conditions on parameters, we show that such a sequence converges strongly to a fixed point of the mapping which solves a variational inequality. We also present some convergence results for the implicit viscosity approximation method in complete ℝ-trees without the endpoint condition.

Approximation of endpoints for multivalued nonexpansive mappings in geodesic spaces

Let [Formula: see text] be a complete CAT(0) space, [Formula: see text] be a closed and convex subset of [Formula: see text] and [Formula: see text] be a multivalued nonexpansive mapping. We prove that the sequence of Ishikawa iteration converges to an endpoint of [Formula: see text]. This improves, extends and unifies some recently announced results of the current literature.

Three-Step Fixed Point Iteration for Generalized Multivalued Mapping in Banach Spaces

The convergence of three-step fixed point iterative processes for generalized multivalued nonexpansive mapping was considered in this paper. Under some different conditions, the sequences of three-step fixed point iterates strongly or weakly converge to a fixed point of the generalized multivalued nonexpansive mapping. Our results extend and improve some recent results.