New iterative schemes for solving variational inequality and fixed points problems involving demicontractive and quasi-nonexpansive mappings in Banach spaces

In this paper, we suggest and analyze a new iterative method for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of fixed points of a demicontractive mapping which is the unique solution of some variational inequality problems involving accretive operators in a Banach space. We prove the strong convergence of the proposed iterative scheme without imposing any compactness condition on the mapping or the space. Finally, applications of our theorems to some constrained convex minimization problems are given.

A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces

We consider a new subgradient extragradient iterative algorithm with inertial extrapolation for approximating a common solution of variational inequality problems and fixed point problems of a multivalued demicontractive mapping in a real Hilbert space. We established a strong convergence theorem for our proposed algorithm under some suitable conditions and without prior knowledge of the Lipschitz constant of the underlying operator. We present numerical examples to show that our proposed algorithm performs better than some recent existing algorithms in the literature.

Some Convergence Theorems for Contractive Type Mappings inCAT(0) Spaces

We establish theorems of strong convergence, for the Ishikawa-type (or two step;cf. Ishikawa, 1974) iteration scheme, to a fixed point of a uniformlyL-Lipschitzian asymptotically demicontractive mapping and a uniformlyL-Lipschitzian hemicontractive mapping inCAT(0) space. Moreover, we will propose some open problems.