New semi-implicit midpoint rule for zero of monotone mappings in Banach spaces

The Implicit Midpoint Rule for Nonexpansive Mappings in Banach Spaces

The implicit midpoint rule (IMR) for nonexpansive mappings is established in Banach spaces. The IMR generates a sequence by an implicit algorithm. Weak convergence of this algorithm is proved in a uniformly convex Banach space which either satisfies Opial’s property or has a Fréchet differentiable norm. Consequently, this algorithm applies in both `p and Lp for 1 < p < ∞.

Modified semi-implicit midpoint rule for nonexpansive mappings

where h >  is a stepsize. It is known that if f :RN →RN is Lipschitz continuous and sufficiently smooth, then the sequence {xn} generated by (.) converges to the exact solution of (.) as h→  uniformly over t ∈ [, t̄] for any fixed t̄ > . If we write the function f in the form f (t) = g(t) – t, then differential equation (.) becomes x′ = g(t)– t. Then the equilibrium problem associated w...

The viscosity iterative algorithms for the implicit midpoint rule of nonexpansive mappings in uniformly smooth Banach spaces

The aim of this paper is to introduce a viscosity iterative algorithm for the implicit midpoint rule of nonexpansive mappings in uniformly smooth spaces. Under some appropriate conditions on the parameters, we prove some strong convergence theorems. As applications, we apply our main results to solving fixed point problems of strict pseudocontractive mappings, variational inequality problems in...

New Generalized Mixed Equilibrium Problem with Respect to Relaxed Semi-Monotone Mappings in Banach Spaces

We introduce the new generalized mixed equilibrium problem with respect to relaxed semimonotone mappings. Using the KKM technique, we obtain the existence of solutions for the generalizedmixed equilibriumproblem in Banach spaces. Furthermore, we also introduce a hybrid projection algorithm for finding a common element in the solution set of a generalized mixed equilibrium problem and the fixed ...

Convergence of an Implicit Iteration for Affine Mappings in Normed and Banach Spaces