In this paper, we propose a new iteration method based on the hybrid steepest descent method and Ishikawa-type method for seeking a solution of a variational inequality involving a Lipschitz continuous and strongly monotone mapping on the set of common fixed points for a finite family of Lipschitz continuous and quasi-pseudocontractive mappings in a real Hilbert space.
In this paper, we introduce a new explicit iteration method based on the steepest descent method and Krasnoselskii-Mann type method for finding a solution of a variational inequality involving a Lipschitz continuous and strongly monotone mapping on the set of common fixed points for a finite family of nonexpansive mappings in a real Hilbert space.
A Lipschitz continuous and strongly monotone mapping is a strongly monotone mapping.
(1) For the structure of Banach spaces, we extend the duality mapping to a more general case, that is, a convex, continuous and strongly coercive Bregman function which is bounded on bounded subsets, and uniformly convex and uniformly smooth on bounded subsets.
Corollary 2.1 improves Theorem 3.1 in [43], in the following aspects: (1) For the structure of Banach spaces, we extend the normalized duality mapping to a more general case, that is, a convex, continuous, and strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets.
After that, Ceng et al. [6] introduced the following algorithm: u_{n+1}=P_{W}bigl[a_{n}rho g u_{n})+ I-a_{n} mu F)S(u_{n})bigr], quadforall ngeq0, (1.4) where F is a Lipschitz continuous and strongly monotone mapping, g is a Lipschitz continuous mapping.
Theorem 3.5 Let E be a real normed linear space, and T : E → E be a uniformly continuous and strongly ϕ-quasi-accretive mapping.
In the next section we present a general regularization method for VI (1.1) with the regularizer being a Lipschitz continuous and strongly monotone operator.
Then dom ( T r ) = E. Lemma 5.2 Let E be a reflexive Banach space and let g : E → R be a convex, continuous and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed and convex subset of E and let f : C × C → R be a bifunction satisfying (A1 - A4).
Let E be a reflexive Banach space and let (f : E rightarrow R) be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let (F : C times C rightarrow R) be a bifunction satisfying (A1 - A4).
