Gang Cai

Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces

Abstract In this paper, we study a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of variational inequalities for two inverse strongly accretive mappings and the set of common fixed points of an infinite family of nonexpansive mappings in a real 2-uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the corresponding results announced by many others.

Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces

In this paper, we introduce two iterative algorithms for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of finite variational inequalities for inverse strongly monotone mappings and the set of common fixed points of an asymptotically k-strictly pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems by using the proposed algorithms under some suitable conditions. Our results improve and extend the corresponding results announced by many others.

Hybrid algorithm for generalized mixed equilibrium problems and variational inequality problems and fixed point problems

In this paper, we introduce a new iterative algorithm by hybrid method for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of a general variational inequality problem for finite inverse strongly monotone mappings and the set of common fixed points of infinite family of strictly pseudocontractive mappings in a real Hilbert space. Then we prove strong convergence of the scheme to a common element of the three above described sets. Our result improves and extends the corresponding results announced by many others.

Convergence theorems for equilibrium problems and fixed point problems of a finite family of asymptotically k-strictly pseudocontractive mappings in the intermediate sense

In this paper, we introduce a modified explicit iterative process for finding a common element of the solutions of an equilibrium problem and the set of common fixed points of a finite family of asymptotically k-strictly pseudocontractive mappings in the intermediate sense in the framework of Hilbert spaces. We get a weak convergence theorem for finding a common element of the above two sets and then we modify these algorithms to have a strong convergence theorem by using a hybrid method in the mathematical programming. Our results improve and extend the recent ones announced by many others.

Modified extragradient methods for variational inequality problems and fixed point problems for an infinite family of nonexpansive mappings in Banach spaces

In this paper, we introduce a new general iterative algorithm for finding a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general variational inequality for two inverse-strongly accretive mappings in Banach space. We obtain some strong convergence theorems by a modified extragradient method under suitable conditions. Our results extend the recent results announced by many others.

Approximation of common fixed points of a countable family of continuous pseudocontractions in a uniformly smooth Banach space

Abstract In this paper, we introduce a new implicit iterative algorithm for finding a common element of a countable family of continuous pseudocontractions in a uniformly smooth Banach space. We obtain some strong convergence theorems under suitable conditions. Our results extend the recent results announced by many others.

Well-posedness of second order degenerate integro-differential equations in vector-valued function spaces

Abstract We consider the well-posedness of the second order degenerate integrodifferential equations (P2): , (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu′)(0) = (Mu′)(2π), in periodic Lebesgue-Bochner spaces Lp(𝕋, X), periodic Besov spaces (𝕋, X) and and periodic Triebel-Lizorkin spaces (𝕋, X), where A and M are closed linear operators on a Banach space X satisfying D(A) ⊂ D(M), a ∊ L1 (ℝ+) and α is a scalar number. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for the well-posedness of (P2) in the above three function spaces.

Strong convergence theorems for variational inequality problems and fixed point problems in uniformly smooth and uniformly convex Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of solutions of a general variational inequality problem for finite inverse-strongly accretive mappings and the set of common fixed points for a nonexpansive mapping in a uniformly smooth and uniformly convex Banach space. We obtain a strong convergence theorem under some suitable conditions. Our results improve and extend the recent ones announced by many others in the literature.

A viscosity approximation scheme for finite mixed equilibrium problems and variational inequality problems and fixed point problems

Abstract In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of finite mixed equilibrium problems, the set of solutions of variational inequalities for two cocoercive mappings, the set of common fixed points of an infinite family of nonexpansive mappings and the set of common fixed points of a nonexpansive semigroup in Hilbert space. Then we prove a strong convergence theorem under some suitable conditions. The results obtained in this paper extend and improve many recent ones announced by many others.

A viscosity scheme for mixed equilibrium problems, variational inequality problems and fixed point problems

In this paper, we study a new general iterative scheme for finding a common element of the set of solutions of finite general mixed equilibrium problems, the set of solutions of finite variational inequalities for cocoercive mappings, the set of solutions of common fixed points of an infinite family of nonexpansive mappings and the set of solutions of fixed points of a nonexpansive semigroup in Hilbert space. Strong convergence theorems are obtained under appropriate conditions.