Solving variational inequalities involving nonexpansive type mappings (original) (raw)
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Fixed Point Theory and Applications, 2014
In this paper, we present relaxed and composite viscosity methods for computing a common solution of a general systems of variational inequalities, common fixed points of infinitely many nonexpansive mappings and zeros of accretive operators in real smooth and uniformly convex Banach spaces. The relaxed and composite viscosity methods are based on Korpelevich's extragradient method, the viscosity approximation method and the Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for relaxed and composite viscosity algorithms not only in the setting of a uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results given in the literature.
IJNNA
ln this paper, a new approach for finding common element of the set of solutions of the variational inequality problem for accretive mappings and the set of fixed points for asymptotically nonexpansive mappings is introduced and studied. Consequently, strong convergence results for finite families of asymptotically nonexpansive mappings and variational inequality problems are established in the setting of uniformly convex Banach space and 2-uniformly smooth Banach space. Furthermore, we prove that a slight modification of our novel scheme could be applied in finding common element of solution of variational inequality problems in Hilbert space. Our results improve, extend and generalize several recently announced results in literature.. . .
Viscosity iterative method for a new general system of variational inequalities in Banach spaces
Journal of Inequalities and Applications, 2013
In this paper, we study a new iterative method for finding a common element of the set of solutions of a new general system of variational inequalities for two different relaxed cocoercive mappings and the set of fixed points of a nonexpansive mapping in real 2-uniformly smooth and uniformly convex Banach spaces. We prove the strong convergence of the proposed iterative method without the condition of weakly sequentially continuous duality mapping. Our result improves and extends the corresponding results announced by many others. MSC: 46B10; 46B20; 47H10; 49J40 Keywords: a new general system of variational inequalities; relaxed cocoercive mapping; strong convergence Ax-Ay ≤ L xy , ∀x, y ∈ C; (ii) accretive if there exists j(xy) ∈ J(xy) such that Ax-Ay, j(xy) ≥ , ∀x, y ∈ C; (iii) α-inverse strongly accretive if there exist j(xy) ∈ J(xy) and α > such that Ax-Ay, j(xy) ≥ α Ax-Ay , ∀x, y ∈ C;
Approximating solutions of variational inequalities for asymptotically nonexpansive mappings
Applied Mathematics and Computation, 2009
Purpose: Our purpose in this paper is studying the strong convergence of implicit and explicit iterative schemes for approximating solutions of some variational inequalities on the sets of common fixed points for a semigroup of asymptotically nonexpansive mappings. Methods: We prove strong convergence theorems of such iterative scheme in a uniformly convex and uniformly smooth Banach spaces under certain conditions. Results: We obtain two convergence theorems of such iterative scheme for approximating a common fixed point, which is a unique solutions of the variational inequality. Conclusions: Our results extend and improve those of Zegeye et al., Sunthrayuth and Kumam, and Li et al. The methods of the proof given in this paper are also different.
Fixed Point Theory and Applications, 2012
In this paper, we prove a strong convergence theorem for finding a common solution of a general system of finite variational inequalities for finite different inverse-strongly accretive operators and solutions of fixed point problems for a nonexpansive semigroup in a Banach space based on a viscosity approximation method by using weak contraction mappings. Moreover, we can apply the above results to find the solutions of the class of k-strictly pseudocontractive mappings and apply a general system of finite variational inequalities into a Hilbert space. The results presented in this paper extend and improve the corresponding results of Ceng et al. (2008), Katchang and Kumam (2011), Wangkeeree and Preechasilp (2012), Yao et al. (2010) and many other authors. MSC: Primary 47H05; 47H10; 47J25
SIAM Journal on Control and Optimization
We study in this paper the existence and approximation of solutions of variational inequalities involving generalized pseudo-contractive mappings in Banach spaces. The convergence analysis of a proposed hybrid iterative method for approximating common zeros or fixed points of a possibly infinitely countable or uncountable family of such operators will be conducted within the conceptual framework of the "viscosity approximation technique" in reflexive Banach spaces. This technique should make existing or new results in solving variational inequalities more applicable.