Q-uniformly Smooth Banach Spaces Research Articles

Approximation methods for solutions of generalized multi-valued mixed quasi-variational inclusion systems

Abstract The purpose of this paper is to introduce new approximation methods for solutions of generalized non-accretive multi-valued mixed quasi-variational inclusion systems involving ( A , η ) -accretive mappings in q-uniformly smooth Banach spaces and, by using the new resolvent operator technique associated with ( A , η ) -accretive mappings, Nadler’s fixed point theorem and Liu’s inequality, we prove some existence theorems of solutions for our systems by constructing the new Mann iterative algorithm. Further, we study the stability of the iterative sequence generated by the perturbed iterative algorithms. The results presented in this paper improve and generalize the corresponding results of recent works given by some authors.

Synchronal algorithm for a countable family of strict psedocontractions in q-uniformly smooth Banach spaces

Let E be a real q-uniformly smooth Banach space whose duality map is weakly sequentially continuous and C be a nonempty, closed and convex subset of E. Let { Ti} ∞=1 : C → E be a family of k-strict pseudocontractions for k ∈ (0, 1) such that � ∞ i=1 F (Ti) � ∅, f be a contraction with coefficient β ∈ (0, 1) and { λi} ∞=1 be a real sequence in (0, 1) such that � ∞=1 λi = 1. Let G : C → E be an η-strongly accretive and L-Lipschitzian operator with L> 0, η> 0. Let { αn} and { βn} be sequences in (0, 1) satisfying some conditions. For some positive real numbers γ, µ appropriately chosen, let { xn} be a sequence de“ned by

Approximating fixed points of a countable family of strict pseudocontractions in Banach spaces

We prove the strong convergence of the modified Mann-type iterative scheme for a countable family of strict pseudocontractions in q-uniformly smooth Banach spaces. Our results mainly improve and extend the results announced in (Y. Yao, H. Zhou, Y.-C. Liou, Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings, J. Appl. Math. Comput. 29 (2009), 383-389).

Graph convergence for the H ( ⋅ , ⋅ ) -mixed mappingwith an application for solving the system of generalized variationalinclusions

In this paper, we investigate a class of accretive mappings called theH(⋅,⋅)-mixed mappingsin Banach spaces. We prove that the proximal-point mapping associated with theH(⋅,⋅)-mixed mapping issingle-valued and Lipschitz continuous. Some examples are given to justify thedefinition of H(⋅,⋅)-mixed mapping.Further, a concept of graph convergence concerned with theH(⋅,⋅)-mixed mapping isintroduced in Banach spaces and some equivalence theorems betweengraph-convergence and proximal-point mapping convergence for theH(⋅,⋅)-mixed mappingssequence are proved. As an application, we consider a system of generalizedvariational inclusions involving H(⋅,⋅)-mixed mappingsin real q-uniformly smooth Banach spaces. Using the proximal-pointmapping method, we prove the existence and uniqueness of solution and suggest aniterative algorithm for the system of generalized variational inclusions.Furthermore, we discuss the convergence criteria for the iterative algorithmunder some suitable conditions.MSC: 47J19, 49J40, 49J53.

A new general iterative algorithm with Meir-Keeler contractions for variational inequality problems in q-uniformly smooth Banach spaces

Abstract In this paper, we generalize the iterative scheme and extend the space studied in (Fixed Point Theory and Applications 2012:46, 2012). Further, we prove some strong convergence theorems of the new iterative scheme for variational inequality problems in q-uniformly smooth Banach spaces under very mild conditions. Our results improve and extend corresponding ones announced by many others. MSC:47H09, 47H10.

General convergence analysis of projection methods for a system of variational inequalities in q-uniformly smooth Banach spaces

In this paper, we introduce and consider a system of variational inequalities involving two different operators in q-uniformly smooth Banach spaces. We suggest and analyze a new explicit projection method for solving the system under some more general conditions. Our results extend and unify the results of Verma (Appl. Math. Lett. 18:1286-1292, 2005) and Yao, Liou and Kang (J. Glob. Optim., 2011, doi:10.1007/s10898-011-9804-0) and some other previously known results.

Combination of the Hybrid Steepest-Descent Method and the Viscosity Approximation

In the context of a general q-uniformly smooth Banach space, we will prove the convergence of a scheme which covers Yamada’s hybrid steepest-descent method and Moudafi’s viscosity approximation in their complete forms. Moreover, we present an application to the minimization over a set of fixed points.

Synchronal and cyclic algorithms for fixed point problems and variational inequality problems in Banach spaces

In this paper, we study synchronal and cyclic algorithms for finding a common fixed point of a finite family of strictly pseudocontractive mappings, which solve the variational inequality where f is a contraction mapping, G is an η-strongly accretive and L-Lipschitzian operator, is a positive integer, are arbitrary fixed constants, and are N-strict pseudocontractions. Furthermore, we prove strong convergence theorems of such iterative algorithms in a real q-uniformly smooth Banach space. The results presented extend, generalize and improve the corresponding results recently announced by many authors. MSC:47H06, 47H09, 47H10, 47J05, 47J20, 47J25.

Iterative algorithm with mixed errors for solving a new system of generalized nonlinear variational-like inclusions and fixed point problems in Banach spaces

A new system of generalized nonlinear variational-like inclusions involving A-maximal m-relaxed η-accretive (so-called, (A, η)-accretive in [36]) mappings in q-uniformly smooth Banach spaces is introduced, and then, by using the resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings due to Lan et al., the existence and uniqueness of a solution to the aforementioned system is established. Applying two nearly uniformly Lipschitzian mappings S 1 and S 2 and using the resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings, we shall construct a new perturbed N-step iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping Q = (S 1, S 2) which is the unique solution of the aforesaid system. We also prove the convergence and stability of the iterative sequence generated by the suggested perturbed iterative algorithm under some suitable conditions. The results presented in this paper extend and improve some known results in the literature.

Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive mappings

Let q>1 and let K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Let T:K→CB(K) be a multi-valued strictly pseudo-contractive map with a nonempty fixed point set. A Krasnoselskii-type iteration sequence { x n } is constructed and proved to be an approximate fixed point sequence of T, i.e., lim n → ∞ d( x n ,T x n )=0. This result is then applied to prove strong convergence theorems for a fixed point of T under additional appropriate conditions. Our theorems improve several important well-known results.MSC:47H04, 47H06, 47H15, 47H17, 47J25.

Strong convergence of a CQ method for k-strictly asymptotically pseudocontractive mappings

Let E be a real q-uniformly smooth Banach space, which is also uniformly convex (for example, or spaces, ), and C be a nonempty bounded closed convex subset of E. Let be a k-strictly asymptotically pseudocontractive map with a nonempty fixed point set. A hybrid algorithm is constructed to approximate fixed points of such maps. Furthermore, strong convergence of the proposed algorithm is established.

A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces

In this paper, we introduce a 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 systems of variational inequalities for two inverse strongly accretive mappings in a q-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as improvement, supplementation, development and extension of the corresponding results in some references to a great extent.

Explicit averaging cyclic algorithm for common fixed points of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces

Let E be a real q-uniformly smooth Banach space which is also uniformly convex and K be a nonempty, closed and convex subset of E. We obtain a weak convergence theorem of the explicit averaging cyclic algorithm for a finite family of asymptotically strictly pseudocontractive mappings of K under suitable control conditions, and elicit a necessary and sufficient condition that guarantees strong convergence of an explicit averaging cyclic process to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces. The results of this paper are interesting extensions of those known results.

Iterative methods for variational inequality problems and fixed point problems of a countable family of strict pseudo-contractions in a q-uniformly smooth Banach space

In this article, we introduce iterative methods (implicit and explicit) for finding a common fixed point set of a countable family of strict pseudo-contractions, which is a unique solution of some variational inequality. Furthermore, we prove the strong convergence theorems of such iterative scheme in a q-uniformly smooth Banach space which admits a weakly sequentially continuous generalized duality mapping. The results presented in this article extend and generalize the corresponding results announced by Yamada and Ceng et al. from Hilbert spaces to Banach spaces. Mathematics Subject Classification 2010: 47H09; 47J05; 47J20; 65J15.

A new class of general nonlinear random set-valued variational inclusion problems involving A-maximal m-relaxed η-accretive mappings and random fuzzy mappings in Banach spaces

At the present article, we consider a new class of general nonlinear random A- maximal m-relaxed η-accretive equations with random relaxed cocoercive mappings and random fuzzy mappings in q-uniformly smooth Banach spaces. By using the resolvent mapping technique for A-maximal m-relaxed η-accretive mappings due to Lan et al. and Chang's lemma, we construct a new iterative algorithm with mixed errors for finding the approximate solutions of this class of nonlinear random equations. We also verify that the approximate solutions obtained by the our proposed algorithm converge to the exact solution of the general nonlinear random A-maximal m-relaxed η-accretive equations with random relaxed cocoercive mappings and random fuzzy mappings in q-uniformly smooth Banach spaces.Mathematical Subject Classification 2010: Primary, 47B80; Secondary, 47H40, 60H25.

Graphical approximation of common solutions to generalized nonlinear relaxed cocoercive operator equation systems with (A, η)-accretive mappings

Abstract In this paper, we develop a new perturbed iterative algorithm framework with errors based on the variational graphical convergence of operator sequences with (A, η)-accretive mappings in Banach space. By using the generalized resolvent operator technique associated with (A, η)-accretive mappings, we also prove the existence of solutions for a class of generalized nonlinear relaxed cocoercive operator equation systems and the variational convergence of the sequence generated by the perturbed iterative algorithm in q-uniformly smooth Banach spaces. The obtained results improve and generalize some well-known results in recent literatures. 2000 Mathematics Subject Classification: 47H05; 49J40

Resolvent operator technique for solving a system of generalized variational-like inclusions in Banach spaces

In this paper, we apply H(?,?)-?-cocoercive operator introduced in [2] for solving a system of generalized variational-like inclusions in q-uniformly smooth Banach spaces. By using the approach of resolvent operator associated with H(?,?)-?-cocoercive operator, an iterative algorithm for solving a system of generalized variational-like inclusions is constructed. We prove the existence of solutions of system of generalized variational-like inclusions and convergence of iterative sequences generated by the algorithm. An example through Matlab programming is constructed.

An iterative algorithm for a general system of variational inequalities and fixed point problems in q-uniformly smooth Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of common fixed points of an infinite family of notself strict pseudocontractions and the set of solutions of a general variational inequality problem for finite inverse-strongly accretive mappings in q-uniformly smooth Banach space. We obtain some strong convergence theorems under suitable conditions. Our results improve and extend the recent results announced by Qin et al. (J Comput Appl Math 233:231–240, 2009), Yao et al. (Acta Appl Math 110:1211–1244, 2010) and many others.

A system of generalized variational inclusions involving generalized H(·, ·)-accretive mapping in real q-uniformly smooth Banach spaces

In this paper, we consider a class of accretive mappings called generalized H(·, ·)-accretive mappings in Banach spaces. We prove that the proximal-point mapping of the generalized H(·, ·)-accretive mapping is single-valued and Lipschitz continuous. Further, we consider a system of generalized variational inclusions involving generalized H(·, ·)-accretive mappings in real q-uniformly smooth Banach spaces. Using proximal-point mapping method, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as a refinement and improvement of some known results in the literature.

Convergence of hybrid steepest descent method for variational inequalities in Banach spaces

Let E be a q-uniformly smooth real Banach space with constant d q , q > 1. Let T i : E → E, i = 1, 2, … , r be a finite family of nonexpansive mappings with K ≔ ∩ i = 1 r Fix ( T i ) ≠ ∅ and K = Fix( T r T r−1 … T 1) = Fix( T 1 T r … T 2) = ⋯ = Fix( T r−1 T r−2 … T r ). Let G : E → E be an η-strongly accretive map which is also κ-Lipschitzian. A hybrid steepest descent method introduced by Yamada [25] and studied by various authors is proved to converge strongly to the unique solution of the variational inequality problem VI( G, K) in q-uniformly smooth real Banach space, in particular, in L p spaces 1 < p < ∞.