Real Q-uniformly Smooth Banach Space Research Articles

Viscosity iterative algorithm for variational inequality problems and fixed point problems in a real q-uniformly smooth Banach space

The purpose of this paper is to study a viscosity iterative algorithm for finding a common element of the set of solutions of a general variational inequality problem for two inverse strongly accretive operators and the set of fixed points of a δ-strict pseudocontraction in a real q-uniformly smooth Banach space. Some strong convergence theorems are obtained under appropriate conditions. As an application, we prove some strong convergence theorems for fixed point problems and variational inequality problems or equilibrium problems in Hilbert spaces. These results improve and extend the corresponding results announced by many others.

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

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.

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.

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.

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.

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.

A SYSTEM OF VARIATIONAL INCLUSIONS IN BANACH SPACES

A system of variational inclusions with (A;;m )-accretive operators in real q-uniformly smooth Banach spaces is introduced. Us- ing the resolvent operator technique associated with (A;;m )-accretive operators, we prove the existence and uniqueness of solutions for this system of variational inclusions and propose a Mann type iterative algo- rithm for approximating the unique solution for the system of variational inclusions.

Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces

In this paper, we continue to discuss the properties of iterates generated by a strict pseudocontraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336–349 (2007)]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51–60 (2005)] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.

Convergence of path and an iterative method for families of nonexpansive mappings

Let E be a real q-uniformly smooth Banach space with q ≥ 1 + d q . Let K be a closed, convex and nonempty subset of E. Let be a family of nonexpansive self-mappings of K. For arbitrary fixed δ ∈ (0, 1) define a family of nonexpansive maps by S i := (1 − δ)I + δT i where I is the identity map of K. Let Assume either at least one of the T i 's is demicompact or E admits weakly sequentially continuous duality map. It is proven that the fixed point sequence {z t n } converges strongly to a common fixed point of the family where and {t n } is a sequence in (0, 1), satisfying appropriate conditions. As an application, it is proven that the iterative sequence {x n } defined by: x 0 ∈ K, converges strongly to a common fixed point of the family where {α n } and {σ i,n } are sequences in (0, 1) satisfying appropriate conditions.

Weak and strong convergence of an explicit iteration scheme with perturbed mapping for nonexpansive mappings

In this paper, we consider an explicit iteration scheme with perturbed mapping for nonexpansive mappings in real q-uniformly smooth Banach spaces. Some weak and strong convergence theorems for this explicit iteration scheme are established. In particular, necessary and sufficient confiditions for strong convergence of this explicit iteration scheme are obtained. At last, some useful corollaries for strong convergence of this explicit iteration scheme are given.

The p-step iterative algorithm for a system of generalized mixed quasi-variational inclusions with [formula omitted]-accretive operators in q-uniformly smooth banach spaces

In this paper, we introduce and study a new system of generalized mixed quasi-variational inclusions with ( A , η ) -accretive operators in q-uniformly smooth Banach spaces. By using the resolvent operator technique associated with ( A , η ) -accretive operators, we construct a new p-step iterative algorithm for solving this system of generalized mixed quasi-variational inclusions in real q-uniformly smooth Banach spaces. We also prove the existence of solutions for the generalized mixed quasi-variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.

A system of variational inclusions with [formula omitted]– [formula omitted]-accretive operators

In this paper, we introduce and study a system of variational inclusions with P – η -accretive operators in real q-uniformly smooth Banach spaces. By using the resolvent operator technique associated with P – η -accretive operators, we prove the existence and uniqueness of solutions for this system of variational inclusions and construct a Mann iterative algorithm to approximate the unique solution. The results in this paper extend and improve some known results in the literature.

The existence and uniqueness of solution and the convergence of a multi-step iterative algorithm for a system of variational inclusions with (A, η, m)-accretive operators

In this paper, we introduce and study a new system of variational inclusions with (A, η, m)-accretive operators which contains variational inequalities, variational inclusions, systems of variational inequalities and systems of variational inclusions in the literature as special cases. By using the resolvent technique for the (A, η, m)-accretive operators, we prove the existence and uniqueness of solution and the convergence of a new multi-step iterative algorithm for this system of variational inclusions in real q-uniformly smooth Banach spaces. The results in this paper unifies, extends and improves some known results in the literature.

Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps

Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, L p or ℓ p spaces, 1 < p < ∞ ), and K a nonempty closed convex (not necessarily bounded) subset of E. Let T : K → K be a k-strictly asymptotically pseudocontractive map with a nonempty fixed-point set. It is proved that ( I − T ) is demiclosed at 0. Furthermore, weak and strong convergence of an averaging iteration method to a fixed point of T are proved.

STRONG CONVERGENCE THEOREMS OF ISHIKAWA ITERATION PROCESS WITH ERRORS FOR FIXED POINTS OF LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES

Let q > 1 and E be a real q-uniformly smooth Banach space, K be a nonempty closed convex subset of E and T : K → K be a Lipschitz continuous mapping. Let {un} and {vn} be bounded sequences in K and {αn} and {βn} be real sequences in [0, 1] satisfying some restrictions. Let {xn} be the sequence generated from an arbitrary x1 ∈ K by the Ishikawa iteration process with errors: yn = (1 − βn)xn + βnTxn + vn, xn+1 = (1−αn)xn +αnTyn +un, n ≥ 1. Sufficient and necessary conditions for the strong convergence {xn} to a fixed point of T is established.

THREE-STEP ITERATIVE CONVERGENCE THEOREMS WITH ERRORS IN BANACH SPACES

Let q > 1 and E be a real q-uniformly smooth Banach space, K be a nonempty closed convex subset of E and T : K → K be a single-valued mapping. Let {un}∞ n=1, {vn}∞ n=1, {wn}∞ n=1 be three sequences in K and {αn}∞ n=1, {βn}∞ n=1 and {γn}∞ n=1 be real sequences in [0, 1] satisfying some restrictions. Let {xn} be the sequence generated from an arbitrary x1 ∈ K by the three-step iteration process with errors: xn+1 = (1−αn)xn+αnTyn+un, yn = (1 − βn)xn + βnTzn + vn, zn = (1 − γn)xn + γnTxn + wn, n ≥ 1. Sufficient and necessary conditions for the strong convergence {xn} to a fixed point of T is established. We also derive the corresponding new results on the strong convergence of the three-step iterative process.

Fixed point theorems for asymptotically pseudocontactive maps in certain Banach Spaces

Let E be a real q-uniformly smooth Banach space (for example, Lp or lp spaces ,1‹p‹∞) and let T: E→E be asymptotically pseudocontractive mapping with nonempty fixed point set F(T) and a real sequence {kn} n=1∞ . Let {xn}n≥1 be the sequence generated from an arbitrary x1 in E by xn+1=(1-αn) xn + αnT nxn, n≥1. If the range of T is bounded and there exists a strictly increasing function Φ : [ o , ∞) →[ o , ∞) with Φ(0)=0 such that (Tnxn-p, j(xn-p)) ≤ kn||xn-p||2 _ Φ(||xn-p||), p∈F(T), then {xn}∞n=1 converges strongly to p, provided that {kn} and {αn} satisfy certain properties. This result compliments the results of Chang, Park and Cho (2000), by dropping the Lipschitz condition on T. 2000 Mathematics Subject Classification : 47H09 47J05. KEY WORDS: Fixed points, asymptotically pseudocontractive maps, modified Mann iterative methods, q-uniformly smooth spaces. Global Journal of Mathematical Sciences Vol.3(2) 2004: 137-142

Strong convergence of approximants to fixed points of Lipschitzian pseudocontractive maps

Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, L p or l p spaces, 1 < p < ∞) and K be a nonempty closed convex and bounded subset of E with φ ≠ int ( K). Let T: K → K be a Lipschitzian pseudocontractive mapping such that for z ϵ int ( K), ∥ z − Tz∥ < ∥ x − Tx∥,for all x ϵ ∂( K). Then for z 0 ϵ K arbitrary, the iteration process { z n } defined by z n + 1 ≔ (1 − μ n + 1) z + μ n + 1 y n ; y n ≔ ( 1 −- α n) z n + α n Tz n converges strongly to a fixed point of T, provided that {μ n } and {α n }satisfy certain conditions. Moreover, if T is strictly pseudocontractive with a nonempty fixed-point set, then it is proved that the Mann type iteration scheme converges strongly to a fixed point of T.

Global Iterative Schemes for Accretive Operators

Let E be a real q-uniformly smooth Banach space and A:E→2E be an m-accretive operator which satisfies a linear growth condition of the form ‖Ax‖≤c(1+‖x‖) for some constant c>0 and for all x∈E. It is proved that if two real sequences {λn} and {θn} satisfy appropriate conditions, the sequence {xn} generated from arbitrary x0∈E by xn+1=xn−λn(un+θn(xn−z)); un∈Axnn≥0, converges strongly to some x*∈A−1(0). Furthermore, if E is a reflexive Banach space with a uniformly Gâteaux differentiable norm, and if every weakly compact convex subset of E has the fixed-point property for nonexpansive mappings and A:D(A)≔E→2E is m-accretive, then for arbitrary, z,x0∈E the sequence {xn} defined by xn+1+λn(un+1+θn(xn+1−z))=xn+en, for un∈Axn, where en∈E is such that ∑‖en‖<∞∀n≥0, converges strongly to some x*∈A−1(0).