K-strictly Pseudo-contractive Mappings Research Articles

On the Convergence of Two Iterative Methods for k-strictly Pseudo-contractive Mappings in Cat(0) Spaces

In this paper, we prove the demiclosedness principle for k -strictly pseudo-contractive mappings and establish the Δ - convergence theorem of the cyclic algorithm for such mappings in CAT(0) spaces. Also, we give the strong convergence theorem of the modified Halpern iteration for k -strictly pseudo-contractive mappings in CAT(0) spaces. Our results extend and improve the corresponding recent results announced by many authors in the literature.

Convergence analysis for split hierachical monotone variational inclusion problem in Hilbert spaces

Abstract In this paper, we introduce a new iterative algorithm for approximating a common solution of Split Hierarchical Monotone Variational Inclusion Problem (SHMVIP) and Fixed Point Problem (FPP) of k-strictly pseudocontractive mappings in real Hilbert spaces. Our proposed method converges strongly, does not require the estimation of operator norm and it is without imposing the strict condition of compactness; these make our method to be potentially more applicable than most existing methods in the literature. Under standard and mild assumption of monotonicity of the SHMVIP associated mappings, we establish the strong convergence of the iterative algorithm.We present some applications of our main result to approximate the solution of Split Hierarchical Convex Minimization Problem (SHCMP) and Split Hierarchical Variational Inequality Problem (SHVIP). Some numerical experiments are presented to illustrate the performance and behavior of our method. The result presented in this paper extends and complements some related results in literature.

A Modified Iterative Method for a Finite Collection of Non-self Mappings and a Family of Variational Inequality Problems

This paper deals with an iterative algorithm with the help of averaged mappings for the common solution of a fixed point problem of a finite collection of k-strictly pseudocontractive non-self mappings and a system of variational inequality problems in the setting of a real Hilbert space. Under some suitable conditions, the sequence generated by the proposed algorithm converges strongly to the common solution of the above said problems. Also, a numerical example is given to establish the superiority of the proposed algorithm over some existing methods.

On approximation of fixed points of multivalued pseudocontractive mappings in Hilbert spaces

AbstractIt is proved that if a multivalued k-strictly pseudocontractive mapping S of Chidume et al. (Abstr. Appl. Anal. 2013:629468, 2013) is of type one, then $I-S$ I − S is demiclosed at zero. Also, under this condition, the Mann (respectively, Ishikawa) sequence weakly (respectively, strongly) converges to a fixed point of a multivalued k-strictly pseudocontractive (respectively, pseudocontractive) mapping S without the condition that the fixed point set of S is strict, where S is of type one if for any pair $r,g \in D(S)$ r , g ∈ D ( S ) , $$\|u-v\|\leq\Phi(Sr,Sg) \quad\mbox{for all } u\in P_{S}r, v\in P_{S}g, $$ ∥ u − v ∥ ≤ Φ ( S r , S g ) for all u ∈ P S r , v ∈ P S g , and Φ denotes the Hausdorff metric. The results obtained give a partial answer to the problem of the removal of the strict fixed point set condition, which is usually imposed on multivalued mappings. Thus, the results extend, complement, and improve the results on multivalued and single-valued mappings in the contemporary literature.

A New Iterative Scheme for Common Solution of Equilibrium Problems, Variational Inequalities and Fixed Point of k-strictly Pseudo-contractive Mappings in Hilbert Spaces

In this paper, we present a new iterative scheme for finding a common point among the set of solution of equilibrium problems, the set of solution to a variational inequality problem and the fixed point set of k-strictly pseudo-contractive mappings in a real Hilbert space. We then prove that the proposed scheme converges strongly to a common element which is the solution of a variational inequality problem, system of equilibrium problems, and a fixed point of k-strictly pseudo-contractive mappings. These results improve and generalize recent works in this direction.

Iterative algorithms for monotone inclusion problems, fixed point problems and minimization problems

We introduce new implicit and explicit iterative schemes for finding a common element of the set of fixed points of k-strictly pseudocontractive mapping and the set of zeros of the sum of two monotone operators in a Hilbert space. Then we establish strong convergence of the sequences generated by the proposed schemes to a common point of two sets, which is a solution of a certain variational inequality. Further, we find the unique solution of the quadratic minimization problem, where the constraint is the common set of two sets mentioned above. As applications, we consider iterative schemes for the Hartmann-Stampacchia variational inequality problem and the equilibrium problem coupled with fixed point problem.MSC:47H05, 47H09, 47H10, 47J05, 47J07, 47J25, 47J20, 49M05.

Strong convergence theorems for solving a general system of finite variational inequalities for finite accretive operators and fixed points of nonexpansive semigroups with weak contraction mappings

Abstract 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:47H05, 47H10, 47J25.

Strong convergence theorems for variational inequalities and fixed points of a countable family of nonexpansive mappings

A new general iterative method for finding a common element of the set of solutions of variational inequality and the set of common fixed points of a countable family of nonexpansive mappings is introduced and studied. A strong convergence theorem of the proposed iterative scheme to a common fixed point of a countable family of nonexpansive mappings and a solution of variational inequality of an inverse strongly monotone mapping are established. Moreover, we apply our main result to obtain strong convergence theorems for a countable family of nonexpansive mappings and a strictly pseudocontractive mapping, and a countable family of uniformly k-strictly pseudocontractive mappings and an inverse strongly monotone mapping. Our main results improve and extend the corresponding result obtained by Klin-eam and Suantai (J Inequal Appl 520301, 16 pp, 2009). Mathematics Subject Classification (2000): 47H09, 47H10

Some results on a general iterative method for k-strictly pseudo-contractive mappings

AbstractLet H be a Hilbert space, C be a closed convex subset of H such that C ± C ⊂ C, and T : C → H be a k-strictly pseudo-contractive mapping with F(T) ≠ ∅ for some 0 ≤ k < 1. Let F : C → C be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0 and f : C → C be a contraction with the contractive constant α ∈ (0, 1). Let "Equation missing", "Equation missing" and τ < 1. Let {α n } and {β n } be sequences in (0, 1). It is proved that under appropriate control conditions on {α n } and {β n }, the sequence {x n } generated by the iterative scheme xn+1= α n γf(x n ) + β n x n + ((1 - β n )I - α n μF)P C Sx n , where S : C → H is a mapping defined by Sx = kx + (1 - k)Tx and P C is the metric projection of H onto C, converges strongly to q ∈ F(T), which solves the variational inequality 〈μFq - γf(q), q - p〉 ≤ 0 for p ∈ F(T).MSC: 47H09, 47H05, 47H10, 47J25, 49M05

A general iterative scheme for k-strictly pseudo-contractive mappings and optimization problems

In this paper, we introduce a new general iterative scheme for finding fixed points of a strictly pseudo-contractive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a fixed point of the mapping, which is a solution of a certain optimization problem related to a strongly positive bounded linear operator. Additional results of the main result are also obtained.

Strong convergence of iterative methods for k-strictly pseudo-contractive mappings in Hilbert spaces

In this paper, we introduce composite iterative schemes for finding fixed points of k-strictly pseudo-contractive mappings for some 0 ⩽ k < 1 in Hilbert spaces. Then, under certain different control conditions, we establish strong convergence theorems on the composite iterative schemes. The main theorems improve and generalize the recent corresponding results of Cho et al. [5] and Marino and Xu [9] as well as Halpern [6], Wittmann [12], Moudafi [10] and Xu [14].

Mann iteration of weak convergence theorems in Banach space

In this paper, by using Mann’s iteration process we will establish several weak convergence theorems for approximating a fixed point of k-strictly pseudocontractive mappings with respect to p in p-uniformly convex Banach spaces. Our results answer partially the open question proposed by Marino and Xu, and extend Reich’s theorem from nonexpansive mappings to k-strict pseudocontractive mappings.