General Iterative Method Research Articles

Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space

Let be N strictly pseudononspreading mappings defined on closed convex subset C of a real Hilbert space H. Consider the problem of finding a common fixed point of these mappings and introduce cyclic algorithms based on general viscosity iteration method for solving this problem. We will prove the strong convergence of these cyclic algorithm. Moreover, the common fixed point is the solution of the variational inequality .

A Hybrid Gradient‐Projection Algorithm for Averaged Mappings in Hilbert Spaces

It is well known that the gradient‐projection algorithm (GPA) is very useful in solving constrained convex minimization problems. In this paper, we combine a general iterative method with the gradient‐projection algorithm to propose a hybrid gradient‐projection algorithm and prove that the sequence generated by the hybrid gradient‐projection algorithm converges in norm to a minimizer of constrained convex minimization problems which solves a variational inequality.

The General Iterative Methods for Asymptotically Nonexpansive Semigroups in Banach Spaces

We introduce the general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups which is a unique solution of some variational inequalities. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature.

A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions

We study strong convergence of the sequence generated by implicit and explicit general iterative methods for a one‐parameter nonexpansive semigroup in a reflexive Banach space which admits the duality mapping Jφ, where φ is a gauge function on [0, ∞). Our results improve and extend those announced by G. Marino and H.‐K. Xu (2006) and many authors.

A general extending and constraining procedure for linear iterative methods

Algebraic reconstruction techniques (ARTs), on both their successive and simultaneous formulations, have been developed since the early 1970s as efficient ‘row-action methods’ for solving the image-reconstruction problem in computerized tomography. In this respect, two important development directions were concerned with, first, their extension to the inconsistent case of the reconstruction problem and, second, their combination with constraining strategies, imposed by the particularities of the reconstructed image. In the first part of this paper, we introduce extending and constraining procedures for a general iterative method of an ART type and we propose a set of sufficient assumptions that ensure the convergence of the corresponding algorithms. As an application of this approach, we prove that Cimmino's simultaneous reflection method satisfies this set of assumptions, and we derive extended and constrained versions for it. Numerical experiments with all these versions are presented on a head phantom widely used in the image reconstruction literature. We also consider hard thresholding constraining used in sparse approximation problems and apply it successfully to a 3D particle image-reconstruction problem.

The modified general iterative methods for nonexpansive semigroups in banach spaces†

In this paper, we introduce the modified general iterative approximation methods for finding a common fixed point of nonexpansive semigroups which is a unique solution of some variational inequalities. The strong convergence theorems are established in the framework of a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature. Mathematics Subject Classification (2000) 47H05, 47H09, 47J25, 65J15

A general iterative method for finding common solutions of system of equilibrium problems, system of variational inequalities and fixed point problems

In this paper, we introduce a general iterative algorithm for finding the common element of the set of common fixed points of an infinite family of nonexpansive mappings, the set of solutions of systems of equilibrium problems and the set of solutions of systems of variational inequalities for two strongly monotone mappings in a real Hilbert space. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions. Our results improve and extend the corresponding results announced by Colao and Marino [V. Colao and G. Marino, strong convergence for a minimization problem on points of equilibrium and common fixed points of an infinite family of nonexpansive mappings, Nonlinear Anal. (2010) 3513–3524] and many others.

A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems

In this article, we introduce a new general composite iterative scheme for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality problem for an inverse-strongly monotone mapping in Hilbert spaces. It is shown that the sequence generated by the proposed iterative scheme converges strongly to a common element of the above three sets under suitable control conditions, which solves a certain optimization problem. The results of this article substantially improve, develop, and complement the previous well-known results in this area.2010 Mathematics Subject Classifications: 49J30; 49J40; 47H09; 47H10; 47J20; 47J25; 47J05; 49M05.

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

RELATIVISTIC GRAVITATIONAL COLLAPSE IN COMOVING COORDINATES: THE POST-QUASISTATIC APPROXIMATION

A general iterative method proposed some years ago for the description of relativistic collapse is presented here in comoving coordinates. For doing that we redefine the basic concepts required for the implementation of the method for comoving coordinates. In particular, the definition of the post-quasistatic approximation in comoving coordinates is given. We write the field equations, the boundary conditions and a set of ordinary differential equations (the surface equations) which play a fundamental role in the algorithm. As an illustration of the method, we show how to build up a model inspired by the well-known Schwarzschild interior solution. Both the adiabatic and the nonadiabatic case are considered.

Viscosity methods for common solutions for equilibrium and hierarchical fixed point problems

Implicit and explicit viscosity methods for finding common solutions of equilibrium and hierarchical fixed points are presented. These methods are used to solve systems of equilibrium problems and variational inequalities where the involving operators are complements of nonexpansive mappings. The results here are situated on the lines of the research of the corresponding results of Moudafi [Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Probl. 23 (2007), pp. 1635–1640; Weak convergence theorems for nonexpansive mappings and equilibrium problems, to appear in JNCA], Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-points problems, Fixed Point Theory Appl. Art ID 95453 (2006), 10 pp.; Strong convergence of an iterative method for hierarchical fixed point problems, Pac. J. Optim. 3 (2007), pp. 529–538; Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems, to appear in JNCA], Yao and Liou [Weak and strong convergence of Krasnosel'skiĭ–Mann iteration for hierarchical fixed point problems, Inverse Probl. 24 (2008), 015015 8 pp.], S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006), pp. 506–515], Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, preprint.], Combettes and Hirstoaga [Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), pp. 117–136] and Plubtieng and Pumbaeang [A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007), pp. 455–469.].

An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings

Let H be a Hilbert space, { T i } i ∈ N a family of nonexpansive mappings from H into itself, G i : C × C → R a finite family of equilibrium functions ( i ∈ { 1 , 2 , … , K } ) , A a strongly positive bounded linear operator with coefficient γ ̄ and f an α -contraction on H . Let W n be the mapping generated by { T i } and { λ n } as in (1.5), let S r k , n k be the resolvent generated by G k and r k , n as in Lemma 2.4. Moreover, let { r k , n } k = 1 K , { ϵ n } and { λ n } satisfy appropriate conditions and F ≔ ( ⋂ k = 1 K S E P ( G k ) ) ∩ ( ⋂ n ∈ N F i x ( T n ) ) ≠ 0̸ ; we introduce an explicit scheme which defines a suitable sequence as follows: z n + 1 = ϵ n γ f ( z n ) + ( I − ϵ n A ) W n S r 1 , n 1 S r 2 , n 2 ⋯ S r K , n K z n ∀ n ∈ N and { z n } strongly converges to x ∗ ∈ F which satisfies the variational inequality 〈 ( A − γ f ) x ∗ , x − x ∗ 〉 ≥ 0 for all x ∈ F . The results presented in this paper mainly extend and improve some recent results in [Vittorio Colao, et al., An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, Nonlinear Anal. 71 (2009) 2708–2715; S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007) 455–469; S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515].

Convergence Theorems of a General Composite Iterative Method for Nonexpansive Semigroups in Banach Spaces

We introduce a general composite iterative scheme for nonexpansive semigroups in Banach spaces. We establish some strong convergence theorems of the general iteration scheme under different control conditions. The results presented in this paper improve and extend the corresponding results of Marino and Xu (2006), and others, from Hilbert spaces to Banach spaces.

A general iterative method for obtaining an infinite family of strictly pseudo-contractive mappings in Hilbert spaces

In this work, we consider a general composite iterative method for obtaining an infinite family of strictly pseudo-contractive mappings in Hilbert spaces. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of fixed points, which solves the variational inequality 〈 ( γ f − μ F ) q , p − q 〉 ≤ 0 , for p ∈ ∩ i = 1 ∞ F ( T i ) . Our results improve and extend corresponding ones announced by many others.

Convergence of General Nonstationary Iterative Methods for Solving Singular Linear Equations

In this paper, we analyze the convergence of the general nonstationary iterative methods for solving consistent singular linear equations (in particular, singular Hermitian positive semidefinite linear systems), and we discuss relations of general stationary results and ours. We utilize the quotient convergence to prove the convergence of the two-stage iterative algorithms for solving the consistent singular Hermitian positive semidefinite linear equations.

The General Hybrid Approximation Methods for Nonexpansive Mappings in Banach Spaces

We introduce two general hybrid iterative approximation methods (one implicit and one explicit) for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Marino and Xu (2006), Wangkeeree et al. (in press), and Ceng et al. (2009).

A General Iterative Algorithm for Generalized Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities

We introduce a new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibrium problems, and the set of solutions of the variational inclusion for a β‐inverse‐strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (2006), Su et al. (2008), Klin‐eam and Suantai (2009), Tan and Chang (2011), and some other authors.

Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E*. Let 𝒮 = {T(s) : 0 ≤ s < ∞} be a nonexpansive semigroup on E such that Fix(𝒮): = ⋂t≥0Fix(T(t)) ≠ ∅, and f is a contraction on E with coefficient 0 < α < 1. Let F be δ‐strongly accretive and λ‐strictly pseudocontractive with δ + λ > 1 and γ a positive real number such that . When the sequences of real numbers {αn} and {tn} satisfy some appropriate conditions, the three iterative processes given as follows: xn+1 = αnγf(xn)+(I − αnF)T(tn)xn, n ≥ 0, yn+1 = αnγf(T(tn)yn)+(I − αnF)T(tn)yn, n ≥ 0, and zn+1 = T(tn)(αnγf(zn)+(I − αnF)zn), n ≥ 0 converge strongly to , where is the unique solution in Fix(𝒮) of the variational inequality , x ∈ Fix(𝒮). Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others.

A New General Iterative Method for Solution of a New General System of Variational Inclusions for Nonexpansive Semigroups in Banach Spaces

We introduce a new general system of variational inclusions in Banach spaces and propose a new iterative scheme for finding common element of the set of solutions of the variational inclusion with set‐valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of nonexpansive semigroups in a uniformly convex and 2‐uniformly smooth Banach space. Furthermore, strong convergence theorems are established under some certain control conditions. As applications, finding a common solution for a system of variational inequality problems and minimization problems is given.