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Three-step iterations for nonexpansive mappings and inverse-strongly monotone mappings
Abstract
This paper introduces a three-step iteration for finding a common element of the set of fixed points of a nonexpansive mapping
and the set of solutions of the variational inequality for an inverse-strongly monotone mapping by viscosity approximation
methods in a Hilbert space. The authors show that the iterative sequence converges strongly to a common element of the two
sets, which solves some variational inequality. Subsequently, the authors consider the problem of finding a common fixed point
of a nonexpansive mapping and a strictly pseudo-contractive mapping and the problem of finding a common element of the set
of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping. The results obtained
in this paper extend and improve the corresponding results announced by Nakajo, Takahashi, and Toyoda.
... Some efforts have been done to extended and generalize Theorem 1.1 in various prospects (cf. [15,16,18,19,20,22,21,25,28,27,30,32,33,36,35,43,41,42] ...
... Thus by (33) we get ...
In this paper, we employ an inertial method to approximate a common point of two sets, the set of zeros of an inverse strongly monotone mapping and the set of fixed points of a quasi-nonexpansive mapping in a Hilbert space. This common point turns out to be a solution of variational inequality problem. As applications of our results a number of new algorithms are suggested to find solutions of various important problems in analysis.
... Ax, x ≥ ∥x∥ 2 , x ∈ H. In 2009, motivated and inspired by above results, Meijuan Shang, Yongfu SU, Xiaolong Qin [41], introduced a general three-step iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping by viscosity approximation methods in a Hilbert space. They showed that the iterative sequence converges strongly to a common element of two sets, which solves some variational inequality. ...
... Theorem 3.1.18: [41] Let C be a closed convex subset of a real Hilbert space H. Let f: C → C be a contracton with coefficient k (0 < k < 1). Let A be an α-inverse-strongly-monotone mapping of C into H and let S be a nonexpansive mapping of C into itself such that F(S) ∩ VI(C, A) ≠ ф. ...
The variational inequality problem provides a broad unifying setting for the study of optimization, equilibrium and related problems and serves as a useful computational framework for the solution of a host of problems in very diverse applications. Variational inequalities have been a classical subject in mathematical physics, particularly in the calculus of variations associated with the minimization of infinite-dimensional functionals. This paper presents a survey of main results related to variational inequalities and fixed point problems defined on real Hilbert spaces and Banach spaces. Keywords: Fixed Point Problem, Inverse-Strongly-Monotone Mappings, Monotone Mappings, Projection Mappings, Variational Inequality Problem.
... u ∈ C is a solution of the variational inequality (1.1) if and only if u ∈ C is a fixed point of the mapping P C (I − λA), where I is the identity mapping and λ > 0 is a constant. Recently, iterative methods have been applied to approximate common elements in the fixed point set of nonexpansive mappings and in the solution set of variational inequality (1.1), see [3,4,5,6,12,13,15,19,21] and the reference therein. Noor and Huang [13] considered a three-step iterative method for finding a common element in the set of fixed points of a nonexpansive mapping and in the set of solutions of the variational inequality problem (1.1) in a real Hilbert space. ...
... Recently, three-step iterative algorithms were studied by many authors; see [3,4,5,6,7,9,11,12,13,14,16,17,19,20,21]. In 1989, Glowinski and Le Tallec [7] used three-step iterative schemes to find the approximate solutions of the elastoviscoplasticity problem, liquid crystal theory, and eigenvalue computation, and they showed that three-step approximations perform better numerically. ...
In this paper, a three-step iterative method is considered for finding a common element in the set of fixed points of a non-expansive mapping and in the set of solutions of a variational inclusion problem in a real Hilbert space.
... Chen et al. [13] incorporated viscosity approximation methods for fnding the common elements to monotone and nonexpansive mappings. Numerous algorithms use viscosity approximation methods to fnd the common element of variational inequality problem and fxed point problem such as Ceng and Yao's [14] strong convergence result by combining the extragradient method and viscosity approximation method such that the two sequences generated by the algorithm converge strongly to the common element, a general three-step iterative process by Shang et al. [15] in which two projections are calculated onto C in frst two steps, and in the third step, the third projection onto C is combined using viscosity approximation method, a generalized viscosity type extragradient method by Anh et al. [16] which uses a strongly positive linear bounded operator to converge to the common element for variational inequality problem, fxed point problem and equilibrium problem, and two-step extragradient-viscosity method by Hieu et al. [17] in which frst step calculates three projections onto C and second step combines the projections using viscosity approximation method. ...
This paper aims to introduce an iterative algorithm based on an inertial technique that uses the minimum number of projections onto a nonempty, closed, and convex set. We show that the algorithm generates a sequence that converges strongly to the common solution of a variational inequality involving inverse strongly monotone mapping and fixed point problems for a countable family of nonexpansive mappings in the setting of real Hilbert space. Numerical experiments are also presented to discuss the advantages of using our algorithm over earlier established algorithms. Moreover, we solve a real-life signal recovery problem via a minimization problem to demonstrate our algorithm’s practicality.
... Some more recent progresses on the investigation of the implicit and explicit schemes (1.10) and (1.12) can be found in [3][4][5]9,10,[12][13][14][15][16][19][20][21][22]24]. ...
The set S of fixed points of a nonexpansive mapping T in a Hilbert space is always closed convex. Assume the set is also nonempty. It is then of interest to find the element x † of this set with least norm; that is, the minimum-norm fixed point of T. In this article we provide two methods (one implicit and one explicit) for finding x †. As a matter of fact, we will consider a more general problem of finding a point in S which solves a variational inequality problem. Applications to convex minimization problems and convexly constrained linear inverse problems are included.
In this paper, we study some implicit algorithms to approximate a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an inverse strongly monotone mapping in Hilbert spaces. Using shrinking projection method, we obtain couple of strong convergence results. As applications of our results a new algorithm is derived to find a common fixed point of two mappings. We also suggest an algorithm to approximate solution of split feasibility problem.
The purpose of this paper is to study convex feasibility problems in the setting of a real Hilbert space. The approximation of common elements of solution set of variational inequality problems and fixed point set of nonexpansive mappings is considered. Strong convergence theorems are established in the framework of Hilbert spaces.
Two algorithms have been constructed for finding the minimum-norm fixed point of a λ-strict pseudo-contraction T in Hilbert space. It is shown that the proposed algorithms strongly converge to the minimum-norm fixed point of T.
Today, complex networks have attracted increasing attention from various fields of science and engineering. It has been demonstrated that many complex networks display various synchronization phenomena. In this paper, we introduce a time-varying complex dynamical network model. We then further investigate its synchronization phenomenon and prove several network synchronization theorems. Especially, we show that synchronization of such a time-varying dynamical network is completely determined by the inner-coupling matrix, and the eigenvalues and the corresponding eigenvectors of the coupling configuration matrix of the network.
Iterative algorithms for nonexpansive mappings and maximal monotone operators are investigated. Strong convergence theorems
are proved for nonexpansive mappings, including an improvement of a result of Lions. A modification of Rockafellar’s proximal
point algorithm is obtained and proved to be always strongly convergent. The ideas of these algorithms are applied to solve
a quadratic minimization problem.
The minimization of the drag functional for the stationary, isentropic, compressible Navier–Stokes equations (NSE) in three spatial dimensions is considered. In order to establish the existence of an optimal shape, the general result on compactness of families of generalized solutions to the NSE is established within in the framework of the modern theory of nonlinear PDEs [P. L. Lions,]. The family of generalized solutions to the NSE is constructed over a family of admissible domains U ad . Any admissible domain Ω = B \ S contains an obstacle S, e.g., a wing profile. Compactness properties of the family of admissible domains in the form of the condition (H Ω) is imposed. Roughly speaking, the condition (H Ω) is satisfied, provided that for any sequence of admissible domains {Ωn} ⊂ U ad there is a subsequence convergent both in Hausdorff metrics and in the sense of Kuratowski and Mosco. The analysis is performed for the adiabatic constant γ > 1 in the pressure law p(ρ) = ρ γ and it is based on the technique used in [P. I. Plotnikov and J. Sokolowski, Comm. Math. Phys., 258 (2005), pp. 567–608] in the case of discretized NSE. The result is a generalization to the stationary equations with γ > 1 of the results obtained in [E.78] for evolution equations within the range γ > 3/2 for the adiabatic ratio.
In this paper, we investigate the locally and globally adaptive synchronization problem for an uncertain complex dynamical
network with time-varying coupling delays based on the decentralized control. The coupling terms here are bounded by high-order
polynomials with known gains that are ubiquitous in a large class of complex dynamical networks. We generalize the usual technology
of searching for an appropriate coordinates transformation to change the network dynamics into a series of decoupled lower-dimensional
systems. Several adaptive synchronization criteria are derived by constructing the Lyapunov-Krasovskii functional and Barbalat
lemma, and the proposed criteria are simple in form and convenient for the practical engineering design. Numerical simulations
illustrated by a nearest-neighbor coupling network verify the effectiveness of the proposed synchronization scheme.
We use viscosity approximation methods to obtain strong convergence to common fixed points of monotone mappings and a countable family of nonexpansive mappings. Let C be a nonempty closed convex subset of a Hilbert space H and P
C is a metric projection. We consider the iteration process {x
n} of C defined by x
1 = x ∈ C is arbitrary and where f is a contraction on C, {S
n} is a sequence of nonexpansive self-mappings of a closed convex subset C of H, and A is an inverse-strongly-monotone mapping of C into H. It is shown that {x
n} converges strongly to a common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping which solves some variational inequality. Finally, the ideas of our results are applied to find a common element of the set of equilibrium problems and the set of solutions of the variational inequality problem, a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. The results of this paper extend and improve the results of Chen, Zhang and Fan.
We consider some new iterative methods for solving general monotone mixed variational inequalities by using the updating technique of the solution. The convergence analysis of these new methods is considered and the proof of convergence is very simple. These new methods are versatile and are easy to implement. Our results differ from those of B. S. He [Appl. Math. Optimization 35, No. 1, 69-76 (1997; Zbl 0865.90119)], M. V. Solodov and P. Tseng [SIAM J. Control Optimization 34, No. 5, 1814-1830 (1996; Zbl 0866.49018)], and M. A. Noor [J. Math. Anal. Appl. 229, No. 1, 330-343 (1999; Zbl 0927.49004); Appl. Math. Lett. 11, No. 4, 109-113 (1998; Zbl 0941.49005); Math. Comput. Modelling 29, No. 3, 95-100 (1999; Zbl 0994.47061)] for solving monotone variational inequalities.
In this paper, we suggest and analyze three-step iteration methods for finding the common element of the set of fixed points of a nonexpansive mappings and the set of solutions of the variational inequalities. We also study the convergence criteria of three-step iterative method under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results.
It is well known that the mixed variational inequalities are equivalent to the fixed point problems and the resolvent equations. Using this equivalence, we suggest and consider a new three-step iterative method for solving mixed variational inequalities. The new iterative method is obtained by using three-steps under suitable conditions. We prove that the new method is globally convergent. Our results can be viewed as significant extensions of the previously known results for mixed variational inequalities. Since mixed variational inequalities include variational inequalities as special cases, our method appears to be a new one for solving variational inequalities. Preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method.
In this paper, we suggest and consider a class of new three-step approximation schemes for general variational inequalities. Our results include Ishikawa and Mann iterations as special cases. We also study the convergence criteria of these schemes.
In this paper, we introduce a new three-step iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality using the technique of updating the solution. We show that the sequence converges strongly to a common element of two sets under some control conditions. Results proved in this paper may be viewed as an improvement and refinement of the recent results of Noor and Huang [M. Aslam Noor, Z. Huang, Three-step methods for nonexpansive mappings and variational inequalities, Appl. Math. Comput., in press] and Yao and Yao [Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput., in press].
In this paper, we suggest and analyze some three-step iterative schemes for finding the common elements of the set of the solutions of the Noor variational inequalities involving two nonlinear operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of the suggested iterative schemes under some mild conditions. Since the Noor variational inequalities include variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.
This letter reports the finding of a new chaotic attractor in a simple three-dimensional autonomous system, which connects the Lorenz attractor and Chen's attractor and represents the transition from one to the other.
The Morse decomposition theory for single-valued dynamical systems is extended to general dynamical systems and differential inclusions. Relations between chain recurrent sets, chain recurrent components (chain control sets), and Morse decompositions for open-loop systems are discussed. The robustness of feedback laws of closed-loop systems with respect to small time delays and sample-hold controls is also addressed.
We revisit a classic design problem for a plate of variable thickness under the model of Kirchhoff. Our main contribution has two goals. One is to provide a rather general existence result under a main assumption on the structure of the tensor of material constants. The other focuses on providing a minimal number of additional design variables for a relaxation of the problem when that assumption on the tensor of elastic constants does not hold. In both situations, the cost functional can be pretty general.
This note further investigates the locally and globally adaptive synchronization of an uncertain complex dynamical network. Several network synchronization criteria are deduced. Especially, our hypotheses and designed adaptive controllers for network synchronization are rather simple in form. It is very useful for future practical engineering design. Moreover, numerical simulations are also given to show the effectiveness of our synchronization approaches.
Today, complex networks have attracted increasing attention from various fields of science and engineering. It has been demonstrated that many complex networks display various synchronization phenomena. In this note, we introduce a time-varying complex dynamical network model. We then further investigate its synchronization phenomenon and prove several network synchronization theorems. Especially, we show that synchronization of such a time-varying dynamical network is completely determined by the inner-coupling matrix, and by the eigenvalues and the corresponding eigenvectors of the coupling configuration matrix of the network.
The proximal point algorithm is investigated in a more general form where the requirement for exact minimization at each iteration is weakened, and the subdifferential is replaced by an arbitrary maximal monotone operator T. Convergence is established under several criteria amenable to implementation. The rate of convergence is shown to be ″typically″ linear with an arbitrarily good modulus if c//k stays large enough, in fact superlinear if c//k approaches infinity . The case of T equals the subdifferential is treated in extra detail. Application is also made to a related case corresponding to minimax problems.
In this paper, we introduce an iteration process of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for an inverse strongly-monotone mapping, and then obtain a weak convergence theorem. Using this result, we obtain a weak convergence theorem for a pair of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping.
LetH be a nonempty closed convex subset of a topological vector spaceE andF be a real-valued function onH H Then, we prove that, under some conditions, there existsx
0H such thatF(x
0,y)0 for ally H. Furthermore, we obtain a necessary and sufficient condition that a finite system of convex inequalities is irreducibly inconsistent.
Viscosity approximation methods for nonexpansive mappings are studied. Consider the iteration process {xn}, where x0∈C is arbitrary and xn+1=αnf(xn)+(1−αn)SPC(xn−λnAxn), f is a contraction on C, S is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H. It is shown that {xn} converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly-monotone mapping which solves some variational inequality.
In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.