- Research
- Open Access
An algorithm for finding common solutions of various problems in nonlinear operator theory
- Eric U Ofoedu^{1},
- Jonathan N Odumegwu^{1},
- Habtu Zegeye^{2} and
- Naseer Shahzad^{3}Email author
https://doi.org/10.1186/1687-1812-2014-9
© Ofoedu et al.; licensee Springer. 2014
- Received: 27 August 2013
- Accepted: 21 November 2013
- Published: 9 January 2014
Abstract
In this paper, it is our aim to prove strong convergence of a new iterative algorithm to a common element of the set of solutions of a finite family of classical equilibrium problems; a common set of zeros of a finite family of inverse strongly monotone operators; the set of common fixed points of a finite family of quasi-nonexpansive mappings; and the set of common fixed points of a finite family of continuous pseudocontractive mappings in Hilbert spaces on assumption that the intersection of the aforementioned sets is not empty. Moreover, the common element is shown to be the metric projection of the initial guess on the intersection of these sets.
MSC:47H06, 47H09, 47J05, 47J25.
Keywords
- classical equilibrium problem
- generalized mixed equilibrium problem
- η-inverse strongly monotone mapping
- maximal monotone operator
- nonexpansive mappings
- real Hilbert space
- pseudocontractive mappings
- variational inequality problem
1 Introduction
If $L\in [0,1)$, then T is called strict contraction or simply a contraction; and if $L=1$, then T is called nonexpansive. A point $x\in D(T)$ is called a fixed point of an operator T if and only if $Tx=x$. The set of fixed points of an operator T is denoted by $Fix(T)$, that is, $Fix(T):=\{x\in D(T):Tx=x\}$.
Every nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive. The following examples show that the converse is not true.
Example 1.1 (see [1])
The concept of quasi-nonexpansive mappings was essentially introduced by Diaz and Metcalf [3]. Although Examples 1.1 and 1.2 guarantee the existence of a quasi-nonexpansive mapping which is not nonexpansive, we must note that a linear quasi-nonexpansive mapping defined on a subspace of a given vector space is nonexpansive on that subspace.
Another important generalization of the class of nonexpansive mappings is the class of pseudocontractive mappings. These mappings are intimately connected with the important class of nonlinear accretive operators. This connection will be made precise in what follows.
It now follows trivially from (1.3) that every nonexpansive mapping is pseudocontractive. We note immediately that the class of pseudocontractive mappings is larger than that of nonexpansive mappings. For examples of pseudocontractive mappings which are not nonexpansive, the reader may see [5].
It is easy to see from inequalities (1.3) and (1.4) that an operator A is monotone if and only if the mapping $T:=(I-A)$ is pseudocontractive. Consequently, the fixed point theory for pseudocontractive mappings is intimately connected with the zero of monotone mappings. For the importance of monotone mappings and their connections with evolution equations, the reader may consult any of the references [5, 6].
Due to the above connection, fixed point theory of pseudocontractive mappings became a flourishing area of intensive research for several authors.
The classical equilibrium problem (EP) includes as special cases the monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, Nash equilibria in noncooperative games. Furthermore, there are several other problems, for example, the complementarity problems and fixed point problems, which can also be written in the form of the classical equilibrium problem. In other words, the classical equilibrium problem is a unifying model for several problems arising from engineering, physics, statistics, computer science, optimization theory, operations research, economics and countless other fields. For the past 20 years or so, many existence results have been published for various equilibrium problems (see, e.g., [7–10]). Approximation methods for such problems thus become a necessity.
Iterative approximation of fixed points and zeros of nonlinear mappings has been studied extensively by many authors to solve nonlinear mapping equations as well as variational inequality problems and their generalizations (see, e.g., [11–19]). Most published results on nonexpansive mappings (for example) focus on the iterative approximation of their fixed points or approximation of common fixed points of a given family of this class of mappings.
where ${P}_{C}$ denotes the metric projection from H onto a closed convex subset C of H. They proved that if the sequence ${\{{\alpha}_{n}\}}_{n\ge 0}$ is bounded away from 1, then ${\{{x}_{n}\}}_{n\ge 0}$ defined by (1.7) converges strongly to ${P}_{F(T)}({x}_{0})$.
Formulations similar to (1.7) for different classes of nonlinear problems had been presented by Kim and Xu [21], Nilsrakoo and Saejung [22], Ofoedu et al. [23], Yang and Su [24], Zegeye and Shahzad [25–27].
In this paper, motivated by the results of the authors mentioned above, it is our aim to prove strong convergence of a new iterative algorithm to a common element of the set of solutions of a finite family of classical equilibrium problems; a common set of zeros of a finite family of inverse strongly monotone mappings; a set of common fixed points of a finite family of quasi-nonexpansive mappings; and a set of common fixed points of a finite family of continuous pseudocontractive mappings in Hilbert spaces on assumption that the intersection of the aforementioned sets is not empty. Moreover, the common element is shown to be the metric projection of the initial guess on the intersection of these sets. Our theorems complement the results of the authors mentioned above and those of several other authors.
2 Preliminary
In what follows, we shall make use of the following lemmas.
Lemma 2.1 (see, e.g., Kopecka and Reich [28])
Lemma 2.2 Let C be a closed convex nonempty subset of a real Hilbert space H; and let ${P}_{C}:H\to C$ be the metric projection of H onto C. Let $x\in H$, then ${x}_{0}={P}_{C}x$ if and only if $\u3008z-{x}_{0},x-{x}_{0}\u3009\le 0$ for all $z\in C$.
Lemma 2.4 (see Zegeye [29])
Lemma 2.5 (see Zegeye [29])
- (1)
${F}_{r}$ is single-valued;
- (2)${F}_{r}$ is firmly nonexpansive type mapping, i.e., for all $x,y\in H$,${\parallel {F}_{r}x-{F}_{r}y\parallel}^{2}\le \u3008{F}_{r}x-{F}_{r}y,x-y\u3009;$
- (3)
$Fix({F}_{r})$ is closed and convex; and $Fix({F}_{r})=Fix(T)$ for all $r>0$.
In the sequel, we shall require that the bifunction $f:C\times C\to \mathbb{R}$ satisfies the following conditions:
(A1) $f(x,x)=0$, $\mathrm{\forall}x\in C$;
(A2) f is monotone in the sense that $f(x,y)+f(y,x)\le 0$ for all $x,y\in C$;
(A3) ${lim\hspace{0.17em}sup}_{t\to {0}^{+}}f(tz+(1-t)x,y)\le f(x,y)$ for all $x,y,z\in C$;
(A4) the function $y\mapsto f(x,y)$ is convex and lower semicontinuous for all $x\in C$.
Lemma 2.6 (see, e.g., [7, 30])
- (1)
${G}_{r}$ is single-valued for all $r>0$;
- (2)${G}_{r}$ is firmly nonexpansive, that is, for all $x,z\in H$,${\parallel {G}_{r}x-{G}_{r}z\parallel}^{2}\le \u3008{G}_{r}x-{G}_{r}z,x-z\u3009;$
- (3)
$Fix({G}_{r})=\mathit{EP}(f)$ for all $r>0$;
- (4)
$\mathit{EP}(f)$ is closed and convex.
Lemma 2.7 (see Ofoedu [31])
Lemma 2.8 (Compare with Lemma 13 of Ofoedu [31])
3 Main results
where ${A}_{n}={A}_{n(modd)}$, ${S}_{n}={S}_{n(modl)}$; $\{{r}_{n}\}\subset (0,\mathrm{\infty})$ such that ${lim}_{n\to \mathrm{\infty}}{r}_{n}={r}_{0}>0$; ${\{{\alpha}_{n}\}}_{n\ge 1}$ a sequence in $(0,1)$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$; ${\{{\beta}_{i}\}}_{i=1}^{m},{\{{\xi}_{h}\}}_{h=1}^{t}\subset (0,1)$ such that ${\sum}_{i=1}^{m}{\beta}_{i}=1={\sum}_{h=1}^{t}{\xi}_{h}$; $\eta \in (0,1)$ and $\{{\lambda}_{n}\}$ is a sequence in $[a,b]$ for some $a,b\in \mathbb{R}$ such that $0<a<b<2\gamma $, $\gamma ={min}_{1\le j\le d}\{{\gamma}_{j}\}$.
Lemma 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${T}_{1},{T}_{2},\dots ,{T}_{m}:C\to C$ be m continuous pseudocontractive mappings; let ${S}_{1},{S}_{2},\dots ,{S}_{l}:C\to C$ be l continuous quasi-nonexpansive mappings; let ${A}_{1},{A}_{2},\dots ,{A}_{d}:C\to H$ be d ${\gamma}_{j}$-inverse strongly monotone mappings with constants ${\gamma}_{j}\in (0,1)$, $j=1,2,\dots ,d$; let ${f}_{1},{f}_{2},\dots ,{f}_{t}:C\times C\to \mathbb{R}$ be t bifunctions satisfying conditions (A1)-(A4). Let $F:={\bigcap}_{i=1}^{m}Fix({T}_{i})\cap {\bigcap}_{j=1}^{d}{A}_{j}^{-1}(0)\cap {\bigcap}_{k=1}^{l}Fix({S}_{k})\cap {\bigcap}_{h=1}^{t}\mathit{EP}({f}_{h})\ne \mathrm{\varnothing}$. Let $\{{x}_{n}\}$ be a sequence defined by (3.1), then the sequence $\{{x}_{n}\}$ is well defined for each $n\ge 0$.
So, $p\in {C}_{k+1}$. This implies, by induction, that $F\subset {C}_{n}$ so that the sequence generated by (3.1) is well defined for all $n\ge 0$. □
Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${T}_{1},{T}_{2},\dots ,{T}_{m}:C\to C$ be m continuous pseudocontractive mappings; let ${S}_{1},{S}_{2},\dots ,{S}_{l}:C\to C$ be l continuous quasi-nonexpansive mappings; let ${A}_{1},{A}_{2},\dots ,{A}_{d}:C\to H$ be d ${\gamma}_{j}$-inverse strongly monotone mappings with constants ${\gamma}_{j}\in (0,1)$, $j=1,2,\dots ,d$; let ${f}_{1},{f}_{2},\dots ,{f}_{t}:C\times C\to \mathbb{R}$ be t bifunctions satisfying conditions (A1)-(A4). Let $F:={\bigcap}_{i=1}^{m}Fix({T}_{i})\cap {\bigcap}_{j=1}^{d}{A}_{j}^{-1}(0)\cap {\bigcap}_{k=1}^{l}Fix({S}_{k})\cap {\bigcap}_{h=1}^{t}\mathit{EP}({f}_{h})\ne \mathrm{\varnothing}$. Let $\{{x}_{n}\}$ be a sequence defined by (3.1). Then the sequence ${\{{x}_{n}\}}_{n\ge 0}$ converges strongly to the element of F nearest to ${x}_{0}$.
and hence $\parallel {x}_{n}-{w}_{n}\parallel \le \parallel {x}_{n}-{x}_{n+1}\parallel +\parallel {x}_{n+1}-{w}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$, which implies that ${w}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$.
and hence ${z}_{n}\to {x}^{\ast}$ as $n\to \mathrm{\infty}$.
Continuing, we obtain that ${S}_{k}{x}^{\ast}={x}^{\ast}$, $k=3,\dots ,l$. Hence, ${x}^{\ast}\in {\bigcap}_{k=1}^{l}F({S}_{k})$.
Similarly, we have that ${A}_{j}{x}^{\ast}=0$ for $j=2,\dots ,d$. Thus, ${x}^{\ast}\in {\bigcap}_{j=1}^{d}{A}_{i}^{-1}(0)$.
Now, by Lemma 2.2 we have that ${x}^{\ast}={P}_{F}({x}_{0})$. This completes the proof. □
Remark 3.3 We note that ${x}^{\ast}={P}_{F}({x}_{0})$ makes sense since it could be easily shown that F is closed and convex. In fact, it is enough to show that the set of zeros of γ-inverse monotone mappings and a fixed point set of continuous quasi-nonexpansive mappings are convex sets. Closure of the two sets simply follows from the continuity of the mappings involved.
then it could be easily checked that Γ is a bifunction and satisfies properties (A1)-(A4). Thus, the so-called generalized mixed equilibrium problem reduces to the classical equilibrium problem for the bifunction Γ. Thus, consideration of the so-called generalized mixed equilibrium problem in place of the classical equilibrium problem studied in this paper leads to no further generalization.
4 Application (convex differentiable optimization)
In Section 1, we defined a Lipschitz continuous mapping and an inverse strongly monotone mapping. Inverse strongly monotone mappings arise in various areas of optimization and nonlinear analysis (see, for example, [32–38]). It follows from the Cauchy-Schwarz inequality that if a mapping $A:D(A)\subseteq H\to R(A)\subseteq H$ is $\frac{1}{L}$-inverse strongly monotone, then A is L-Lipschitz continuous. The converse of this statement, however, fails to be true. To see this, take for instance $A=-I$, where I is the identity mapping on H, then A is L-Lipschitz continuous (with $L=1$) but not $\frac{1}{L}$-inverse strongly monotone (that is, not firmly nonexpansive in this case).
Baillon and Haddad [39] showed in 1977 that if $D(A)=H$ and A is the gradient of a convex functional on H, then A is $\frac{1}{L}$-inverse strongly monotone if and only if A is L-Lipschitz continuous. This remarkable result, which has important applications in optimization theory (see, for example, [40–42]), has become known as the Baillon-Haddad theorem. In fact, we have the following theorem.
Theorem 4.1 (Baillon-Haddad) (see Corollary 10 of [39])
Let $\varphi :H\to \mathbb{R}$ be a convex Fréchet-differentiable functional on H such that ∇ϕ is L-Lipschitz continuous for some $L\in (0,+\mathrm{\infty})$, then ∇ϕ is a $\frac{1}{L}$-inverse strongly monotone mapping (where ∇ϕ denotes the gradient of the functional ϕ).
Now, let us turn to the problem of minimizing a continuously Fréchet-differentiable convex functional with minimum norm in Hilbert spaces.
It is easy to see that if $K=H$, then optimality condition (4.2) is equivalent to $z\in \mathrm{\Omega}$ if and only if $z\in {(\mathrm{\nabla}\varphi )}^{-1}(0)$.
Thus, we obtain the following as a corollary of Theorem 3.2.
where ${(\mathrm{\nabla}\varphi )}_{n}={(\mathrm{\nabla}\varphi )}_{n(modd)}$, ${S}_{n}={S}_{n(modl)}$; $\{{r}_{n}\}\subset (0,\mathrm{\infty})$ such that ${lim}_{n\to \mathrm{\infty}}{r}_{n}={r}_{0}>0$; ${\{{\alpha}_{n}\}}_{n\ge 1}$ a sequence in $(0,1)$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1-{\alpha}_{n})>0$; ${\{{\beta}_{i}\}}_{i=1}^{m},{\{{\xi}_{h}\}}_{h=1}^{t}\subset (0,1)$ such that ${\sum}_{i=1}^{m}{\beta}_{i}=1={\sum}_{h=1}^{t}{\xi}_{h}$; $\eta \in (0,1)$ and $\{{\lambda}_{n}\}$ is a sequence in $[a,b]$ for some $a,b\in \mathbb{R}$ such that $0<a<b<\frac{2}{L}$, $L={max}_{1\le j\le d}\{{L}_{j}\}$. Then the sequence ${\{{x}_{n}\}}_{n\ge 0}$ converges strongly to the element of F nearest to ${x}_{0}$.
Proof Since, by our hypothesis, ${(\mathrm{\nabla}\varphi )}_{j}$ is ${L}_{j}$-Lipschitz continuous for some ${L}_{j}\in (0,+\mathrm{\infty})$, $j=1,2,\dots ,d$, we obtain from Theorem 4.1 that ${(\mathrm{\nabla}\varphi )}_{j}$ is $\frac{1}{{L}_{j}}$-inverse strongly monotone, $j=1,2,\dots ,d$; and since $L={max}_{1\le j\le d}\{{L}_{j}\}$, it is then easy to see that ${(\mathrm{\nabla}\varphi )}_{j}$ is $\frac{1}{L}$-inverse strongly monotone, $j=1,2,\dots ,d$. The rest, therefore, follows as in the proof of Theorem 3.2 with $\gamma =\frac{1}{L}$. This completes the proof. □
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support.
Authors’ Affiliations
References
- Kim GE: Weak and strong convergence for quasi-nonexpansive mappings in Banach spaces. Bull. Korean Math. Soc. 2012, 49(4):799–813. 10.4134/BKMS.2012.49.4.799View ArticleMathSciNetGoogle Scholar
- Dotson WG: On the Mann iterative process. Trans. Am. Math. Soc. 1970, 149: 63–73.View ArticleMathSciNetGoogle Scholar
- Diaz JB, Metcalf FT: On the set of subsequential limit points of successive approximations. Trans. Am. Math. Soc. 1969, 135: 459–485.MathSciNetGoogle Scholar
- Kato T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn. 1967, 19: 508–520. 10.2969/jmsj/01940508View ArticleGoogle Scholar
- Chidume CE Lecture Notes in Mathematics 1965. Geometric Properties of Banach Spaces and Nonlinear Iterations 2009.Google Scholar
- Ofoedu EU, Zegeye H: Further investigation on iteration processes for pseudocontractive mappings with application. Nonlinear Anal. TMA 2012, 75: 153–162. 10.1016/j.na.2011.08.015View ArticleMathSciNetGoogle Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63(1–4):123–145.MathSciNetGoogle Scholar
- Katchang P, Jitpeera T, Kumam P: Strong convergence theorems for solving generalized mixed equilibrium problems and general system of variational inequalities by the hybrid method. Nonlinear Anal. Hybrid Syst. 2010. 10.1016/j.nahs.2010.07.001Google Scholar
- Zegeye, H, Shahzad, N: Strong convergence theorems for a solution of finite families of equilibrium and variational inequality problems. Optimization, 1–17, iFirst (2012)Google Scholar
- Zegeye H, Shahzad N: Convergence theorems for a common point of solutions of equilibrium and fixed point of relatively nonexpansive multi-valued mapping problems. Abstr. Appl. Anal. 2012., 2012: Article ID 859598Google Scholar
- Censor Y, Gibali A, Reich S, Sabach S: Common solutions to variational inequalities. Set-Valued Var. Anal. 2012, 20: 229–247. 10.1007/s11228-011-0192-xView ArticleMathSciNetGoogle Scholar
- Censor Y, Gibali A, Reich S: A von Neumann alternating method for finding common solutions to variational inequalities. Nonlinear Anal. 2012, 75: 4596–4603. 10.1016/j.na.2012.01.021View ArticleMathSciNetGoogle Scholar
- Ishikawa S: Fixed point by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5View ArticleMathSciNetGoogle Scholar
- Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleGoogle Scholar
- Martines-Yanes C, Xu HK: Strong convergence of CQ method for fixed point iteration. Nonlinear Anal. 2006, 64: 2400–2411. 10.1016/j.na.2005.08.018View ArticleMathSciNetGoogle Scholar
- Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings. J. Math. Anal. Appl. 2003, 279: 372–379. 10.1016/S0022-247X(02)00458-4View ArticleMathSciNetGoogle Scholar
- Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 1979, 67: 274–276. 10.1016/0022-247X(79)90024-6View ArticleMathSciNetGoogle Scholar
- Zegeye H, Shahzad N: Strong convergence theorems for a common zero of countably infinite family of α -inverse strongly accretive mappings. Nonlinear Anal. 2009, 71: 531–538. 10.1016/j.na.2008.10.091View ArticleMathSciNetGoogle Scholar
- Zegeye H: Strong convergence theorems for maximal monotone mappings in Banach spaces. J. Math. Anal. Appl. 2008, 343: 663–671. 10.1016/j.jmaa.2008.01.076View ArticleMathSciNetGoogle Scholar
- Bauschke HH, Matouskova E, Reich S: Projections and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 2004, 56: 715–738. 10.1016/j.na.2003.10.010View ArticleMathSciNetGoogle Scholar
- Kim TH, Xu HK: Strong convergence of modified Mann iteration for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 2006, 64: 1140–1152. 10.1016/j.na.2005.05.059View ArticleMathSciNetGoogle Scholar
- Nilsrakoo W, Saejug S: Weak and strong convergence theorems for countable Lipschitzian mappings and its applications. Nonlinear Anal. TMA 2008, 69(8):2695–2708. 10.1016/j.na.2007.08.044View ArticleGoogle Scholar
- Ofoedu EU, Shehu Y, Ezeora JN: Solution by iteration of nonlinear variational inequalities involving finite family of asymptotically nonexpansive mappings and monotone mappings. Panam. Math. J. 2008, 18(4):61–75.MathSciNetGoogle Scholar
- Yang L, Su Y: Strong convergence theorems for countable Lipschitzian mappings and its applications in equilibrium and optimization problems. Fixed Point Theory Appl. 2009., 2009: Article ID 462489 10.1155/2009/462489Google Scholar
- Zegeye H, Shahzad N: Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 2007. 10.1016/j.na.2007.11.005Google Scholar
- Zegeye H, Shahzad N: A hybrid approximation method for equilibrium, variational inequality and fixed point problems. Nonlinear Anal. Hybrid Syst. 2010, 4: 619–630. 10.1016/j.nahs.2010.03.005View ArticleMathSciNetGoogle Scholar
- Zegeye H, Shahzad N: Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces. Optim. Lett. 2011, 5(4):691–704. 10.1007/s11590-010-0235-5View ArticleMathSciNetGoogle Scholar
- Kopecka E, Reich S: A note on alternating projections in Hilbert space. J. Fixed Point Theory Appl. 2012, 12: 41–47. 10.1007/s11784-013-0097-4View ArticleMathSciNetGoogle Scholar
- Zegeye H: An iterative approximation method for a common fixed point of two pseudo-contractive mappings. ISRN Math. Anal. 2011., 2011: Article ID 621901 10.5402/2011/621901Google Scholar
- Kassay G, Reich S, Sabach S: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 2011, 21: 1319–1344. 10.1137/110820002View ArticleMathSciNetGoogle Scholar
- Ofoedu EU: A general approximation scheme for solutions of various problems in fixed point theory. Int. J. Anal. 2013., 2013: Article ID 762831 10.1155/2013/762831Google Scholar
- Byrne CL: Applied Iterative Methods. AK Peters, Wellesley; 2008.Google Scholar
- Combettes PL: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 2004, 53: 475–504. 10.1080/02331930412331327157View ArticleMathSciNetGoogle Scholar
- Dunn JC: Convexity, monotonicity and gradient processes in Hilbert space. J. Math. Anal. Appl. 1976, 53: 145–158. 10.1016/0022-247X(76)90152-9View ArticleMathSciNetGoogle Scholar
- Lions PL, Mercier B: Splitting algorithms for the sum two nonlinear operators. SIAM J. Numer. Anal. 1979, 16: 964–979. 10.1137/0716071View ArticleMathSciNetGoogle Scholar
- Liu F, Nashed MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal. 1998, 6: 313–344. 10.1023/A:1008643727926View ArticleMathSciNetGoogle Scholar
- Tseng P: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim. 1991, 29: 119–138. 10.1137/0329006View ArticleMathSciNetGoogle Scholar
- Zhu DL, Marcotte P: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim. 1996, 6: 714–726. 10.1137/S1052623494250415View ArticleMathSciNetGoogle Scholar
- Baillon JB, Haddad G: Quelques propriétés des opérateurs angle-bornés et n -cycliquement monotones. Isr. J. Math. 1977, 26: 137–150. 10.1007/BF03007664View ArticleMathSciNetGoogle Scholar
- Combettes PL, Wajs VR: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 2005, 4: 1168–1200. 10.1137/050626090View ArticleMathSciNetGoogle Scholar
- Yamada I, Ogura N: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 2004, 25: 619–655.View ArticleMathSciNetGoogle Scholar
- Masad E, Reich S: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 2007, 8: 367–371.MathSciNetGoogle Scholar
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