The purpose of this paper is to propose some hybrid extragradientArmijo algorithms for finding a common element of the set of solutions of a finite family of pseudomonotone equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings in real Hilbert spaces. The proposed methods combine extragradient and Mann’s iterative methods as well as Armijo linesearch and hybrid techniques. The strong convergence of the proposed methods are established without assumption on the Lipschitztype condition of the bifunctions involved
Hybrid Extragradient-Armijo Methods for Finite Families of Pseudomonotone Equilibrium problems and Nonexpansive Mappings∗ Le Q. Thuy†, Pham K. Anh‡, Le D. Muu§ Abstract. The purpose of this paper is to propose some hybrid extragradient-Armijo algorithms for finding a common element of the set of solutions of a finite family of pseudomonotone equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings in real Hilbert spaces. The proposed methods combine extragradient and Mann’s iterative methods as well as Armijo line-search and hybrid techniques. The strong convergence of the proposed methods are established without assumption on the Lipschitz-type condition of the bifunctions involved. Keywords: Equilibrium problem; Pseudomonotone bifuction; Nonexpansive mapping; Hybrid method; Armijo line search. AMS Subject Classifications 47 H09, 47 H10, 47 J25, 65 K10, 65 Y05, 90 C25, 90 C33. 1. Introduction Let C be a nonempty closed convex subset of a real Hilbert space H and f be a bifunction from C × C to R ∪ {+∞} satisfying condition f (x, x) = 0 for every x ∈ C. Such a bifunction is called an equilibrium bifuction. We consider the following problem: Finding x∗ ∈ C such that f (x∗ , y) ≥ 0, ∀y ∈ C. (1.1) Problem (1.1) is refered to as the equilibrium problem or Ky Fan inequality [8] and its solution set is denoted by Sol(C, f ). The equilibrium problem (1.1) provides a unified framework for a wide class of problems, such as convex optimization, variational inequality, nonlinear complementarity, Nash equilibrium and fixed point problems. The existence and solution methods for equilibrium problems have been extensively studied (see, e.g. [3], [11], [16, 18], [20] and the references therein). A mapping T : C → C is said to be nonexpansive if T (x) − T (y) ≤ x − y for all x, y ∈ C. The set of fixed points of T is denoted by F (T ). In recent year, the problem of finding a common solution of equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings in Hilbert spaces has attracted attention of several authors (see e.g. [1], [5], [10], [12], [17], [19], [21], ..., and the references therein). The common approach in these papers is the use of the proximal point method for handling monotone equilibrium problems. However, for pseudomonotone equilibrium problems, the auxiliary regularized subproblems may not strongly monotone, even not pseudomonotone, hence they cannot be solved by available methods requiring the monotonicity of these subproblems. In this article we propose three hybrid extragradient-Armijo algorithms for finding a common element of the set of solutions of a finite family of pseudomonotone equilibrium problems {fi }N i=1 and the set of fixed points of a finite family of nonexpansive mappings ∗ This paper was complete during the authors’ stay at the Vietnam Institute for Advanced Study in Mathematics (VIASM). † School of Applied Mathematics and Informatics Ha Noi University of Science and Technology, Hanoi, Vietnam (E-mail: thuy.lequang@hust.edu.vn). ‡ Department of Mathematics, Vietnam National University, Hanoi, Vietnam (E-mail: anhpk@vnu.edu.vn) § Institute of Mathematics, VAST, Hanoi, Vietnam (E-mail: ldmuu@math.ac.vn). 1 Le Q. Thuy, Pham K. Anh and Le D. Muu 2 {Sj }M j=1 in Hilbert spaces, without assuming the Lipschitz-type conditions on the bifunctions fi , i = 1, . . . , N. We combine the extragradient method and Armijo-type line search techniques for handling pseudomonotone equilibrium problems [1,7], and Mann’s iterative scheme for finding fixed points of nonexpansive mappings [13]. This paper is organized as follows: In Section 2, we recall some definitions and preliminary results. Section 3 deals with the convergence analysis of the proposed hybrid extragradient-Armijo methods. 2. Preliminaries We recall some definitions and results that will be used in the next section. Let C be a nonempty closed convex of a real Hilbert space H with an inner product ., . and the induced norm . . Let T : C → C be a nonexpansive mapping with the set of fixed points F (T ). We begin by recalling the following properties of nonexpansive mappings. Lemma 2.1. [9] Assume that T : C → C is a nonexpansive mapping. Then (i) I − T is demiclosed, i.e., whenever {xn } is a sequence in C weakly converging to some x ∈ C and the sequence {(I − T )xn } strongly converges to some y , it follows that (I − T )x = y. (ii) if T has a fixed point, then F (T ) is a closed convex subset of H. It is well known that if C is a nonempty closed and convex subset of H, then for every x ∈ H, there exists a unique element PC x, defined by PC x = arg min { y − x : y ∈ C} . The mapping PC : H → C is called the metric (orthogonal) projection of H onto C. It enjoys the following remarkable properties: (i) PC is firmly nonexpansive, or 1-inverse strongly monotone (1-ism), i.e., PC x − PC y, x − y ≥ PC x − PC y (ii) x − PC x 2 + PC x − y 2 2 for all x, y ∈ C. ≤ x−y 2 . (2.1) (2.2) (iii) z = PC x if only if x − z, z − y ≥ 0, ∀y ∈ C. (2.3) A bifunction f is called monotone on C if f (x, y) + f (y, x) ≤ 0, ∀x, y ∈ C; f is pseudomonotone on C if f (x, y) ≥ 0 ⇒ f (y, x) ≤ 0, ∀x, y ∈ C. It is obvious that any monotone bifunction is a pseudomonotone one, but not vice versa. Throughout this paper we consider bifunctions with the following assumptions: A1. f is pseudomonotone on C; A2. f is weakly continuous on C × C; A3. f (x, .) is convex and subdifferentiable on C, for every fixed x ∈ C. The following results will be used in the next section. Le Q. Thuy, Pham K. Anh and Le D. Muu 3 Lemma 2.2. [6] Let C be a convex subset of a real Hilbert space H and ϕ : C → R be a convex and subdifferentiable function on C. Then, x∗ is a solution to the convex problem min {ϕ(x) : x ∈ C} if only if 0 ∈ ∂ϕ(x∗ ) + NC (x∗ ), where ∂ϕ(x∗ ) denotes the subdifferential of ϕ and NC (x∗ ) is the normal cone of C at x∗ . Lemma 2.3. [15] Let X be a uniformly convex Banach space, r be a positive number and Br (0) ⊂ X be a closed ball centered at the origin with radius r. Then, for any given subset {x1 , x2 , . . . , xN } ⊂ Br (0) and for any positive numbers λ1 , λ2 , . . . , λN with N i=1 λi = 1, there exists a continuous, strictly increasing, and convex function g : [0, 2r) → [0, ∞) with g(0) = 0 such that, for any i, j ∈ {1, 2, . . . , N} with i < j, 2 N N ≤ λk xk λk xk 2 − λi λj g( xi − xj ). k=1 k=1 Lemma 2.4. [2] Let C ⊂ H be a closed convex subset and f : C × C → R ∪ {+∞} be an equilibrium bifuntion satisfying Assumptions A1 − A3. If the solution set Sol(C, f ) is nonempty, then it is weakly closed and convex. 3. Main results Throughout this section we assume that the common solution set is nonempty, i.e., F = N M i=1 j=1 ∩ Sol(C, fi ) ∩ F (Sj ) = ∅, and that each bifunction fi (i = 1, . . . , N) satisfies Assumptions A1 − A3. Since F = ∅, all the sets F (Sj ) j = 1, . . . , M and Sol(C, fi ) i = 1, . . . , N are nonempty, hence according to Lemmas 2.1, 2.4, they are closed and convex and their intersection F is a nonempty closed and convex subset of C. Thus given any fixed element x0 ∈ C there exists a unique element xˆ := PF (x0 ). Algorithm 3.1. Choose positive numbers β > 0, σ ∈ (0, β2 ), γ ∈ (0, 1) and the sequence {αn } ⊂ (0, 1) satisfying the condition lim sup αn < 1. Let x0 ∈ C and set n := 0. n→∞ Step 1. Solve N strong convex programs yni = argmin{fi (xn , y) + β xn − y 2 2 : y ∈ C} i = 1, . . . , N. and set di (xn ) = xn − yni . Step 2. Let I(xn ) = {i ∈ {1, 2, . . . , N} : di (xn ) = 0}. • For all i ∈ I(xn ), find the smallest positive integer number min such that i fi xn − γ mn di (xn ), yni ≤ −σ di (xn ) 2 . Define Vni := {x ∈ H : wni , x − z¯ni ≤ 0}, i where z¯ni = xn − γ mn di (xn ), wni ∈ ∂2 fi (¯ zni , z¯ni ). Compute zni = PC∩Vni (xn ), • For all i ∈ / I(xn ), set zni = xn . (3.1) Le Q. Thuy, Pham K. Anh and Le D. Muu 4 Step 3. Find in = argmax{ zni − xn : i = 1, . . . , N}, and set z¯n := znin . Step 4. Compute ujn = αn xn + (1 − αn )Sj z¯n , j = 1, . . . , M. Step 5. Find jn = argmax{ ujn − xn : j = 1, . . . , M}, and set u¯n := ujnn . Step 6. Compute xn+1 = PCn ∩Qn (x0 ), where Cn = {v ∈ C : u¯n − v ≤ xn − v }, Qn = {v ∈ C : x0 − xn , v − xn ≤ 0}. Increase n by 1 and go back to Step 1. We now prove the strong convergence of Algorithm 3.1. Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Suppose that {fi }N i=1 is a finite family of bifunctions satisfying conditions A1 − A3 and M {Sj }j=1 is a finite family of nonexpansive mappings on C. Moreover, suppose that the solution set F = N M i=1 j=1 ∩ Sol(C, fi ) ∩ F (Sj ) is nonempty. Then, the sequence {xn } generated by Algorithm 3.1 converges strongly to xˆ = PF x0 . Proof. Consider two cases. • Case 1. For any natural number k there exists a number n > k such that I(xn ) = ∅. • Case 2. There exists a number n0 such that I(xn ) = ∅; ∀n ≥ n0 . We begin with Case 1 by dividing the proof into several steps. Step 1. We prove that the linesearch is finite for every i ∈ I(xn ), i.e., there exists the smallest nonnegative integer min satisfying i fi (xn − γ mn di (xn ), yni ) ≤ −σ di (xn ) 2 ∀n, i = 1, . . . , N. Indeed, assuming by contradiction that for every nonnegative integer m, one has fi xn − γ m di (xn ), yni + σ di (xn ) 2 > 0. Letting m → ∞, we obtain fi xn , yni + σ di (xn ) 2 ≥ 0. (3.2) On the other hand, since yni is the unique solution of the strongly convex problem min{fi (xn , y) + β y − xn 2 2 : y ∈ C}, we have β i β y − xn 2 ≥ fi (xn , yni ) + yn − xn 2 , ∀y ∈ C. 2 2 With y = xn , the last inequality becomes fi (xn , y) + fi (xn , yni ) + β i d (xn ) 2 2 ≤ 0. Combining (3.2) with (3.3) yields σ di (xn ) 2 ≥ β i d (xn ) 2 . 2 (3.3) Le Q. Thuy, Pham K. Anh and Le D. Muu 5 Hence it must be either di (xn ) = 0 or σ ≥ β2 . The first case contradicts di (xn ) = 0, while the second one contradicts the choice σ < β2 . Step 2. We show that F ⊆ C ∩ Vni , xn ∈ Vni ∀i ∈ I(xn ). Indeed, let x∗ ∈ F . Since fi (x∗ , x) ≥ 0 for all x ∈ C, by pseudomonotonicity of fi , we have fi (¯ zni , x∗ ) ≤ 0. (3.4) It follows from wni ∈ ∂2 fi (¯ zni , z¯ni ) that f (¯ zni , x∗ ) =f (¯ zni , x∗ ) − f (¯ zni , z¯ni ) ≥ wni , x∗ − z¯ni . (3.5) Combining (3.4) and (3.5), we get wni , x∗ − z¯ni ≤ 0. On the other hand, from the definition of Vni , we have x∗ ∈ Vni . Thus F ⊆ C ∩ Vni . i Now we show that xn ∈ / Vni . In fact, from z¯ni = xn − γ mn di (xn ), it follows that i yni − z¯ni 1 − γ mn i = (¯ zn − xn ). γ min Then using (3.1) and the assumption fi (x, x) = 0 for all x ∈ C, we have 0 > −σ di (xn ) 2 ≥ fi (¯ zni , yni ) = fi (¯ zni , yni ) − fi (¯ zni , z¯ni ) ≥ wni , yni − z¯ni i = 1 − γ mn i z¯n − xn , wni . γ min Hence xn − z¯ni , wni > 0, which implies xn ∈ / Vni . Step 3. (Solodov-Svaiter) For all i ∈ I(xn ), we claim that zni = PC∩Vni (¯ yni ), where y¯ni = PVni (xn ). Indeed, let K := {x ∈ H : t, x − x0 ≤ 0} with t = 0. It is easy to check that t, y − x0 PK (y) = y − t, t 2 Hence, y¯ni = PVni (xn ) wni , xn − z¯ni i wn = xn − wni 2 i γ mn wni , di (xn ) i = xn − wn . wni 2 Note that, for every y ∈ C ∩ Vni , there exists λi ∈ (0, 1) such that xˆ = λi xn + (1 − λi )y ∈ C ∩ ∂Vni , Le Q. Thuy, Pham K. Anh and Le D. Muu 6 where ∂Vni = {x ∈ H : wni , x − z¯ni = 0}. Since xn ∈ C, xˆ ∈ ∂Vni and y¯ni = PVni (xn ), we have y − y¯ni ≥ (1 − λi )2 y − y¯ni 2 = xˆ − λi xn − (1 − λi )¯ yni 2 = (ˆ x − y¯ni ) − λi (xn − y¯ni ) 2 = xˆ − y¯ni 2 + λ2i xn − y¯ni 2 − 2λi xˆ − y¯ni , xn − y¯ni = xˆ − y¯ni 2 + λ2i xn − y¯ni 2 ≥ xˆ − y¯ni 2 . 2 (3.6) At the same time xˆ − xn 2 = = = xˆ − y¯ni + y¯ni − xn 2 xˆ − y¯ni 2 − 2 xˆ − y¯ni , xn − y¯ni + y¯ni − xn xˆ − y¯ni 2 + y¯ni − xn 2 . 2 Using zni = PC∩Vn (xn ) and the Pythagorean theorem, we can write xˆ − y¯ni 2 = ≥ = xˆ − xn 2 − y¯ni − xn 2 zni − xn 2 − y¯ni − xn 2 zni − y¯ni 2 . (3.7) From (3.6) and (3.7), it follows that zni − y¯ni ≤ y − y¯ni , ∀y ∈ C ∩ Vni . Hence yni ). zni = PC∩Vn (¯ Step 4. For all i = 1, 2, . . . , N, we show that zni − x∗ 2 ≤ xn − x∗ 2 − zni − xn 2 . (3.8) z¯n − x∗ 2 ≤ xn − x∗ 2 − z¯n − xn 2 . (3.9) It is clear that these inequalities hold true for all i ∈ I(xn ). If i ∈ / I(xn ), by Step 2 of Algorithm 3.1, one has zni = PC∩Vni (xn ), i.e., xn − zni , z − zni ≤ 0, ∀z ∈ C ∩ Vni . Substituting z = x∗ ∈ F ⊆ C ∩ Vni by Step 2, we have xn − zni , x∗ − zni ≤ 0 ⇔ xn − zni , x∗ − xn + xn − zni ≤ 0, which implies that zni − xn , xn − x∗ ≤ − zni − xn 2 . Hence zni − x∗ 2 = = ≤ = (zni − xn ) + (xn − x∗ ) 2 zni − xn 2 + xn − x∗ 2 + 2 zni − xn , xn − x∗ zni − xn 2 + xn − x∗ 2 − 2 zni − xn 2 xn − x∗ 2 − zni − xn 2 . Le Q. Thuy, Pham K. Anh and Le D. Muu 7 Thus (3.9) follows from the definition of in . Step 5. It holds that F ⊂ Cn ∩ Qn for every n ≥ 0. In fact, for each x∗ ∈ F , by convexity of . 2 , the nonexpansiveness of Sj , and (3.9), we can write u¯n − x∗ 2 = αn xn + (1 − αn )Sjn z¯n − x∗ 2 ≤ αn xn − x∗ 2 + (1 − αn ) Sjn z¯n − x∗ ≤ αn xn − x∗ 2 + (1 − αn ) z¯n − x∗ 2 ≤ αn xn − x∗ 2 + (1 − αn ) xn − x∗ 2 2 ≤ xn − x∗ 2 . (3.10) which implies u¯n − x∗ ≤ xn − x∗ or x∗ ∈ Cn . Hence F ⊂ Cn for all n ≥ 0. Now we show that F ⊂ Cn Qn by induction. Indeed, it is clear that F ⊂ C0 . Besides, F ⊂ C = Q0 , hence F ⊂ C0 Q0 . Assume that F ⊂ Cn−1 Qn−1 for some n ≥ 1. Then from xn = PCn−1 Qn−1 x0 and (2.3), we get xn − z, x0 − xn ≥ 0, ∀z ∈ Cn−1 Qn−1 . Since F ⊂ Cn−1 Qn−1 , we have xn − z, x0 − xn ≥ 0 for all z ∈ F , which together with definition of Qn imply that F ⊂ Qn . Hence F ⊂ Cn Qn for all n ≥ 1. Since F and Cn ∩ Qn are nonempty closed convex subsets, PF x0 and xn+1 := PCn ∩Qn (x0 ) are well-defined. Step 6. There hold the relations lim xn − Sj xn = 0, and n→∞ lim xn+1 − xn = lim xn − ujn = lim xn − zni = lim xn − yni = 0. n→∞ n→∞ n→∞ n→∞ Indeed, from the definition of Qn and (2.2), we see that xn = PQn x0 . Therefore, for every u ∈ F ⊂ Qn , we get xn − x0 2 ≤ u − x0 2 . (3.11) This implies that the sequence {xn } is bounded. From (3.10), the sequence {¯ un }, and j hence, the sequence {un } are also bounded. Observing that xn+1 = PCn Qn x0 ∈ Qn and xn = PQn x0 , applying (2.2) with x = x0 and y = xn+1 , we have xn − x0 2 ≤ xn+1 − x0 2 − xn+1 − xn 2 ≤ xn+1 − x0 2 . (3.12) Thus, the sequence { xn − x0 } is decreasing, hence it has a finite limit as n approaches infinity. From (3.12) we obtain xn+1 − xn 2 ≤ xn+1 − x0 2 − xn − x0 2 , and thus lim xn+1 − xn = 0. n→∞ (3.13) Since xn+1 ∈ Cn , u¯n − xn+1 ≤ xn+1 − xn . Thus u¯n − xn ≤ u¯n − xn+1 + xn+1 − xn ≤ 2 xn+1 −xn . Combining the last inequality with (3.13) we find that u¯n −xn → 0 as n → ∞. From the definition of jn , we conclude that lim ujn − xn = 0 n→∞ (3.14) Le Q. Thuy, Pham K. Anh and Le D. Muu 8 for all j = 1, . . . , M. Moreover, by Step 4, for any fixed x∗ ∈ F, we have ujn − x∗ 2 = αn xn + (1 − αn )Sj z¯n − x∗ 2 ≤ αn xn − x∗ 2 + (1 − αn ) Sj z¯n − x∗ ≤ αn xn − x∗ 2 + (1 − αn ) z¯n − x∗ ≤ αn xn − x∗ 2 + (1 − αn ) = xn − x∗ 2 2 2 xn − x∗ 2 − z¯n − xn 2 − (1 − αn ) z¯n − xn 2 . Thus (1 − αn ) z¯n − xn ≤ xn − x∗ = 2 2 − ujn − x∗ 2 xn − x∗ + ujn − x∗ xn − x∗ − ujn − x∗ ≤ xn − ujn xn − x∗ + ujn − x∗ . (3.15) Using the last inequality together with (3.14) and taking into account the boundedness of the two sequences {ujn }, {xn } as well as the condition lim supn→∞ αn < 1, we obtain limn→∞ z¯n − xn = 0. By the definition of in , we get lim zni − xn = 0 (3.16) n→∞ for all i = 1, . . . , N. On the other hand, since ujn = αn xn + (1 − αn )Sj z¯n , we have ujn − xn = (1 − αn ) Sj z¯n − xn = (1 − αn ) (Sj xn − xn ) + (Sj z¯n − Sj xn ) ≥ (1 − αn ) ( Sj xn − xn − Sj z¯n − Sj xn ) ≥ (1 − αn ) ( Sj xn − xn − z¯n − xn ) . Therefore 1 uj − xn . 1 − αn n The last inequality together with (3.14), (3.16) and the condition lim supn→∞ αn < 1 implies that lim Sj xn − xn = 0 (3.17) Sj xn − xn ≤ z¯n − xn + n→∞ for all j = 1, . . . , M. Since {xn } is bounded, there exists a subsequence of {xn } converging weakly to x¯. For the sake of simplicity, we denote the weakly convergent subsequence again by {xn } , i.e., xn ⇀ x¯. N Step 7. We show that x¯ ∈ F = M Sol(C, fi ) F (Sj ) . i=1 j=1 Indeed, from (3.17) and the demiclosedness of I − Sj , we have x¯ ∈ F (Sj ). Hence, x¯ ∈ M j=1 F (Sj ). Observing that yni = argmin{fi (xn , y) + β xn − y 2 2 : y ∈ C}, from Lemma 2.2, we obtain 0 ∈ ∂2 fi (xn , y) + β xn − y 2 2 (yni ) + NC (yni ). Le Q. Thuy, Pham K. Anh and Le D. Muu 9 Thus, there exists w ∈ ∂2 fi (xn , yni ) and w¯ ∈ NC (yni ) such that w + β(xn − yni ) + w¯ = 0. (3.18) Since w¯ ∈ NC (yni ), we have w, ¯ y − yni ≤ 0 for all y ∈ C, which together with (3.18) implies that w, y − yni ≥ β yni − xn , y − yni (3.19) for all y ∈ C. Since w ∈ ∂2 fi (xn , yni ), fi (xn , y) − fi (xn , yni ) ≥ w, y − yni , ∀y ∈ C. (3.20) From (3.19) and (3.20), it follows that fi (xn , y) − fi (xn , yni ) ≥ β yni − xn , y − yni , ∀y ∈ C. (3.21) Recalling that xn ⇀ x¯ and xn − yni → 0 as n → ∞, we get yni ⇀ x¯. Letting n → ∞ in (3.21) and using assumptions A2, we conclude that fi (¯ x, y) ≥ 0 for all y ∈ C N (i = 1, . . . , N). Thus, x¯ ∈ Sol(C, fi ), hence x¯ ∈ F . i=1 Step 8. We show that the sequence {xn } converges strongly to xˆ := PF x0 . Indeed, from xˆ ∈ F and (3.11), we obtain xn − x0 ≤ xˆ − x0 . The last inequality together with xn ⇀ x¯ and the weak lower semicontinuity of the norm . implies that x¯ − x0 ≤ lim inf n→∞ xn − x0 ≤ lim supn→∞ xn − x0 ≤ xˆ − x0 . By the definition of xˆ we find x¯ = xˆ and limn→∞ xn −x0 = xˆ−x0 . Thus limn→∞ xn = xˆ , and together with the fact that xn ⇀ xˆ , we conclude xn → xˆ as n → ∞. Finally, suppose that {xm } is another weakly convergent subsequence of {xn }. By a similar argument as above, we can conclude that {xm } converges strongly to xˆ := PF x0 . Therefore, the sequence {xn } generated by the Algorithm 3.1 converges strongly to PF x0 . Now we consider Case 2, when I(xn ) = ∅ for all n ≥ n0 . If xn+1 = xn , then Algorithm 3.1 terminates at a finite iteration n < ∞, and xn is a N M i=1 j=1 common element of two sets ∩ Sol(C, fi ) and ∩ F (Sj ), i.e., xn ∈ F . un − xn || ≤ Indeed, we have xn = xn+1 = PCn ∩Qn (x0 ) ∈ Cn . By the definition of Cn , ||¯ ||xn − xn || = 0, hence u¯n = xn . From the definition of jn , we obtain ujn = xn , ∀j = 1, . . . , M, (3.22) which together with ujn = αn xn + (1 − αn )Sj z¯n and 0 < αn < 1 imply that xn = Sj z¯n . Let x∗ ∈ F. By the nonexpansiveness of Sj , we get ||xn − x∗ ||2 = ||Sj z¯n − x∗ ||2 ≤ ||¯ zn − x∗ ||2 ≤ ||xn − x∗ ||2 − z¯n − xn 2 . From the last inequality and the definition of d(xinn ), we obtain xn = ynin = z¯n . Thus xn = Sj z¯n = Sj xn or xn ∈ F (Sj ) for all j = 1, . . . , M. Moreover, from the equality xn = z¯n and definition of in , we also get xn = zni for all i = 1, . . . , N. Combining this fact with definition of d(xin ) we see that xn = yni for all i = 1, . . . , N. Thus, 1 xn = argmin{ρfi (xn , y) + ||xn − y||2 : y ∈ C}. 2 Le Q. Thuy, Pham K. Anh and Le D. Muu 10 By [14, Proposition 2.1], from the last relation we conclude that xn ∈ Sol(C, fi ) for all i = 1, . . . , N, hence xn ∈ F . Otherwise xn+1 = xn for all n, similarly as in the proof of Steps 6 and 8, the sequence {xn } converges strongly to PF x0 . The proof of Theorem 3.1 is complete. Replacing Mann’s iteration in Step 4 of Algorithm 3.1 by Halpern’s one, we come to the following algorithm with modified sets Cn . Algorithm 3.2. Initialize: x0 ∈ C, β > 0; σ ∈ (0, β2 ), γ ∈ (0, 1), n := 0 and the nonnegative sequences {αn,l } (l = 0, . . . , M) satisfying the conditions: 0 ≤ αn,j ≤ 1, M αn,j = 1, lim inf αn,0 αn,l > 0 for all l = 1, . . . , M. n→∞ j=0 Step 1. Solve N strong convex programs β xn − y 2 yni = argmin{fi (xn , y) + 2 : y ∈ C}, i = 1, . . . , N. and set di (xn ) = xn − yni . Step 2. Let I(xn ) = {i ∈ {1, 2, . . . , N} : di (xn ) = 0}. • For all i ∈ I(xn ), find the smallest positive integer number min such that i fi xn − γ mn di (xn ), yni ≤ −σ di (xn ) 2 . Compute zni = PC∩Vni (xn ), where Vni = {x ∈ H : wni , x − z¯ni ≤ 0}. with z¯ni = xn − γ mn di (xn ) and wni ∈ ∂2 fi (¯ zni , z¯ni ) • For all i ∈ / I(xn ), set zni = xn . Step 3. Find in = argmax{ zni − xn : i = 1, . . . , N}, and set z¯n := znin . Step 4. Compute ujn = αn x0 + (1 − αn )Sj z¯n , j = 1, . . . , M. Step 5. Find jn = argmax{ ujn − xn : j = 1, . . . , M}, and set u¯n := ujnn . Step 6. Compute xn+1 = PCn ∩Qn (x0 ), where Cn = {v ∈ C : u¯n − v 2 ≤ αn x0 − v Qn = {v ∈ C : x0 − xn , v − xn ≤ 0}. 2 + (1 − αn ) xn − v 2 }, Increase n by 1 and go back to Step 1. Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Suppose that {fi }N i=1 is a finite family of bifunctions satisfying assumptions A1 − A3, M and {Sj }j=1 is a finite family of nonexpansive mappings on C. Moreover, suppose that the solution set F is nonempty. Then, the sequence {xn } generated by Algorithm 3.2 converges strongly to xˆ = PF x0 . Le Q. Thuy, Pham K. Anh and Le D. Muu 11 Proof. Arguing similarly as in the proof of Theorem 3.1, we conclude that F ⊂ Cn ∩ Qn for all n ≥ 0. Moreover, the sequence {xn } is bounded and lim xn+1 − xn = 0. (3.23) n→∞ Since xn+1 ∈ Cn+1 , u¯n − xn+1 2 ≤ αn x0 − xn+1 2 + (1 − αn ) xn − xn+1 2 . Letting n → ∞, from (3.23), limn→∞ αn = 0 and the boundedness of {xn } , {ujn }, we obtain lim u¯n − xn+1 = 0. n→∞ Proving similarly to (3.14) and (3.15), we get lim ujn − xn = 0, j = 1, . . . , M, n→∞ and (1 − αn ) z¯n − xn 2 ≤ xn − ujn xn − x∗ + ujn − x∗ (3.24) for each x∗ ∈ F . Letting n → ∞, from (3.24), one has lim z¯n − xn = 0, j = 1, . . . , N, n→∞ By the definition of in , we get lim zni − xn = 0, i = 1, . . . , N. n→∞ Using ujn = αn x0 + (1 − αn )Sj z¯n , by a straightforward computation, we obtain Sj xn − xn ≤ z¯n − xn + αn 1 ujn − xn + x0 − xn , 1 − αn 1 − αn which implies that limn→∞ Sj xn − xn = 0. The rest of the proof of Theorem 3.2 can be carried out similarly as in Steps 7 and 8 of the proof of Theorem 3.1. Performing an iteration involving a convex combination of the identity operator and the mappings Sj , j = 1, . . . , N, instead of using Mann’s iteration, we come to the following algorithm. Algorithm 3.3. Initialize: x0 ∈ C, β > 0; σ ∈ (0, β2 ), γ ∈ (0, 1), n := 0 and the positive M sequences {αn,l } (l = 0, . . . , M) satisfying the conditions: 0 ≤ αn,j ≤ 1, αn,j = 1, j=0 lim inf αn,0 αn,l > 0 for all l = 1, . . . , M. n→∞ Step 1. Solve N strong convex programs β xn − y 2 yni = argmin{fi (xn , y) + 2 : y ∈ C}, i = 1, . . . , N. and set di (xn ) = xn − yni . Step 2. Let I(xn ) = {i ∈ {1, 2, . . . , N} : di (xn ) = 0}. • For all i ∈ I(xn ), find the smallest positive integer number min such that i fi xn − γ mn di (xn ), yni ≤ −σ di (xn ) 2 . Compute zni = PC∩Vni (xn ), where Vni = {x ∈ H : wni , x − z¯ni ≤ 0}. with z¯ni = xn − γ mn di (xn ) and wni ∈ ∂2 fi (¯ zni , z¯ni ). Le Q. Thuy, Pham K. Anh and Le D. Muu 12 • For all i ∈ / I(xn ), set zni = xn . Step 3. Find in = argmax{ zni − xn : i = 1, . . . , N}, and set z¯n := znin . Step 4. Compute N un = αn,0 xn + αn,j Sj z¯n j=1 xn+1 = PCn ∩Qn (x0 ) where Cn = {v ∈ C : un − v ≤ xn − v }, Qn = {v ∈ C : x0 − xn , v − xn ≤ 0}. Increase n by 1 and go back to Step 1. Theorem 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Suppose that {fi }N i=1 is a finite family of bifunctions satisfying conditions A1 − A3, and M {Sj }j=1 is a finite family of nonexpansive mappings on C. Moreover, suppose that the solution set F is nonempty. Then, the sequence {xn } generated by the Algorithm 3.3 converges strongly to xˆ = PF x0 . Proof. As before we conclude that F ⊂ Cn Qn and lim xn+1 − xn = lim yni − xn = lim zni − xn = lim un − xn = 0 n→∞ n→∞ n→∞ n→∞ (3.25) for all i = 1, . . . , N. For every x∗ ∈ F , by Lemmas 2.3 and (3.8), we have M ∗ 2 un − x αn,j Sj z¯n − x∗ = αn,0 xn + 2 j=1 M ∗ = αn,0 (xn − x ) + αn,j (Sj z¯n − x∗ ) 2 αn,j Sj z¯n − x∗ 2 j=1 M ≤ αn,0 xn − x∗ 2 + − αn,0 αn,l g( Sl z¯n − xn ) j=1 M ∗ 2 ≤ αn,0 xn − x + αn,j z¯n − x∗ 2 − αn,0 αn,l g( Sl z¯n − xn ) αn,j xn − x∗ 2 − αn,0αn,l g( Sl z¯n − xn ) j=1 M ≤ αn,0 xn − x∗ 2 + j=1 ∗ 2 ≤ xn − x − αn,0 αn,l g( Sl z¯n − xn ). Therefore αn,0 αn,l g( Sl z¯n − xn ) ≤ xn − x∗ 2 − un − x∗ 2 ≤ ( xn − x∗ − un − x∗ ) ( xn − x∗ + un − x∗ ) ≤ xn − un ( xn − x∗ + un − x∗ ) . The last inequality together with (3.25), lim inf n→∞ αn,0 αn,l > 0 and the boundedness of {xn } , {un } implies that limn→∞ g( Sl z¯n − xn ) = 0. Hence lim Sl z¯n − xn = 0. n→∞ (3.26) Le Q. Thuy, Pham K. Anh and Le D. Muu 13 Moreover, from (3.25), (3.26) and Sl xn − xn ≤ Sl xn − Sl z¯n + Sl z¯n − xn ≤ xn − z¯n + Sl z¯n − xn we obtain lim Sl xn − xn = 0 ∀l = 1, . . . , M n→∞ The same argument as in the proof of Step 7 and 8 in Theorem 3.1 shows that the sequence {xn } converges strongly to xˆ := PF x0 . The proof of Theorem 3.3 is complete. References [1] P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optimization, 62(2011), 271-283. [2] M. Bianchi, S. Schaible, Generalized monotone bifunctions and equilibrium problems, Journal Optimization Theory Applications, 90(1996), 31-43. [3] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Mathematics Students, 63(1994), 123-145. [4] S. S. Chang, J. K. Kim, X. R. Wang, Modified block iterative algorithm for solving convex feasibility problems in Banach spaces, Journal of Inequalities and Applications (2010), DOI:10.1155/2010/869684. [5] L. C. Ceng, A. Petrusel and J. C. Yao, Iterative approaches to solving equilibrium problems and fixed point problems of inenitely many nonexpansive mappings, Journal Optimization Theory Application, 143(2009), 37-58. [6] P. Daniele, F. Giannessi and A. Maugeri, Equilibrium problems and variational models, Kluwer, 2003. [7] B. V. Dinh, P. G. Hung, L. D. Muu, Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numerical Functional Analysis and Optimization, 35(2014), 539-563. [8] K. Fan, A minimax inequality and applications, in: O. Shisha(Ed.) Inequality, Academic Press, New York, III(1972), 103-113. [9] K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, 28, Cambridge University Press, Cambridge (1990). [10] T. Jitpeera1 , P. Kumam, An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings, Journal of Nonlinear Analysis and Optimization, 1(2010), 71-91. [11] J. Kang, Y. Su, X. Zhang, Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications, Nonlinear Analysis: Hybrid Systems, 4(2010), 755-765. [12] P. Katchang, P. Kumam, An iterative algorithm for common fixed points for nonexpansive semigroups and strictly pseudocontractive mappings with optimization problems, Journal Global Optimization, 56(2013), 1563-1589. [13] W. R. Mann, Mean value methods in iteration, Proceedings of the American Mathematics Society, 4(1953), 506-510. [14] G. Mastroeni, On auxiliary principle for equilibrium problems, Publicatione del Dipartimento di Mathematica dell Universita di Pisa, 3(2000), 1244-1258. [15] S. Plubtieng and K. Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications (2008), Article ID 583082. Le Q. Thuy, Pham K. Anh and Le D. Muu 14 [16] S. Saewan, P. Kumam, The hybrid block iterative algorithm for solving the system of equilibrium problems and variational inequality problems, SpringerPlus (2012), http://www.springerplus.com/content/1/1/8. [17] Y. Shehu, Iterative approximation method for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems, Journal Global Optimization, 51(2011), 69-78. [18] Y. Su, M. Li, H. Zhang, New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators, Applied Mathematics and Computation, 217(2011), 5458-5465. [19] A. Tada, W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, in: W. Takahashi, T. Tanaka (Eds.), Nonlinear Analysis and Convex Analysis, Yokohama Publishers, Yokohama, (2006). [20] S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point in Hilbert space, Journal of Mathematical Analysis and Applications, 331(2007), 506-515. [21] P. T. Vuong, J. J. Strodiot and N. V. Hien, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, Journal Optimization Theory Application, 155(2012), 605-627. [...]... method for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems, Journal Global Optimization, 51(2011), 69-78 [18] Y Su, M Li, H Zhang, New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators, Applied Mathematics and Computation, 217(2011), 5458-5465 [19] A Tada, W Takahashi, Strong convergence theorem for an equilibrium. .. many nonexpansive mappings, Journal of Nonlinear Analysis and Optimization, 1(2010), 71-91 [11] J Kang, Y Su, X Zhang, Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications, Nonlinear Analysis: Hybrid Systems, 4(2010), 755-765 [12] P Katchang, P Kumam, An iterative algorithm for common fixed points for nonexpansive semigroups and strictly pseudocontractive mappings. .. iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications (2008), Article ID 583082 Le Q Thuy, Pham K Anh and Le D Muu 14 [16] S Saewan, P Kumam, The hybrid block iterative algorithm for solving the system of equilibrium problems and variational inequality problems, SpringerPlus (2012), http://www.springerplus.com/content/1/1/8... argument as in the proof of Step 7 and 8 in Theorem 3.1 shows that the sequence {xn } converges strongly to xˆ := PF x0 The proof of Theorem 3.3 is complete References [1] P N Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optimization, 62(2011), 271-283 [2] M Bianchi, S Schaible, Generalized monotone bifunctions and equilibrium problems, Journal Optimization... − xn = 0 The rest of the proof of Theorem 3.2 can be carried out similarly as in Steps 7 and 8 of the proof of Theorem 3.1 Performing an iteration involving a convex combination of the identity operator and the mappings Sj , j = 1, , N, instead of using Mann’s iteration, we come to the following algorithm Algorithm 3.3 Initialize: x0 ∈ C, β > 0; σ ∈ (0, β2 ), γ ∈ (0, 1), n := 0 and the positive M... problem and a nonexpansive mapping, in: W Takahashi, T Tanaka (Eds.), Nonlinear Analysis and Convex Analysis, Yokohama Publishers, Yokohama, (2006) [20] S Takahashi, W Takahashi, Viscosity approximation methods for equilibrium problems and fixed point in Hilbert space, Journal of Mathematical Analysis and Applications, 331(2007), 506-515 [21] P T Vuong, J J Strodiot and N V Hien, Extragradient methods and. .. problems and fixed point problems of inenitely many nonexpansive mappings, Journal Optimization Theory Application, 143(2009), 37-58 [6] P Daniele, F Giannessi and A Maugeri, Equilibrium problems and variational models, Kluwer, 2003 [7] B V Dinh, P G Hung, L D Muu, Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numerical Functional Analysis and Optimization, 35(2014),... optimization and variational inequalities to equilibrium problems, Mathematics Students, 63(1994), 123-145 [4] S S Chang, J K Kim, X R Wang, Modified block iterative algorithm for solving convex feasibility problems in Banach spaces, Journal of Inequalities and Applications (2010), DOI:10.1155/2010/869684 [5] L C Ceng, A Petrusel and J C Yao, Iterative approaches to solving equilibrium problems and fixed... closed convex subset of a real Hilbert space H Suppose that {fi }N i=1 is a finite family of bifunctions satisfying conditions A1 − A3, and M {Sj }j=1 is a finite family of nonexpansive mappings on C Moreover, suppose that the solution set F is nonempty Then, the sequence {xn } generated by the Algorithm 3.3 converges strongly to xˆ = PF x0 Proof As before we conclude that F ⊂ Cn Qn and lim xn+1 − xn... optimization problems, Journal Global Optimization, 56(2013), 1563-1589 [13] W R Mann, Mean value methods in iteration, Proceedings of the American Mathematics Society, 4(1953), 506-510 [14] G Mastroeni, On auxiliary principle for equilibrium problems, Publicatione del Dipartimento di Mathematica dell Universita di Pisa, 3(2000), 1244-1258 [15] S Plubtieng and K Ungchittrakool, Hybrid iterative methods for ... extragradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings, Journal of Nonlinear Analysis and Optimization,... Plubtieng and K Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications... Armijo-type line search techniques for handling pseudomonotone equilibrium problems [1,7], and Mann’s iterative scheme for finding fixed points of nonexpansive mappings [13] This paper is organized