Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 383740, 19 pages doi:10.1155/2010/383740 Research Article A Hybrid Projection Algorithm for Finding Solutions of Mixed Equilibrium Problem and Variational Inequality Problem Filomena Cianciaruso,1 Giuseppe Marino,1 Luigi Muglia,1 and Yonghong Yao2 Dipartimento di Matematica, Universit´ della Calabria, 87036 Arcavacata di Rende (CS), Italy a Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Correspondence should be addressed to Giuseppe Marino, gmarino@unical.it Received June 2009; Accepted 16 September 2009 Academic Editor: Mohamed A Khamsi Copyright q 2010 Filomena Cianciaruso et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We propose a modified hybrid projection algorithm to approximate a common fixed point of a k-strict pseudocontraction and of two sequences of nonexpansive mappings We prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems Introduction In this paper, we define an iterative method to approximate a common fixed point of a kstrict pseudocontraction and of two sequences of nonexpansive mappings generated by two sequences of firmly nonexpansive mappings and two nonlinear mappings Let us recall from that the k-strict pseudocontractions in Hilbert spaces were introduced by Browder and Petryshyn in Definition 1.1 S : C → C is said to be k-strict pseudocontractive if there exists k ∈ 0, such that Sx − Sy ≤ x−y k I−S x− I −S y , ∀x, y ∈ C 1.1 The iterative approximation problems for nonexpansive mappings, asymptotically nonexpansive mappings, and asymptotically pseudocontractive mappings were studied extensively by Browder , Goebel and Kirk , Kirk , Liu , Schu , and Xu 8, Fixed Point Theory and Applications in the setting of Hilbert spaces or uniformly convex Banach spaces Although nonexpansive mappings are 0-strict pseudocontractions, iterative methods for k-strict pseudocontractions are far less developed than those for nonexpansive mappings The reason, probably, is that the second term appearing in the previous definition impedes the convergence analysis for iterative algorithms used to find a fixed point of the k-strict pseudocontraction S However, k-strict pseudocontractions have more powerful applications than nonexpansive mappings in solving inverse problems In the recent years the study of iterative methods like Mann’s like methods and CQ-methods has been extensively studied by many authors 1, 10–13 and the references therein If C is a closed and convex subset of a Hilbert space H and F : C × C → R is a bi-function we call equilibrium problem Find x ∈ C s.t F x, y ≥ 0, ∀y ∈ C, 1.2 and we will indicate the set of solutions with EP F If A : C → H is a nonlinear mapping, we can choose F x, y Ax, y − x , so an equilibrium point i.e., a point of the set EP F is a solution of variational inequality problem VIP Find x ∈ C s.t Ax, y − x ≥ 0, ∀y ∈ C 1.3 We will indicate with V I C, A the set of solutions of VIP The equilibrium problems, in its various forms, found application in optimization problems, fixed point problems, convex minimization problems; in other words, equilibrium problems are a unified model for problems arising in physics, engineering, economics, and so on see 10 As in the case of nonexpansive mappings, also in the case of k-strict pseudocontraction mappings, in the recent years many papers concern the convergence of iterative methods to a solutions of variational inequality problems or equilibrium problems; see example for, 10, 14–18 Here we prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems Preliminaries Let H be a real Hilbert space and let C be a nonempty closed convex subset of H We denote by PC the metric projection of H onto C It is well known 19 that x − PC x , PC x − y ≥ 0, ∀x ∈ H and y ∈ C 2.1 Lemma 2.1 (see [20]) Let X be a Banach space with weakly sequentially continuous duality mapping J, and suppose that xn n∈N converges weakly to x0 ∈ X, then for any x ∈ X, lim inf xn − x0 ≤ lim inf xn − x n→∞ n→∞ Moreover if X is uniformly convex, equality holds in 2.2 if and only if x0 2.2 x Fixed Point Theory and Applications Recall that a point u ∈ C is a solution of a VIP if and only if u PC I − λA u ∀λ > 0, that is, u ∈ V I C, A ⇐⇒ u ∈ Fix PC I − λA , ∀λ > 2.3 Definition 2.2 An operator A : C → H is said to be α-inverse strongly monotone operator if there exists a constant α > such that Ax − Ay, x − y ≥ α Ax − Ay ∀x, y ∈ C 2.4 If α we say that A is firmly nonexpansive Note that every α-inverse strongly monotone operator is also 1/α Lipschitz continuous see 21 Lemma 2.3 (see [2]) Let C be a nonempty closed convex subset of a real Hilbert space H and let S : C → C be a k-strict pseudocontractive mapping Then St : tI − t S with t ∈ k, is a Fix S nonexpansive mapping with Fix St Main Theorem Theorem 3.1 Let C be a closed convex subset of a real Hilbert space H Let i A be an α-inverse strongly monotone mapping of C into H, ii B a β-inverse strongly monotone mapping of C into H, iii Tn n∈N and Vn n∈N two sequences of firlmy nonexpansive mappings from C to H Let S : C → C be a k-strict pseudocontraction Fix S / ∅ − k S and let us define the sequence xn n∈N as follows: Set Sk kI x1 ∈ C, C1 C, un Tn I − rn A xn zn Vn I − λn B un , yn Cn αn xn w ∈ Cn : xn where i αn n∈N ⊂ 0, a with a < 1; ii λn n∈N ⊂ b, c ⊂ 0, 2β ; iii rn n∈N ⊂ d, e ⊂ 0, 2α 3.1 − αn Sk zn , yn − w ≤ xn − w PCn x1 , ∀n ∈ N, , Fixed Point Theory and Applications Moreover suppose that i F : Fix S ii ∩n Fix Vn I − λn B ∩n Fix Tn I − rn A / ∅; Tn I − rn A n∈N pointwise converges in C to an operator R and Vn I − λn B pointwise converges in C to an operator W; ∩n Fix Vn I − λn B iii Fix W Then xn n∈N ∩n Fix Tn I − rn A and Fix R strongly converges to x∗ n∈N PF x1 Proof We begin to observe that the mappings Tn I − rn A and Vn I − λn B are nonexpansive for all n ∈ N since they are compositions of nonexpansive mappings see 22, page 419 As a rule, if p ∈ F un − p ≤ xn − p zn − p ≤ un − p , 3.2 ≤ xn − p Now we divide the proof in more steps Step Cn is closed and convex for each n ∈ N Indeed Cn is the intersection of Cn with the half space w ∈ H : w, xn − yn ≤ L , where L xn − yn 3.3 /2 Step F ⊆ Cn for each n ∈ N For each w ∈ F we have yn − w αn xn − αn Sk zn − w ≤ αn xn − w αn xn − w ≤ αn xn − w αn xn − w ≤ αn xn − w − αn zn − w − αn Vn I − λn B un − w − αn un − w − αn Tn I − rn A xn − w − αn xn − w xn − w So the claim immediately follows by induction 3.4 Fixed Point Theory and Applications Step limn → ∞ xn − x1 exists and xn n∈N is asymptotically regular, that is, limn → xn Since xn PCn x1 , xn PCn x1 , and Cn ⊆ Cn , by 2.1 choosing y xn , x C Cn , we have ≤ x1 − xn , xn − xn xn − x1 and x1 − xn , xn − x1 ≤ − x1 − xn ∞ x1 − xn x1 − xn 3.5 x1 − xn , that is, xn − x1 ≤ xn − x1 By xn PCn x1 and F ⊆ Cn , we have x1 − xn ≤ x1 − PF x1 Then limn → ∞ xn − x1 exists and xn 2 xn − x1 − x1 xn − x1 2 xn − x1 , x1 − xn xn − x1 xn − x1 2 xn − xn , x1 − xn − xn − x1 ≤ xn 1 − x1 − xn − x1 by 3.5 , and consequently limn → ∞ xn x1 − xn is bounded Moreover xn xn − xn n∈N 3.6 − xn 3.7 Step limn → ∞ xn − yn and limn → By xn ∈ Cn , it follows xn − Sk zn ∞ yn − xn yn − xn ≤ yn − xn ≤ xn − xn xn 1 , − xn ≤ xn − xn −→ 3.8 Moreover yn − xn and by boundedness of αn n∈N , − αn xn − Sk zn , it follows that limn → ∞ xn − Sk zn 3.9 6 Fixed Point Theory and Applications Step limn → ∞ Bun − Bw For w ∈ F, we have 0, for each w ∈ F ≤ αn xn − w − αn Sk zn − w ≤ αn xn − w − αn zn − w ≤ αn xn − w − αn Vn I − λn B un − Vn I − λn B w ≤ αn xn − w − αn I − λn B un − I − λn B w αn xn − w − αn un − w ≤ αn xn − w yn − w − αn un − w ≤ xn − w 2 2 λ2 Bun − Bw n − λn 2β − λn − 2λn Bun − Bw, un − w Bun − Bw − αn λn λn − 2β Bun − Bw 3.10 Consequently − αn λn 2β − λn Bun − Bw ≤ xn − w − yn − w xn − w − yn − w ≤ and by Step 4, the assumptions on αn xn − yn n∈N xn − w xn − w and λn n∈N , yn − w yn − w 3.11 , we obtain the claim of Step Step limn → ∞ un − zn Since Vn is firmly nonexpansive, for any w ∈ F, we have zn − w ≤ I − λn B un − I − λn B w, zn − w I − λn B un − I − λn B w zn − w − I − λn B un − I − λn B w − zn − w ≤ un − w − λn 2β − λn Bun − Bw zn − w 3.12 − un − zn − λn Bun − Bw ≤ 2 un − w zn − w − un − zn − λn Bun − Bw un − w zn − w − un − zn 2λn un − zn , Bun − Bw − λ2 Bun − Bw n 2 Fixed Point Theory and Applications which implies ≤ un − w − un − zn 2λn un − zn , Bun − Bw ≤ xn − w zn − w − un − zn 2λn un − zn 3.13 Bun − Bw Consequently yn − w 2 ≤ αn xn − w ≤ xn − w − αn zn − w − − αn un − zn 2 − αn λn un − zn 3.14 Bun − Bw which implies − αn un − zn ≤ xn − w ≤ − yn − w − αn λn un − zn xn − w − yn − w − αn λn un − zn ≤ xn − yn By the assumptions on αn un n∈N the claim follows xn − w yn − w Bun − Bw xn − w n∈N , Bun − Bw yn − w − αn λn un − zn Steps and 6, and the boundedness of xn n∈N Bun − Bw 3.15 yn n∈N and Step limn → ∞ xn − un and limn → ∞ xn − Sk xn Since Tn is firmly nonexpansive, for each p ∈ ∩n Fix Tn I − rn A , we have un − p Tn I − rn A xn − Tn I − rn A p ≤ un − p, I − rn A xn − I − rn A p I − rn A xn − I − rn A p − 2 un − p I − rn A xn − I − rn A p − un − p xn − p − rn 2α − rn Axn − Ap − xn − un − rn Axn − Ap ≤ xn − p un − p −rn Axn − Ap 2 2 un − p − xn − un 2rn xn − un , Axn − Ap , 3.16 Fixed Point Theory and Applications and consequently un − p ≤ xn − p − xn − un 2rn xn − un Axn − Ap 3.17 Then, for each w ∈ F, we have yn − w ≤ αn xn − w ≤ xn − w 2 − αn un − w − − αn xn − un − αn rn xn − un 3.18 Axn − Aw by 3.17 , consequently − αn xn − un ≤ xn − w ≤ xn − yn − yn − w xn − w 2 − αn rn xn − un yn − w Axn − Aw − αn rn xn − un Axn − Aw , 3.19 and by the assumptions on αn n∈N , Step and the boundedness of xn n∈N and yn follows that xn − un → as n → ∞ By Step we note that also xn − zn → Finally xn − Sk xn ≤ xn − Sk zn n∈N it Sk zn − Sk xn ≤ xn − Sk zn zn − xn ≤ xn − Sk zn zn − un 3.20 un − xn , and by previous steps, it follows that xn − Sk xn → as n → ∞ Step The set of weak cluster points of xn n∈N is contained in F We will use three times the Opial’s Lemma 2.1 Let p be a weak cluster point of xn n∈N and let xnj j∈N be a subsequence of xn n∈N such that xnj p We prove that p ∈ Fix S Fix Sk We suppose for absurd that p / Sk p By Opial’s Lemma 2.1 and xn − Sk xn → as n → ∞, we obtain lim inf xnj − p < lim inf xnj − Sk p j→ ∞ j→ ∞ lim inf xnj − Sk xnj − Sk xnj − Sk p ≤ lim inf j→ ∞ j→ ∞ lim inf xnj − p j→ ∞ which is a contradiction xnj − Sk xnj Sk xnj − Sk p 3.21 Fixed Point Theory and Applications Since Fix R we note that ∩n Fix Tn I − rn A it is enough to prove that p ∈ Fix R Now if p / Rp lim inf xnj − p < lim inf xnj − Rp j→ ∞ j→ ∞ ≤ lim inf j→ ∞ xnj − Tnj I − rnj A xnj Tnj I − rnj A xnj − Tnj I − rnj A p ≤ lim inf j→ ∞ xnj − unj Tnj I − rnj A p − Rp Tnj I − rnj A p − Rp xnj − p lim inf xnj − p j→ ∞ 3.22 This leads to a contraddiction again By the hypotheses and Step the claim follows By the same idea and using Step 6, we prove that p ∈ Fix W ∩n Fix Vn I − λn B Step xn → x∗ PF x1 Since x∗ PF x1 ∈ Cn and xn PCn x1 , we have x1 − xn ≤ x1 − x∗ Let xnj j∈N be a subsequence of xn n∈N such that xnj 3.23 p By Step 8, p ∈ F Thus x1 − x∗ ≤ x1 − p ≤ lim inf x1 − xnj j→ ∞ ≤ lim sup x1 − xnj j→ ∞ ≤ x1 − x∗ 3.24 Therefore we have x1 − x∗ x1 − p lim x1 − xnj j→ ∞ 3.25 Since H has the Kadec-Klee property, then xnj → p as j → ∞ x1 − p and by the uniqueness of the projection PF x1 , it Moreover, by x1 − x∗ follows that p x∗ PF x1 Thence every subsequence xnj j∈N converges to x∗ as j → ∞ and consequently xn → x∗ , as n → ∞ Remark 3.2 Let us observe that one can choose Tn n∈N and Vn n∈N as sequences of γn inverse strongly monotone operators and ηn -inverse strongly monotone operators provided γn ≥ 1, ηn ≥ for all n ∈ N 10 Fixed Point Theory and Applications The hypotheses ii and iii in the main Theorem 3.1 seem very strong but, in the sequel, we furnish two cases in which ii and iii are satisfied Let us remember that the metric projection on a convex closed set PC is a firmly nonexpansive mapping see 19 so we claim that have the following proposition r > and A an α-inverse strongly Proposition 3.3 If rn n∈N ⊂ 0, ∞ is such that limn rn monotone, then PC I − rn A realizes conditions (ii) and (iii) with R PC I − rA Proof To prove ii we note that for each x ∈ C, PC I − rn A x − PC I − rA x ≤ I − rn A x − I − rA x ≤ |rn − r| Ax 3.26 Moreover, iii follows directly by 2.2 Now we consider the mixed equilibrium problem Find x ∈ C : f x, y Ax, y − x ≥ 0, h x, y ∀y ∈ C 3.27 In the sequel we will indicate with MEP f, h, A the set of solution of our mixed equilibrium problem If A we denote MEP f, h, with MEP f, h We notice that for h and A the problem is the well-known equilibrium problem 23–25 If h and A is an α-inverse strongly monotone operator we have the equilibrium problems studied firstly in 26 and then in 18, 22, 27 If h x, y ϕ y − ϕ x and A we refound the mixed equilibrium problem studied in 16, 28, 29 Definition 3.4 A bi-function g : C × C → R is monotone if g x, y A function G : C → R is upper hemicontinuous if lim sup G tx t→0 g y, x ≤ for all x, y ∈ C 1−t y ≤G y Next lemma examines the case in which A Lemma 3.5 Let C be a convex closed subset of a Hilbert space H Let f : C × C → R be a bi-function such that f1 f x, x for all x ∈ C; f2 f is monotone and upper hemicontinuous in the first variable; f3 f is lower semicontinuous and convex in the second variable Let h : C × C → R be a bi-function such that h1 h x, x for all x ∈ C; h2 h is monotone and weakly upper semicontinuous in the first variable; h3 h is convex in the second variable 3.28 Fixed Point Theory and Applications 11 Moreover let us suppose that H for fixed r > and x ∈ C, there exists a bounded set K ⊂ C and a ∈ K such that for all z ∈ C \ K, −f a, z h z, a 1/r a − z, z − x < 0, for r > and x ∈ H let Tr : H → C be a mapping defined by z ∈ C : f z, y Tr x y − z, z − x ≥ 0, r h z, y ∀y ∈ C , 3.29 called resolvent of f and h Then Tr x / ∅; Tr x is a single value; Tr is firmly nonexpansive; Fix Tr and it is closed and convex MEP f, h Proof Let x0 ∈ H For any y ∈ C define z ∈ C : −f y, z Gr,x0 y y − z, z − x ≥ r h z, y 3.30 We will prove that, by KKM’s lemma, ∩y∈C Gr,x0 y is nonempty First of all we claim that Gr,x0 is a KKM’s map In fact if there exists {y1 , , yN } ⊂ C does not appartiene to Gr,x0 yi for any i 1, , N then such that y i αi yi with i αi −f yi , y h y, yi yi − y, y − x0 < 0, r ∀i 3.31 By the convexity of f and h and the monotonicity of f, we obtain that f y, y ≤ y − y, y − x0 r h y, y αi f y, yi i αi h y, yi i ≤− αi f yi , y αi h y, yi i αi −f yi , y i that is absurd i h y, yi r αi yi − y, y − x0 i r 3.32 αi yi − y, y − x0 i yi − y, y − x0 r < 0, 12 Fixed Point Theory and Applications Now we prove that Gr,x0 · , the relation w Gr,x0 We recall that, by the weak lower semicontinuity of lim sup y − zm , zm − x0 ≤ y − z, z − x0 3.33 m w holds Let z ∈ Gr,x0 y and let zm We want to prove that m be a sequence in Gr,x0 y such that zm −f y, z h z, y z y − z, z − x0 ≥ r 3.34 Since f is lower semicontinuous and convex in the second variable and h is weakly upper semicontinuous in the first variable, then ≤ lim sup −f y, zm m ≤ lim sup −f y, zm m ≤ −lim inf f y, zm m ≤ −f y, z h z, y y − z, z − x0 r lim sup y − z, z − x0 lim sup h zm , y r m m lim sup y − z, z − x0 lim sup h zm , y r m m y − z, z − x0 r h zm , y 3.35 w Now we observe that Gr,x0 y Gr,x0 y is weakly compact for at least a point y ∈ C In fact by hypothesis H there exist a bounded K ⊂ C and a ∈ K, such that for all z ∈ C \ K it results z / Gr,x0 a Then Gr,x0 a ⊂ K, that is, it is bounded It follows that Gr,x0 a is weakly ∈ compact Then by KKM’s lemma ∩y∈C Gr,x0 y is nonempty However if z ∈ ∩y∈C Gr,x0 then −f y, z h z, y y − z, z − x0 ≥ 0, r ∀y ∈ C 3.36 As in 24, Lemma , since f is upper hemicontinuous and convex in the first variable and monotone, we obtain that 3.36 is equivalent to claim that z is such that f z, y h z, y y − z, z − x0 ≥ 0, r ∀y ∈ C, 3.37 that is, z ∈ Tr x0 This prove To prove and we consider z1 ∈ Tr x1 and z2 ∈ Tr x2 They satisfy the relations f z1 , z2 h z1 , z2 f z2 , z1 h z2 , z1 z2 − z1 , z1 − x1 ≥ 0, r z1 − z2 , z2 − x2 ≥ r 3.38 Fixed Point Theory and Applications 13 By the monotonicity of f and h, summing up both the terms, 0≤ z2 − z1 , z1 − x1 − z2 − z1 , z2 − x2 r z2 − z1 , z1 − x1 − z2 x2 r − z2 − z1 z2 − z1 , x2 − x1 r 3.39 so we conclude z2 − z1 ≤ z2 − z1 , x2 − x1 3.40 that means simultaneously that z1 z2 if x1 x2 and Tr is firmly nonexpansive To prove , it is enough to follow iii and iv in 25, Lemma 2.12 Remark 3.6 We note that if h 0, our lemma reduces to 25, Lemma 2.12 The coercivity condition H is fulfilled Moreover our lemma is more general than 16, Lemma 2.2 In fact i our hypotheses on f are weaker f weak upper semicontinuous implies f upper hemicontinuous ; ii if ϕ satisfies the condition in Lemma 2.2 , choosing h x, y ϕ y − ϕ x one has that h is concave and upper semicontinuous in the first variable and convex and lower semicontinous in the second variable; iii the coercivity condition H by the equivalence of 3.36 and 3.37 is the same Lemma 3.7 Let us suppose that (f1)–(f3), (h1)–(h3) and (H) hold Let x, y ∈ H, r1 , r2 > Then Tr2 y − Tr1 x ≤ y − x Proof By Lemma 3.5, defining u1 r2 − r1 r2 Tr2 y − y 3.41 Tr1 x and u2 : Tr2 y, we know that f u2 , z h u2 , z f u1 , z h u1 , z z − u2 , u2 − y ≥ 0, r2 z − u1 , u1 − x ≥ 0, r1 ∀z ∈ C, 3.42 ∀z ∈ C In particular, f u2 , u1 h u2 , u1 f u1 , u2 h u1 , u2 u1 − u2 , u2 − y ≥ 0, r2 u2 − u1 , u1 − x ≥ r1 3.43 14 Fixed Point Theory and Applications Hence, summing up this two inequalities and using the monotonicity of f and h, u − u1 , u1 − x u − y − r1 r2 ≥ 3.44 We derive from 3.44 that u2 − u1 , u1 − u2 − x u2 − r1 u2 − y r2 ≥ 0, 3.45 and so − u2 − u1 u2 − u1 , u2 − y 1− r1 r2 y−x ≥ 3.46 , 3.47 Then, u2 − u1 ≤ u2 − u1 y−x 1− r1 r2 u2 − y and thus the claim holds Proposition 3.8 Let us suppose that f and h are two bi-functions satisfying the hypotheses of Lemma 3.5 Let Tr be the resolvent of f and h Let A be an α-inverse strongly monotone operator Let us suppose that rn n∈N ⊂ 0, ∞ is such that limn rn r > Then Trn I − rn A realize (ii) and (iii) in Theorem 3.1 Proof Let x be in a bounded closed convex subset K of C To prove i it is enough to observe that by Lemma 3.7 Trn I − rn A x − Tr I − rA x ≤ |rn − r| Ax |rn − r| Tr I − rA x − I − rA x r 3.48 When n → ∞, by boundedness of the terms that not depend on n, we obtain ii To prove iii let W Tr I − rA the pointwise limit of Trn I − rn A It is necessary to prove only that Fix W ⊂ ∩n Fix Trn I − rn A Let x ∈ Fix W We want to prove that x ∈ MEP f, h, A Let wn Trn I − rn A x Thus, by definition of Trn , wn is the unique point such that f wn , y h wn , y y − wn , wn − I − rn A x ≥ 0, rn ∀y 3.49 By monotonicity of f and h this implies h wn , y y − wn , wn − I − rn A x ≥ f y, wn rn 3.50 Fixed Point Theory and Applications 15 Passing to the limit on n, by f3 and h2 we obtain y − x, Ax ≥ f y, x , h x, y Let now u ty ∀y − t x with t ∈ 0, Then by the convexity of f and h h u, y − t f u, x h u, y u − x, Ax t f u, y f u, u h u, u ≤ t f u, y ≤ t f u, y 3.51 h u, y y − x, Ax h u, x 3.52 Passing t → we obtain by f1 and h1 f x, y h x, y Ax, y − x ≥ 3.53 That is, x ∈ MEP f, h, A At this point we observe that from the definitions of MEP f, h, A Fix Trn I − rn A and Trn , one has MEP f, h, A By Propositions 3.3 and 3.8 we can exhibit iterative methods to approximate fixed points of the k-strict pseudo contraction that are also solution of a system of two variational inequalities VI C,A and VI C,B PC ; solution of a system of two mixed equilibrium problems Tn Trn and Vn solution of a mixed equilibrium problem and a variational inequality Tn Vn PC Vn Tn T λn ; Trn and However when the properties of the mapping Tn and Vn are well known, one can prove convergence theorems like Theorem 3.1 without use of Opial’s lemma In next theorem our purpose is to prove a strong convergence theorem to approximate a fixed point of S that is also a solution of a mixed equilibrium problem and a solution of a variational inequality V I C, B One can note that we relax the hypotheses on the convergence of the sequences rn n∈N and λn n∈N Theorem 3.9 Let C be a closed convex subset of a real Hilbert space H, let f, h : C × C → R be two bi-functions satisfying (f1)–(f3),(h1)–(h3), and (H) Let S : C → C be a k-strict pseudocontraction Let A be an α-inverse strongly monotone mapping of C into H and let B be a β-inverse strongly monotone mapping of C into H Let us suppose that F Fix S ∩ MEP f, h, A ∩ V I C, B / ∅ 16 Fixed Point Theory and Applications Set Sk − k S, one defines the sequence xn kI n∈N as follows: x1 ∈ C, C1 f un , y y − un , un − xn rn zn PC I − λn B un , h un , y yn − w ≤ xn − w w ∈ Cn : xn PCn x1 , Axn , y − un ≥ 0, 3.54 − αn Sk zn , αn xn yn Cn C, , ∀n ∈ N, where ⊂ 0, a with a < 1; i αn n∈N ii λn n∈N ⊂ b, c ⊂ 0, 2β ; iii rn n∈N ⊂ d, e ⊂ 0, 2α Then xn n∈N strongly converges to x∗ PF x1 Proof First of all we observe that by Lemma 3.5 we have that un Trn I − rn A xn We can follow the proof of Theorem 3.1 from Steps 1–7 We prove only the following Step 10 The set of weak cluster points of xn n∈N is contained in F Let p be a cluster point of xn ; we begin to prove that p ∈ MEP f, h, A We know that f un , y h un , y y − un , un − xn ≥ 0, rn Axn , y − un ∀y ∈ C, 3.55 and by f2 h un , y Axn , y − un y − un , un − xn ≥ f y, un , rn ∀y ∈ C 3.56 Let xnj j∈N be a subsequence of xn n∈N weakly convergent to p, then by Step unj j → ∞ Let ρt : ty − t p, t ∈ 0, Then by 3.56 ρt − unj , Aρt ρt − unj , Aρt − Axnj ≥ ρt − unj , Aρt − Axnj p as Axnj , ρt − unj f y, unj − h unj , y ρt − unj , Aρt − Aunj y − unj , unj − xnj rnj ρt − unj , Aunj − Axnj f y, unj − h unj , y − − y − unj , unj − xnj rnj ≥ ρt − unj , Aunj − Axnj f y, unj − h unj , y − y − unj , unj − xnj rnj 3.57 Fixed Point Theory and Applications 17 Since A is Lipschitz continuous and unj − xnj → as j → ∞, we have Aunj − Axnj → as j → ∞ By condition f3 , for x ∈ H fixed, the function f x, · is lower semicontinuos and convex, and thus weakly lower semicontinuous 30 Since xn −un → 0, as n → ∞ and by the assumption on rn we obtain unj −xnj /rnj → Then we obtain by h2 ρt − p, Aρt ≥ f y, p − h p, y 3.58 Using f1 , f3 , h1 , h3 we obtain f ρt , ρt − t f ρt , p h ρt , ρt ≤ tf ρt , y ≤ tf ρt , y th ρt , y ≤ tf ρt , y th ρt , y th ρt , y − t f ρt , p − h p, ρt − t ρt − p, Aρt t f ρt , y h ρt , y − t h ρt , p 3.59 − t y − p, Aρt Consequently f ρt , y − t y − p, Aρt ≥ h ρt , y 3.60 by f2 and h2 , as t → 0, we obtain p ∈ MEP f, h, A Now we prove that p ∈ V I C, B We define the maximal monotone operator Tx Bx NC x, if x ∈ C, ∅, se x / C, ∈ 3.61 where NC x is the normal cone to C at x, that is, NC x {w ∈ H : x − u, w ≥ 0, ∀u ∈ C} 3.62 Since zn ∈ C, by the definition of NC we have x − zn , y − Bx ≥ x − zn , zn − I − λn B un ≥ 0, But zn 3.63 3.64 PC I − λn B un , then and hence x − zn , zn − un λn Bun ≥ 3.65 18 Fixed Point Theory and Applications By 3.63 , 3.65 , and by the β-inverse monotonicity of B, we obtain x − znj , y ≥ x − znj , Bx ≥ x − znj , Bx − x − znj , By xn − zn → as n → p as j → ∞ Then znj x − znj , znj − unj λnj λnj Bunj x − znj , Bznj − Bunj x − znj , Bx − Bznj − znj − unj 3.66 ∞ immediately consequence of Steps and , it follows that x − p, y ≥ 0, 3.67 moreover, since T is a maximal operator, ∈ T p, that is, p ∈ V I C, B Finally, to prove that p ∈ Fix S Fix Sk we follow Step as in Theorem 3.1 Since also Step can be followed as in Theorem 3.1, we obtain the claim References G L Acedo and H.-K Xu, “Iterative methods for strict 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inequality V I C, B One can note that... Journal of Mathematical Analysis and Applications, vol 20, pp 197–228, 1967 F E Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the