Thông tin tài liệu
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 840319, 25 pages doi:10.1155/2011/840319 Research Article The Shrinking Projection Method for Common Solutions of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Strictly Pseudocontractive Mappings Thanyarat Jitpeera and Poom Kumam Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th Received 21 September 2010; Revised 14 December 2010; Accepted 20 January 2011 Academic Editor: Jewgeni Dshalalow Copyright q 2011 T. Jitpeera and P. Kumam. 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 introduce the shrinking hybrid projection method for finding a common element of the set of fixed points of strictly pseudocontractive mappings, the set of common solutions of the variational inequalities with inverse-strongly monotone mappings, and the set of common solutions of generalized mixed equilibrium problems in Hilbert spaces. Furthermore, we prove strong convergence theorems for a new shrinking hybrid projection method under some mild conditions. Finally, we apply our results to Convex Feasibility Problems CFP. The results obtained in this paper improve and extend the corresponding results announced by Kim et al. 2010 and the previously known results. 1. Introduction Let H be a real Hilbert space with inner product ·, · and norm ·,andletE be a nonempty closed convex subset of H.LetT : E → E be a mapping. In the sequel, we will use FT to denote the set of fixed points of T,thatis,FT{x ∈ E : Tx x}.Wedenoteweak convergence and strong convergence by notations and → , respectively. Let S : E → E be a mapping. Then S is called 1 nonexpansive if Sx − Sy ≤ x − y , ∀x, y ∈ E, 1.1 2 Journal of Inequalities and Applications 2 strictly pseudocontractive with the coefficient k ∈ 0, 1 if Sx − Sy 2 ≤ x − y 2 k I − S x − I − S y 2 , ∀x, y ∈ E, 1.2 3 pseudocontractive if Sx − Sy 2 ≤ x − y 2 I − S x − I − S y 2 , ∀x, y ∈ E. 1.3 The class of strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings. Within the past several decades, many authors have been devoted to the studies on the existence and convergence of fixed points for strictly pseudocontractive mappings. In 2008, Zhou 1 considered a convex combination method to study strictly pseudocontractive mappings. More precisely, take k ∈ 0, 1, and define a mapping S k by S k x kx 1 − k Sx, ∀x ∈ E, 1.4 where S is strictly pseudocontractive mappings. Under appropriate restrictions on k,itis proved that the mapping S k is nonexpansive. Therefore, the techniques of studying nonex- pansive mappings can be applied to study more general strictly pseudocontractive mappings. Recall that letting A : E → H be a mapping, then A is ca lled 1 monotone if Ax − Ay, x − y ≥ 0, ∀x, y ∈ E, 1.5 2 β-inverse-strongly monotone if there exists a constant β>0suchthat Ax − Ay, x − y ≥ β Ax − Ay 2 , ∀x, y ∈ E. 1.6 The domain of the function ϕ : E → ∪{∞} is the set dom ϕ {x ∈ E : ϕx < ∞}. Let ϕ : E → ∪{∞} be a proper extended real-valued function and let F be a bifunction of E × E into such that E ∩ dom ϕ / ∅,where is the set of real numbers. There exists the generalized mixed equilibrium problem for finding x ∈ E such that F x, y Ax, y − x ϕ y − ϕ x ≥ 0, ∀y ∈ E. 1.7 The set of solutions of 1.7 is denoted by GMEPF, ϕ, A,thatis, GMEP F, ϕ, A x ∈ E : F x, y Ax, y − x ϕ y − ϕ x ≥ 0, ∀y ∈ E . 1.8 Journal of Inequalities and Applications 3 We see that x is a solution of a problem 1.7 which implies that x ∈ dom ϕ {x ∈ E : ϕx < ∞}. In particular, if A ≡ 0, then the problem 1.7 is reduced into the m ixed equilibrium problem 2 for finding x ∈ E such that F x, y ϕ y − ϕ x ≥ 0, ∀y ∈ E. 1.9 The set of solutions of 1.9 is denoted by MEPF, ϕ. If A ≡ 0andϕ ≡ 0, then the problem 1.7 is reduced into the equilibrium problem 3 for finding x ∈ E such that F x, y ≥ 0, ∀y ∈ E. 1.10 The set of solutions of 1.10 is denoted by EPF. This pro blem contains fixed point problems and includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem; please consult 4, 5. If F ≡ 0andϕ ≡ 0, then the problem 1.7 is reduced into the Hartmann-Stampacchia variational inequality 6 for finding x ∈ E such that Ax, y − x ≥ 0, ∀y ∈ E. 1.11 The set of solutions of 1.11 is denoted by VIE, A. The variational inequality has been extensively studied in the literature. See, for example, 7–10 and the references therein. Many authors solved the problems GMEPF, ϕ, A,MEPF, ϕ,andEPF based on iterative methods; see, for instance, 4, 5, 11–25 and reference therein. In 2007, Tada and Takahashi 26 intr oduced a hybrid method for finding the common element of the set of fixed point of nonexpansive mapping and the set of s olutions of equilibrium problems in Hilbert spaces. Let {x n } and {u n } be sequences generated by the following iterative algorithm: x 1 x ∈ H, F u n ,y 1 r n y − u n ,u n − x n ≥ 0, ∀y ∈ E, w n 1 − α n x n α n Su n , E n { z ∈ H : w n − z ≤ x n − z } , D n { z ∈ H : x n − z, x − x n ≥ 0 } , x n1 P E n ∩D n x, ∀n ≥ 1. 1.12 Then, they proved that, under certain appropriate conditions imposed on {α n } and {r n },the sequence {x n } generated by 1.12 converges strongly to P FS∩EPF x. In 2009, Qin and Kang 27 introduced an explicit viscosity approximation method for finding a common element of the set of fixed point of strictly pseudocontractive mappings 4 Journal of Inequalities and Applications and the set of solutions of variational inequalities with inverse-strongly monotone mappings in Hilbert spaces: x 1 ∈ E, z n P E x n − μ n Cx n , y n P E x n − λ n Bx n , x n1 n f x n β n x n γ n α 1 n S k x n α 2 n y n α 3 n z n , ∀n ≥ 1. 1.13 Then, they proved that, under certain appropriate conditions imposed on { n }, {β n }, {γ n }, {α 1 n }, {α 2 n },and{α 3 n },thesequence{x n } generated by 1.13 converges strongly to q ∈ FS ∩ VIE, B ∩ VIE, C,whereq P FS∩VIE,B∩VIE,C fq. In 2010, Kumam and Jaiboon 28 introduced a new method for finding a common element of the set of fixed point of strictly pseudocontractive mappings, the set of common solutions of variational inequalities with inverse-strongly monotone mappings, and the set of common solutions of a system of generalized mixed equilibrium problems in Hilbert spaces. Then, they proved that, under certain a ppropriate conditions imposed on { n }, {β n },and {α i n },wherei 1, 2, 3, 4, 5. The sequence {x n } converges strongly to q ∈ Θ : FS∩VIE, B∩ VIE, C ∩ GMEPF 1 ,ϕ,A 1 ∩ GMEPF 2 ,ϕ,A 2 ,whereq P Θ I − A γfq. In this paper, motivate, by Tada and Takahashi 26, Qin and Kang 27,andKumam and Jaiboon 28, we introduce a new shrinking projection method for finding a common element of the set of fixed points of strictly pseudocontractive mappings, the set of common solutions of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities for inverse-strongly monotone mappings in Hilbert spaces. Finally, we apply our results to Convex Feasibility Problems CFP. The results obtained in this paper improve and extend the corresponding results announced by the previously known results. 2. Preliminaries Let H be a real Hilbert space, and let E be a nonempty closed convex subset of H.Inareal Hilbert space H,itiswellknownthat λx 1 − λ y 2 λ x 2 1 − λ y 2 − λ 1 − λ x − y 2 , 2.1 for all x, y ∈ H and λ ∈ 0, 1. For any x ∈ H,thereexistsaunique nearest point in E, denoted by P E x,suchthat x − P E x ≤ x − y , ∀y ∈ E. 2.2 The mapping P E is called the metric projection of H onto E. It is well known that P E is a firmly nonexpansive mapping of H onto E,thatis, x − y, P E x − P E y ≥ P E x − P E y 2 , ∀x, y ∈ H. 2.3 Journal of Inequalities and Applications 5 Moreover, P E x is characterized by the following properties: P E x ∈ E and x − P E x, y − P E x ≤ 0, x − y 2 ≥ x − P E x 2 y − P E x 2 2.4 for all x ∈ H, y ∈ E. Lemma 2.1. Let E be a nonempty closed convex subset of a real Hilbert space H.Givenx ∈ H and z ∈ E,then, z P E x ⇐⇒ x − z, y − z ≤ 0, ∀y ∈ E. 2.5 Lemma 2.2. Let H be a Hilbert space, let E be a nonempty closed convex subset of H,andletB be a mapping of E into H.Letu ∈ E.Then,forλ>0, u ∈ VI E, B ⇐⇒ u P E u − λBu , 2.6 where P E is the metric projection of H onto E. Lemma 2.3 see 1. Let E be a nonempty closed convex subset of a real H ilbert space H,andlet S : E → E be a k-strictly pseudocontractive mapping with a fixed point. Then FS is closed a nd convex. Define S k : E → E by S k kx 1 − kSx for each x ∈ E.ThenS k is nonexpansive such that FS k FS. Lemma 2.4 see 29. Let E be a closed convex subset of a real Hilbert space H,andletS : E → E be a nonexpansive m apping. Then I − S is demiclosed at zero; that is, x n x, x n − Sx n −→ 0 2.7 implies x Sx. Lemma 2.5 see 30. Each Hilbert space H satisfies the Kadec-Klee property, for any sequence {x n } with x n xand x n →x together implying x n − x→0. Lemma 2.6 see 31. Let E be a closed convex subset of H.Let{x n } be a bounded sequence in H. Assume that 1 the weak ω-limit set ω w x n ⊂ E, 2 for each z ∈ E, lim n →∞ x n − z exists. Then {x n } is weakly convergent to a point in E. Lemma 2.7 see 32. Let E be a closed convex subset of H.Let{x n } be a sequence in H and u ∈ H. Let q P E u.If{x n } is ω w x n ⊂ E and satisfies the condition x n − u ≤ u − q 2.8 for all n,thenx n → q. 6 Journal of Inequalities and Applications Lemma 2.8 see 33. Let E be a nonempty closed convex subset of a strictly convex Banach space X.Let{T n : n ∈ } be a sequence of nonexpansive mappings on E. Suppose ∞ n1 FT n is nonempty. Let δ n be a sequence of positive number with ∞ n1 δ n 1. Then a mapping S on E defined by Sx ∞ n1 δ n T n x 2.9 for x ∈ E is well defined, nonexpansive, and FS ∞ n1 FT n holds. For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction F, the function A,andthesetE: A1 Fx, x0forallx ∈ E A2 F is monotone, that is, Fx, yFy, x ≤ 0forallx, y ∈ E A3 for each x, y, z ∈ E, lim t →0 Ftz 1 − tx, y ≤ Fx, y A4 for each x ∈ E, y → Fx, y is convex and lower semicontinuous A5 for each y ∈ E, x → Fx, y is weakly upper semicontinuous B1 for each x ∈ H and r>0, there exists a bounded subset D x ⊆ E and y x ∈ E such that, for any z ∈ E \ D x , F z, y x ϕ y x − ϕ z 1 r y x − z, z − x < 0, 2.10 B2 E is a bounded set. By similar argument as i n the proof of Lemma 2.9 in 34, we have the following lemma appearing. Lemma 2.9. Let E be a nonempty closed convex subset of H.LetF : E × E → be a bifunction that satisfies (A1)–(A5), and let ϕ: E → ∪{∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and x ∈ H, define a mapping T F r : H → E as follows: T F r x z ∈ E : F z, y ϕ y − ϕ z 1 r y − z, z − x ≥ 0, ∀y ∈ E , 2.11 for all z ∈ H. Then, the following hold: 1 for each x ∈ H, T F r x / ∅, 2 T F r is single valued, 3 T F r is firmly nonexpansive, that is, for any x, y ∈ H, T F r x − T F r y 2 ≤ T F r x − T F r y, x − y , 2.12 4 FT F r MEPF, ϕ, 5 MEPF, ϕ is closed and convex. Journal of Inequalities and Applications 7 Lemma 2.10. Let H be a Hilbert space, let E be a nonempty closed convex subset of H,andlet A : E → H be ρ-inverse-strongly monotone. If 0 <r≤ 2ρ,thenI − ρA is a nonexpansive mapping in H. Proof. For all x, y ∈ E and 0 <r≤ 2ρ,wehave I − rA x − I − rA y 2 x − y − r Ax − Ay 2 x − y 2 − 2r x − y, Ax − Ay r 2 Ax − Ay 2 ≤ x − y 2 − 2rρ Ax − Ay r 2 Ax − Ay 2 x − y 2 r r − 2ρ Ax − Ay 2 ≤ x − y 2 . 2.13 So, I − ρA is a nonexpansive mapping of E into H. 3. Main Results In this section, we prove a strong convergence theorem of the new shrinking p rojection method for finding a common element of the set of fixed points of strictly pseudocontractive mappings, the set of common solutions of generalized mixed equilibrium problems and the set of common solutions of the variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Theorem 3.1. Let E be a nonempty closed convex subset of a real Hilbert space H.LetF 1 and F 2 be two bifunctions from E × E to satisfying (A1)–(A5), and let ϕ : E → ∪{∞} be a pr oper lower semicontinuous and convex function with either (B1) or (B2). Let A 1 , A 2 , B, C be four ρ, ω, β, ξ-inverse-strongly monotone mappings of E into H, respectively. Let S : E → E be a k-strictly pseudocontractive mapping with a fixed point. Define a mapping S k : E → E by S k x kx1−kSx, for all x ∈ E. Suppose that Θ : F S ∩ GMEP F 1 ,ϕ,A 1 ∩ GMEP F 2 ,ϕ,A 2 ∩ VI E, B ∩ VI E, C / ∅. 3.1 Let {x n } be a sequence generated by the following iterative algorithm: x 0 ∈ H, E 1 E, x 1 P E 1 x 0 ,u n ∈ E, v n ∈ E, F 1 u n ,u ϕ u − ϕ u n A 1 x n ,u− u n 1 r n u − u n ,u n − x n ≥ 0, ∀u ∈ E, F 2 v n ,v ϕ v − ϕ v n A 2 x n ,v− v n 1 s n v − v n ,v n − x n ≥ 0, ∀v ∈ E, y n P E x n − λ n Bx n ,z n P E x n − μ n Cx n , 8 Journal of Inequalities and Applications t n α 1 n S k x n α 2 n y n α 3 n z n α 4 n u n α 5 n v n , E n1 { w ∈ E n : t n − w ≤ x n − w } , x n1 P E n1 x 0 , ∀n ≥ 0, 3.2 where {α i n } are sequences in 0, 1,wherei 1, 2, 3, 4, 5, r n ∈ 0, 2ρ, s n ∈ 0, 2ω,and{λ n }, {μ n } are positive sequences. Assume that the control sequences satisfy the following restrictions: C1 5 i1 α i n 1, C2 lim n →∞ α i n α i ∈ 0, 1,wherei 1, 2, 3, 4, 5, C3 a ≤ r n ≤ 2ρ and b ≤ s n ≤ 2ω,wherea, b are two positive constants, C4 c ≤ λ n ≤ 2β and d ≤ μ n ≤ 2ξ,wherec, d are two positive constants, C5 lim n →∞ |λ n1 − λ n | lim n →∞ |μ n1 − μ n | 0. Then, {x n } converges strongly to P Θ x 0 . Proof. Letting p ∈ Θ and by Lemma 2.9,weobtain p P E p − λ n Bp P E p − μ n Cp T F 1 r n I − r n A 1 p T F 2 s n I − s n A 2 p. 3.3 Note that u n T F 1 r n I − r n A 1 x n ∈ dom ϕ and v n T F 2 s n I − s n A 2 x n ∈ dom ϕ,thenwehave u n − p T F 1 r n I − r n A 1 x n − T F 1 r n I − r n A 1 p≤x n − p, v n − p T F 2 s n I − s n A 2 x n − T F 2 s n I − s n A 2 p ≤ x n − p . 3.4 Next, we will divide the proof into six steps. Step 1. We show that {x n } is well defined and E n is closed and convex for any n ≥ 1. From the assumption, we see that E 1 E is closed and convex. Suppose that E k is closed and convex for some k ≥ 1. Next, we show that E k1 is closed and convex for some k. For any p ∈ E k ,weobtain t k − p ≤ x k − p 3.5 is equivalent to t k − p 2 2 t k − x k ,x k − p ≤ 0. 3.6 Thus, E k1 is closed and convex. Then, E n is closed and convex for any n ≥ 1. This implies that {x n } is well defined. Journal of Inequalities and Applications 9 Step 2. We show that Θ ⊂ E n for each n ≥ 1. From the assumption, we see that Θ ⊂ E E 1 . Suppose Θ ⊂ E k for some k ≥ 1. For any p ∈ Θ ⊂ E k ,sincey n P E x n − λ n Bx n and z n P E x n − μ n Cx n ,foreachλ n ≤ 2β and μ n ≤ 2ξ by Lemma 2.10,wehaveI − λ n B and I − μ n C are nonexpansive. Thus, we obtain y n − p P E x n − λ n Bx n − P E p − λ n Bp ≤ x n − λ n Bx n − p − λ n Bp I − λ n B x n − I − λ n B p ≤ x n − p , z n − p P E x n − μ n Cx n − P E p − μ n Cp ≤ x n − μ n Cx n − p − μ n Cp I − μ n C x n − I − μ n C p ≤ x n − p . 3.7 From Lemma 2.3,wehaveS k is nonexpansive with FS k FS. It follows that t n − p α 1 n S k x n α 2 n y n α 3 n z n α 4 n u n α 5 n v n − p ≤ α 1 n S k x n − p α 2 n y n − p α 3 n z n − p α 4 n u n − p α 5 n v n − p ≤ α 1 n x n − p α 2 n x n − p α 3 n x n − p α 4 n x n − p α 5 n x n − p x n − p . 3.8 It follows that p ∈ E k1 . This implies that Θ ⊂ E n for each n ≥ 1. Step 3. We claim that lim n →∞ x n1 − x n 0 and lim n →∞ x n − t n 0. From x n P E n x 0 ,weget x 0 − x n ,x n − y ≥ 0 3.9 for each y ∈ E n .UsingΘ ⊂ E n ,wehave x 0 − x n ,x n − p ≥ 0foreachp ∈ Θ,n∈ . 3.10 10 Journal of Inequalities and Applications Hence, for p ∈ Θ,weobtain 0 ≤ x 0 − x n ,x n − p x 0 − x n ,x n − x 0 x 0 − p − x 0 − x n ,x 0 − x n x 0 − x n ,x 0 − p ≤− x 0 − x n 2 x 0 − x n x 0 − p . 3.11 It follows that x 0 − x n ≤ x 0 − p , ∀p ∈ Θ,n∈ . 3.12 From x n P E n x 0 and x n1 P E n1 x 0 ∈ E n1 ⊂ E n ,wehave x 0 − x n ,x n − x n1 ≥ 0. 3.13 For n ∈ ,wecompute 0 ≤ x 0 − x n ,x n − x n1 x 0 − x n ,x n − x 0 x 0 − x n1 − x 0 − x n ,x 0 − x n x 0 − x n ,x 0 − x n1 ≤− x 0 − x n 2 x 0 − x n ,x 0 − x n1 ≤− x 0 − x n 2 x 0 − x n x 0 − x n1 , 3.14 and then x 0 − x n ≤ x 0 − x n1 , ∀n ∈ . 3.15 Thus, the sequence {x n −x 0 } is a bounded and nondecreasing sequence, so lim n →∞ x n −x 0 exists; that is, there exists m such that m lim n →∞ x n − x 0 . 3.16 [...]... P Katchang and P Kumam, “A general iterative method of fixed points for mixed equilibrium problems and variational inclusion problems, ” Journal of Inequalities and Applications, vol 2010, Article ID 370197, 25 pages, 2010 22 W Kumam, C Jaiboon, P Kumam, and A Singta, “A shrinking projection method for generalized mixed equilibrium problems, variational inclusion problems and a finite family of quasinonexpansive... and P Kumam, “Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities,” Journal of Inequalities and Applications, vol 2010, Article ID 728028, 43 pages, 2010 15 C Jaiboon, P Kumam, and U W Humphries, “Weak convergence theorem by an extragradient method for variational inequality, equilibrium and fixed point problems, ” Bulletin of the. .. Malaysian Mathematical Sciences Society, vol 32, no 2, pp 173–185, 2009 16 J S Jung, “Strong convergence of composite iterative methods for equilibrium problems and fixed point problems, ” Applied Mathematics and Computation, vol 213, no 2, pp 498–505, 2009 17 J K Kim, S Y Cho, and X Qin, “Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings,” Journal of Inequalities... convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces,” Nonlinear Analysis Theory, Methods & Applications, vol 73, no 6, pp 1466–1480, 2010 20 J.-W Peng and J.-C Yao, “Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems, ” Mathematical and Computer Modelling, vol... iterative algorithm for mixed- equilibrium problems in Hilbert spaces,” Journal of Inequalities and Applications, vol 2008, Article ID 454181, 23 pages, 2008 13 C Jaiboon and P Kumam, “A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Fixed Point Theory and Applications, vol 2009, Article ID 374815,... mappings,” Journal of Inequalities and Applications, vol 2010, Article ID 458247, 25 pages, 2010 23 Y Yao, Y.-C Liou, and J.-C Yao, “A new hybrid iterative algorithm for fixed -point problems, variational inequality problems, and mixed equilibrium problems, ” Fixed Point Theory and Applications, vol 2008, Article ID 417089, 15 pages, 2008 Journal of Inequalities and Applications 25 24 H Zegeye and N Shahzad,... theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol 133, no 3, pp 359–370, 2007 27 X Qin and S M Kang, “Convergence theorems on an iterative method for variational inequality problems and fixed point problems, ” Bulletin of the Malaysian Mathematical Sciences Society, vol 33, no 1, pp 155–167, 2010 28 P Kumam and C Jaiboon, “A system of generalized. .. spaces,” Nonlinear Analysis Theory, Methods & Applications, vol 69, no 2, pp 456–462, 2008 2 L.-C Ceng and J.-C Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, ” Journal of Computational and Applied Mathematics, vol 214, no 1, pp 186–201, 2008 3 E Blum and W Oettli, “From optimization and variational inequalities to equilibrium problems, ” The Mathematics Student, vol... this paper, and the second author was supported by the Commission on Higher Education and the Thailand Research Fund under Grant MRG5380044 Moreover, they also would like to thank the National Research University Project of Thailand’s Office of the Higher Education Commission for financial support under NRU-CSEC Project no 54000267 References 1 H Zhou, “Convergence theorems of fixed points for κ-strict... fixed -point sets of nonexpansive mappings in Banach spaces,” Transactions of the American Mathematical Society, vol 179, pp 251–262, 1973 34 J.-W Peng and J.-C Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems, ” Taiwanese Journal of Mathematics, vol 12, no 6, pp 1401–1432, 2008 35 P L Combettes, The convex feasibility . approximation method for finding a common element of the set of fixed point of strictly pseudocontractive mappings 4 Journal of Inequalities and Applications and the set of solutions of variational. pseudocontractive mappings, the set of common solutions of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities for inverse-strongly monotone. Corporation Journal of Inequalities and Applications Volume 2011, Article ID 840319, 25 pages doi:10.1155/2011/840319 Research Article The Shrinking Projection Method for Common Solutions of Generalized Mixed Equilibrium Problems
Ngày đăng: 21/06/2014, 05:20
Xem thêm: Báo cáo hóa học: " Research Article The Shrinking Projection Method for Common Solutions of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Strictly Pseudocontractive Mappings" ppt, Báo cáo hóa học: " Research Article The Shrinking Projection Method for Common Solutions of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Strictly Pseudocontractive Mappings" ppt
