روش ترکیبی شیب اضافی برای مشکلات تعادل عمومی و مشکلات نقطه ثابت در فضای هیلبرت

In this paper, we introduce an iterative scheme by the hybrid methods for finding a common element of the set of fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of a variational inequality problem in a Hilbert space. Then, we prove the strongly convergent theorem by a hybrid extragradient method to the common element of the set of fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of a variational inequality problem. Our results extend and improve the results of Bnouhachem et al. [A. Bnouhachem, M. Aslam Noor, Z. Hao, Some new extragradient iterative methods for variational inequalities, Nonlinear Analysis (2008) doi:10.1016/j.na.2008.02.014] and many others.

Throughout this paper, we always assume that HH is a real Hilbert space with inner product 〈.,.〉〈.,.〉 and norm ‖.‖‖.‖, respectively, and CC is a nonempty closed convex subset of HH. Let FF be a bifunction of C×CC×C into RR, where RR are real numbers. The equilibrium problem for F:C×C→RF:C×C→R is to find x∈Cx∈C such that equation(1.1) View the MathML sourceF(x,y)≥0for all y∈C. Turn MathJax on The set of solutions of (1.1) is denoted by View the MathML sourceEP(F). Given a mapping T:C→HT:C→H, let F(x,y)=〈Tx,y−x〉F(x,y)=〈Tx,y−x〉 for all x,y∈Cx,y∈C. Then, View the MathML sourcez∈EP(F) if and only if 〈Tz,y−z〉≥0〈Tz,y−z〉≥0 for all y∈Cy∈C. Numerous problems in physics, optimization, and economics reduce to find a solution of (2.1) (see [1] and [2]). In 1997, Combettes and Hirstoaga [2] introduced an iterative scheme of finding the best approximation to the initial data when View the MathML sourceEP(F) is nonempty and proved a strong convergence theorem. Let A:C⟶HA:C⟶H be a mapping. The classical variational inequality, denoted by View the MathML sourceV I(C,A), is to find x∗∈Cx∗∈C such that 〈Ax∗,v−x∗〉≥0〈Ax∗,v−x∗〉≥0 Turn MathJax on for all v∈Cv∈C. The variational inequality has been extensively studied in the literature. See, e.g. [3] and the references therein. Let B:C→HB:C→H be a nonlinear mapping. Then, we consider the following generalized equilibrium problem: Find equation(1.2) View the MathML sourcez∈Csuch that F(z,y)+〈Bz,y−z〉≥0,∀y∈C. Turn MathJax on The set of solutions of (1.2) is denoted by View the MathML sourceGEP, i.e., View the MathML sourceGEP={z∈C:F(z,y)+〈Bz,y−z〉≥0,∀y∈C}. In the case of View the MathML sourceB≡0,GEP is denoted by View the MathML sourceEP(F). In the case of View the MathML sourceF≡0,GEP is also denoted by View the MathML sourceV I(C,A). A mapping AA of CC into HH is called αα-inverse-strongly monotone [4] if there exists a positive real number αα such that 〈Au−Av,u−v〉≥α‖Au−Av‖2〈Au−Av,u−v〉≥α‖Au−Av‖2 Turn MathJax on for all u,v∈Cu,v∈C. Recently, Takahashi and Toyoda [5], Yao et al. [6] and Plubtieng and Punpaeng [7] introduced an iterative method for finding an element of View the MathML sourceV I(C,A)∩F(S), where A:C→HA:C→H is an View the MathML sourceα-inverse-stronglymonotone mapping. We recall that, a mapping A:C⟶HA:C⟶H is said to be View the MathML sourcemonotone if View the MathML source〈Au−Av,u−v〉≥0,for all u,v∈C; Turn MathJax on AA is said to be View the MathML sourcek-Lipschitz continuous if there exists a positive real number kk such that View the MathML source‖Au−Av‖≤k‖u−v‖,for all u,v∈C; Turn MathJax on Remark 1.1. It is obvious that any αα-inverse-strongly monotone mapping AA is monotone and Lipschitz continuous. It is well known that if AA is a strongly monotone and Lipschitz continuous mapping on CC, then the variational inequality problem has a unique solution. How to actually find a solution of the variational inequality problem is one of the most important topics in the study of the variational inequality problem. The variational inequality has been extensively studied in the literature. See, e.g., [3] and [8] and the references therein. In 1976, Korpelevich [9] introduced the following so-called extragradient method: equation(1.3) View the MathML source{x0=x∈C,yn=PC(xn−λAxn),xn+1=PC(xn−λAyn) Turn MathJax on for all n≥0n≥0, where View the MathML sourceλ∈(0,1k),C is a nonempty closed convex subset of RnRn and AA is a monotone and kk-Lipschitz continuous mapping of CC into RnRn. He proved that if View the MathML sourceV I(C,A) is nonempty, then the sequences {xn}{xn} and View the MathML source{x̄n}, generated by (1.3), converge to the same point View the MathML sourcez∈V I(C,A). In 2003, Takahashi and Toyoda [5], introduced the following iterative scheme: equation(1.4) View the MathML source{x1=x∈Cchosen arbitrary ,xn+1=αnxn+(1−αn)SPC(xn−λnAxn),∀n≥1, Turn MathJax on where {αn}{αn} is a sequence in (0,1)(0,1), and {λn}{λn} is a sequence in (0,2α)(0,2α). They proved that if View the MathML sourceF(S)∩V I(C,A)≠0̸, then the sequence {xn}{xn} generated by (1.4) converges weakly to some View the MathML sourcez∈F(S)∩V I(C,A). Recently, Zeng and Yao [10] proved the following iterative scheme: equation(1.5) View the MathML source{x0=x∈C,yn=PC(xn−λnAxn),xn+1=αnx0+(1−αn)SPC(xn−λnAyn),∀n≥0, Turn MathJax on where {λn}{λn} and {αn}{αn} satisfy the following conditions: (i) λnk⊂(0,1−δ)λnk⊂(0,1−δ) for some δ∈(0,1)δ∈(0,1) and (ii) View the MathML sourceαn⊂(0,1),∑n=1∞αn=∞,limn⟶∞αn=0. They proved that the sequence {xn}{xn} and {yn}{yn} converges strongly to the same point View the MathML sourcePF(S)∩V I(C,A)x0 provided that limn⟶∞‖xn+1−xn‖=0limn⟶∞‖xn+1−xn‖=0. In 2007, Takahashi et al. [11] introduced the modified Mann iteration method for a family of nonexpansive mappings {Tn}{Tn}. Let x0∈Hx0∈H. For C1=CC1=C and u1=PC1x0u1=PC1x0, define a sequence {un}{un} of CC as follows: equation(1.6) View the MathML source{yn=αnun+(1−αn)Tnun,Cn+1={z∈Cn:‖yn−z‖≤‖un−z‖},un+1=PCn+1x0,n∈N, Turn MathJax on where 0≤αn≤a<10≤αn≤a<1 for all n∈Nn∈N. Then we prove that the sequence {un}{un} converges strongly to z0=PF(T)x0z0=PF(T)x0. In 2008, Bnouhachem et al. [12] introduced the following new extragradient iterative method for finding an element of View the MathML sourceF(S)∩V I(C,A). Let CC be a closed convex subset of a real Hilbert space H,AH,A be αα-inverse strongly monotone mapping of CC into HH and let SS be a nonexpansive mapping of CC into itself such that View the MathML sourceF(S)∩V I(C,A)≠0̸. Let the sequences {xn},{yn}{xn},{yn} be given by equation(1.7) View the MathML source{x1,u∈Cchosen arbitrary ,yn=PC(xn−λnAxn),xn+1=βnxn+(1−βn)S(αnu+(1−αn)PC(xn−λnAyn)),∀n≥1, Turn MathJax on where {αn},{βn},{λn}⊆(0,1){αn},{βn},{λn}⊆(0,1) satisfy some parameters controlling conditions. They proved that the sequence {xn}{xn} defined by (1.7) converges strongly to a common element of View the MathML sourceF(S)∩V I(C,A). In this paper, motivated and inspired by the results of Bnouhachem et al. [12] and Takahashi et al. [11], we introduce a new iterative scheme by the hybrid extragradient method, as follows: x0=x∈HC1=C,x1=PCx0x0=x∈HC1=C,x1=PCx0 and let equation(1.8) View the MathML source{un∈C,F(un,y)+〈Bxn,y−un〉+1rn〈y−un,un−xn〉≥0,∀y∈Cyn=PC(un−λnAun),zn=αnxn+(1−αn)S(βnxn+(1−βn)PC(un−λnAyn)),Cn+1={z∈Cn:‖zn−z‖≤‖xn−z‖},xn+1=PCn+1x0,n∈N, Turn MathJax on where {αn},{βn},{λn}⊆(0,1){αn},{βn},{λn}⊆(0,1) satisfy some parameters controlling conditions. We will prove that {xn}{xn} and {un}{un} in (1.8) strongly converge to a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for nonexpansive mappings in a Hilbert space. Our results extend and improved that the corresponding ones announced by Bnouhachem et al. [12].