• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Sensitivity Analysis for a New System of Parametric Generalized Mixed Implicity Equilibrium Problems in Banach Spaces

    2014-10-09 01:20:02DINGXieping

    DING Xieping

    (College of Mathematics and Software Science,Sichuan Normal University,Chengdu 610066,Sichuan)

    1 Introduction

    In recent years,much attention has been devoted to developing general methods for the sensitivity analysis of solution set of various variational inclusions and equilibrium problems.From the mathematical and engineering points of view,sensitivity properties of various variational inclusions and equilibrium problems can provide new insight concerning the problem being studied and can stimulate ideas for solving these problems.The sensitivity analysis of solution set for various parametric variational inclusions have been studied extensively by many authors using quite different methods,see [1-21] and the references therein.It is worth mentioning that most of the results in the direction have been obtained in the setting of Hilbert spaces.

    Recently,Kazmi and Khan[22]studied sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving P-η-accretive mappings and Ding[23]studied sensitivity analysis for a system of parametric generalized implicit quasi-variational-like inclusions involving H-η-monotone mappings in uniformly smooth Banach spaces respectively.By using the Yosida approximation and Wiener-Hopf equation technique,Moudafi[24]and Huang et al[25]studied the sensitivity analysis of solutions for generalized mixed implicit equilibrium problems in Hilbert spaces.By using the technique of the system of auxiliary equation problems,Ding[26]studied sensitivity analysis for a system of generalized mixed implicity equilibrium problems in uniformly smooth Banach spaces.

    Inspired and motivated by the above works,we shall introduce and study a new system of parametric generalized mixed implicit equilibrium problems involving non-monotone set-valued mappings in real Banach spaces.First,an auxiliary mixed equilibrium problem(AMEP) is introduced.The existence and uniqueness of solutions of the AMEP is proved under quite mild assumptions without any coercive conditions.Next,by using the solution mapping of the AMEP,a system of parametric generalized equation problems(SPGEP) is considered and its equivalence with the SPGMIEP is also proved.By using a fixed point formulation of the SPGEP,we study the behavior and sensitivity analysis of solution set of the SPGMIEP.Under suitable assumptions,we prove that the solution set of the SPGMIEP is nonempty,closed and Lipschitz continuous with respect to the parameters.Our results are new,which improve and generalize some known results in this field.

    2 Preliminaries

    Let B be a real Banach space with dual space B*and let‖·‖denote the norm of B and B*and〈· ,· 〉 denote the duality pairing between B*and B.Let R=(-∞,+∞)and C(B)be the family of all nonempty compact subsets of B.

    Definition 2.1Let C be a closed convex subset of a Hausdorff topological vector space E.A real valued bifunction F:C×C→(-∞,∞)is said to be

    (i) monotone if

    (ii) α-strongly monotone if there exists a real α>0 such that

    (iii) δ-Lipschitz continuous if there exists a real δ>0 such that

    Remark 2.1Clearly,strong monotonicity of F implies monotonicity of F.

    Definition 2.2A mapping η:B ×B→B*is said to be

    (i)monotone if

    (ii)δ-strongly monotone if there exists a δ >0 such that

    (iii) τ-Lipschitz continuous if there exists a constant τ>0 such that

    (iv) affine in first argument if

    Definition 2.3The bifunction φ:B ×B→(-∞,+∞]is said to be skew-symmetric if

    The skew-symmetric bifunctions have the properties which can be considered an analogue of monotonicity of gradient and nonnegativity of a second derivative for the convex function.For the properties and applications of the skew-symmetric bifunction,the reader may consult Antipin[27].

    The following result is a direct consequence of Theorem 1 of Ding and Tan[28](also see Lemma 2.2 of Ding[29]).

    Lemma 2.1Let C be a nonempty convex subset of a topological vector space and let f:C×C→[-∞,+∞]be such that

    (i) f(x,x)≥0 for each x∈C;

    (ii) for each y∈C,x→f(x,y) is upper semicontinuous;

    (iii) for each x∈C,y→f(x,y) is convex;

    (iv) there exist a nonempty compact subset K of C and y∈K such that f(x,y) <0,?x∈C\K.

    Then there exists a∈K such that f(,y)≥0 for all y∈C.

    Lemma 2.2Let C be a closed convex subset of a reflexive Banach space B with intC≠?.Let F:C×C→R and φ:B ×B→R be two bifunctions, η:B ×B→B*be a mapping and ρ >0 be a positive number.Suppose the following conditions are satisfied:

    (i) F is monotone and δ-Lipschitz continuous such that F(x,x)≥0 for each x∈C;

    (ii) for each y∈C,x→F(x,y) is upper semicontinuous under weak topology and for each x∈C,y→F(x,y) is convex;

    (iii) ηis σ-strongly monotone and τ-Lipschitz continuous with η(x,y) +η(y,x) =0,?x,y∈B;

    (iv) ηis affine in first argument and continuous from weak topology in B to weak*topology in B*in second argument;

    (v) φis skew symmetric and weakly continuous,and φis proper convex in the first argument.

    Then for each x∈B,the following auxiliary mixed equilibrium problem (AMEP):find z∈C such that

    ProofFor each x∈B,define a bifunction f:C×C→R by

    Since F(z,z)≥0 for each z∈C and η(x,y) +η(y,x) =0,?x,y∈B,we have η(z,z) =0 for all z∈B,by the definition of f,we have that f(z,z)≥0,?z∈C.The condition (i) of Lemma 2.1 is satisfied.Note that for each y∈C,x→F(x,y) is upper semicontinuous under weak topology,φis weakly continuous,and ηis continuous from weak topology in B to weak*topology in B*in second argument,we have for each y∈C,z→f(z,y) is weakly upper semicontinuous and so the condition (ii) of Lemma 2.2 is satisfied.Since for each z∈C,y→F(z,y) is convex and φis convex in first argument,and ηis affine in first argument,we have that for each z∈C,y→f(z,y) is convex.The condition (iii) of Lemma 2.1 is satisfied.By (v),for each y∈C,z→φ(z,y) is proper convex,weakly continuous and int{y∈C:φ(y,y) <∞} =intC≠?.Take y*∈int{y∈C:φ(y,y) <∞}.By Proposition I.2.6 of Pascali and Sburlan[30], φ(· ,y*) is subdifferential at y*.Hence,we have

    Noting that φ(·,· ) is skew symmetric,we have

    Since F is δ-Lipschitz continuous,and ηis σ-strongly monotone and τ-Lipschitz continuous,we have

    Let R =[ρ(δ+‖r‖) +τ‖y*-x‖]/σand K ={z∈C:‖z-y*‖≤R}.Then K is a weakly compact convex subset in C and y*∈K.It follows from (2)that f(z,y*) <0 for all z∈C\K and hence the condition (iv) of Lemma 2.1 is satisfied.For each x∈B,by Lemma 2.1,there exists a point∈C such that f(,y)≥0,?y∈C.By the definition of f,we obtain that for each x∈B,

    i.e.,∈C is a solution of the AMEP(1).Now,we prove the uniqueness of solutions of the AMEP (1).For each x∈B,let z1,z2∈C be any two solutions of the AMEP,then from the condition (i) we have

    Noting η(z1,z2)+η(z2,z1) =0,taking y=z2in (3)and y=z1in(4) and adding these two inequalities,we obtain

    Assume z1≠z2.Noting that F is monotone,φis skew symmetric and ηis σ-strongly monotone,it follows from (5) that

    which is a contraction.Therefore,we must have z1=z2.This completes the proof.

    Remark 2.21)Lemma 2.2 improves Lemma 2.2 of Ding and Ho[21]in the following way:the coercive conditions of Lemma 2.2 in [21] is removed.

    2)By Lemma 2.2,we also obtain that for each x∈B,there exists a uniquesuch that

    and hence the solution mapping:X → C of the AMEP (1) is a well-defined single-valued mapping.

    Theorem 2.1In the assumptions of Lemma 2.2,if further assume that F is θ-strongly monotone,then the solution mapping:X →C of the AMEP

    (1) is τ/(σ+ρθ)-Lipschitz continuous.

    Remark 2.3Theorems 2.1 improves and generalizes Theorem 3.1 of Kazmi and Khan[31]and Theorem 2.1 of Ding and Ho[21]in following way:

    1)from Hilbert spaces to reflexive Banach spaces;

    2) the AMEP (1) is more general than the models in [21,31-33];

    3)the coercive conditions is removed.

    3 System of parametric generalized mixed implicit equilibrium problems

    In what follows,unless other specified,let R=( -∞,+∞).For each i∈{1,2},let Bibe a real reflexive Banach space with norm‖·‖iand the dual spacebe the dual pair betweenand Bi, Λiand Ωibe two open subsets of Biin which parameters λiand ωitakes the values,C(Bi) denotes the family of all nonempty compact subsets of Bi,and(· ,· ) be the Hausdorff metric on C(Bi) defined by

    For each i∈{1,2},let Cibe a nonempty closed convex subset of Biwith intCi≠?,Fi:Ci×Ci×Λi→R and φi:Ci×Ci→R be functions.For each i∈{1,2},let gi:Ci×Λi→Ciwith gi(Ci,λi) =Ci,?λi∈Λi, ηi:Bi×Bi×Ωi→and Mi:C1×C2×B1×B2×Ω1×Ω2→Bibe single-valued mappings,and Ti:C1×Ω1→C(B1) and Si:C2×Ω2→C(B2) be set-valued mappings.

    We consider the following system of parametric generalized mixed implicit equilibrium problems(SPGMIEP):for i∈{1,2} and given (λi,ωi)∈Λi×Ωi,find (x1,x2)∈C1×C2, (u1,v1)∈T1(x1,ω1)×S1(x2,ω2), (u2,v2)∈T2(x1,ω1) ×S2(x2,ω2)such that

    Special cases:

    (I) If for i∈{1,2},Let M1(x1,x2,u1,v1,ω1,ω2) =G1(x1,x2,ω1) +N1(u1,v1,ω1)and M2(x1,x2,u2,v2,ω1,ω2) =G2(x1,x2,ω2) +N2(u2,v2,ω2),where Gi:C1×C2×Ωi→Biand Ni:B1×B2×Ωi→Biand φi≡0,then the SPGMIEP (1) reduces to the following parametric problem:for given (λi,ωi)∈Λi×Ωi,find (x1,x2)∈C1×C2, (u1,v1)∈T1(x1,ω1) ×S1(x2,ω2), (u2,v2)∈T2(x1,ω1) ×S2(x2,ω2) such that

    (II) If for i∈{1,2},let Bi=B,Ci=C, Λi=Λ, Ωi=Ω,Fi=F,Mi=M,Ti=T,Si=S, ηi=η,gi=g and φi=φ,then the SPGMIEP (1) reduces to the following parametric problem:for given (λ,ω)∈Λ×Ω,find x∈C, (u,v)∈T(x,ω) ×S(x,ω) such that

    The problems(2) and (3) include many(parametric)generalized mixed equilibrium problems as special cases,for examples,see [21-26,31-33]and the references therein.

    Now,for i∈{1,2} and fixed (λi,ωi)∈Λi×Ωi,we assume that Fi(· ,· ,λi), η(· ,· ,ωi)and φisatisfy all conditions of Lemma 2.2.Related to SPGMIEP (1),we consider the following system of parametric equation problems (SPEP):find (x1,x2)∈C1×C2, (u1,v1)∈T1(x1,ω1) ×S1(x2,ω2),(u2,v2)∈T2(x1,ω1) ×S2(x2,ω2),such that

    Lemma 3.1For fixed (λi,ωi)∈Λi×Ωi,(x1,x2,u1,v1,u2,v2) with (x1,x2)∈C1×C2, (u1,v1)∈T1(x1,ω1) ×S1(x2,ω2), (u2,v2)∈T2(x1,ω1)×S2(x2,ω2) is a solution of the SPEP (4) if and only if(x1,x2,u1,v1,u2,v2) with (x1,x2)∈C1×C2, (u1,v1)∈T1(x1,ω1) ×S1(x2,ω2), (u2,v2)∈T2(x1,ω1)×S2(x2,ω2) is a solution of the SGMIEP (1).

    ProofFor fixed (λi,ωi)∈Λi×Ωi,if(x1,x2,u1,v1,u2,v2)with (x1,x2)∈C1×C2, (u1,v1)∈T1(x1,ω1) ×S1(x2,ω2), (u2,v2)∈T2(x1,ω1) ×S2(x2,ω2) is a solution of the SPEP (4),then we have

    Hence (x1,x2,u1,v1,u2,v2) with (x1,x2)∈C1×C2,(u1,v1)∈T1(x1) ×S1(x2), (u2,v2)∈T2(x1) ×S2(x2) is a solution of the SGMIEP (1).

    Conversely,for fixed (λi,ωi)∈Λi×Ωi,if(x1,x2,u1,v1,u2,v2) with (x1,x2)∈C1×C2, (u1,v1)∈T1(x1,ω1) ×S1(x2,ω2), (u2,v2)∈T2(x1,ω1) ×S2(x2,ω2) is a solution of the SGMIEP (1),then the system of inequalities (6) holds.For ρ1,ρ2>0,it follows from (6) that

    Let z1=g1(x1,λ1) -ρ1M1(x1,x2,u1,v1,ω1)∈B1and z2=g2(x2,λ2) -ρ2M2(x1,x2,u2,v2,ω2)∈B2,then we have

    Remark 3.1Lemma 3.1 improves Lemma 3.1 of Ding and Ho[21]and generalizes Lemma 2.3 of Huang et al[25]and Lemma 3.1 of Kazmi and Khan[31]in the following ways:

    1)from Hilbert spaces to Reflexive Banach spaces;

    2)from a generalized mixed equilibrium problem to the more general system of generalized mixed implicit equilibrium problems.

    Now,by Lemma 3.1,for each i∈{1,2} and given (λi,ωi)∈Λi×Ωi,we can define the solution set S(λ1,λ2,ω1,ω2) of the SPGMIEP (1) as follows:

    For i∈{1,2},we define the mappings Φ1:C1×C2×Λ1×Ω1×Ω2→2C1and Φ2:C1×C2×Λ2×Ω1×Ω2→2C2as follows:

    Again define a mapping Ψ:C1×C2×Λ1×Λ2×Ω1×Ω2→2B1×B2as follows:

    Lemma 3.2For i∈{1,2} and given (λi,ωi)∈Λi×Ωi, (x1,x2) is a fixed point of Ψif and only if(x1,x2)∈S(λ1,λ2,ω1,ω2),i.e,there exist(u1,v1)∈T1(x1,ω1) ×S1(x2,ω2), (u2,v2)∈T2(x1,ω1) ×S2(x2,ω2) such that(x1,x2,u1,u2,v1,v2) is a solution of the SPGMIEP (1).

    ProofFor each fixed (λi,ωi)∈Λi×Ωi,by the definition of Ψ, (x1,x2)∈C1×C2is a fixed point of Ψif and only if there exist(u1,v1)∈T1(x1,ω1) ×S1(x2,ω2) and (u2,v2)∈T2(x1,ω1) ×S2(x2,ω2)such that

    By Lemma 3.1,the relation (14) holds if and only if(x1,x2,u1,u2,v1,v2)is a solution of the SPGMIEP(1).This completes the proof.

    Definition 3.1For i=1,2,Mi:C1×C2×B1×B2×Ω1×Ω2→Biis said to be (m(i,1),m(i,2),m(i,3),m(i,4),m(i,5),m(i,6))-mixed Lipschitz continuous in first to six arguments,if there exist constants m(i,1),m(i,2),m(i,3),m(i,4),m(i,5),m(i,6)>0 such that

    Definition 3.2For i∈{1,2},let Ti:B1×Ω1→C(B1) be a set-valued mapping.

    (i) Tiis said to be-μi-Lipschitz continuous in first argument,if there exists a constant μi>0 such that

    (ii) Tiis said to be-ti-Lipschitz continuous in second argument,if there exists a constant ti>0 such that

    Similarly,we can define the Lipschitz continuity of the mappings Si:C2×Ω2→C(B2).

    The modulus of smoothness of a Banach space B is the function ρB:[0,∞)→[0,∞) defined by

    Definition 3.3Let K be a closed convex subset of a Banach space B.A mapping g:K→K is said to be γ-strongly accretive if,for any x,y∈K,there exist j(x-y)∈J(x-y) and a constant γ>0 such that

    where J:B→2B*is the normalized duality mapping defined by

    Lemma 3.3[34]Let B be a uniformly smooth Banach space and J be the normalized duality mapping from B into B*.Then,for all x,y∈B,we have

    Lemma 3.4[35]Let(X,d) be a complete metric space and T1,T2:X→C(X) be two set-valued contractive mappings with same contractive constants θ∈(0,1),i.e.,

    Theorem 3.1For each i∈{1,2},let Cibe a nonempty closed convex subset of a uniformly smooth Banach space Biwith ρBi(t)≤Dit2for some Di>0.For fixed (λi,ωi)∈Λi×Ωi,Fi:Ci×Ci×Λi→R,ηi:Bi×Bi×Ωi→Biand φi:Ci×Ci→R satisfy all conditions of Theorem 2.1 where F, θ, δ, η, τ, σ and φare replaced by Fi, θi, δi, ηi, τi, σiand φi.Let Mi:C1×C2×B1×B2×Ω1×Ω2→Bibe(m(i,1),m(i,2),m(i,3),m(i,4))-mixed Lipschitz continuous in first to fourth arguments,Ti:B1×Ω1→C(B1) be-μi-Lipschitz continuous in first argument,Si:B2×Ω2→CB(B2) be-si-Lipschitz continuous in first argument and gi:Ki×Λi→Kibe γi-strongly accretive and βi-lipschitz continuous in first argument.If the following conditions hold for ρ1,ρ2>0:

    Then for any (x1,x2,λ1,λ2,ω1,ω2)∈C1×C2×Λ1×Λ2×Ω1×Ω2, Ψ(x1,x2,λ1,λ2,ω1,ω2)∈C(B1×B2) and Ψis a uniform*-contraction mapping with respect to (λi,ωi)∈Λi×Ωi,i=1,2,where*is a Hausdorff metric on C(B1×B2).Moreover,for each fixed (λi,ωi)∈Λi×Ωi,i=1,2,the solution set S(λ1,λ2,ω1,ω2) of the SPGMIEP (1) is nonempty closed.

    ProofLet(x1,x2,λ1,λ2,ω1,ω2) be an arbitrary element in C1×C2×Λ1×Λ2×Ω1×Ω2.Since for i=1,2,Tiand Siare all compact-valued,and,giand Miare all continuous.By the definitions of Φ1and Φ2,we have Φ1(x1,x2,λ1,ω1,ω2)∈C(B1) and Φ2(x1,x2,λ2,ω1,ω2)∈C(B2)and hence Ψ(x1,x2,λ1,λ2,ω1,ω2)∈C(B1×B2).For any fixed(x1,x2,λ1,λ2,ω1,ω2),(1,2,λ1,λ2,ω1,ω2)∈C1×C2×Λ1×Λ2×Ω1×Ω2and for each(a1,a2)∈Ψ(x1,x2,λ1,λ2,ω1,ω2),by definitions of Ψ and Φi,i∈{1,2},we have that there exist(u1,v1)∈T1(x1,ω1) ×S1(x2,ω2) and (u2,v2)∈T2(x1,ω1) ×S2(x2,ω2) such that

    This shows that the solution set S(λ1,λ2,ω1,ω2)) of the SPGIQVLI(1) is a*-Lipschitz continuous mapping from Λ1×Λ2×Ω1×Ω2into B1×B2.This completes the proof.

    [1] Dafermos S.Sensitivity analysis in variational inequalities[J].Math Oper Res,1988,13:421-434.

    [2] Mukherjee R N,Verma H L.Sensitivity analysis of generalized variational inequalities[J].J Math Anal Appl,1992,167:299-304.

    [3] Noor M A.Sensitivity analysis for quasi-variational inequalities[J].J Optim Theory Appl,1997,95:399-407.

    [4] Yen N D.Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint [J].Math Oper Res,1995,20:695-708.

    [5] Verma R U.Sensitivity analysis for generalized strongly monotone variational inclusions based on the (A,η)-resolvent operator technique[J].Appl Math Lett,2006,19:1409-1413.

    [6] Robinson S M.Sensitivity analysis for variational inequalities by normal-map technique[C]//Giannessi F,Maugeri A.Variational Inequalities and Network Equilibrium Problems.New York:Plenum Press,1995.

    [7] Adly S.Perturbed algorithms and sensitivity analysis for a general class of variational inclusions [J].J Math Anal Appl,1996,201:609-630.

    [8] Noor M A,Noor K I.Sensitivity analysis for quasi-variational inclusions[J].J Math Anal Appl,1999,236:290-299.

    [9] Agarwal R P,Cho Y J,Huang N J.Sensitivity analysis for strongly nonlinear quasi-variational inclusions[J].Appl Math Lett,2000,13(6):19-24.

    [10] Ding X P,Lou C L.On parametric generalized quasivariational inequalities[J].J Optim Theory Appl,1999,100(1):195-205.

    [11] Liu Z,Debnath L,Kang S M,et al.Sensitivity analysis for parametric completely generalized nonlinear implicit quasivariational inclusions[J].J Math Anal Appl,2003,277:142-154.

    [12] Salahuddin.Parametric generalized set-valued variational inclusions and resolvent equations[J].J Math Anal Appl,2004,198:146-156.

    [13] Park J Y,Jeong J U.Parametric generalized mixed variational inequalities[J].Appl Math Lett,2004,17:43-48.

    [14] Ding X P.Sensitivity analysis for generalized nonlinear implicit quasi-variational inclusions[J].Appl Math Lett,2004,17:225-235.

    [15] Ding X P.Sensitivity analysis of solution set for a new class of generalized implicit quasi-variational inclusions[J].Fixed Point Theory and Applications,2006,7:81-94.

    [16] Ding X P.Parametric completely generalized nonlinear implicit quasi-variational inclusions involving h-maximal monotone mappings[J].J Comput Appl Math,2005,182(2):252-289.

    [17] Peng J W,Long X L.Sensitivity analysis for parametric completely generalized strongly nonlinear implicit quasi-variational inclusions[J].Comput Math Appl,2005,50:869-880.

    [18] Agarwal R P,Huang N J,Tan M Y.Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions[J].Appl Math Lett,2004,17:345-352.

    [19] Ding X P,Yao J C.Sensitivity analysis for a system of parametric mixed quasi-variational inclusions[J].J Nonlinear Convex A-nal,2007,8(2):211-225.

    [20] Ding X P,Wang Z B.Sensitivity analysis for a system of parametric generalized mixed quasi-variational inclusions involving (K,η)-monotone mappings[J].Appl Math Comput,2009,214:318-327.

    [21] Ding X P,Ho J L.New Iterative algorithm for solving a system of generalized mixed implicit equilibrium problems in Banach spaces[J].Taiwan J Math.2011,15(2):673-695.

    [22]Kazmi K R,Khan E A.Sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving P-η-accretive mappings[J].J Math Anal Appl.2008,337:1198-1211.

    [23] Ding X P.System of parametric generalized implicit quasi-variational-like inclusions involving H-η-monotone operators in Banach spaces[J].J Sichuan Normal Univ:Natural Sci,2010,33(6):1-11.

    [24] Moudafi A.Mixed equilibrium problems:sensitivity analysis and algorithmic aspects[J].Comput Math Appl,2002,44:1099-1108.

    [25]Huang N J,Lan H Y,Cho Y J.Sensitivity analysis for nonlinear generalized mixed implicit equilibrium problems with non-monotone set-valued mappings[J].J Comput Appl Math,2006,196:608-618.

    [26]Ding X P.Sensitivity analysis for a system of generalized mixed implicit equilibrium problems in uniformly smooth Banach spaces[J].Nonlinear Anal,2010,73:1264-1276.

    [27] Antipin A S.Iterative gradient prediction-type methods for computing fixed-point of extremal mappings[C]//Guddat J,Jonden H Th,Nizicka F,et al.Parametric Optimization and Related Topics IV.Frankfurt Main:Peter Lang,1997:11-24.

    [28] Ding X P,Tan K K.A minimax inequality with applications to existence of equilibrium point and fixed point theorems [J].Colloq Math,1992,63:233-247.

    [29] Ding X P.Existence and algorithm of solutions for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces[J].Acta Math Sinica,2012,28(3):503-514.

    [30] Pascali D,Surian S.Nonlinear Mappings of Monotone Type[M].Alphen aan den Rijn:Sijthoff and Noordhoff International Publishers,1978.

    [31] Kazmi K R,Khan F A.Existence and iterative approximation of solutions of generalized mixed equilibrium problems [J].Comput Math Appl,2008,56:1314-1321.

    [32] Ding X P.Existence and algorithm of solutions for a system of generalized mixed implicit equilibrium problems in Banach spaces[J].Appl Math Mech,2010,31(9):1049-1062.

    [33] Ding X P.Auxiliary principle and approximation solvability for system of new generalized mixed equilibrium problems in reflexive Banach spaces[J].Appl Math Mech,2011,32(2):231-240.

    [34]Petryshyn W V.A characterization of strictly convexity of Banach spaces and other uses of duality mappings [J].J Funct Anal,1970,6:282-291.

    [35] Lim T C.On fixed point stability for set-valued contractive mappings with application to generalized differential equations [J].J Math Anal Appl,1985,110:436-441.

    [36] Nadler S B.Multivalued contraction mapping[J].Pacific J Math,1969,30:475-488.

    天美传媒精品一区二区| 床上黄色一级片| 日本色播在线视频| 国产 一区精品| 日日啪夜夜撸| 午夜福利高清视频| 精品国产一区二区三区久久久樱花 | 五月伊人婷婷丁香| 午夜福利在线观看免费完整高清在| 亚洲av中文字字幕乱码综合| 亚洲精华国产精华液的使用体验| 国产色爽女视频免费观看| 亚洲天堂国产精品一区在线| 成人亚洲精品av一区二区| 亚洲人成网站在线播| 观看免费一级毛片| 日韩欧美在线乱码| 亚洲欧美成人精品一区二区| 国产免费男女视频| 欧美日韩综合久久久久久| 午夜福利在线观看免费完整高清在| 国产极品精品免费视频能看的| 国产成人精品久久久久久| 亚洲精品aⅴ在线观看| 国产 一区精品| 亚洲国产精品专区欧美| 丰满人妻一区二区三区视频av| 91av网一区二区| 免费无遮挡裸体视频| 免费av观看视频| 亚洲成av人片在线播放无| 日日摸夜夜添夜夜添av毛片| 又黄又爽又刺激的免费视频.| 韩国高清视频一区二区三区| 韩国av在线不卡| 国产淫片久久久久久久久| 一级毛片我不卡| 嫩草影院入口| 简卡轻食公司| 成年av动漫网址| 夜夜爽夜夜爽视频| 国产精品不卡视频一区二区| 三级经典国产精品| 亚洲中文字幕日韩| 久久99精品国语久久久| 男女下面进入的视频免费午夜| 国产成人91sexporn| 欧美性猛交黑人性爽| 国产免费视频播放在线视频 | 国产精品久久久久久精品电影| 亚洲成人中文字幕在线播放| 亚洲欧美成人综合另类久久久 | 麻豆乱淫一区二区| 成人二区视频| 免费黄网站久久成人精品| 亚洲av成人精品一区久久| 精品国内亚洲2022精品成人| 秋霞伦理黄片| 成人午夜精彩视频在线观看| 国产伦理片在线播放av一区| 精品国产露脸久久av麻豆 | 精品久久久久久久久久久久久| 欧美一级a爱片免费观看看| 午夜视频国产福利| 久久久久久久久久久免费av| 边亲边吃奶的免费视频| 日本免费一区二区三区高清不卡| 麻豆久久精品国产亚洲av| 亚洲精品久久久久久婷婷小说 | 国产黄a三级三级三级人| 长腿黑丝高跟| 最近视频中文字幕2019在线8| 岛国在线免费视频观看| 能在线免费看毛片的网站| 欧美xxxx性猛交bbbb| 干丝袜人妻中文字幕| 国产精品伦人一区二区| 国产精品蜜桃在线观看| 久久久久久久久久久丰满| 99热全是精品| 亚洲不卡免费看| 精品人妻一区二区三区麻豆| 亚洲人成网站在线观看播放| 国产成人午夜福利电影在线观看| 免费av观看视频| 色网站视频免费| 亚洲精品久久久久久婷婷小说 | 精品久久久久久久末码| 青春草国产在线视频| 国产成人a区在线观看| 成年版毛片免费区| 午夜日本视频在线| 久久久久精品久久久久真实原创| 国产一区二区在线av高清观看| or卡值多少钱| 国模一区二区三区四区视频| 亚洲欧美精品综合久久99| 十八禁国产超污无遮挡网站| 不卡视频在线观看欧美| 免费av观看视频| 小蜜桃在线观看免费完整版高清| 精品酒店卫生间| 我的老师免费观看完整版| 日日啪夜夜撸| 少妇丰满av| 91aial.com中文字幕在线观看| 亚洲av.av天堂| a级毛片免费高清观看在线播放| 一夜夜www| 丰满少妇做爰视频| 毛片一级片免费看久久久久| 欧美xxxx黑人xx丫x性爽| 禁无遮挡网站| 少妇人妻精品综合一区二区| 免费看av在线观看网站| 日本黄色片子视频| 国产在线一区二区三区精 | 免费看光身美女| 夜夜看夜夜爽夜夜摸| 久久热精品热| 精华霜和精华液先用哪个| 在线观看美女被高潮喷水网站| 蜜桃亚洲精品一区二区三区| 看十八女毛片水多多多| 国产高清视频在线观看网站| 麻豆av噜噜一区二区三区| 成人高潮视频无遮挡免费网站| 久久韩国三级中文字幕| 别揉我奶头 嗯啊视频| 不卡视频在线观看欧美| 91久久精品电影网| 久久精品夜夜夜夜夜久久蜜豆| 成年女人永久免费观看视频| 久久精品久久精品一区二区三区| 能在线免费看毛片的网站| 亚洲aⅴ乱码一区二区在线播放| 国产在视频线精品| 亚洲欧美清纯卡通| 国产精品伦人一区二区| 大又大粗又爽又黄少妇毛片口| 国产在线男女| 亚洲国产日韩欧美精品在线观看| 波野结衣二区三区在线| 男人舔奶头视频| 五月伊人婷婷丁香| 国产男人的电影天堂91| 久久久久久大精品| 天美传媒精品一区二区| 天天躁日日操中文字幕| 简卡轻食公司| 高清毛片免费看| 一级毛片aaaaaa免费看小| 精品国产一区二区三区久久久樱花 | 久久久久久大精品| 精品久久久久久成人av| 亚洲精品乱码久久久久久按摩| 听说在线观看完整版免费高清| 中文字幕制服av| av福利片在线观看| 男女国产视频网站| 热99re8久久精品国产| 国产免费男女视频| 天堂√8在线中文| 午夜激情欧美在线| 精品国产一区二区三区久久久樱花 | 午夜精品在线福利| 国产亚洲精品av在线| www日本黄色视频网| 国产在视频线精品| 亚洲精华国产精华液的使用体验| 亚洲欧美成人综合另类久久久 | 成人三级黄色视频| 国产片特级美女逼逼视频| 亚洲欧美成人精品一区二区| 爱豆传媒免费全集在线观看| 桃色一区二区三区在线观看| 可以在线观看毛片的网站| 亚洲精品日韩在线中文字幕| 亚洲精品影视一区二区三区av| 国产在视频线在精品| 大话2 男鬼变身卡| 亚洲国产精品成人综合色| 十八禁国产超污无遮挡网站| 黄片无遮挡物在线观看| 亚洲欧美中文字幕日韩二区| 国语自产精品视频在线第100页| 亚洲av福利一区| 成年免费大片在线观看| 国产免费福利视频在线观看| 久久6这里有精品| 亚洲av成人精品一区久久| av在线观看视频网站免费| 天天躁日日操中文字幕| 青春草视频在线免费观看| 日本五十路高清| 性插视频无遮挡在线免费观看| 欧美一级a爱片免费观看看| 欧美日韩在线观看h| 一边摸一边抽搐一进一小说| 国产午夜精品论理片| 亚洲av.av天堂| 亚洲欧美一区二区三区国产| 插逼视频在线观看| 青青草视频在线视频观看| 天堂影院成人在线观看| 男的添女的下面高潮视频| 免费看美女性在线毛片视频| 久久久a久久爽久久v久久| 村上凉子中文字幕在线| 久久国内精品自在自线图片| 伊人久久精品亚洲午夜| 男女边吃奶边做爰视频| 99久久人妻综合| 欧美日韩综合久久久久久| 久久久精品欧美日韩精品| 亚洲图色成人| 亚洲自拍偷在线| 久久国内精品自在自线图片| 尾随美女入室| 黄片wwwwww| 亚洲国产精品成人久久小说| 人妻制服诱惑在线中文字幕| 国产亚洲91精品色在线| 久久精品国产鲁丝片午夜精品| 男女啪啪激烈高潮av片| 最近手机中文字幕大全| 岛国毛片在线播放| 国产精品女同一区二区软件| 秋霞伦理黄片| 五月玫瑰六月丁香| 日本三级黄在线观看| 亚洲精品乱码久久久久久按摩| 欧美成人一区二区免费高清观看| 内射极品少妇av片p| 亚洲伊人久久精品综合 | 边亲边吃奶的免费视频| 久久久久九九精品影院| 久久久久久久久久久免费av| 校园人妻丝袜中文字幕| 中文字幕av在线有码专区| 九九久久精品国产亚洲av麻豆| 久久人人爽人人片av| 国产精品久久久久久久电影| 天天躁日日操中文字幕| 日韩,欧美,国产一区二区三区 | 国产毛片a区久久久久| 99在线人妻在线中文字幕| 免费黄网站久久成人精品| 国产三级中文精品| 老司机影院成人| 黄色配什么色好看| 偷拍熟女少妇极品色| 国产免费又黄又爽又色| 菩萨蛮人人尽说江南好唐韦庄 | 日本免费一区二区三区高清不卡| 国内精品宾馆在线| 99视频精品全部免费 在线| 天美传媒精品一区二区| 亚洲欧美成人精品一区二区| 国产av不卡久久| 国产精品久久电影中文字幕| 深夜a级毛片| 国产真实乱freesex| 久久久久久久久久久免费av| 亚洲av一区综合| 国产欧美日韩精品一区二区| 亚州av有码| 中文字幕制服av| 2022亚洲国产成人精品| 成人性生交大片免费视频hd| 97超视频在线观看视频| 日本三级黄在线观看| 欧美日本亚洲视频在线播放| 久99久视频精品免费| 亚洲av二区三区四区| 我要搜黄色片| 日日摸夜夜添夜夜添av毛片| 亚洲精品国产成人久久av| 国产亚洲av片在线观看秒播厂 | 爱豆传媒免费全集在线观看| 精品人妻一区二区三区麻豆| 国产伦精品一区二区三区视频9| 欧美3d第一页| 蜜臀久久99精品久久宅男| 一本一本综合久久| 天堂中文最新版在线下载 | 黄片wwwwww| 久久精品国产亚洲网站| 国产老妇女一区| 日日摸夜夜添夜夜爱| 国产又色又爽无遮挡免| 亚洲欧洲日产国产| 国产探花在线观看一区二区| 免费不卡的大黄色大毛片视频在线观看 | 成年av动漫网址| 亚洲欧美精品专区久久| a级毛色黄片| 人妻夜夜爽99麻豆av| 亚洲国产最新在线播放| 国产黄片美女视频| 毛片一级片免费看久久久久| 亚洲在久久综合| 国产精品爽爽va在线观看网站| 成人综合一区亚洲| 亚洲av成人av| 精品国内亚洲2022精品成人| 国产av不卡久久| 国产 一区 欧美 日韩| 亚洲欧洲国产日韩| 久久久久久久国产电影| 26uuu在线亚洲综合色| 亚洲一区高清亚洲精品| 亚洲成人精品中文字幕电影| a级毛色黄片| 中文字幕av成人在线电影| 美女cb高潮喷水在线观看| 国产精品国产高清国产av| av在线天堂中文字幕| 99久久精品一区二区三区| 丰满人妻一区二区三区视频av| 亚洲图色成人| 久久久成人免费电影| 久久久久久久国产电影| 日本色播在线视频| 18禁动态无遮挡网站| 国产精品无大码| 国产中年淑女户外野战色| 欧美一区二区精品小视频在线| 亚洲四区av| 全区人妻精品视频| 一本久久精品| 一级黄色大片毛片| 最后的刺客免费高清国语| 网址你懂的国产日韩在线| 午夜亚洲福利在线播放| 少妇熟女aⅴ在线视频| 久久综合国产亚洲精品| 欧美日本视频| 国产精品三级大全| 国产探花在线观看一区二区| 亚洲国产欧美人成| a级一级毛片免费在线观看| 久久精品影院6| 男女那种视频在线观看| 亚洲18禁久久av| 中文在线观看免费www的网站| 国内揄拍国产精品人妻在线| 99久久人妻综合| 亚洲欧美成人综合另类久久久 | 国产av不卡久久| 午夜精品一区二区三区免费看| 七月丁香在线播放| 亚洲av成人精品一区久久| 国产精品国产三级国产av玫瑰| 精品人妻一区二区三区麻豆| 免费av毛片视频| 九九爱精品视频在线观看| 欧美日韩精品成人综合77777| 欧美3d第一页| 黄片wwwwww| 亚洲av不卡在线观看| 日本色播在线视频| 蜜桃久久精品国产亚洲av| 少妇被粗大猛烈的视频| 成年av动漫网址| 日韩中字成人| 国产精品久久电影中文字幕| 色综合亚洲欧美另类图片| 久久精品国产亚洲av天美| 色噜噜av男人的天堂激情| 中文字幕免费在线视频6| 色噜噜av男人的天堂激情| 亚洲av电影不卡..在线观看| www.色视频.com| 久久精品国产鲁丝片午夜精品| 九九久久精品国产亚洲av麻豆| 日本三级黄在线观看| 九九久久精品国产亚洲av麻豆| 99久久人妻综合| www.色视频.com| 中文字幕人妻熟人妻熟丝袜美| 精品久久久久久电影网 | 成人特级av手机在线观看| 国产在线男女| 我要搜黄色片| 伦理电影大哥的女人| 毛片一级片免费看久久久久| 国产av码专区亚洲av| 国产在视频线精品| 日韩欧美 国产精品| 国产人妻一区二区三区在| ponron亚洲| 能在线免费观看的黄片| av在线观看视频网站免费| 啦啦啦啦在线视频资源| 26uuu在线亚洲综合色| 国产精品久久电影中文字幕| 我的女老师完整版在线观看| 天堂网av新在线| 长腿黑丝高跟| 国产精品人妻久久久久久| 国产高清国产精品国产三级 | 丝袜美腿在线中文| 美女脱内裤让男人舔精品视频| 丰满少妇做爰视频| 插逼视频在线观看| 亚洲av免费高清在线观看| 纵有疾风起免费观看全集完整版 | 中文字幕亚洲精品专区| 亚洲伊人久久精品综合 | 亚洲自拍偷在线| 亚洲欧美精品综合久久99| 最新中文字幕久久久久| 亚洲欧美精品综合久久99| 精品国产三级普通话版| 国产大屁股一区二区在线视频| 日韩三级伦理在线观看| 精品久久久久久久人妻蜜臀av| 我要看日韩黄色一级片| www.av在线官网国产| 亚洲精品影视一区二区三区av| 九九在线视频观看精品| 国产精品一区www在线观看| 神马国产精品三级电影在线观看| 婷婷色av中文字幕| 99久久精品热视频| 中文字幕久久专区| 亚洲最大成人av| 免费人成在线观看视频色| 免费播放大片免费观看视频在线观看 | 毛片一级片免费看久久久久| 亚洲成av人片在线播放无| 久久人人爽人人爽人人片va| 亚洲图色成人| 精华霜和精华液先用哪个| 人妻少妇偷人精品九色| 精品不卡国产一区二区三区| 两个人视频免费观看高清| 国产色爽女视频免费观看| 久久99蜜桃精品久久| 美女cb高潮喷水在线观看| 国产精品人妻久久久久久| 一本—道久久a久久精品蜜桃钙片 精品乱码久久久久久99久播 | 91久久精品电影网| 成人午夜高清在线视频| 国产伦一二天堂av在线观看| 国产成人a∨麻豆精品| 老司机影院毛片| 99久久精品国产国产毛片| 午夜福利高清视频| 久久久国产成人精品二区| 久久这里只有精品中国| 久久久久久久亚洲中文字幕| 色综合站精品国产| 成人高潮视频无遮挡免费网站| 国模一区二区三区四区视频| 午夜福利高清视频| 国产精品伦人一区二区| 亚洲av男天堂| 国产淫语在线视频| 一级二级三级毛片免费看| 白带黄色成豆腐渣| 日韩亚洲欧美综合| 久久99蜜桃精品久久| 九九爱精品视频在线观看| 国产欧美另类精品又又久久亚洲欧美| 天堂av国产一区二区熟女人妻| 免费大片18禁| 夫妻性生交免费视频一级片| 久99久视频精品免费| 精品人妻视频免费看| 高清在线视频一区二区三区 | 中文字幕人妻熟人妻熟丝袜美| 少妇的逼好多水| 高清视频免费观看一区二区 | 老师上课跳d突然被开到最大视频| 国产黄a三级三级三级人| 高清视频免费观看一区二区 | 丝袜喷水一区| 噜噜噜噜噜久久久久久91| 精品酒店卫生间| 大香蕉97超碰在线| 欧美日本亚洲视频在线播放| av黄色大香蕉| 亚洲欧洲日产国产| 亚洲国产精品合色在线| 春色校园在线视频观看| 久久久亚洲精品成人影院| 国产v大片淫在线免费观看| 我的女老师完整版在线观看| 免费av观看视频| 少妇猛男粗大的猛烈进出视频 | 老司机影院毛片| 国产精品一区www在线观看| 国产 一区精品| 日韩欧美国产在线观看| 国产精品爽爽va在线观看网站| www.av在线官网国产| 99久国产av精品国产电影| 亚洲精品,欧美精品| 国产精品永久免费网站| 国产一区二区在线av高清观看| 国产成人freesex在线| 亚洲欧美清纯卡通| 国产精品福利在线免费观看| videos熟女内射| 日韩强制内射视频| 老司机影院毛片| 51国产日韩欧美| 免费播放大片免费观看视频在线观看 | 欧美三级亚洲精品| 麻豆av噜噜一区二区三区| 久久久久久伊人网av| 亚洲人成网站在线观看播放| 性插视频无遮挡在线免费观看| 午夜福利高清视频| 欧美高清性xxxxhd video| 又黄又爽又刺激的免费视频.| av.在线天堂| 精品久久久久久久久亚洲| 色视频www国产| 日产精品乱码卡一卡2卡三| 亚洲人与动物交配视频| 2022亚洲国产成人精品| 精品无人区乱码1区二区| 日本免费一区二区三区高清不卡| 久久久国产成人精品二区| 热99在线观看视频| 天堂网av新在线| 一级二级三级毛片免费看| 亚洲最大成人中文| 国内精品宾馆在线| 亚洲精品自拍成人| 欧美成人午夜免费资源| 国产成人精品婷婷| 亚洲,欧美,日韩| av国产久精品久网站免费入址| ponron亚洲| 99久国产av精品| 日日撸夜夜添| 国产精品人妻久久久久久| 天天躁日日操中文字幕| 久久热精品热| 直男gayav资源| 亚洲内射少妇av| 特级一级黄色大片| 久久国内精品自在自线图片| 黄片无遮挡物在线观看| 级片在线观看| 我要看日韩黄色一级片| 美女脱内裤让男人舔精品视频| 欧美xxxx性猛交bbbb| 26uuu在线亚洲综合色| 中文字幕久久专区| 最近中文字幕高清免费大全6| 天堂中文最新版在线下载 | 国产人妻一区二区三区在| 免费观看a级毛片全部| 国产又黄又爽又无遮挡在线| 中国国产av一级| 人体艺术视频欧美日本| 啦啦啦啦在线视频资源| 亚洲av福利一区| 欧美日韩国产亚洲二区| 亚洲欧美清纯卡通| 国产精品福利在线免费观看| 久久久a久久爽久久v久久| 久久久精品欧美日韩精品| 亚洲成人av在线免费| 青春草国产在线视频| 高清日韩中文字幕在线| 久久亚洲国产成人精品v| 一区二区三区乱码不卡18| 日产精品乱码卡一卡2卡三| 最近最新中文字幕大全电影3| 免费大片18禁| 亚洲电影在线观看av| 超碰97精品在线观看| 黑人高潮一二区| 不卡视频在线观看欧美| 久久久久网色| 欧美bdsm另类| 亚洲av中文字字幕乱码综合| 精品久久久久久久末码| 汤姆久久久久久久影院中文字幕 | 免费一级毛片在线播放高清视频| 久久韩国三级中文字幕| 男女啪啪激烈高潮av片| 在线播放国产精品三级| 我要看日韩黄色一级片| 国产免费福利视频在线观看| 欧美成人精品欧美一级黄| 99在线人妻在线中文字幕| 国产白丝娇喘喷水9色精品| 久久精品夜色国产| 美女国产视频在线观看| 色5月婷婷丁香| 亚洲av免费高清在线观看| 亚洲精品成人久久久久久| 一级毛片aaaaaa免费看小| 欧美成人a在线观看| 日本五十路高清| 国产成人a区在线观看| 日日干狠狠操夜夜爽| 中文字幕av成人在线电影| 国产av一区在线观看免费| 青青草视频在线视频观看| 国产极品精品免费视频能看的| 久热久热在线精品观看| 国产成人福利小说| 免费av不卡在线播放| 久久99热6这里只有精品| 五月玫瑰六月丁香| 色综合色国产| 99热6这里只有精品|