• <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 ?

    Some Random Coincidence Point and Common Fixed Point Results in Cone Metric Spaces Over Banach Algebras

    2020-01-07 06:24:22JIANGBinghuaCAIZelinCHENJinyang

    JIANG Bing-hua, CAI Ze-lin, CHEN Jin-yang

    (School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China)

    Abstract: In this paper, we obtain some tripled common random fixed point and tripled random fixed point theorems with several generalized Lipschitz constants in such spaces. We consider the obtained assertions without the assumption of normality of cones. The presented results generalize some coupled common fixed point theorems in the existing literature.

    Key words: Tripled random fixed point; Tripled random coincidence point; Cone metric space over Banach algebra; Generalized Lipschitz constant; Tripled common random fixed point

    §1. Introduction

    Fixed point theory plays a basic role in applications of many branches of mathematics.Finding the fixed point of contractive mapping becomes the center of strong research activity(see [3, 8-10, 19, 21, 30, 35]). In 2007 Huang and Zhang[15]introduced cone metric space and proved some fixed point theorems of contractive mappings in such spaces. Since then,some authors proved lots of fixed point theorems for contractive or expansive mappings in cone metric spaces that expanded certain fixed point results in metric spaces, (see[1, 2, 5, 11, 13, 20,26, 31]). Hassen[7]introduced tripled fixed point of w-compatible mappings in abstract metric spaces and coupled coincidence point and common coupled fixed point results in cone metric spaces. However, latterly, some authors made a conclusion that fixed point results in cone metric spaces are just equivalent to those in metric spaces (see [4, 6, 22, 32]). But fortunately,very recently, Liu and Xu[24]introduced the concept of cone metric space over Banach algebra and proved the non-equivalence of fixed point results in these new spaces and usual metric spaces. As a result, it is essentially necessary to investigate fixed point results in cone metric spaces over Banach algebras. Random coincidence point theorems are stochastic generalizations of classical coincidence point theorems, and play an important role in the theory of random differential and integral equations. Random fixed point theorems for contractive mapping on complete separable metric space have been proved by several authors (see [17, 23]). ′Ciri′c[12]and Zhu [37]proved some coupled random fixed point and coupled random coincidence results in partially ordered metric spaces. Afterwards, many coupled random coincidence results in partially ordered metric spaces were considered (see [16, 34]). In this paper, we obtain tripled common random fixed point and tripled random fixed point theorems with several generalized Lipschitz constants in cone metric spaces over Banach algebras by omitting the normality of cones. The presented results improve the main results of [7]in a large extent.

    §2. Preliminaries

    Let A be a Banach algebra with a unit e, and θ the zero element of A. A nonempty closed convex subset P of A is called a cone if

    (i) {θ,e}?P;

    (ii ) P2=PP ?P,P(?P)={θ};

    (iii) λP +μP ?P for all λ,μ≥0.

    On this basis, we define a partial orderingwith respect to P by xy if and only if y ?x ∈P. We shall write xy to indicate that xy but xy, while xwill indicate that y ?x ∈intP, where intP stands for the interior of P. Writeas the norm on A. A cone P is called normal if there is a number M >0 such that for all x,y ∈A,

    The least positive number satisfying above is called the normal constant of P.

    In the following we always suppose that A is a Banach algebra with a unit e. P is a cone in A with intP?, andis a partial ordering with respect to P.

    Definition 2.1[24]Let X be a nonempty set and A a Banach algebra. Suppose that the mapping d:X ×X →A satisfies:

    (i) θ ?d(x,y) for all x,y ∈X withand d(x,y)=θ if and only if x=y;

    (ii) d(x,y)=d(y,x) for all x,y ∈X;

    Then d is called a cone metric on X, and (X,d) is called a cone metric space over Banach algebra A.

    Example 2.2[24]Let A =MnR = {a = (aij)n×n|aij∈R} for all 1 ≤i,j ≤n be the algebra of all n-square real matrices, and define the norm

    Then A is a real Banach algebra with the unit e, the identity matrix. Let P ={a ∈≥0 for all 1 ≤i,j ≤n}. Then P ?A is a normal cone with a normal constant M = 1. Let X =MnR, and define the metric d=X ×X ?→A by

    Then (X,d) is a cone metric space with a Banach algebra A.

    Definition 2.3[36]Let (X,d) be a cone metric space over Banach algebra, x ∈X and{xn} a sequence in X. Then

    (i) {xn} converges to x whenever, for every c ∈E withthere is a natural number N such that d(xn,x)c for all n ≥N. We denote this by=x or xn→x(n →∞).

    (ii) {xn} is a Cauchy sequence whenever, for every c ∈E withthere is a natural number N such that d(xn,xm)c for all n,m ≥N.

    (iii) (X,d) is complete if every Cauchy sequence is convergent.

    The following lemmas are often used (in particular when dealing with cone metric spaces in which the cones need not be normal).

    Lemma 2.4[36]Let (X,d) be a cone metric space over Banach algebra A and P a cone in A. Then the following properties are often used.

    Lemma 2.5[29]Let A be a Banach algebra with a unit e, x ∈A, thenexists and the spectral radius ρ(x) satisfies

    If ρ(x)<|λ|, then λe ?x invertible in A, moreover,

    where λ is a complex constant.

    Lemma 2.6[29]Let A be a Banach algebra with a unit e,a,b ∈A. If a commutes with b,then

    Definition 2.7[33]An element (x,y,z) ∈X3is said to be a tripled fixed point of the mapping F :X3→X if F(x,y,z)=x,F(y,z,x)=y, and F(z,x,y)=z.

    Note that if (x,y,z) is a tripled fixed point of F, then (y,z,x) and (z,x,y) are tripled fixed points of F too.

    Definition 2.8[33]An element (x,y,z)∈X3is called

    (1)a tripled coincidence point of the mapping F :X3→X and g :X →X if F(x,y,z)=gx,F(y,z,x)=gy,F(z,x,y)=gz, and (gx,gy,gz) is called a tripled point of coincidence;

    (2) a common tripled fixed point of mapping F : X3→X and g : X →X if F(x,y,z) =

    gx=x, F(y,z,x)=gy =y and F(z,x,y)=gz =z.

    Definition 2.9[2]The mapping F :X3→X and g :X →X are called w-compatible provided that gF(x,y,z)=F(gx,gy,gz)whenever F(x,y,z)=gx,F(y,z,x)=gy and F(z,x,y)=gz.

    Let (?,Σ) be a measurable space with Σ a sigma algebra of subsets of ? and let (X,d)be a metric space. A mapping T : ? →X is called Σ-measurable if for any open subset U of X, T?1(U) = {ω : T(ω) ∈U} ∈Σ. In what follows, when we speak of measurability we shall mean Σ-measurability. A mapping T :?×X →X is called a random operator if for any x ∈X,T(·,x) is measurable. A measurable mapping ξ :? →X is called a random fixed point of a random operator T :?×X →X, if ξ(ω)=T(ω,ξ(ω)) for every ω ∈?.

    Definition 2.10[18]Let (X,d) be a separable metric space and (?,Σ) be a measurable space. Then F :?×X3→X and g :?×X →X are said to be w-compatible random operators if

    whenever F(ω,(x,y,z)) = g(ω,x),F(ω,(y,z,x)) = g(ω,y),F(ω,(z,x,y)) = g(ω,z) for all ω ∈? and x,y,z ∈X are satisfied.

    Lemma 2.11[36]Let P be a cone in a Banach algebra A and k ∈P be a given vector. Let{un} be a sequence in P. If for eachthere exist N1such thatfor all n>N1,then for each, there exist N2such that kunfor all n>N2.

    Now, we state our main results as follows.

    §3. Main Results

    In this section, we prove some tripled random coincidence and tripled random fixed point theorems for contractive mappings with several generalized Lipschitz constants in the setting of cone metric spaces over Banach algebras by deleting the normality of cones.

    Lemma 3.1[36]Let A be a Banach algebra and k ∈A. If ρ(k)<1, then.

    Remark 3.2If<1, it is natural that ρ(k)<1, yet, the converse is not true.

    Lemma 3.3[36]Let A be a Banach algebra with a unit e, {xn} a sequence in A. If {xn}converges to x in A , and for any n ≥1, {xn} commutes with x, then ρ(xn)→ρ(x) as n →∞.

    Theorem 3.4Let(X,d)be a separable cone metric space over Banach algebra A, P be a cone in A and (?,Σ) be a measurable space. Suppose that the mappings and F :?×X3→X and g :?×X →X satisfy the following contractive condition:

    for all x,y,z,u,v,w ∈X, where ai∈P,aiaj= ajai(i,j = 1,...,15), aiare generalized Lipschitz constants with ρ(a1+a2+a3)+ρ(a4+···+a9)+ρ(a10+a11+a12)<1. Let F(·,v),g(·,x)be measurable for v ∈X3and x ∈X, respectively. Suppose that F(ω×X3) ?g(ω×X) and g(ω ×X) is a complete subspace of X for each ω ∈?. Then there exist mappings ξ,η,θ :? →X such that F(ω,(ξ(ω),η(ω),θ(ω))) = g(ω,ξ(ω)), F(ω,(η(ω),θ(ω),ξ(ω))) = g(ω,η(ω))and F(ω,(θ(ω),ξ(ω),η(ω)))=g(ω,θ(ω)) for all ω ∈?, that is, F and g have a tripled random coincidence point.

    ProofLet Θ = {η : ? →X} be a family of measurable mappings. We construct three sequences of measurable mappings {ξn}, {ηn}, {θn} in Θ and three sequences {g(ω,ξn(ω))},{g(ω,ηn(ω))}, {g(ω,θn(ω))} in X as follows.

    Let ξ0,η0,θ0∈Θ. Since F(ω,(ξ0(ω),η0(ω),θ0(ω))) ∈F(ω×X3) ?g(ω×X), by a sort of Filippov measurable implicit function theorem(see[35]),there is ξ1∈Θ such that g(ω,ξ1(ω))=F(ω,(ξ0(ω),η0(ω),θ0(ω))). Similarly as F(ω,η0(ω), θ0(ω)ξ0(ω))) ∈g(ω×X), there is η1∈Θ such that g(ω,η1(ω))=F(ω,(η0(ω),θ0(ω)ξ0(ω))),F(ω,(θ0(ω),ξ0(ω),η0(ω)))∈g(ω×X),there is θ1∈Θ such that g(ω,θ1(ω)) = F(ω,(θ0(ω),ξ0(ω),η0(ω))). Continuing this process we can construct sequences {ξn(ω)}, {ηn(ω)} and {θn(ω)} in X such that

    for all n ∈N. According to (3.1), we have

    Further, we have

    Hence, we obtain that

    Similarly, we can prove that

    and

    Put

    Uniting (3.3)-(3.5), ones assert that

    Furthermore,

    Then

    Accordingly, it is clear that

    Similarly, we can prove that

    and

    Uniting (3.7)-(3.9), one gets that

    By using (3.6) and (3.10), it is easy to see that

    Then by Lemma 2.5 and Lemma 2.6, it follows that 2e ?k is invertible. Furthermore,To multiply in both side of (3.11) by (2e ?k)?1, we obtain

    Denote h=(2e ?k)?1k, then by (3.12) we get

    Note by Lemma 2.6 that

    so by Lemma 3.3 it leads to

    which establishes that e ?h is invertible and→0 as n →∞. Thus for all m > n ≥1,ones have

    Now, by (3.13) and sρ(h)<1, it follows that

    Owing to

    we have (e ?h)?1hnd0→θ (n →∞).

    According to Lemma 2.4, and for anythere exists N0such that for all n >N0, (e ?h)?1hnd0Furthermore, from (3.14) and for any m > n > N0, it follows that d(g(ω,ξn(ω)),g(ω,ξm(ω)))+d(g(ω,ηn(ω)),g(ω,ηm(ω)))+d(g(ω,θn(ω)),g(ω,θm(ω)))which implies that d(g(ω,ξn(ω)),g(ω,ξm(ω)))c,d(g(ω,ηn(ω)),g(ω,ηm(ω)))d(g(ω,θn(ω)),g(ω,θm(ω))). Hence, {g(ω,ξn(ω))}, {g(ω,ηn(ω))}, {g(ω,θn(ω))} are Cauchy sequences in g(X). Since g(X) is complete, there exist ξ?(ω),η?(ω) and θ?(ω) ∈X for all ω ∈? such that g(ω,ξn(ω)) →g(ω,ξ?(ω)),g(ω,ηn(ω)) →g(ω,η?(ω)),g(ω,θn(ω)) →g(ω,θ?(ω)) as n →∞.Moreover, note that

    Hence, we get that

    Similarly, it is easily obtain that

    and

    Put

    On view of (3.15)-(3.17), we get

    Then

    where A = a1+a2+a3+a10+a11+a12, B = a4+a5+a6+10+a11+12, C = a4+a5+a6+a7+a8+a9, ρ(A) ≤ρ(a1+a2+a3)+ρ(a4+···+a9)+ρ(a10+a11+a12) < 1. Since g(ω,ξn(ω))→g(ω,ξ?(ω)),g(ω,ηn(ω))→g(ω,η?(ω)),g(ω,θn(ω))→g(ω,θ?(ω)),it follows that for any cθ, there exists N0such that for n>N0. Then by Lemma 2.11 we have

    6

    Hence,

    Now, according to Lemma 2.4, it follows that δ =θ, that is,

    which implies that

    Thus,

    Therefore (ξ?(ω),η?(ω),θ?(ω)) is a tripled coincidence point of F and g for all ω ∈?.

    Corollary 3.5Let (X,d) be a separable cone metric space over Banach algebra A and P be a cone in A, (?,Σ) be a measurable space. Suppose that the mappings F :?×X3→X,g :?×X →X satisfy the following contractive condition:

    for all x,y,z,u,v,w ∈X,where k,l,t ∈P are generalized Lipschitz constant with ρ(k+l+t)<1,F(.,v),g(.,x) are measurable for v ∈X3and x ∈X, respectively, F(ω ×X3) ?g(ω ×X)and g(ω ×X) is complete subspace of X for each ω ∈?, then there are mappings ξ,η,θ :? →X, such that F(ω,(ξ(ω),η(ω),θ(ω))) = g(ω,ξ(ω)), F(ω,(η(ω),θ(ω),ξ(ω))) = g(ω,η(ω)),F(ω,(θ(ω),ξ(ω),η(ω))) = g(ω,θ(ω)) for all ω ∈?, that is F and g have a tripled random coincidence point.

    Corollary 3.6Let (X,d) be a separable cone metric space over Banach algebra A, P be a cone in A and(?,Σ)be a measurable space. Suppose that the mappings F :?×X3→X,g :?×X →X satisfy the following contractive condition:

    for all x,y,z,u,v,w ∈X,where k,l ∈P are generalized Lipschitz constants with ρ(k)+ρ(l)<1,F(·,v),g(·,x) are measurable for v ∈X3and x ∈X, respectively, F(ω ×X3) ?g(ω ×X)and g(ω ×X) is complete subspace of X for each ω ∈?, then there are mappings ξ,η,θ :? →X, such that F(ω,(ξ(ω),η(ω),θ(ω))) = g(ω,ξ(ω)), F(ω,(η(ω),θ(ω),ξ(ω))) = g(ω,η(ω)),F(ω,(θ(ω),ξ(ω),η(ω))) = g(ω,θ(ω)) for all ω ∈?, that is, F and g have a tripled random coincidence point.

    The conditions of Theorem 3.4 are not enough to prove the existence of a common tripled fixed point for the mappings F and g. By restricting to w-compatibility for F and g, we obtain the following theorem.

    Theorem 3.7In addition to hypotheses of Theorem 3.4, if F and g are w-compatible,then F and g have a unique tripled common fixed point. Moreover, a tripled common random fixed point of F and g is of the form (ξ?(ω),ξ?(ω),ξ?(ω))∈X for all ω ∈?.

    ProofBy Theorem 3.4,F and g have tripled random coincidence point(ξ?(ω),η?(ω),θ?(ω)).Then (g(ω,ξ?(ω)),g(ω,η?(ω)),g(ω,(θ?(ω))) is a tripled random point of coincidence of F and g such that

    First, we shall show that the tripled random point of coincidence is unique. Suppose that F and g have another tripled random point of coincidence(g(ω,ξ??(ω)),g(ω,η??(ω)),g(ω,θ??(ω)))such that

    where (ξ??(ω),η??(ω),θ??(ω))∈X3for all ω ∈?. Then we have

    Hence,

    Similarly, we have

    and

    By combining (3.19)-(3.21), we get

    Set α=a7+···+a12, γ =d(g(ω,ξ?(ω)),g(ω,ξ??(ω)))+d(g(ω,η?(ω)),g(ω,η??(ω)))+d(g(ω,θ?(ω)),g(ω,θ??(ω))), we haveNow that ρ(α)<1,

    which leads to αn→θ (n →∞), we claim that, for each c, there exists n0(c) such that αnc (n>n0(c)). Consequently by Lemma 2.11,

    d(g(ω,ξ?(ω)),g(ω,ξ??(ω)))+d(g(ω,η?(ω)),g(ω,η??(ω)))+d(g(ω,θ?(ω)),g(ω,θ??(ω)))=θ.

    Hence,

    that is,

    which implies the uniqueness of the tripled random point of coincidence of F and g. By a similar way, someone can prove that

    In view of (3.22)-(3.24), one can assert

    In other words, the unique tripled random point of coincidence of F and g is(g(ω,ξ?(ω)),g(ω,η?(ω)),g(ω,θ?(ω))). Let u(ω) = g(ω,ξ?(ω)) = F(ω,(ξ?(ω)),η?(ω)),θ?(ω))).Since F and g are w-compatible, then we have

    Thus (g(ω,u(ω)),g(ω,u(ω)),g(ω,u(ω))) is a tripled random point of coincidence. We also have (u(ω),u(ω),u(ω)) is a tripled random point of coincidence. Note that the uniqueness of the tripled random point of coincidence implies that g(ω,u(ω)) = u(ω). Therefore u(ω) =g(ω,u(ω)) = F(ω,(u(ω),u(ω),u(ω))). Hence (u(ω),u(ω),u(ω)) is the unique tripled common random fixed point of F and g for all ω ∈?. This completes the proof.

    Putting g(ω,·)=I(ω,·)(identity mapping )in Theorem 3.4, we obtain the following result.The following example illustrates our conclusions.

    Example 3.8Let X = R3, A = R3. For each x = (x1,x2,x3) ∈X, let|x1|+|x2|+|x3|, P ={(x1,x2,x3)∈R3|x1≥0,x2≥0,x3≥0}. The multiplication is defined by

    Then one can easily verify that A is a Banach algebra with unit e=(1,0,0). Can be observed,P is a cone in A. A metric d on X is defined by

    Easy to know (x,d) is a complete cone metric space over the Banach algebra A. Consider the following mapping g :?×X →X,g(ω,x)=((x1,2x2,3x3), for each (ω,(x1,x2,x3))∈?×X.Then g is a surjection. Define f :?×X3→X by

    Firstly, easy to verify

    (1) d(F(ω,(x,y,z)),F(ω,(u,v,w)))d(f(ω,(x,y,z)),f(ω,(u,v,w))).

    (2) d(f(ω,(x,y,z)),f(ω,(x,y,z)))

    On the side, the following inequality naturally holds,

    By (1),(2) and (3) have

    Get through theorem 3.4, we commandthat g,F immediately satisfy (3.1), and aiaj=ajai, (i,j =1,2...12), on the side,

    By Theorem 3.4 F and g have a tripled random coincidence point.

    Remark 3.9Our main results mainly generalize the recent results. In fact, they never consider the normality of cones, which may offer us more applications since there exist lots of non-normal cones (see [28]). Moreover, we establish the contractive mappings with several generalized Lipschitz constants,where the constants are all vectors but not usual real constants.Thus they are different from some ordinary results and more interesting.

    Remark 3.10Our theorems deal not only with common fixed point results with random process, but also with them from usual coupled fixed point to tripled fixed point. Therefore,our results greatly improve and extend some results in the literature (see [7]).

    Remark 3.11Our results are mainly related to tripled random coincidence point and common fixed point results of generalized Lipschitz mappings in cone metric spaces over Banach algebras. Our tripled random coincidence point and common fixed point results cannot reduced to the counterparts of the results with one variable. In other words, the method of [22]cannot be utilized to our main results. This is because the generalized Lipchitz constants from our results are vectors. Moreover, the multiplication of the vectors do not satisfy the combinative law. Hence we cannot use a method of reducing our tripled results to the respective results for mappings with one variable.

    Authors’ contributionsAll authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

    在线观看66精品国产| 日韩欧美在线二视频| 婷婷精品国产亚洲av在线| 婷婷精品国产亚洲av在线| 99久久精品热视频| 99热只有精品国产| 极品教师在线免费播放| 99国产精品一区二区蜜桃av| 日本一二三区视频观看| 久久中文字幕一级| 99国产精品一区二区蜜桃av| 黄片大片在线免费观看| 在线观看免费日韩欧美大片| 成人手机av| tocl精华| 成人亚洲精品av一区二区| 桃色一区二区三区在线观看| 免费看美女性在线毛片视频| 欧美zozozo另类| 成人av在线播放网站| 欧美乱妇无乱码| 狂野欧美白嫩少妇大欣赏| 国产成人欧美在线观看| 欧洲精品卡2卡3卡4卡5卡区| 啦啦啦韩国在线观看视频| 国产熟女xx| 久久精品亚洲精品国产色婷小说| av免费在线观看网站| 夜夜看夜夜爽夜夜摸| 亚洲午夜精品一区,二区,三区| 成人国产综合亚洲| 男男h啪啪无遮挡| 午夜激情福利司机影院| 精品熟女少妇八av免费久了| 久久这里只有精品19| 久久久久性生活片| 日本一区二区免费在线视频| 十八禁人妻一区二区| 亚洲av第一区精品v没综合| 变态另类丝袜制服| 少妇被粗大的猛进出69影院| 丁香欧美五月| 欧美一区二区精品小视频在线| 777久久人妻少妇嫩草av网站| 可以在线观看毛片的网站| 可以在线观看毛片的网站| 九色国产91popny在线| www.熟女人妻精品国产| 国产探花在线观看一区二区| 日韩大尺度精品在线看网址| 少妇熟女aⅴ在线视频| 一级黄色大片毛片| 神马国产精品三级电影在线观看 | 国产精品亚洲一级av第二区| 国产99久久九九免费精品| 欧美日韩瑟瑟在线播放| 欧美乱色亚洲激情| 国产激情偷乱视频一区二区| 白带黄色成豆腐渣| 国产精品一区二区精品视频观看| 男人舔女人的私密视频| 亚洲美女黄片视频| 少妇被粗大的猛进出69影院| 91av网站免费观看| av视频在线观看入口| 日韩三级视频一区二区三区| 少妇裸体淫交视频免费看高清 | 午夜老司机福利片| √禁漫天堂资源中文www| 成人欧美大片| 757午夜福利合集在线观看| 嫩草影视91久久| 好男人电影高清在线观看| 草草在线视频免费看| 国产精品自产拍在线观看55亚洲| 免费看十八禁软件| 最新在线观看一区二区三区| 精品欧美一区二区三区在线| 久久人妻av系列| 搡老熟女国产l中国老女人| 在线a可以看的网站| 亚洲va日本ⅴa欧美va伊人久久| 精品高清国产在线一区| www日本在线高清视频| 亚洲av片天天在线观看| 黄色毛片三级朝国网站| 成人国产综合亚洲| 亚洲人成网站在线播放欧美日韩| 一区二区三区激情视频| 一区二区三区高清视频在线| 不卡一级毛片| 成人三级黄色视频| ponron亚洲| 国产99白浆流出| 精品一区二区三区视频在线观看免费| 999久久久精品免费观看国产| 亚洲成人免费电影在线观看| 老熟妇仑乱视频hdxx| 97人妻精品一区二区三区麻豆| 亚洲国产欧洲综合997久久,| 久久精品国产亚洲av高清一级| 久久久国产成人免费| www.精华液| 9191精品国产免费久久| 一二三四在线观看免费中文在| 亚洲全国av大片| 久久这里只有精品中国| 亚洲性夜色夜夜综合| 欧美色视频一区免费| 可以在线观看的亚洲视频| 麻豆一二三区av精品| 波多野结衣高清无吗| 国产精品自产拍在线观看55亚洲| 久久久久国产一级毛片高清牌| 国产精品一区二区三区四区免费观看 | 一个人免费在线观看电影 | 天堂影院成人在线观看| 男人舔女人的私密视频| 国产成人精品无人区| 成人18禁在线播放| 成人av一区二区三区在线看| 最近最新中文字幕大全免费视频| 99热这里只有是精品50| 最好的美女福利视频网| 18禁裸乳无遮挡免费网站照片| 午夜精品一区二区三区免费看| 亚洲国产高清在线一区二区三| 亚洲真实伦在线观看| 久久婷婷人人爽人人干人人爱| 亚洲18禁久久av| 露出奶头的视频| 成人手机av| 国产激情久久老熟女| 好看av亚洲va欧美ⅴa在| av免费在线观看网站| 男女午夜视频在线观看| 国产av一区二区精品久久| 99精品欧美一区二区三区四区| 99国产精品一区二区蜜桃av| 亚洲免费av在线视频| 午夜日韩欧美国产| 老司机靠b影院| 国产精品免费一区二区三区在线| 免费看十八禁软件| 国产又黄又爽又无遮挡在线| 在线a可以看的网站| 黄色 视频免费看| 18禁国产床啪视频网站| 日韩欧美精品v在线| 亚洲电影在线观看av| 草草在线视频免费看| 日韩欧美免费精品| 亚洲五月婷婷丁香| 久久精品国产99精品国产亚洲性色| 亚洲天堂国产精品一区在线| 欧美高清成人免费视频www| 国产欧美日韩精品亚洲av| 精品久久久久久久毛片微露脸| 精品熟女少妇八av免费久了| 搡老熟女国产l中国老女人| 久久国产乱子伦精品免费另类| 制服丝袜大香蕉在线| 国产av不卡久久| 制服人妻中文乱码| 国内精品久久久久久久电影| cao死你这个sao货| 在线a可以看的网站| 日本撒尿小便嘘嘘汇集6| 免费人成视频x8x8入口观看| 老鸭窝网址在线观看| www.www免费av| 国产成人影院久久av| 免费在线观看完整版高清| 五月玫瑰六月丁香| 五月玫瑰六月丁香| 久久久久久国产a免费观看| 国产一区二区三区在线臀色熟女| 男女下面进入的视频免费午夜| 男人舔奶头视频| 久久欧美精品欧美久久欧美| 国产高清视频在线播放一区| 国产熟女xx| 国产亚洲精品一区二区www| av免费在线观看网站| 国产99白浆流出| 欧美又色又爽又黄视频| 美女午夜性视频免费| 国产在线观看jvid| 欧美+亚洲+日韩+国产| 亚洲国产日韩欧美精品在线观看 | 亚洲人成网站高清观看| 99精品久久久久人妻精品| 美女 人体艺术 gogo| 欧美色欧美亚洲另类二区| 亚洲第一欧美日韩一区二区三区| 亚洲avbb在线观看| 麻豆国产av国片精品| 国产精品自产拍在线观看55亚洲| 国产精品爽爽va在线观看网站| 1024香蕉在线观看| 亚洲精品一区av在线观看| 麻豆成人av在线观看| 久久国产精品人妻蜜桃| av福利片在线| 国产亚洲欧美在线一区二区| 琪琪午夜伦伦电影理论片6080| avwww免费| 亚洲专区国产一区二区| 国产伦在线观看视频一区| 欧美日韩瑟瑟在线播放| 变态另类成人亚洲欧美熟女| 一进一出抽搐gif免费好疼| 日韩精品青青久久久久久| 日本免费一区二区三区高清不卡| 757午夜福利合集在线观看| 亚洲欧美精品综合一区二区三区| 一级毛片高清免费大全| 久久 成人 亚洲| 中文字幕av在线有码专区| 看片在线看免费视频| 在线观看舔阴道视频| 老司机在亚洲福利影院| 亚洲成av人片在线播放无| 欧美丝袜亚洲另类 | 老司机深夜福利视频在线观看| 久久久久国产精品人妻aⅴ院| 我的老师免费观看完整版| 精华霜和精华液先用哪个| 一a级毛片在线观看| ponron亚洲| 亚洲电影在线观看av| 搞女人的毛片| 色老头精品视频在线观看| 在线观看www视频免费| 又粗又爽又猛毛片免费看| 色综合站精品国产| 日本一区二区免费在线视频| 日日爽夜夜爽网站| 动漫黄色视频在线观看| 岛国视频午夜一区免费看| 亚洲欧美日韩高清专用| 亚洲 欧美一区二区三区| 不卡av一区二区三区| 国产亚洲精品一区二区www| 午夜福利在线在线| 亚洲激情在线av| 亚洲七黄色美女视频| 亚洲第一欧美日韩一区二区三区| 日韩国内少妇激情av| 亚洲自拍偷在线| 一本久久中文字幕| 亚洲人与动物交配视频| 亚洲成av人片免费观看| 淫妇啪啪啪对白视频| 亚洲国产高清在线一区二区三| 国产精品久久久久久精品电影| 色精品久久人妻99蜜桃| 久久国产精品人妻蜜桃| 亚洲中文字幕一区二区三区有码在线看 | 成人国语在线视频| 国产激情偷乱视频一区二区| 日韩欧美 国产精品| 变态另类丝袜制服| 亚洲人成伊人成综合网2020| 欧美日韩国产亚洲二区| 美女大奶头视频| 国产成人精品无人区| 国产精品永久免费网站| 小说图片视频综合网站| 狂野欧美激情性xxxx| 蜜桃久久精品国产亚洲av| 欧美绝顶高潮抽搐喷水| 日韩欧美 国产精品| 亚洲精品美女久久av网站| 一本一本综合久久| 免费在线观看日本一区| 成人国产一区最新在线观看| 久久久精品欧美日韩精品| 成人亚洲精品av一区二区| 久热爱精品视频在线9| 99热只有精品国产| 日韩大码丰满熟妇| 亚洲avbb在线观看| 人妻久久中文字幕网| 亚洲午夜精品一区,二区,三区| 在线观看免费视频日本深夜| 岛国视频午夜一区免费看| 一本久久中文字幕| 怎么达到女性高潮| 午夜影院日韩av| 国产伦人伦偷精品视频| 精品久久久久久久久久久久久| 亚洲av五月六月丁香网| 色综合亚洲欧美另类图片| 搡老妇女老女人老熟妇| 无人区码免费观看不卡| 一级毛片高清免费大全| 亚洲真实伦在线观看| 国产高清激情床上av| 亚洲欧美日韩无卡精品| 大型av网站在线播放| 欧美极品一区二区三区四区| 日本熟妇午夜| 91麻豆精品激情在线观看国产| 两性夫妻黄色片| 国产在线精品亚洲第一网站| 日韩欧美 国产精品| 丰满的人妻完整版| 欧美日韩一级在线毛片| 国产伦一二天堂av在线观看| 巨乳人妻的诱惑在线观看| 久久久久久久精品吃奶| 国产精品永久免费网站| 久久欧美精品欧美久久欧美| 性色av乱码一区二区三区2| 成年免费大片在线观看| 国内精品一区二区在线观看| 老司机在亚洲福利影院| 特级一级黄色大片| 久久精品aⅴ一区二区三区四区| 中亚洲国语对白在线视频| 亚洲在线自拍视频| 一进一出抽搐gif免费好疼| 美女黄网站色视频| tocl精华| 看片在线看免费视频| 日韩欧美精品v在线| 国产精品久久久久久人妻精品电影| 正在播放国产对白刺激| 久久久久久人人人人人| 黄频高清免费视频| 一本精品99久久精品77| 久久精品夜夜夜夜夜久久蜜豆 | 好看av亚洲va欧美ⅴa在| 九九热线精品视视频播放| 人妻丰满熟妇av一区二区三区| 国产精品久久久人人做人人爽| 欧美日韩亚洲国产一区二区在线观看| av欧美777| 亚洲欧美精品综合久久99| 女人高潮潮喷娇喘18禁视频| 最新在线观看一区二区三区| 国产成人系列免费观看| 精品日产1卡2卡| 亚洲成人精品中文字幕电影| 久久久精品国产亚洲av高清涩受| 精品人妻1区二区| 亚洲一卡2卡3卡4卡5卡精品中文| 午夜视频精品福利| 精品熟女少妇八av免费久了| 久久久久久九九精品二区国产 | 天天躁狠狠躁夜夜躁狠狠躁| 性欧美人与动物交配| 亚洲乱码一区二区免费版| 国产日本99.免费观看| 久久久久亚洲av毛片大全| 三级毛片av免费| 伊人久久大香线蕉亚洲五| 别揉我奶头~嗯~啊~动态视频| 亚洲中文字幕一区二区三区有码在线看 | 18禁裸乳无遮挡免费网站照片| 亚洲av成人一区二区三| 国产精品,欧美在线| 亚洲第一欧美日韩一区二区三区| 中文资源天堂在线| 不卡一级毛片| 亚洲精品中文字幕一二三四区| 日日夜夜操网爽| 国产一区二区在线观看日韩 | 亚洲av片天天在线观看| 一区福利在线观看| 亚洲人成电影免费在线| 曰老女人黄片| 国产午夜精品久久久久久| 91麻豆av在线| 国产爱豆传媒在线观看 | 成年版毛片免费区| 亚洲av电影在线进入| 亚洲第一电影网av| 日韩欧美国产一区二区入口| a在线观看视频网站| 欧美 亚洲 国产 日韩一| 久久久精品欧美日韩精品| 精品国内亚洲2022精品成人| 日本精品一区二区三区蜜桃| 国内久久婷婷六月综合欲色啪| 无遮挡黄片免费观看| 色综合婷婷激情| 在线永久观看黄色视频| av片东京热男人的天堂| 三级国产精品欧美在线观看 | 国产真实乱freesex| 欧美不卡视频在线免费观看 | 日本熟妇午夜| 黄色视频,在线免费观看| 可以在线观看毛片的网站| 久久婷婷人人爽人人干人人爱| 成人国语在线视频| av天堂在线播放| 久久久国产成人免费| av免费在线观看网站| 国产黄色小视频在线观看| 人人妻人人看人人澡| 欧美黄色片欧美黄色片| 亚洲男人的天堂狠狠| 欧美av亚洲av综合av国产av| 午夜免费激情av| 变态另类成人亚洲欧美熟女| 99国产精品一区二区蜜桃av| 成人精品一区二区免费| 美女黄网站色视频| 午夜成年电影在线免费观看| 欧美黑人欧美精品刺激| 狂野欧美激情性xxxx| 人人妻,人人澡人人爽秒播| 国产精品美女特级片免费视频播放器 | 久久 成人 亚洲| 亚洲熟妇熟女久久| 亚洲第一电影网av| 日韩 欧美 亚洲 中文字幕| 18禁黄网站禁片午夜丰满| 国产午夜福利久久久久久| 午夜福利在线观看吧| 亚洲一区二区三区不卡视频| 久久精品综合一区二区三区| 精品久久久久久久久久免费视频| 国产精品av视频在线免费观看| 欧美中文日本在线观看视频| 国产高清视频在线播放一区| 亚洲精品久久国产高清桃花| 国语自产精品视频在线第100页| 丝袜美腿诱惑在线| 精品国内亚洲2022精品成人| 99久久精品热视频| 精品欧美一区二区三区在线| 午夜免费成人在线视频| 精品熟女少妇八av免费久了| 一本综合久久免费| 午夜福利18| 欧美人与性动交α欧美精品济南到| av免费在线观看网站| 免费人成视频x8x8入口观看| 欧美三级亚洲精品| 久久久精品国产亚洲av高清涩受| 国产真人三级小视频在线观看| 国产片内射在线| 国产欧美日韩精品亚洲av| 黑人欧美特级aaaaaa片| 国产免费av片在线观看野外av| 可以免费在线观看a视频的电影网站| 欧美日本亚洲视频在线播放| 久久伊人香网站| 在线免费观看的www视频| 一级毛片精品| 国产黄色小视频在线观看| 国产精品免费视频内射| 在线观看日韩欧美| 村上凉子中文字幕在线| 在线播放国产精品三级| 黄色视频不卡| 欧美性长视频在线观看| 午夜免费激情av| 91字幕亚洲| 国产69精品久久久久777片 | 国产精品九九99| 国产熟女xx| 国产精品久久久久久人妻精品电影| 亚洲熟女毛片儿| 99精品久久久久人妻精品| 欧美+亚洲+日韩+国产| 18禁黄网站禁片午夜丰满| 大型黄色视频在线免费观看| 久久香蕉激情| 男人舔女人的私密视频| 天堂动漫精品| 夜夜夜夜夜久久久久| 午夜视频精品福利| 国产熟女午夜一区二区三区| 国产高清视频在线播放一区| 黄色片一级片一级黄色片| 嫩草影视91久久| 亚洲免费av在线视频| 国产精品久久久av美女十八| 最新在线观看一区二区三区| 一级a爱片免费观看的视频| 老司机午夜十八禁免费视频| 婷婷精品国产亚洲av在线| 国产日本99.免费观看| 国产三级在线视频| 99久久综合精品五月天人人| 亚洲国产看品久久| 熟女少妇亚洲综合色aaa.| 国产不卡一卡二| 欧美另类亚洲清纯唯美| 亚洲精品久久成人aⅴ小说| 国产精品 欧美亚洲| 久久精品aⅴ一区二区三区四区| 国产人伦9x9x在线观看| 男女床上黄色一级片免费看| 丁香六月欧美| 精品国产乱码久久久久久男人| netflix在线观看网站| 国产亚洲精品久久久久5区| 在线观看舔阴道视频| 欧美性猛交╳xxx乱大交人| 精品久久久久久久人妻蜜臀av| 国产欧美日韩精品亚洲av| 亚洲中文日韩欧美视频| 国产精品精品国产色婷婷| 国产免费男女视频| av视频在线观看入口| 777久久人妻少妇嫩草av网站| 一本久久中文字幕| 老汉色∧v一级毛片| 免费在线观看亚洲国产| 少妇人妻一区二区三区视频| 成年女人毛片免费观看观看9| 亚洲成人中文字幕在线播放| 久久伊人香网站| 亚洲,欧美精品.| 淫妇啪啪啪对白视频| 久久精品综合一区二区三区| 久久精品国产综合久久久| 精品免费久久久久久久清纯| 老司机在亚洲福利影院| 国产精华一区二区三区| 国产一级毛片七仙女欲春2| 在线观看免费午夜福利视频| 欧美+亚洲+日韩+国产| 美女 人体艺术 gogo| 又粗又爽又猛毛片免费看| 国产97色在线日韩免费| 国产精品久久久av美女十八| 校园春色视频在线观看| 91麻豆精品激情在线观看国产| 最近最新中文字幕大全电影3| 国产一区二区在线观看日韩 | 久久久久久九九精品二区国产 | 亚洲精品一区av在线观看| 性欧美人与动物交配| 一a级毛片在线观看| 色尼玛亚洲综合影院| 岛国视频午夜一区免费看| 免费看日本二区| 久久精品aⅴ一区二区三区四区| 国产人伦9x9x在线观看| 一进一出好大好爽视频| 丰满人妻熟妇乱又伦精品不卡| 亚洲,欧美精品.| 中文字幕久久专区| 老司机午夜十八禁免费视频| 久久久国产欧美日韩av| 婷婷丁香在线五月| 日韩精品免费视频一区二区三区| 欧美中文综合在线视频| 亚洲av美国av| 一本大道久久a久久精品| 午夜精品在线福利| 美女免费视频网站| 黄色毛片三级朝国网站| 色精品久久人妻99蜜桃| 国产精品久久久久久亚洲av鲁大| 黄色丝袜av网址大全| 俄罗斯特黄特色一大片| 日本黄色视频三级网站网址| 成人国语在线视频| 哪里可以看免费的av片| av免费在线观看网站| 国产日本99.免费观看| 亚洲男人天堂网一区| 搞女人的毛片| 在线观看日韩欧美| 丰满的人妻完整版| 无限看片的www在线观看| 1024手机看黄色片| 亚洲美女视频黄频| 91在线观看av| 后天国语完整版免费观看| 国产一区在线观看成人免费| 波多野结衣巨乳人妻| 久久天躁狠狠躁夜夜2o2o| 国产野战对白在线观看| 怎么达到女性高潮| a级毛片在线看网站| 国产亚洲精品第一综合不卡| 高潮久久久久久久久久久不卡| 69av精品久久久久久| 成人国产一区最新在线观看| 久久国产精品人妻蜜桃| 亚洲aⅴ乱码一区二区在线播放 | 人妻久久中文字幕网| 亚洲精品粉嫩美女一区| 在线观看免费视频日本深夜| 亚洲第一欧美日韩一区二区三区| 久久久久久国产a免费观看| 一边摸一边做爽爽视频免费| 国产成人欧美在线观看| 久久久久久国产a免费观看| 熟女电影av网| 成熟少妇高潮喷水视频| 亚洲无线在线观看| 国产99白浆流出| 日韩欧美在线二视频| 亚洲九九香蕉| ponron亚洲| 九色国产91popny在线| 99国产精品一区二区蜜桃av| 午夜免费成人在线视频| 中文字幕久久专区| 国模一区二区三区四区视频 | 亚洲av熟女| 欧美又色又爽又黄视频| 99热这里只有是精品50| 欧美在线一区亚洲| 午夜影院日韩av|