• <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 ?

    Fixed Point Theorems in Relational Metric Spaces with an Application to Boundary Value Problems

    2021-05-25 07:14:30PRASADGopiandKHANTWALDeepak
    關(guān)鍵詞:先導(dǎo)性組織路線偉大事業(yè)

    PRASAD Gopiand KHANTWAL Deepak

    1 Department of Mathematics,HNB Garhwal University,Srinagar Garhwal,India.

    2 Department of Mathematics,Graphic Era Hill University,Dehradun,India.

    Abstract. In this paper, we establish fixed point theorems for generalized nonlinear contractive mappings using the concept of w-distance in relational metric spaces.Thus we generalize the recent results of Senapati and Dey[J.Fixed Point Theory Appl. 19,2945-2961(2017)]and many other important results relevant to this literature.In order to revel the usefulness of such investigations, an application to first order periodic boundary value problem are given. Moreover, we furnish a non-trivial example to demonstrate the validity of our generalization over previous existing results.

    Key Words: Binary relation;R-lower semi-continuity;relational metric spaces.

    1 Introduction

    The classical Banach contraction principle (Bcp)has many inferences and huge applicability in mathematical theory and because of this,Bcp has been improved and generalized in various metric settings.One such interesting and important setting is to establish fixed point results in metric spaces equipped with an arbitrary binary relation. Utilizing the notions of various kind of binary relations such as partial order,strict order,near order,tolerance etc. on metric spaces,many researcher are doing their research during several years(see[1-16])and attempting to obtain new extensions of the celebrated Bcp.Among these extensions,we must quote the one due to Alam and Imdad [8], where some relation theoretic analogues of standard metric notions(such as continuity and completeness)were used.Further,Ahmadullah et al. [14]extended the above setting for nonlinear contractions and obtained an extension of the Boyd-Wong[17] fixed point theorem to such spaces.

    On the other hand, recently Senapati and Dey [11] improved and refined the main result of Alam and Imdad [8], Ahmadullah et al. [14] and many others, by utilizing the notion ofw-distance in relational metric spaces,that is,metric spaces endowed with an arbitrary binary relation. Moreover, for further motivation of research in this direction, we refer some important recent generalizations ofw-distance with applications to boundary value problem as well(see,e.g.,[19-21]). It is our aim in this paper to give an extension of these results to nonlinear?-contraction and explore the possibility of their application in finding a solution of first order periodic boundary value problem too.

    2 Preliminaries

    Throughout this chapter,R stands for a non-empty binary relation,N0stands for the the set of whole numbers,i.e.,N0=N∪{0}andRfor the set of all real numbers.

    Definition 2.1.([8]).LetRbe a binary relation on a non-empty set X and x,y∈X. We say that x and y areR-comparative if either(x,y)∈Ror(y,x)∈R. We denote it by[x,y]∈R.

    Definition 2.2.([8]).Let X be a non-empty set andRa binary relation on X. A sequence{xn}?X is called anR-preserving if(xn,xn+1)∈Rfor all n∈N0.

    Definition 2.3.([8]).Let X be a non-empty set and T a self-mapping on X. A binary relationRon X is called T-closed if for any x,y∈X,(x,y)∈Rimplies(Tx,Ty)∈R.

    Definition 2.4.([14]).Let(X,d)be a metric space andRa binary relation on X. We say that(X,d)isR-complete if everyR-preserving Cauchy sequence in X converges.

    The following notion is a generalization ofd-self-closedness of a partial order relation()(defined by Turinici[5-6]).

    Definition 2.5.([8]).Let(X,d)be a metric space. A binary relationRon X is called d-self-closed if for anyR-preserving sequence{xn}such that,there exists a subsequence{xnk}of{xn}with[xnk,x]∈Rfor all k∈N0.

    Definition 2.6.([14]).Let(X,d)be a metric space,Ra binary relation on X and x∈X. A self-mapping T on X is calledR-continuous at x if for anyR-preserving sequence{xn}suchthat,we have. Moreover,T is calledR-continuous if it isR-continuousat each point of X.

    The notion of R-lower semi-continuity (briefly, R-LSC) of a function is defined by Senapati and Dey[11]as follows:

    Definition 2.7.Let(X,d)be a metric space and R be a binary relation defined on X. A function f:X→R∪{?∞,+∞}is said to beR-LSC at x if for everyR-preserving sequence xn converging to x,we haveliminfn→+∞f(xn)≥f(x).

    By presenting examples the respective authors explained that the R-LSC is weaker than R-continuity as well as lower semi-continuity(see for details[11]) and modify the definition of w-distance ( Definition 2.8) and the corresponding Lemma 1 presented in[18] in the context of metric spaces endowed with an arbitrary binary relation R as follows:

    Definition 2.8.Let(X,d)be a metric space andRbe a binary relation on X. A function p:X×X→[0,+∞)is said to be a w-distance on X if

    (w1)p(x,z)≤p(x,y)+p(y,z)for any x,y,z∈X;

    (w2)for any x∈X,p(x,.):X→[0,∞)isR-lower semi-continuous;

    (w3)for any ?>0,there exists δ>0,such that p(z,x)≤δ and p(z,y)≤δ imply d(x,y)≤?.

    Let Φ be the family of all mappings?: [0,+∞)→[0,+∞) satisfying the following properties

    1.?is increasing;

    Recall that,the necessary condition of any real convergent series ∑nanis that

    The following two lemmas are required in our subsequent discussion.

    Lemma 2.1.([14]). Let ?∈Φ.Then for all t>0,we have ?(t)

    Lemma 2.2.([11]). Let(X,d)be a metric space endowed with binary relationRand p:X×X→[0,+∞)be a w-distance. Suppose(xn)and(yn)are twoR-preserving sequences in X and x,y,z∈X. Let(un)and(vn)be sequences of positive real numbers converging to0. Then, we have the followings:

    (L1)If p(xn,y)≤un and p(xn,z)≤vn for all n∈N, then y=z.Moreover, if p(x,y)=0and p(x,z)=0,then y=z.

    (L2)If p(xn,yn)≤un and p(xn,z)≤vn for all n≤N,then yn→z.

    (L3)If p(xn,xm)≤un for all m>n,then(xn)is anR-preserving Cauchy sequence in X.

    (L4)If p(xn,y)≤un for all n∈N,then(xn)is anR-preserving Cauchy sequence in X.

    Given a binary relation R and a self-mappingTon a nonempty setX, we use the following notations:

    (i)F(T):=the set of all fixed points ofT,

    (ii)X(T,R):={x∈X:(x,Tx)∈R}.

    3 Main Results

    In this section,we first consider the existence of fixed points for mappings in relational metric spaces.

    Theorem 3.1.Let(X,d)be a metric space with a w-distance′p′and a binary relation′R′on X.Let T be a self-mapping on X satisfying the following assumptions:

    (a) there exists Y?X with T(X)?Y such that(Y,d)isR-complete,

    (b)Ris T-closed,

    (c) either T isR-continuous orR|Y is d-self-closed,

    (d) X(T,R)is non-empty,

    for all x,y∈X with(x,y)∈R. Then T has a fixed point.

    Proof.In the light of assumption(d), letx0be an arbitrary element ofX(T,R).Define a sequence{xn}of Picard iterates with initial pointx0,i.e,

    Since(x0,Tx0)∈R and R isT-closed,we have

    so that

    Thus the sequence{xn}is R-preserving.Applying the contractive condition(e),we have

    By mathematical induction and the property(Φ1),we obtainp(xn,xn+1)≤?n(p(x0,x1)),for alln∈N0.Now,for allm,n∈N0withm≥n,we have

    Therefore,by(L3),of Lemma 2.2 we have {xn} is an R-preserving Cauchy sequence inY. As(Y,d)is an R-complete,we must havexn→xasn→+∞for somex∈Y.

    Next we claim thatxis a fixed point ofT. At first,we consider thatTis R-continuous.Since{xn}is an R-preserving sequence with,R-continuity ofTimplies that

    會議指出,干部教育培訓(xùn)是干部隊(duì)伍建設(shè)的先導(dǎo)性、基礎(chǔ)性、戰(zhàn)略性工程,在進(jìn)行偉大斗爭、建設(shè)偉大工程、推進(jìn)偉大事業(yè)、實(shí)現(xiàn)偉大夢想中具有不可替代的重要地位和作用。制定實(shí)施好干部教育培訓(xùn)規(guī)劃是全黨的一件大事,對貫徹落實(shí)新時(shí)代黨的建設(shè)總要求和新時(shí)代黨的組織路線、培養(yǎng)造就忠誠干凈擔(dān)當(dāng)?shù)母咚刭|(zhì)專業(yè)化干部隊(duì)伍、確保黨的事業(yè)后繼有人具有重大而深遠(yuǎn)的意義。

    Using the uniqueness of the limit,we obtainTx=x,i.e,xis a fixed point ofT.

    Alternately, let us assume that R|Yisd-self-closed. So there exists a subsequence{xnk}of{xn}with[xnk,x]∈R for allk∈N0. By using the fact that[xnk,x]∈R,contractive assumption(e)and R-lower-semi-continuity ofp,we have

    Since R isT-closed and(xnk,x)∈R,so

    Finally,owing to condition(L1)of Lemma 2.2,we must haveTx=x,i.e.,xis a fixed point ofT.

    3.1 Uniqueness result

    We state the uniqueness related result as follows:

    Theorem 3.2.In addition to the hypotheses of Theorem 3.1,suppose that any of the assumptions(u1)or(u2)holds:

    (u1)For every x,y∈T(X)there exists z∈T(X)such that(z,x), (z,y)∈R.

    (u2)R|T(X)is complete.

    Then T has a unique fixed point.

    Proof.In addition to the hypotheses of Theorem 3.1, suppose that condition (u1) hold.Then,for any two fixed pointsx,yofT,there exists an elementz∈T(X),such that

    Since R isT-closed,we have

    Applying contractive condition(e),we have

    Let us considerun=?n(p(z,x)) andvn=?n(p(z,y)). Clearly, {un} and {vn} are two sequences of real numbers converging to 0.Hence,by(L1)of Lemma 2.2,we havex=y,i.e.,Thas a unique fixed point.

    Secondly, suppose that in addition to the hypotheses of Theorem 3.1 condition (u2)hold.Supposex,yare any two fixed points ofT. Then we must have(x,y)∈R or(y,x)∈R. For(x,y)∈R,we have

    which is a contradiction. Hence, we must havex=y. Similarly, if (y,x)∈R, we havex=y.

    Example 3.1.LetX=[0,+∞) equipped with usual metricd. Then (X,d) is a complete metric space. Define a binary relation (x,y)∈R impliesxyonXand the mappingT:X→Xby

    Then R isT-closed. Define?:[0,∞)→[0,∞) byfor allt∈[0,∞), and awdistancep:X×X→Xbyp(x,y)=y.Now for allx,y∈Xwith(x,y)∈R,we have

    so thatTand?satisfy assumption (e) of Theorem 3.1. Observe that all the other conditions of Theorem 3.1 are also satisfied. Therefore,Thas a unique fixed point (namelyx=0).

    Remark 3.1.It is interesting to note that the mappingTin above example does not satisfy the contractive condition of Theorem 2.1 in Senapati and Dey [11]. For example, if we considerx=0 andy=?where?is arbitrary small but positive. Clearly,(0,?)∈R and if we take a constantλsuch thatp(T(x),T(y))≤λp(x,y), i.e.thenwhich amounts to say thatλ≥1 so thatλ[0,1). Thus Example 3.1 vindicate the utility of Theorem 3.1 over the results of Sanapati and Dey[11]and many others.

    Remark 3.2.If we take?(t)=λt, in our main result Theorem 3.1, then we obtain the Theorem 2.1 of Senapati and Dey[11]and if we setp(x,y)=d(x,y),and?(t)=λt,in our main result,we obtain the Theorem 3.1 of Alam and Imdad[8]. Hence our main result is an improved and generalized version of relation-theoretic metrical fixed-point theorems of Alam and Imdad[8],Senapati and Dey[11]and many others.

    4 An application

    As an application, we present a unique solution for the first order periodic boundary value problem equipped with an arbitrary binary relation,wherein our main results are applicable. We consider the following first order periodic boundary value problem:

    whereT>0 andf:I×R→Ris a continuous function.

    LetC(I) denote the space of all continuous functions defined onI. We recall the following definitions.

    Definition 4.1.([9]).A function α∈C1(I)is called a lower solution of(4.1),if

    Definition 4.2.([9]).A function α∈C1(I)is called a upper solution of(4.1),if

    Theorem 4.1.In addition to the problem(4.1), suppose that there exist λ>0such that for all x,y∈R with x≤y.

    Then the existence of a lower solution or an upper solution of problem(4.1)ensures the existence and uniqueness of a solution of problem(4.1).

    Proof.Problem(4.1)can be rewritten as

    This problem is equivalent to the integral equation

    where Define a mappingT:C(I)→C(I)by

    and a binary relation

    (i)Note thatC(I)equipped with the sup-metric,i.e.,d(x,y)=sup|x(t)?y(t)|fort∈Iandx,y∈C(I) is complete metric space and hence(C(I),d)is R-complete.

    (ii)Choose an R-preserving sequence{xn}such that.Then for allt∈I,we get

    and convergence tox(t) implies thatxn(t)≤z(t) for allt∈I,n∈N0, which amounts to saying that[xn,z]∈R for alln∈N0. Hence,R isd-self-closed.(iii)For any(x,y)∈R,i.e.x(t)≤y(t)then by(4.2),we have

    andG(t,s)>0 for(t,s)∈I×I,we have

    which implies that(Tx,Ty)∈R,i.e.,R isT-closed.

    (iv)Letα∈C1(I)be a lower solution of(4.1),then we must have

    Multiplying both sides byeλt,we have

    which implies that

    Asα(0)≤α(T),we have

    therefore

    By using(4.3)and(4.4),we have

    so that

    for allt∈I,i.e.,(α(t),Tα(t))∈R for allt∈Iwhich implies thatX(T,R)≠φ.

    (v)For all(x,y)∈R,

    so that

    Now,if we setp(x,y)=d(x,y),then we have

    where?∈Φ.Hence all the conditions of Theorem 3.1 are satisfied,consequentlyThas a fixed point. Finally following the proof of our earlier Theorem 3.2,Thas a unique fixed point,which is in fact a unique solution of the problem(4.1).

    Acknowledgement

    The authors thank the referees for their careful reading of the manuscript and useful comments.

    猜你喜歡
    先導(dǎo)性組織路線偉大事業(yè)
    關(guān)于新時(shí)代黨的組織路線的研究述評
    黨政論壇(2023年1期)2023-04-15 06:14:56
    黨的“組織路線”概念是如何提出的?(上)
    準(zhǔn)確理解組織路線的科學(xué)內(nèi)涵
    譜寫新時(shí)代中國特色社會主義偉大事業(yè)新篇章
    一流本科教育建設(shè)下的公共基礎(chǔ)課程與后續(xù)專業(yè)課程融合度的探索與研究
    扎實(shí)踐行新時(shí)代黨的組織路線——我省書寫新時(shí)代組織工作壯美畫卷
    “四個(gè)偉大”是一個(gè)頂層設(shè)計(jì)
    “四個(gè)偉大”:治國理政的大邏輯
    前線(2017年10期)2017-11-09 09:12:39
    氣象科普在公共氣象服務(wù)中的重要作用論述
    科技視界(2017年12期)2017-09-11 19:21:29
    淺談中學(xué)歷史課時(shí)效性教學(xué)法
    科技資訊(2015年7期)2015-07-02 20:55:04
    国产精品亚洲一级av第二区| 97超级碰碰碰精品色视频在线观看| 成人18禁在线播放| 91九色精品人成在线观看| 免费在线观看影片大全网站| 欧美日本中文国产一区发布| 国产欧美日韩一区二区精品| 日日爽夜夜爽网站| 此物有八面人人有两片| 亚洲av电影在线进入| 精品熟女少妇八av免费久了| 免费一级毛片在线播放高清视频 | 久久久久久大精品| 99久久国产精品久久久| 久久久久久久久免费视频了| 久久香蕉精品热| 一区二区日韩欧美中文字幕| 久久精品成人免费网站| 两个人看的免费小视频| 丝袜在线中文字幕| 琪琪午夜伦伦电影理论片6080| 久久九九热精品免费| 亚洲国产中文字幕在线视频| 免费无遮挡裸体视频| 国产在线观看jvid| av网站免费在线观看视频| 久久久久国产一级毛片高清牌| 精品久久久精品久久久| 天天躁狠狠躁夜夜躁狠狠躁| 高清黄色对白视频在线免费看| 亚洲成国产人片在线观看| 精品卡一卡二卡四卡免费| 女人被躁到高潮嗷嗷叫费观| 精品久久久久久久毛片微露脸| 国产极品粉嫩免费观看在线| 人人澡人人妻人| 变态另类成人亚洲欧美熟女 | 亚洲av电影不卡..在线观看| 少妇 在线观看| 日本 av在线| 亚洲欧美一区二区三区黑人| av福利片在线| 国产aⅴ精品一区二区三区波| 一二三四在线观看免费中文在| 一a级毛片在线观看| 51午夜福利影视在线观看| 女人精品久久久久毛片| 亚洲国产精品合色在线| 日韩精品免费视频一区二区三区| 欧美在线一区亚洲| 亚洲国产精品久久男人天堂| 精品国产亚洲在线| 国产一区二区激情短视频| 亚洲一卡2卡3卡4卡5卡精品中文| 99国产精品99久久久久| 久久国产乱子伦精品免费另类| 久久中文字幕人妻熟女| 欧美另类亚洲清纯唯美| 亚洲三区欧美一区| 欧美激情极品国产一区二区三区| 久9热在线精品视频| 欧美一级a爱片免费观看看 | 国产一级毛片七仙女欲春2 | bbb黄色大片| 看片在线看免费视频| 男人舔女人的私密视频| 免费女性裸体啪啪无遮挡网站| 女人被狂操c到高潮| 丁香六月欧美| 欧美最黄视频在线播放免费| 人妻丰满熟妇av一区二区三区| 亚洲国产精品久久男人天堂| 夜夜躁狠狠躁天天躁| 黄色毛片三级朝国网站| 成人永久免费在线观看视频| 亚洲欧美激情在线| 亚洲精品一区av在线观看| 久久亚洲精品不卡| 99在线人妻在线中文字幕| 一夜夜www| 欧美成狂野欧美在线观看| 可以免费在线观看a视频的电影网站| 制服人妻中文乱码| 亚洲午夜理论影院| 国产一级毛片七仙女欲春2 | 91国产中文字幕| 黄色a级毛片大全视频| 亚洲精品国产区一区二| 黄色视频不卡| 精品久久蜜臀av无| 欧美成人午夜精品| 欧美激情极品国产一区二区三区| www国产在线视频色| 亚洲va日本ⅴa欧美va伊人久久| 一级a爱片免费观看的视频| 久久精品国产亚洲av高清一级| www.精华液| 一边摸一边抽搐一进一出视频| 国产成人系列免费观看| 91国产中文字幕| 啦啦啦韩国在线观看视频| 国产精品免费视频内射| 国产成年人精品一区二区| 69精品国产乱码久久久| 亚洲 欧美一区二区三区| 亚洲精品在线美女| 久久香蕉国产精品| 国产成人av教育| 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| 国产成人精品久久二区二区91| 熟女少妇亚洲综合色aaa.| 一级黄色大片毛片| 亚洲人成77777在线视频| 国产单亲对白刺激| 欧美精品亚洲一区二区| 欧美激情 高清一区二区三区| 亚洲七黄色美女视频| 成年女人毛片免费观看观看9| 国产成人av激情在线播放| 欧美黑人欧美精品刺激| 久久精品aⅴ一区二区三区四区| 1024香蕉在线观看| 99在线人妻在线中文字幕| 亚洲欧美一区二区三区黑人| 老司机午夜十八禁免费视频| 无限看片的www在线观看| 一区福利在线观看| 国产色视频综合| 国产成人免费无遮挡视频| 欧美日本亚洲视频在线播放| 高清毛片免费观看视频网站| 啦啦啦韩国在线观看视频| 婷婷精品国产亚洲av在线| 免费女性裸体啪啪无遮挡网站| 精品久久久久久成人av| 久久香蕉精品热| 国产午夜精品久久久久久| 黄色片一级片一级黄色片| 美国免费a级毛片| 国产一区在线观看成人免费| 又大又爽又粗| 9191精品国产免费久久| 日韩欧美一区二区三区在线观看| 国产成人一区二区三区免费视频网站| 久热爱精品视频在线9| 十八禁网站免费在线| 免费在线观看亚洲国产| 欧美成人午夜精品| 亚洲性夜色夜夜综合| 国产亚洲欧美精品永久| 午夜视频精品福利| 免费人成视频x8x8入口观看| 可以在线观看毛片的网站| 男女午夜视频在线观看| 亚洲成av人片免费观看| 不卡av一区二区三区| 久久天堂一区二区三区四区| 97超级碰碰碰精品色视频在线观看| 亚洲色图av天堂| 99热只有精品国产| 亚洲成国产人片在线观看| 亚洲五月色婷婷综合| 亚洲精品粉嫩美女一区| 精品午夜福利视频在线观看一区| 妹子高潮喷水视频| 搡老妇女老女人老熟妇| 亚洲欧美日韩高清在线视频| 美女午夜性视频免费| 亚洲人成电影观看| 精品卡一卡二卡四卡免费| 日韩精品免费视频一区二区三区| 精品国产一区二区久久| av免费在线观看网站| 精品一区二区三区四区五区乱码| 变态另类成人亚洲欧美熟女 | 久久人妻av系列| 日韩欧美国产一区二区入口| 18禁裸乳无遮挡免费网站照片 | 两性午夜刺激爽爽歪歪视频在线观看 | 韩国av一区二区三区四区| 男女下面进入的视频免费午夜 | 亚洲人成伊人成综合网2020| 国产免费男女视频| av视频免费观看在线观看| 国产av又大| 欧美乱色亚洲激情| 欧美日本中文国产一区发布| 国产单亲对白刺激| 国产99白浆流出| 精品国产乱码久久久久久男人| 成年女人毛片免费观看观看9| 精品国产亚洲在线| 欧美在线一区亚洲| 欧美乱码精品一区二区三区| 国产欧美日韩精品亚洲av| 久久精品国产99精品国产亚洲性色 | 亚洲国产看品久久| 久久午夜亚洲精品久久| 欧美不卡视频在线免费观看 | 精品人妻在线不人妻| 欧美日韩福利视频一区二区| 在线观看66精品国产| 久久久国产精品麻豆| 成人亚洲精品av一区二区| 老司机靠b影院| 18禁美女被吸乳视频| 怎么达到女性高潮| 他把我摸到了高潮在线观看| 国产精品秋霞免费鲁丝片| 欧美日韩精品网址| 国产av精品麻豆| 久久久久国产一级毛片高清牌| 黄片播放在线免费| 丝袜人妻中文字幕| 久久久久久亚洲精品国产蜜桃av| 91精品国产国语对白视频| 亚洲伊人色综图| 18禁美女被吸乳视频| 精品日产1卡2卡| 一本综合久久免费| 国产男靠女视频免费网站| 88av欧美| 国产三级黄色录像| 在线观看免费午夜福利视频| 黄色a级毛片大全视频| 日韩精品青青久久久久久| 狠狠狠狠99中文字幕| 欧美大码av| 久久久久亚洲av毛片大全| 最近最新中文字幕大全免费视频| 啦啦啦韩国在线观看视频| 久久国产乱子伦精品免费另类| 欧美成人一区二区免费高清观看 | 成人永久免费在线观看视频| 91精品三级在线观看| 两人在一起打扑克的视频| 欧美国产精品va在线观看不卡| 1024香蕉在线观看| 精品久久蜜臀av无| 国产成人欧美| 国产精品 国内视频| 99精品在免费线老司机午夜| 婷婷六月久久综合丁香| 免费不卡黄色视频| 国产一级毛片七仙女欲春2 | 日韩大码丰满熟妇| 午夜福利影视在线免费观看| 久久久水蜜桃国产精品网| 午夜久久久在线观看| 亚洲成人久久性| 亚洲国产欧美一区二区综合| 99国产精品一区二区三区| 日韩三级视频一区二区三区| 两个人看的免费小视频| 丰满的人妻完整版| 免费久久久久久久精品成人欧美视频| 天天躁夜夜躁狠狠躁躁| 后天国语完整版免费观看| 欧美激情久久久久久爽电影 | 色老头精品视频在线观看| 亚洲无线在线观看| 怎么达到女性高潮| 88av欧美| 黑人欧美特级aaaaaa片| 丰满的人妻完整版| 日本在线视频免费播放| 国产xxxxx性猛交| 精品国产一区二区久久| 色哟哟哟哟哟哟| 免费人成视频x8x8入口观看| 亚洲最大成人中文| 午夜亚洲福利在线播放| 在线观看舔阴道视频| 黄色女人牲交| 在线观看日韩欧美| 国产成人精品久久二区二区91| 国产野战对白在线观看| 黑丝袜美女国产一区| 欧美日韩一级在线毛片| 一级毛片高清免费大全| 亚洲自拍偷在线| 国产精品久久久久久人妻精品电影| 高清在线国产一区| 动漫黄色视频在线观看| 久久久国产精品麻豆| 久久久国产成人精品二区| 成人国产综合亚洲| 黄片小视频在线播放| 国产熟女午夜一区二区三区| 国产亚洲精品av在线| 精品国产一区二区三区四区第35| 99riav亚洲国产免费| 久久人妻av系列| 欧美+亚洲+日韩+国产| 久久精品国产亚洲av高清一级| 国产亚洲欧美精品永久| 日本黄色视频三级网站网址| 久热这里只有精品99| 成人手机av| 久久国产精品人妻蜜桃| 国产99白浆流出| 波多野结衣av一区二区av| 国产av一区二区精品久久| 18禁观看日本| 琪琪午夜伦伦电影理论片6080| 黄色视频,在线免费观看| 久久午夜亚洲精品久久| 久久热在线av| 9热在线视频观看99| 国产免费av片在线观看野外av| 好男人在线观看高清免费视频 | 国产在线精品亚洲第一网站| 成人国产一区最新在线观看| 国产亚洲精品久久久久5区| 欧美另类亚洲清纯唯美| 激情视频va一区二区三区| 久久久久久国产a免费观看| 亚洲免费av在线视频| 亚洲av五月六月丁香网| 国产精品久久久久久人妻精品电影| 精品久久久精品久久久| 黑丝袜美女国产一区| 国产1区2区3区精品| 亚洲成av片中文字幕在线观看| 色尼玛亚洲综合影院| 18禁观看日本| 色精品久久人妻99蜜桃| 久久久久久久久中文| 久久精品国产亚洲av香蕉五月| 18禁黄网站禁片午夜丰满| 欧美国产精品va在线观看不卡| 欧美老熟妇乱子伦牲交| x7x7x7水蜜桃| 九色亚洲精品在线播放| 美女高潮喷水抽搐中文字幕| 美女 人体艺术 gogo| 成年版毛片免费区| 一二三四社区在线视频社区8| 亚洲一区高清亚洲精品| 亚洲中文字幕日韩| 欧美激情极品国产一区二区三区| 色综合欧美亚洲国产小说| 999精品在线视频| 精品国产乱码久久久久久男人| 欧美国产日韩亚洲一区| 午夜视频精品福利| 国产成人精品无人区| 国产99久久九九免费精品| 成人三级黄色视频| 午夜福利视频1000在线观看 | 久久久久久久午夜电影| 黄色毛片三级朝国网站| 露出奶头的视频| 看片在线看免费视频| 国产午夜精品久久久久久| 91大片在线观看| 啦啦啦免费观看视频1| 别揉我奶头~嗯~啊~动态视频| 两性午夜刺激爽爽歪歪视频在线观看 | 亚洲五月婷婷丁香| 欧美日本视频| 在线免费观看的www视频| 免费在线观看亚洲国产| 麻豆成人av在线观看| 成人免费观看视频高清| 成人18禁在线播放| 精品人妻1区二区| 男人的好看免费观看在线视频 | 午夜免费激情av| 色综合婷婷激情| 国产蜜桃级精品一区二区三区| 桃色一区二区三区在线观看| 色在线成人网| 美女扒开内裤让男人捅视频| 精品国产美女av久久久久小说| √禁漫天堂资源中文www| 久久婷婷人人爽人人干人人爱 | 日韩三级视频一区二区三区| 亚洲中文日韩欧美视频| 俄罗斯特黄特色一大片| 老司机深夜福利视频在线观看| 日本免费a在线| 国产成+人综合+亚洲专区| 香蕉丝袜av| 亚洲中文日韩欧美视频| 免费观看人在逋| 成人永久免费在线观看视频| 久久久久久久精品吃奶| 女同久久另类99精品国产91| 好男人在线观看高清免费视频 | 美女高潮到喷水免费观看| 亚洲 欧美一区二区三区| 长腿黑丝高跟| 久久久久久久久中文| 日韩视频一区二区在线观看| 欧美绝顶高潮抽搐喷水| 啪啪无遮挡十八禁网站| 久热爱精品视频在线9| 高清毛片免费观看视频网站| 99久久久亚洲精品蜜臀av| 日韩有码中文字幕| 99国产极品粉嫩在线观看| aaaaa片日本免费| 亚洲全国av大片| 18禁黄网站禁片午夜丰满| 久久人人97超碰香蕉20202| 午夜老司机福利片| av中文乱码字幕在线| 女人精品久久久久毛片| 国产精品香港三级国产av潘金莲| 丰满人妻熟妇乱又伦精品不卡| 亚洲第一青青草原| 脱女人内裤的视频| 90打野战视频偷拍视频| 久久精品亚洲熟妇少妇任你| 电影成人av| 麻豆成人av在线观看| 两性夫妻黄色片| 国产一级毛片七仙女欲春2 | e午夜精品久久久久久久| 老司机靠b影院| 视频在线观看一区二区三区| 女人爽到高潮嗷嗷叫在线视频| 亚洲国产看品久久| 午夜老司机福利片| 精品一区二区三区视频在线观看免费| 日日干狠狠操夜夜爽| 亚洲av成人一区二区三| 天堂√8在线中文| 国产精品一区二区在线不卡| 日本欧美视频一区| 精品一区二区三区四区五区乱码| 亚洲精品国产区一区二| 视频在线观看一区二区三区| 色婷婷久久久亚洲欧美| 极品教师在线免费播放| 亚洲精品av麻豆狂野| 国产精品免费一区二区三区在线| 成年女人毛片免费观看观看9| 成人av一区二区三区在线看| 国产av又大| 韩国精品一区二区三区| 精品日产1卡2卡| 精品国产美女av久久久久小说| 夜夜躁狠狠躁天天躁| 久久精品人人爽人人爽视色| 天天一区二区日本电影三级 | 老汉色∧v一级毛片| av福利片在线| 国产成人系列免费观看| 久久午夜综合久久蜜桃| 色综合站精品国产| 欧美精品啪啪一区二区三区| 桃红色精品国产亚洲av| 国产精品久久久久久亚洲av鲁大| 欧美日韩亚洲国产一区二区在线观看| 黄片小视频在线播放| 看片在线看免费视频| 国产亚洲精品久久久久5区| av免费在线观看网站| 午夜精品在线福利| 涩涩av久久男人的天堂| 亚洲国产精品久久男人天堂| 国产一区二区三区综合在线观看| 午夜福利影视在线免费观看| 亚洲精品久久国产高清桃花| 91字幕亚洲| 狂野欧美激情性xxxx| 日韩欧美一区二区三区在线观看| 免费在线观看视频国产中文字幕亚洲| 天天添夜夜摸| 国产亚洲精品综合一区在线观看 | 9色porny在线观看| 国产一区在线观看成人免费| 亚洲欧美激情综合另类| 丝袜在线中文字幕| 很黄的视频免费| av在线播放免费不卡| 国内久久婷婷六月综合欲色啪| 国产精品久久久久久精品电影 | 亚洲在线自拍视频| 国产精品免费一区二区三区在线| 国产人伦9x9x在线观看| 精品人妻在线不人妻| 高潮久久久久久久久久久不卡| 极品教师在线免费播放| 变态另类成人亚洲欧美熟女 | 巨乳人妻的诱惑在线观看| 国产aⅴ精品一区二区三区波| 国产精品日韩av在线免费观看 | 午夜福利视频1000在线观看 | 丰满人妻熟妇乱又伦精品不卡| 变态另类丝袜制服| 亚洲va日本ⅴa欧美va伊人久久| 午夜免费激情av| 午夜视频精品福利| 好男人在线观看高清免费视频 | 久久久国产欧美日韩av| 美女 人体艺术 gogo| 免费在线观看影片大全网站| 久久九九热精品免费| 久久久久久免费高清国产稀缺| 如日韩欧美国产精品一区二区三区| 麻豆成人av在线观看| 丝袜人妻中文字幕| 亚洲精品一卡2卡三卡4卡5卡| 一级片免费观看大全| 国产精品99久久99久久久不卡| 满18在线观看网站| 99国产精品一区二区三区| 国产伦人伦偷精品视频| 国产亚洲精品一区二区www| 美女高潮到喷水免费观看| 麻豆一二三区av精品| 视频区欧美日本亚洲| 窝窝影院91人妻| 欧美激情极品国产一区二区三区| 亚洲精品av麻豆狂野| 日韩视频一区二区在线观看| 一二三四在线观看免费中文在| 18禁黄网站禁片午夜丰满| 男女下面进入的视频免费午夜 | 99国产精品一区二区三区| 大香蕉久久成人网| 久久国产乱子伦精品免费另类| 国产aⅴ精品一区二区三区波| 亚洲性夜色夜夜综合| 又紧又爽又黄一区二区| 麻豆国产av国片精品| 天天躁夜夜躁狠狠躁躁| 非洲黑人性xxxx精品又粗又长| 亚洲精品国产区一区二| 国产主播在线观看一区二区| 精品乱码久久久久久99久播| 国产激情欧美一区二区| 亚洲av电影在线进入| netflix在线观看网站| 亚洲七黄色美女视频| 亚洲欧美日韩另类电影网站| 午夜精品在线福利| 日韩欧美国产一区二区入口| 久久久久国产一级毛片高清牌| 国产av一区二区精品久久| 欧美久久黑人一区二区| 久久久久久久午夜电影| 久久精品亚洲精品国产色婷小说| 51午夜福利影视在线观看| 精品少妇一区二区三区视频日本电影| 日韩大码丰满熟妇| 久久久久久人人人人人| 日本vs欧美在线观看视频| 国产又爽黄色视频| 成人18禁在线播放| 久久精品国产清高在天天线| 亚洲精品av麻豆狂野| 国产精品av久久久久免费| 亚洲欧美精品综合一区二区三区| 一本久久中文字幕| 一进一出抽搐gif免费好疼| 精品国产一区二区久久| 人人妻,人人澡人人爽秒播| 欧美色欧美亚洲另类二区 | 18美女黄网站色大片免费观看| 身体一侧抽搐| 精品国产一区二区久久| 97人妻精品一区二区三区麻豆 | 一本久久中文字幕| 成人特级黄色片久久久久久久| 给我免费播放毛片高清在线观看| 亚洲精品国产区一区二| 久99久视频精品免费| 免费看a级黄色片| 色综合欧美亚洲国产小说| 国产av一区二区精品久久| 国产在线精品亚洲第一网站| 亚洲精品国产精品久久久不卡| 99久久99久久久精品蜜桃| www.999成人在线观看| 啪啪无遮挡十八禁网站| 91九色精品人成在线观看| 性色av乱码一区二区三区2| 国产精品乱码一区二三区的特点 | 好看av亚洲va欧美ⅴa在| 一个人观看的视频www高清免费观看 | 亚洲人成电影免费在线| av中文乱码字幕在线| 女人被狂操c到高潮| 国产在线观看jvid| 亚洲第一电影网av| 91精品三级在线观看| 国产激情欧美一区二区| 天天添夜夜摸| 国产精品国产高清国产av| 久久精品成人免费网站| 成人av一区二区三区在线看| 日本撒尿小便嘘嘘汇集6| 啦啦啦免费观看视频1| 正在播放国产对白刺激| АⅤ资源中文在线天堂| 午夜福利影视在线免费观看| 精品免费久久久久久久清纯| 精品不卡国产一区二区三区| 国产亚洲av嫩草精品影院| 国产男靠女视频免费网站| 亚洲成国产人片在线观看| 久久狼人影院| 欧美国产日韩亚洲一区| 在线天堂中文资源库| 欧美绝顶高潮抽搐喷水| 一本久久中文字幕| 老司机靠b影院| 无人区码免费观看不卡| 久久人人精品亚洲av| 老司机午夜福利在线观看视频|