亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        Existence of Positive Solutions for Eigenvalue Problems of Fourth-order Elastic Beam Equations

        2017-06-05 15:01:17LUHaixia

        LU Hai-xia

        (School of Arts and Science,Suqian College,Suqian 223800,China)

        Existence of Positive Solutions for Eigenvalue Problems of Fourth-order Elastic Beam Equations

        LU Hai-xia

        (School of Arts and Science,Suqian College,Suqian 223800,China)

        In this paper,we investigate the positive solutions of fourth-order elastic beam equations with both end-points simply supported.By using the approximation theorem of completely continuous operators and the global bifurcation techniques,we obtain the existence of positive solutions of elastic beam equations under some conditions concerning the first eigenvalues corresponding to the relevant linear operators,when the nonlinear term is non-singular or singular,and allowed to change sign.

        elastic beam equations;singular;positive solutions;global bifurcation

        §1.Introduction and Preliminaries

        In this paper,we consider the existence of positive solutions of the following fourth-order two-point boundary value problem

        where λ is a positive parameter,f:[0,1]×R1→R1is continuous and may be singular at t=0,1.

        Fourth-order two-point boundary value problems are useful for material mechanics because the problems usually characterize the deflection of an elastic beam.In mechanics,the problem (1.1)describes the deflection of an elastic beam with both end-points simply supported.The existence of positive solutions for the elastic beam equations has been studied extensively,see for example[19]and references therein,when the nonlinear term satisfies

        And the major methods used are upper and lower solution method,contraction mapping and iterative technique,Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type, topological degree theory and global bifurcation technique.However,when(1.2)is not satisfied, which means the nonlinear term is allowed to change sign,there are only a few papers concerned with the fourth-order boundary value problems.Yao[10]considered the existence of positive solutions of elastic beam equations by constructing control functions and a special cone and using fixed point theorem of cone expansion-compression type.Lu[11]obtained the existence of positive solutions by the topological degree and fixed point theory of nonlinear operator on lattice.

        In this paper,by using the approximation theorem of completely continuous operators and the global bifurcation techniques,we study the existence of positive solutions of the problem (1.1)under some conditions concerning the first eigenvalues corresponding to the relevant linear operators,when the nonlinear term f is non-singular or singular and in the case that(1.2)is not satisfied.The method and results in this paper improve those in[10-11].

        For the remainder of this section,we present some definitions and lemmas which are used in Section 2 and Section 3.

        Let E be a Banach space,P be a cone of E.

        Definition 1.1[12-13]Let B:E→E be a linear operator.B is said to be a u0-bounded operator,if there exists u0∈P{θ},such that for any x∈P{θ},there exist a natural number n and real numbers ζ,η>0,such that

        Lemma 1.1[12-13]Let B be a completely continuous u0-bounded operator,λ1>0 be the first eigenvalue of B,then B must have a positive eigenfunction∈P{θ},corresponding to λ1, and λ1is the unique positive eigenvalue of B corresponding to positive eigenfunction.

        Lemma 1.2[14]Let B be a completely continuous u0-bounded operator,A:E→E be an operator(we don’t suppose A maps P to P).If there exist ?0∈P{θ}and λ>0 such that A?0>B?0,λA?0=?0,then λ<λ1,where λ1>0 is the first eigenvalue of B.

        Let X be a Banach space and{Cn|n=1,2,···}be a family of connected subsets of X, we define

        Lemma 1.3[14]Suppose that the following conditions are satisfied

        (1)There exist zn∈Cn(n=1,2,···)and z?∈X,such that zn→z?;

        (2)rn→+∞(n→∞),where rn=sup{‖x‖|x∈Cn};

        Then there must exist an unbounded connected component C in D and z?∈C.

        §2.Existence of Positive Solutions in the Case That f is Not Singular

        In this section we consider the boundary value problem(1.1)in the case that f is not singular and f(t,u)=a(t)u+F(t,u).

        We assume that

        (H1)a∈C[0,1]with a(t)≥0 on[0,1]and a(t)/≡0 on any subinterval of[0,1];

        (H3)There exists α∈(-∞,+∞),such that

        where it is not supposed that f(t,u)≥0(u≥0).

        Let

        It is easy to verify that G(t,s)is nonnegative continuous and for t,s∈[0,1]×[0,1],

        It is obvious that(1.1)can be converted to the following integral equation

        It is easy to see that A:C[0,1]→C[0,1]is a completely continuous operator.Evidently,the fixed point of λA is the solution of(1.1).

        We see from(H2)that the linearization of the boundary value problem(1.1)is

        By Theorem 2.3 in Ma[15],we have

        Lemma 2.1Suppose(H1),(H2)are satisfied.Then

        (1)Problem(2.3)has an infinite sequence of positive eigenvalues

        (2)To each eigenvalue λkthe algebraic is 1 and there corresponds an essential eigenfunction ?kwhich has exactly k-1 simple zero in(0,1)and is positive near 0.

        Let

        By Rabinowitz[16],Lemma 2 in Sun[17]and Lemma 2.1 we know that

        Lemma 2.2Suppose(H1),(H2)are satisfied,then C+1is an unbounded connected component of((0,+∞)×S+1)∪{(λ1,θ)}in R1×C[0,1].

        Define the linear operator

        Lemma 2.3Operator B defined by(2.4)is a u0-bounded operator.

        Let u0(t)=P(t),t∈[0,1].For any u∈P{θ},we have

        which means the linear operator B is u0-bounded operator.

        Let r(B)and λBdenote the spectral radius and the first eigenvalue of B respectively,then λB=(r(B))?1.

        By(H3),there exists M0>0 such that

        Theorem 2.2Suppose that(H1)~(H3)hold and α≤0 in(H3),then for any λ∈(λ1,+∞),the boundary value problem(1.1)has at least a positive solution.

        §3.Existence of Positive Solutions in the Case That f is Singular

        In this section we consider the boundary value problem(1.1)in the case that f(t,u)= h(t)g(u)and h is allowed to be singular at t=0 or t=1.i.e.,

        We assume that

        Define nonlinear operator A and linear operator B

        Then the fixed point of λA is the solution of(3.1)~(3.2).

        For any natural number n(n≥2),we set

        Then hn:[0,1]→[0,+∞)is continuous and hn(t)≤h(t),t∈(0,1).Let

        then An,Bn:C[0,1]→C[0,1]are continuous.And the boundary value problem(3.5),(3.2)and (3.6),(3.2)can be converted into the following nonlinear integral equation and linear integral equation u(t)=λAnu(t)and u(t)=λBnu(t),respectively.

        Then we have the following lemma.

        Similarly,B:C[0,1]→C[0,1]is completely continuous.

        Lemma 3.2Suppose that(H′3)is satisfied,then operators B and Bndefined by(3.3)and (3.7)are u0-bounded operators.

        ProofBy(2.2)and(H′3)and by the same method as the proof of Lemma 2.3,we know that Lemma 3.2 holds.

        Let λ1and λ1n(n=2,3,···)denote the first eigenvalue of u0-bounded linear operators B and Bnrespectively,then λ1>0 and λ1n>0 and λ1=(r(B))?1,λ1n=(r(Bn))?1(n= 2,3,···),where(r(B))?1and(r(Bn))?1denote the spectral radius of linear operators B and Bnrespectively.

        (i)D is the subset of the solution of the boundary value problem(3.1),(3.2)and

        For any(λ,u)∈D,it follows the definition of D that there exist the subsequence{nk}?{n} and(λnk,unk)∈C+1nk,such that λnk→λ,unk→u.Thus{λnk}and{unk}are bounded.And by the proof of Lemma 3.1 we know Anuniformly converges to A on a bounded set.So

        which means u=λAu.So(i)holds.

        If(3.8)does not hold,then for any δ1:0<δ1<δ,there exist λ>λ1+ε0,u∈K and N2>N1, such that u=λAN2u,0<‖u‖<δ1.By(3.9)we have

        It follows from Lemma 3.2 τBN1,τBN2are u0-bounded linear operators.Let τ?1λ1N1and τ?1λ1N2be the first eigenvalue of τBN1and τBN2respectively.From(3.10)and Lemma 1.2 we know that λ<τ?1λ1N2.Since hN2≥hN1,then τBN2≥τBN1and so τ?1λ1N2≤τ?1λ1N1. Then

        which is a contradiction.Thus(3.8)holds.

        By(3.8)and the definition of D,we know that(ii)holds.

        It follows from Lemma 3.3 that(λ1n,θ)→(λ1,θ).Note that for any n≥2,C+1nis unbounded.Hence,by Lemma 1.3 there exists an unbounded connected component C in D, containing(λ1,θ).From the property(ii)of D we have

        By(3.11)and the same method as the proof of Theorem 2.1,we have

        which means Theorem 3.1 holds.

        It follows from the same method as the proof of Theorem 2.2,we have

        Theorem 3.2Suppose that(H′

        1)~(H′3)are satis fied and β≤0 in(H′2),then for any λ∈(λ1,+∞),the boundary value problem(3.1)~(3.2)has at least a positive solution.

        ExampleConsider the following fourth-order boundary value problem

        [1]AGARWAL R P,Chow Y M.Iterative method for fourth order boundary value problem[J].J Comput App Math,1984,10:203-217.

        [2]GUPTA C P.Existence and uniqueness results for the bending of an elastic beam equation[J].Appl Anal, 1988,26:289-304.

        [3]DALMASSO R.Uniqueness of positive solutions for some nonlinear four-order operators[J].J Math Anal Appl,1996,201:152-168.

        [4]GRAEF R,YANG B.Positive solutions of a nonlinear fourth order boundary value problem[J].Commun Appl Nonl Anal,2007,14:61-73.

        [5]KORMAN P.Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems[J]. Proc Roy Soc Edinburg Sect A,2004,134:179-190.

        [6]YAO Qing-liu.Positive solutions for eigenvalue problems of four-order elastic beam equations[J].Appl Math Lett,2004,17:237-243.

        [7]CUI Yu-jun,ZOU Yu-mei.Existence and uniqueness theorems for fourth-order singular boundary value problems[J].Comput Math Appl,2009,58:1449-1456.

        [8]ZHANG Yu-chuan,ZHOU Zong-fu.Positive solutions for fourth-order delay differential equation of boundary value problem with p-Laplacian[J].Chin Quart J of Math,2014,29:171-179.

        [9]MA Ru-yun,XU Jia.Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem[J].Nonlinear Anal,2010,72:113-122.

        [10]YAO Qing-liu.Existence of n solutions and/or positive solutions to a semipositive elastic beam equation[J]. Nonlinear Anal,2007,66:138-150.

        [11]LU Hai-xia,Sun Li,Sun Jing-xian.Existence of positive solutions to a non-positive elastic beam equation with both ends fixed[J].Boundary Value Problems,2012,56:1-10.

        [12]GUO Da-jun,SUN Jing-xian.Nonlinear Integral Equations[M].Jinan:Shandong Science and Technology Press,1987.

        [13]KRASNOSEL’SKII M A.Topological Methods in the Theory of Nonlinear Integral Equations[M].Oxford: Pergamon Press,1964.

        [14]SUN Jing-xian,LI Hong-yu.Global structure of positive solutions of singular nonlinear Sturm-Liouville problems[J].Acta Mathematica Scientia,2008,28A:424-433.

        [15]MA Ru-yun.Nodal solutions of boundary value problems of fourth-order ordinary differential equations[J]. J Math Anal Appl,2006,319:424-434.

        [16]RABINOWITZ P H.Some global results for nonlinear eigenvalue problems[J].J Functional Anal,1971,7: 487-513.

        [17]SUN Jing-xian.The existence of positive solutions for nonlinear Hammerstein integral equations and their applications[J].Ann Math Ser,1988,9A:90-96.

        tion:34B16,34B18

        :A

        1002–0462(2017)01–0007–09

        date:2016-05-13

        Supported by the National Natural Science Foundation of China(11501260);Supported by the National Natural Science Foundation of Suqian City(Z201444)

        Biography:LU Hai-xia(1976-),female,native of Jianhu,Jiangsu,an associate professor of Suqian College, M.S.D.,engages in nonlinear functional analysis.

        CLC number:O175.8

        国产黑丝美腿在线观看| 久久久国产精品免费无卡顿| 久久丁香花综合狼人| 免费人成在线观看播放视频| 久久亚洲av无码精品色午夜| 四川老熟妇乱子xx性bbw| 国产一级三级三级在线视| 亚洲av午夜福利一区二区国产| 青青手机在线观看视频| 中文字幕一区二区三区人妻少妇| 国产成人精品亚洲午夜| 北岛玲亚洲一区二区三区| 亚洲精品久久国产精品| 真实国产老熟女粗口对白| 熟女人妻丰满熟妇啪啪| 日韩av中文字幕波多野九色| 久久久久88色偷偷| 人人玩人人添人人澡| 美女窝人体色www网站| 日本人妻系列中文字幕| 久久99热狠狠色精品一区| 首页动漫亚洲欧美日韩| 亚洲一区二区视频蜜桃| 午夜天堂一区人妻| 无遮无挡爽爽免费视频| 日韩毛片久久91| 人妻夜夜爽天天爽三区麻豆av| 国产精品久久久国产盗摄| 日本成人一区二区三区| 一区二区三区观看在线视频| 国产精品视频永久免费播放| 亚洲熟女乱色一区二区三区| 国产毛片A啊久久久久| 久久伊人精品中文字幕有尤物 | 亚洲图片日本视频免费| 欧美va免费精品高清在线| 日本精品熟妇一区二区三区| 日本免费视频| 精品人妻系列无码一区二区三区| 国产精品白浆免费观看| 淫片一区二区三区av|