沈文国;

• 1:兰州工业学院基础学科部

摘要(Abstract)：

研究带Riemann-Stieltjes积分边值条件的奇异三阶积分边值问题正解的全局分歧结构.首先,利用相关文献,获得了此类问题的格林函数并推证其满足的性质,同时可获得此类问题等价于一个全连续算子方程;其次,在满足所给的条件时,利用Krein-Rutmann定理建立了此类问题对应的线性问题存在简单的主特征值;最后,当非线性项在零和无穷远处满足非渐进线性增长条件、参数满足不同范围的值时,利用Dancer全局分歧定理、Zeidler全局分歧定理和序列集取极限的方法,建立了此类问题正解的全局结构,进而获得了正解的存在性.

关键词(KeyWords)： 奇异三阶积分边值问题;全局分歧;正解

Abstract:

Keywords:

基金项目(Foundation): 国家自然科学基金(11561038);; 甘肃省自然科学基金(145RJZA087)

作者(Authors): 沈文国;

参考文献(References)：

• [1]Graef J R,Webb J R L.Third order boundary value problems with nonlocal boundary conditions[J].Nonlinear Anal.,2009,71:1542-1551.
• [2]Graef J R,Yang B.Positive solutions of a third order nonlocal boundary value problem[J].Discrete Contin.Dyn.Syst.Ser.,2008,1:89-97.
• [3]Du Z,Lin X,Ge W.Solvability of a third-order nonlocal boundary value problem at resonance[J].Acta Math.Sinica(Chin.Ser.),2006,49:87-94.
• [4]Li S.Positive solutions of nonlinear singular third-order two-point boundary value problem[J].J.Math.Anal.Appl.,2006,323:413-425.
• [5]Sun Y.Positive solutions of singular third-order three-point boundary value problem[J].J.Math.Anal.Appl.,2005,306:589-603.
• [6]Liu Z,Umeb J S,Kang S M.Positive solutions of a singular nonlinear third order two-point boundary value problem[J].J.Math.Anal.Appl.,2007,326(1):589-601.
• [7]Du Z,Ge W,Zhou M.Singular perturbations for third-order nonlinear multi-point boundary value problem[J].J.Differential Equations,2005,218(1):69-90.
• [8]Ma R,An Y.Global structure of positive for superlinear second-order m-point boundary value problems[J].Nonlinear Anal.,2009,34(2):279-290.
• [9]Shen W,He T.Global Structure of Positive Solutions for a Singular Fourth-Order Integral Boundary Value Problem[J].Discrete Dynamics in Nature and Society Volume 2014,Article ID 614376,7 pages.
• [10]Rynne B P.Infinitely many solutions of superlinear fourth order boundary value problems[J].Topol.Methods Nonlinear Anal.,2002,19(2):303-312.
• [11]Ma R,Nodal Solutions for a fourth-Order two-order boundary value problem[J].J.Math.Anal.Appl.,2006,314(1):254-265.
• [12]Shi J,Wang X.On global bifurcation for quasilinear elliptic systems on bounded domains[J].J.Differential Equations,2009,246:2788-2812.
• [13]Shen W.Global structure of nodal solutions for a fourth-order two-point boundary value problem[J].Appl.Math.Comput.,2012,219(1):88-98.
• [14]Dai G,Ma R.Unilateral global bifurcation phenomena and nodal solutions for p-Laplacian[J].J.Differential Equations,2012,252:2448-2468.
• [15]Dai G.Bifurcation and nodal solutions for p-Laplacian problems with non-asymptotic nonlinearity at 0 or∞[J].Appl.Math.Lett.,2013,26:46-50.
• [16]Dai G,Ma R.Unilateral global bifurcation for p-Laplacian with non-p-1-lineariza-tion nonlinearity[J].Discrete contin.dyn.syst.,2015,35(1):99-116.
• [17]Krasnosel′skii M A.Positive Solutions of Operator Equations[M].The Netherlands:P.Noordhoff Ltd.,1964.
• [18]Zhang G,Sun J.Positive solutions of m-point boundary value problems[J].J.Math.Anal.Appl.,2004,291:406-418.
• [19]Guo D,Sun J.Nonlinear Integral Equations[M].Ji′nan:Shandong Science and Technology Press,1987(in Chinese).
• [20]Whyburn G T.Topological Analysis[M].Princeton:Princeton University Press,1958.
• [21]Dancer E.Global solutions branches for positive maps[J].Arch.Rat.Mech.Anal.,1974,55:207-213.
• [22]Zeidler E.Nonlinear Functional Analysis and its Applications:I.Fixed Point Theorems[M].New York:Springer-Verlag,1986.
• [23]Ambrosetti A,Calahorrano R M,Dobarro F R.Global branching for discontinuous problems[J].Comment.Math.Univ.Carolin.,1990,31:13-222.