Journal of Nonlinear Sciences and Applications(JNSA)Journal of Nonlinear Sciences and ApplicationsJNSA 2008-1898 2008-1901International Scientific Research PublicationsJohor, Malaysiainfo@isr-publications.comisr-publications.comisr-publications.com/jnsa10.22436/jnsa.001.01.06MULTIPLE POSITIVE SOLUTIONS FOR NONLINEAR SINGULAR THIRD-ORDER BOUNDARY VALUE PROBLEM IN ABSTRACT SPACESZHANGFANG
School of Mathematics and Physics, Jiangsu Polytechnic University, Changzhou, 213164, PR China.
1503200811364436-44https://www.isr-publications.com/jnsa/1457/download-multiple-positive-solutions-for-nonlinear-singular-third-order-boundary-value-problem-in-abstract-spaceshttps://www.isr-publications.com/jnsa/articles-1457-multiple-positive-solutions-for-nonlinear-singular-third-order-boundary-value-problem-in-abstract-spaces

In this paper, we study the nonlinear singular boundary value problem in abstract spaces: $\begin{cases} u''' + f(t, u) = \theta,\,\,\,\,\, t \in (0, 1),\\ u(0) = u'(0) = \theta, u'(1) = \xi u'(\eta), \end{cases}$ where $$0 < \eta< 1$$ and $$1 < \xi<\frac{1}{\eta}, \theta$$ denotes the zero element of $$E, E$$ is a real Banach space, and $$f(t, u)$$ is allowed to be singular at both end point $$t = 0$$ and $$t = 1$$. We show the existence of at least two positive solutions of this problem.

34G2034B16Singular boundary value problemAbstract spacesPositive solutionsFixed point theorem.
M. GregusThird Order Linear Differential Equations in: Math. Appl., Reidel, Dordrecht1987KlaasenG.Differential inequalities and existence theorems for second and third order boundary value problems10.1016/0022-0396(71)90010-6J. Diff. Equs.197110529537JacksonL. K. Existence and uniqueness of solutions of boundary value problems for third order differential equations J. Diff. Equs. 199313432437D. J. OReganTopological transversality: Application to third order boundary value problems10.1137/0518048SIAM J. Math. Anal.198719630641CabadaA.The method of lower and upper solutions for second, third, fourth and higher order boundary value problems10.1006/jmaa.1994.1250 J. Math. Anal. Appl. 1994185302320A. CabadaThe method of lower and upper solutions for third order periodic boundary value problems10.1006/jmaa.1995.1375 J. Math. Anal. Appl. 1995195568589CabadaA. S. LoisExistence of solution for discontinuous third order boundary value problems10.1016/S0377-0427(99)00199-5J. Comput. Appl. Math.1999110105114CabadaA.Heikkil’a S. Extremality and comparison results for third order functional initial-boundary value problems10.1006/jmaa.2000.7232J. Math. Anal. Appl. 2001255195212CabadaA.Heikkil’a S. Extremality and comparison results for discontinuous implicit third order functional initial-boundary value problems10.1016/S0096-3003(02)00236-9Appl. Math. Comput.2003140 391407YaoQ. Solution and positive solution for a semilinear third-order two-point boundary value problem10.1016/j.aml.2003.09.011Appl. Math. Lett. 20041711711175SunY.Existence of positive solutions for nonlinear third-order three-point boundary value problemJ. Math. Anal. Appl. 2005306589603Guo L.Sun J. Y. ZhaoExistence of positive solutions for nonlinear third-order three-point boundary value problem 10.1016/j.na.2007.03.008Nonlinear Anal.20086831513158K. DeimlingOrdinary differential equations in Banach spaces LNM 886. , Berlin: Springer-Verlag, New York1987LakshmikanthamV.Leela S.Nonlinear differential equations in abstract spacesPergamon, Oxford1981GuoD. V. LakshmikanthamNonlinear Problems in Abstract ConesAcademic Press , Boston, MA1988GuoD.Lakshmikantham V. X. LiuNonlinear Integral Equations in Abstract SpacesKluwer Academic Publishers, Dordrecht, 1996