zbMATH — the first resource for mathematics

Solutions to a three-point boundary value problem. (English) Zbl 1222.34022
Authors’ abstract: By using the fixed-point index theory and the Leggett-Williams fixed-point theorem, we study the existence of multiple solutions to the three-point boundary value problem
\[ u'''(t)+a(t)f(t,u(t),u'(t))=0,\quad 0<t<1, \]
\[ u(0)=u'(0)=0,\quad u'(1)-\alpha u'(\eta)=\lambda, \]
where \(\eta\in(0,\frac12]\), \(\alpha\in[-\frac{1}{2\eta},\frac{1}{\eta})\) are constants, \(\lambda\in(0,\infty)\) is a parameter, and \(a, f\) are given functions. New existence theorems are obtained, which extend and complement some existing results. Examples are also given to illustrate our results.
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI EuDML
[1] Agarwal RP: Focal Boundary Value Problems for Differential and Difference Equations, Mathematics and Its Applications. Volume 436. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998:x+289.
[2] Bisplinghoff RL, Ashley H: Principles of Aeroelasticity. Dover Publications, Mineola, NY, USA; 2002. · Zbl 0114.19802
[3] Henderson J (Ed): Boundary Value Problems for Functional-Differential Equations. World Scientific, River Edge, NJ, USA; 1995:x+306. · Zbl 0834.00035
[4] Anderson, DR, Green’s function for a third-order generalized right focal problem, Journal of Mathematical Analysis and Applications, 288, 1-14, (2003) · Zbl 1045.34008
[5] Avery, RI; Peterson, AC, Three positive fixed points of nonlinear operators on ordered Banach spaces, Computers & Mathematics with Applications, 42, 313-322, (2001) · Zbl 1005.47051
[6] Boucherif, A; Al-Malki, N, Nonlinear three-point third-order boundary value problems, Applied Mathematics and Computation, 190, 1168-1177, (2007) · Zbl 1134.34007
[7] Karakostas, GL; Mavridis, KG; Tsamatos, PC, Triple solutions for a nonlocal functional boundary value problem by Leggett-Williams theorem, Applicable Analysis, 83, 957-970, (2004) · Zbl 1081.34064
[8] Leggett, RW; Williams, LR, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Mathematics Journal, 28, 673-688, (1979) · Zbl 0421.47033
[9] Sun, Y, Positive solutions for third-order three-point nonhomogeneous boundary value problems, Applied Mathematics Letters, 22, 45-51, (2009) · Zbl 1163.34313
[10] Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275. · Zbl 0661.47045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.