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Existence and Uniqueness Results for a Fully Third-Order Boundary Value Problem
Department of Mathematics, Northwest Normal University, Lanzhou, China
Department of Mathematics, Northwest Normal University, Lanzhou, China
- 1 Department of Mathematics, Northwest Normal University, Lanzhou, China
- 2 Department of Mathematics, Northwest Normal University, Lanzhou, China
Advances in Pure Mathematics·Volume 13 (2023)·Pages 495–503·Published 11 August 2023·DOI10.4236/apm.2023.138033
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Abstract
The boundary value problems of the third-order ordinary differential equation have many practical application backgrounds and their some special cases have been studied by many authors. However, few scholars have studied the boundary value problems of the complete third-order differential equations u ′′′( t ) = f ( t , u ( t ), u ′( t ), u ′′( t )) . In this paper, we discuss the existence and uniqueness of solutions and positive solutions of the fully third-order ordinary differential equation on [0,1] with the boundary condition u (0) = u ′(1) = u ′′(1) = 0 . Under some inequality conditions on nonlinearity f some new existence and uniqueness results of solutions and positive solutions are obtained.
KeywordsFully Third-Order BVPSolutionPositive SolutionExistence and UniquenessInequality Conditions
- Greguš, M. (1987) Third Order Linear Differential Equations. Reidel, Dordrecht.
- Greguš, M. (1994) Two Sorts of Boundary-Value Problems of Nonlinear Third Order Differential Equations. Archivum Mathematicum, 30, 285-292. http://dml.cz/dmlcz/107515
- O’Regan, D.J. (1987) Topological Transversality: Application to Third-Order Boundary Value Problems. SIAM Journal on Mathematical Analysis, 18, 630-641. https://doi.org/10.1137/0518048
- Omari, P. and Trombetta, M. (1992) Remarks on the Lower and Upper Solutions Method for Second and Third-Order Periodic Boundary Value Problems. Applied Mathematics and Computation, 50, 1-21. https://doi.org/10.1016/0096-3003(92)90007-N
- Cabada, A. (1994) The Method of Lower and Upper Solutions for Second, Third, Forth, and Higher Order Boundary Value Problems. Journal of Mathematical Analysis and Applications, 185, 302-320. https://doi.org/10.1006/jmaa.1994.1250
- Cabada, A. (1995) The Method of Lower and Upper Solutions for Third-Order Periodic Boundary Value Problems. Journal of Mathematical Analysis and Applications, 195, 568-589. https://doi.org/10.1006/jmaa.1995.1375
- Cabada, A. and Lois, S. (1999) Existence of Solution for Discontinuous Third Order Boundary Value Problems. Journal of Computational and Applied Mathematics, 110, 105-114. https://doi.org/10.1016/S0377-0427(99)00199-5
- Cabada, A., Grossinho, M.R. and Minhos, F. (2003) On the Solvability of Some Discontinuous Third Order Nonlinear Differential Equations with Two Point Boundary Conditions. Journal of Mathematical Analysis and Applications, 285, 174-190. https://doi.org/10.1016/S0022-247X(03)00388-3
- Grossinho, M.R. and Minhós, F.M. (2001) Existence Result for Some Third Order Separated Boundary Value Problems. Nonlinear Analysis: Theory, Methods & Applications, 47, 2407-2418. https://doi.org/10.1016/S0362-546X(01)00364-9
- Grossinho, M.R., Minhos, F.M. and Santos, A.I. (2005) Solvability of Some Third-Order Boundary Value Problems with Asymmetric Unbounded Nonlinearities. Nonlinear Analysis: Theory, Methods & Applications, 62, 1235-1250. https://doi.org/10.1016/j.na.2005.04.029
- Grossinho, M.R., Minhos, F.M. and Santos, A.I. (2005) Existence Result for a Third-Order ODE with Nonlinear Boundary Conditions in Presence of a Sign-Type Nagumo Control. Journal of Mathematical Analysis and Applications, 309, 271-283. https://doi.org/10.1016/j.jmaa.2005.01.030