×

Extremal solutions for third-order nonlinear problems with upper and lower solutions in reversed order. (English) Zbl 1084.34013

The authors study the scalar nonlinear third-order boundary value problem in Laplacian form \[ -\biggl[\varphi\bigl(u''(t)\bigr)\biggr]'= f\bigl(t,u(t)\bigr),\;t\in[a,b];\quad u(a)=A,\;u''(a)=B,\;u''(b)=C. \] Here, \(\varphi:\mathbb{R}\to\mathbb{R}\) is an increasing homeomorphism, and \(f\) is a Carathéodory function. Employing the monotone iterative technique, sufficient conditions are obtained for the existence of minimal and maximal solutions. These extremal solutions are constructed as limits of some monotone sequences of functions. One of the imposed assumptions is the presence of a pair of lower and upper solutions given in the reversed order, that is, the lower solution is over the upper one.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Cabada, A., The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems, J. math. anal. appl., 185, 2, 302-320, (1994) · Zbl 0807.34023
[2] Cabada, A., The method of lower and upper solutions for third-order periodic boundary value problems, J. math. anal. appl., 195, 2, 568-589, (1995) · Zbl 0846.34019
[3] Cabada, A.; Grossinho, M.R.; Minhós, F., On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions, J. math. anal. appl., 285, 174-190, (2003) · Zbl 1048.34033
[4] Cabada, A.; Habets, P.; Pouso, R.L., Optimal existence conditions for \(\phi\)-Laplacian equations with upper and lower solutions in the reversed order, J. differential equations, 166, 385-401, (2000) · Zbl 0999.34011
[5] Cabada, A.; Heikkilä, S., Uniqueness, comparison and existence results for third order initial – boundary value problems, Comput. math. appl., 41, 5-6, 607-618, (2001) · Zbl 0991.34015
[6] Cabada, A.; Heikkilä, S., Extremality and comparison results for discontinuous third order functional initial-boundary value problems, J. math. anal. appl., 255, 195-212, (2001) · Zbl 0976.34009
[7] Cabada, A.; Heikkilä, S., Existence of solutions of third order functional problems with nonlinear boundary conditions, Anziam j., 46, 1, 33-44, (2004) · Zbl 1058.34014
[8] Cabada, A.; Lois, S., Existence of solution for discontinuous third order boundary value problems, J. comput. appl. math., 110, 105-114, (1999) · Zbl 0936.34015
[9] Cabada, A.; Otero-Espinar, V., Existence and comparison results for difference \(\phi\)-Laplacian boundary value problems ith lower and upper solutions in reverse order, J. math. anal. appl., 267, 2, 501-521, (2002) · Zbl 0995.39003
[10] Chen, G., On a kind of nonlinear boundary value problem of third order differential equation, Ann. differential equations (, 4), 381-389, (1988) · Zbl 0679.34014
[11] Greguš, M., Third order linear differential equations, mathematics and its applications, (1987), Reidel Publishing Co. Dordrecht
[12] Grossinho, M.R.; Minhós, F., Existence result for some third order separated boundary value problems, Nonlinear anal., 47, 4, 2407-2418, (2001) · Zbl 1042.34519
[13] Omari, P.; Trombetta, M., Remarks on the lower and upper solutions method for second-and third-order periodic boundary value problems, Appl. math. comput., 50, 1, 1-21, (1992) · Zbl 0760.65078
[14] Rachunková, I., On some three-point problems for third-order differential equations, Math. bohem., 117, 1, 98-110, (1992) · Zbl 0759.34020
[15] Rusnák, J., Existence theorems for a certain nonlinear boundary value problem of the third order, Math. slovaca, 37, 4, 351-356, (1987) · Zbl 0631.34022
[16] Rusnák, J., Constructions of lower and upper solutions for a nonlinear boundary value problem of the third order and their applications, Math. slovaca, 40, 1, 101-110, (1990) · Zbl 0731.34016
[17] Šenkyřík, M., Method of lower and upper solutions for a third-order three-point regular boundary value problem, Acta univ. palack. olomuc. fac. rerum natur. math., 31, 60-70, (1992) · Zbl 0769.34021
[18] Šenkyřík, M., Existence of multiple solutions for a third-order three-point regular boundary value problem, Math. bohem., 119, 2, 113-121, (1994) · Zbl 0805.34018
[19] Wang, J., Existence of solutions of nonlinear two-point boundary value problems for third order nonlinear differential equations, Northeast. math. J., 7, 2, 181-189, (1991) · Zbl 0755.34021
[20] Zhao, W., Existence and uniqueness of solutions for third order nonlinear boundary value problems, Tohoku math. J. (2), 44, 4, 545-555, (1992) · Zbl 0774.34019
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.