×

Nonlinear three-point third-order boundary value problems. (English) Zbl 1134.34007

The paper is concerned with the existence of solutions of the following three-point boundary value problem for a third order ODE:
\[ y'''(t)=f(t,y(t),y'(t), y''(t))\quad 0<t<1,\qquad y(0)=y(a)=y(1)=0. \]
Using a topological transversality theorem of Granas and Schaefer’s theorem, existence of at least one solution is proved under appropriate conditions on the nonlinearity \(f\).

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Boucherif, A., Two-point boundary value problems for third order differential equations, Int. J. Differen. Equat. Appl., 2, 1, 39-45 (2001)
[2] Cabada, A.; Lois, S., Existence of solution for discontinuous third order boundary value problems, J. Comput. Appl. Math., 110, 1, 105-114 (1999) · Zbl 0936.34015
[3] Chyan, C. J.; Henderson, J., A note on global existence for boundary value problems, Int. J. Math. Math. Sci., 12, 3, 615-618 (1989) · Zbl 0681.34021
[4] Constantin, A., A note on a boundary value problem, Nonlinear Anal. T.M.A., 27, 1, 13-16 (1996) · Zbl 0853.34020
[5] Du, Z.; Ge, W.; Lin, X., Existence of solutions for a class of third order nonlinear boundary value problems, J. Math. Anal. Appl., 294, 1, 104-112 (2004) · Zbl 1053.34017
[6] A. Granas, R.B. Guenther, J.W. Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertations Math. 244, P.W.N., Warsaw, 1985.; A. Granas, R.B. Guenther, J.W. Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertations Math. 244, P.W.N., Warsaw, 1985. · Zbl 0615.34010
[7] Jiang, D.; Agarwal, R. P., A uniqueness and existence theorem for a singular third order boundary value problem on \([0, \infty)\), Appl. Math. Lett., 15, 445-451 (2002) · Zbl 1021.34020
[8] Klokov, Y. A., Two-point problems for a third order ordinary differential equation, Differen. Equat., 39, 4, 596-598 (2003) · Zbl 1080.34515
[9] O’ Regan, D., Topological transversality. Application to third order boundary value problems, SIAM J. Math. Anal., 18, 3, 716-726 (1987)
[10] O’Regan, D., Singular and nonsingular third order boundary value problems, Proc. R. Irish Acad., 90A, 1, 29-42 (1990) · Zbl 0725.34024
[11] Wang, J. Z., Boundary value problems for third order differential equations, Ann. Differen. Equat., 10, 4, 424-436 (1994) · Zbl 0819.34022
[12] Yao, Q.; Feng, Y., The existence of solutions for a third order two-point boundary value problem, Appl. Math. Lett., 15, 227-232 (2002) · Zbl 1008.34010
[13] Aftabizadeh, A.; Gupta, C. P.; Xu, J., Existence and uniqueness theorems for three-point boundary value problems, SIAM J. Math. Anal., 20, 3, 716-726 (1989) · Zbl 0704.34019
[14] Aftabizadeh, A.; Deimling, K., A three-point nonlinear boundary value problem, Differen. Integral Equat., 4, 1, 189-194 (1991) · Zbl 0723.34016
[15] Caristi, G., A three-point boundary value problem for a third order differential equation, Boll. UMI, Ser. VI, IV-C, 1, 259-268 (1985) · Zbl 0591.34013
[16] Marano, S. A., A remark on a third order three-point boundary value problem, Bull. Aust. Math. Soc., 49, 1-5 (1994) · Zbl 0808.34019
[17] Murty, K. N.; Prasad, K. R.; Anand, P. V.S., On the use of differential inequalities in three-point boundary value problems, Bull. Inst. Math. Acad. Sinica, 21, 3, 263-275 (1993) · Zbl 0782.34029
[18] Rackunkova, I., On some three-point problems for third order differential equations, Math. Bohem., 117, 1, 98-110 (1992) · Zbl 0759.34020
[19] Agarwal, R. P., On boundary value problems for \(y''' = f(x, y, y^\prime, y'')\), Bull. Inst. Math. Acad. Sinica, 12, 2, 153-157 (1984) · Zbl 0542.34015
[20] L.K. Jackson, Boundary value problems for ordinary differential equations, in: J. Hale, (Ed.), Studies in Differential Equations, in: The Mathematical Association of America, vol. 14, 1977, pp. 93-127.; L.K. Jackson, Boundary value problems for ordinary differential equations, in: J. Hale, (Ed.), Studies in Differential Equations, in: The Mathematical Association of America, vol. 14, 1977, pp. 93-127. · Zbl 0371.34011
[21] Rao, D. R.K.; Murthy, K. M.; Rao, A. S., On three-point boundary value problems associated with third order differential equations, Nonlinear Anal. T.M.A., 5, 6, 669-673 (1981) · Zbl 0485.34011
[22] Hai, D. D.; Schmitt, K., Boundary value problems for higher order differential equations, Nonlinear Anal. T.M.A., 7, 4, 293-305 (1994)
[23] Klokov, Y. A., Upper and lower functions in boundary value problems for a third order ordinary differential equation, Differen. Equat., 36, 12, 1762-1769 (2002) · Zbl 1002.34007
[24] Granas, A.; Guenther, R. B.; Lee, J. W., Existence principles for classical and Carathéodory solutions of nonlinear systems and applications, (Proc. Int. Conf. Theory Appl. Diff. Eq. (1988), Ohio University Press: Ohio University Press Athens, Ohio), 353-364 · Zbl 0717.34022
[25] Grossinho, M. R.; Minhos, F. M., Existence result for some third order separated boundary value problems, Nonlinear Anal. T.M.A., 47, 4, 2407-2418 (2001) · Zbl 1042.34519
[26] Smart, D. R., Fixed Point Theorems (1974), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0297.47042
[27] Granas, A.; Guennoun, Z. E.A., Quelques resultats dans la thé orie de Bernstein-Carathéodory de l’equation \(y'' = f(x, y, y^\prime)\), C.R. Acad. Sci. Paris, t. 306, Serie I, 703-706 (1988) · Zbl 0639.34048
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.