×

zbMATH — the first resource for mathematics

Existence of triple positive solutions of two-point right focal boundary value problems on time scales. (English) Zbl 1122.34015
Summary: We consider the following boundary value problem,
\[ \begin{alignedat}{2} (-1)^{n-1}y^{\Delta n}(t) & =(-1)^{p+1}F(t,y(\sigma^{n-1}(t))), &\quad & t\in[a,b]\cap {\mathbf T},\\ y^{\Delta^i}(a)& =0, & \quad &0\leq i\leq p-1,\\ y^{\Delta^i}(\sigma(b))& =0, &\quad & p\leq i\leq n-1,\end{alignedat} \] where \(n\leq 2\), \(1\leq p \leq n - 1\) is fixed and \(\mathbf T\) is a time scale. By applying fixed-point theorems for operators on a cone, existence criteria are developed for triple positive solutions of the boundary value problem. We also include examples to illustrate the usefulness of the results obtained.

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
39A12 Discrete version of topics in analysis
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bohner, M.; Peterson, A.C., Dynamic equations on time scales. an introduction with applications, (2001), Birkhäuser Philadelphia · Zbl 0978.39001
[2] Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana university math. J., 28, 673-688, (1979) · Zbl 0421.47033
[3] Avery, R.I., A generalization of the Leggett-Williams fixed point theorem, MSR hotline, 2, 9-14, (1998) · Zbl 0965.47038
[4] Gavalas, G.R., Nonlinear differential equations of chemically reacting systems, (1968), Springer-Verlag Boston, MA · Zbl 0174.13401
[5] Leung, A.W., Systems of nonlinear partial differential equations, (1989), Kluwer New York
[6] Erbe, L.H.; Wang, H., On the existence of positive solutions of ordinary differential equations, (), 743-748 · Zbl 0802.34018
[7] Lian, W.; Wong, F.; Yeh, C., On the existence of positive solutions of nonlinear second order differential equations, (), 1117-1126 · Zbl 0857.34036
[8] Wong, P.J.Y., Positive solutions of difference equations with two-point right focal boundary conditions, J. math. anal. appl., 224, 34-58, (1998) · Zbl 0914.39006
[9] Davis, J.M.; Henderson, J.; Wong, P.J.Y., General lidstone problems: multiplicity and symmetry of solutions, J. math. anal. appl., 251, 527-548, (2000) · Zbl 0966.34023
[10] Henderson, J.; Wong, P.J.Y., Double symmetric solutions for discrete lidstone boundary value problems, J. difference equ. appl., 7, 811-828, (2001) · Zbl 1001.39025
[11] Wong, P.J.Y., Multiple symmetric solutions for discrete lidstone boundary value problems, J. difference equ. appl., 8, 765-797, (2002) · Zbl 1010.39005
[12] Agarwal, R.P.; Wong, P.J.Y., Advanced topics in difference equations, (1997), Kluwer Academic Publishers Dordrecht · Zbl 0914.39005
[13] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Dordrecht · Zbl 0923.39002
[14] Erbe, L.H.; Peterson, A.C., Positive solutions for a nonlinear differential equation on a measure chain, Mathl. comput. modelling, 32, 5/6, 571-585, (2000) · Zbl 0963.34020
[15] Agarwal, R.P.; O’Regan, D., Nonlinear boundary value problems on time scales, Nonlinear anal., 44, 527-535, (2001) · Zbl 0995.34016
[16] Agarwal, R.P.; Bohner, M.; O’Regan, D.; Peterson, A.C., Dynamic equations on time scales: A survey, J. comput. appl. math., 141, 1-26, (2002) · Zbl 1020.39008
[17] Wong, P.J.Y., Abel-gontscharoff boundary value problems on measure chains, J. comput. appl. math., 142, 331-355, (2002) · Zbl 1012.34017
[18] Anderson, D.; Avery, R.I.; Davis, J.M.; Henderson, J.; Yin, W., Positive solutions of boundary value problems, (), 189-249
[19] Sun, H.R.; Li, W.T., Positive solutions for nonlinear three-point boundary value problems on time scales, J. math. anal. appl., 299, 508-524, (2004) · Zbl 1070.34029
[20] K.L. Boey and P.J.Y. Wong, On positive solutions of two-point right focal boundary value problems on time scales, Comput. Math. Appl. (to appear). · Zbl 1122.34015
[21] Wong, P.J.Y.; Agarwal, R.P., Multiple positive solutions of two-point right focal boundary value problems, Math. comput. modelling, 28, 3, 41-49, (1998) · Zbl 1098.34523
[22] Wong, P.J.Y.; Agarwal, R.P., Existence of multiple positive solutions of discrete two-point right focal boundary value problems, J. difference equ. appl., 5, 517-540, (1999) · Zbl 0964.39004
[23] K.L. Boey and P.J.Y. Wong, Two-point right focal eigenvalue problems on time scales, Appl. Math. Comput. (to appear). · Zbl 1084.39018
[24] Henderson, J.; Thompson, H.B., Multiple symmetric positive solutions for a second order boundary value problem, (), 2373-2379 · Zbl 0949.34016
[25] Agarwal, R.P.; O’Regan, D., A generalization of the Petryshyn-Leggett-Williams fixed point theorem with applications to integral inclusions, Appl. math. comput., 123, 263-274, (2001) · Zbl 1033.47037
[26] Agarwal, R.P.; O’Regan, D., A fixed point theorem of Leggett-Williams type with applications to single-and multivalued equations, Georgian math. J., 8, 13-25, (2001) · Zbl 0989.47050
[27] Agarwal, R.P.; O’Regan, D., Existence of three solutions to integral and discrete equations via the Leggett Williams fixed point theorem, Rocky mountain J. math., 31, 23-35, (2001) · Zbl 0979.45003
[28] He, X.; Ge, W., Triple solutions for second-order three-point boundary value problems, J. math. anal. appl., 268, 256-265, (2002) · Zbl 1043.34015
[29] Karakostas, G.L.; Mavridis, K.G.; Tsamatos, P.Ch., Triple solutions for a nonlocal functional boundary value problem by Leggett-Williams theorem, Appl. anal., 83, 957-970, (2004) · Zbl 1081.34064
[30] Zima, M., Fixed point theorem of Leggett-Williams type and its application, J. math. anal. appl., 299, 254-260, (2004) · Zbl 1066.47059
[31] Avery, R.I.; Davis, J.M.; Henderson, J., Three symmetric positive solutions for lidstone problems by a generalization of the Leggett-Williams theorem, Electron. J. differential equations, 40, 15, (2000) · Zbl 0958.34020
[32] Anderson, D.; Avery, R.I., Multiple positive solutions to a third-order discrete focal boundary value problem, Computers math. applic., 42, 333-340, (2001) · Zbl 1001.39022
[33] Avery, R.I.; Anderson, D., Existence of three positive solutions to a second-order boundary value problem on a measure chain, J. comput. appl. math., 141, 65-73, (2002) · Zbl 1032.39009
[34] Avery, R.I.; Henderson, J., Existence of three positive pseudo-symmetric solutions for a one-dimensional p-Laplacian, J. math. anal. appl., 277, 395-404, (2003) · Zbl 1028.34022
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.