Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales. (English) Zbl 1155.34012

Summary: We investigate the following nonlinear first-order three-point boundary value problem on time scale \(\mathbb T\):
\[ \begin{aligned} x^\Delta(t)+ p(t)x(\sigma(t))&= f(t,x(\sigma(t))), \quad t\in[0,T]_{\mathbb T},\\ x(0)- \alpha x(\xi)&= \beta x(\sigma(T)). \end{aligned} \]
By using several well-known fixed point theorems, the existence of positive solutions is obtained. Besides, the uniqueness results are obtained by imposing growth restrictions on \(f\). In particular, Green’s function for the above boundary value problem is established.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
39A10 Additive difference equations
Full Text: DOI


[1] Agarwal, R. P.; Bohner, M., Basic calculus on time scales and some of its applications, Results Math., 35, 3-22 (1999) · Zbl 0927.39003
[2] Anderson, D. R., Multiple periodic solutions for a second-order problem on periodic time scales, Nonlinear Anal., 60, 101-115 (2005) · Zbl 1060.34022
[3] Anderson, D. R.; Hoffacker, J., Positive periodic time-scale solutions for functional dynamic equations, Aust. J. Math. Anal. Appl., 3, 1, 1-14 (2006) · Zbl 1098.39009
[4] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[5] Cabada, A., Extremal solutions and Green’s functions of higher order periodic boundary value problems in time scales, J. Math. Anal. Appl., 290, 35-54 (2004) · Zbl 1056.39018
[6] Cabada, A.; Vivero, D. R., Existence of solutions of first-order dynamic equations with nonlinear functional boundary value conditions, Nonlinear Anal., 63, e697-e706 (2005) · Zbl 1224.34291
[7] Hilger, S., Analysis on measure chains—A unified approach to continuous and discrete calculus, Results Math., 18, 18-56 (1990) · Zbl 0722.39001
[8] Kaymakcalan, B.; Lakshmikantham, V.; Sivasundaram, S., Dynamic Systems on Measure Chains (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0869.34039
[9] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), P. Noordhoff Ltd.: P. Noordhoff Ltd. Groningen, The Netherlands · Zbl 0121.10604
[10] Leggett, R. W.; Williams, L. R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28, 673-688 (1979) · Zbl 0421.47033
[11] Ma, Ruyun, Existence and uniqueness of solutions to first-order three-point boundary value problems, Appl. Math. Lett., 15, 211-216 (2002) · Zbl 1008.34009
[12] Otero-Espinar, V.; Vivero, D. R., The existence and approximation of extremal solutions to several first order discontinuous dynamic equations with nonlinear boundary value conditions, Nonlinear Anal. (2007)
[13] Sun, Jian-Ping, Twin positive solutions of nonlinear first-order boundary value problem on time scales, Nonlinear Anal. (2007)
[14] Sun, Jian-Ping; Li, Wan-Tong, Existence of solutions to nonlinear first-order PBVPs on time scales, Nonlinear Anal., 67, 883-888 (2007) · Zbl 1120.34314
[15] Sun, Jian-Ping; Li, Wan-Tong, Existence and multiplicity of positive solutions to nonlinear first-order PBVPs on time scales, Comput. Math. Appl., 54, 861-871 (2007) · Zbl 1134.34016
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.