zbMATH — the first resource for mathematics

An even-order three-point boundary value problem on time scales. (English) Zbl 1056.34013
For \(0\leq i\leq n-1\), the authors study the even-order dynamic equation \[ (-1)^{(n)}x^{(\Delta\nabla)^n}(t)=\lambda h(t)f(x(t)),\quad t\in [a,c],\,n\in\mathbb{N}, \] satisfying the boundary conditions \[ x^{(\Delta\nabla)^i}(a)=0,\qquad x^{(\Delta\nabla)^i}(c)=\beta x^{(\Delta\nabla)^i}(b), \] on a time scale \(\mathbb{T}\), where \(0<\beta(b-a)<c-a\) for \(b\in(a,c)\), \(\beta>0\). The nonlinearity \(f\) is a positive function, and \(h\) is a nonnegative function that is allowed to vanish on some subintervals of \([a,c]\) of the time scale. The existence of at least one positive solution is considered by using the functional type cone expansion-compression fixed-point theorem due to the authors [J. Difference Equ. Appl. 8, 1073–1083 (2002; Zbl 1013.47019)].
Reviewer: Ruyun Ma (Lanzhou)

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
39A10 Additive difference equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
Full Text: DOI
[1] Anderson, D.R, Solutions to second-order three-point problems on time scales, J. differ. equations appl., 8, 673-688, (2002) · Zbl 1021.34011
[2] D.R. Anderson, Eigenvalue intervals for even-order Sturm-Liouville dynamic equations, Int. J. Nonlinear Differ. Equations (2003), in press
[3] D.R. Anderson, Extension of a second-order multi-point problem to time scales, Dynam. Systems Appl. (2003), in press
[4] Anderson, D.R; Hoffacker, J, Green’s function for an even-order mixed derivative problem on time scales, Dynam. systems appl., 12, 1-2, 9-22, (2003) · Zbl 1049.39019
[5] Atici, F.M; Guseinov, G.Sh, On Green’s functions and positive solutions for boundary value problems on time scales, J. comput. appl. math., 141, 75-99, (2002) · Zbl 1007.34025
[6] Avery, R.I; Anderson, D.R, Fixed point theorem of cone expansion and compression of functional type, J. differ. equations appl., 8, 1073-1083, (2002) · Zbl 1013.47019
[7] Bohner, M; Peterson, A, Dynamic equations on time scales: an introduction with applications, (2001), Birkhäuser New York · Zbl 0978.39001
[8] ()
[9] Ma, R, Positive solutions of a nonlinear three-point boundary-value problem, Electronic J. differential equations, 34, 1-8, (1998)
[10] Ma, R, Multiplicity of positive solutions for second-order three-point boundary-value problems, Comput. math. appl., 40, 193-204, (2000) · Zbl 0958.34019
[11] Ma, R, Positive solutions for second-order three-point boundary-value problems, Appl. math. lett., 14, 1-5, (2001) · Zbl 0989.34009
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.