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


Zbl 1013.47019
Full Text: DOI


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