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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)

MSC:
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
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