## 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

Zbl 1013.47019
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### References:

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