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Positive solutions to semi-positone second-order three-point problems on time scales. (English) Zbl 1188.34119
Summary: Using a fixed point theorem of generalized cone expansion and compression we establish the existence of at least two positive solutions for the nonlinear semi-positone three-point boundary value problem on time scales $$u^{\Delta\nabla}(t)+\lambda f(t,u(t))=0,\quad u(a)=0,\quad \alpha u(\eta)=u(T).$$ Here $t\in[a,T]_{\Bbb T}$, where $\Bbb T$ is a time scale, $\alpha>0$, $\eta\in(a,\rho(T))_{\Bbb T}$, $\alpha(\eta-a)<T-a$, and the parameter $\lambda >0$ belongs to a certain interval. These results are new for difference equations as well as for general time scales. An example is provided for differential, difference, and $q$-difference equations.

34N05Dynamic equations on time scales or measure chains
34B10Nonlocal and multipoint boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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