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Existence of multiple positive solutions for even order Sturm-Liouville dynamic equations. (English) Zbl 1196.34117
Summary: We consider the even-order dynamic equation on time scales $$(-1)^n y^{(\Delta\nabla)^n}(t)=f(t,y(t)),\quad t\in [a,b]$$ satisfying the boundary conditions $$\alpha_{i+1}y^{(\Delta\nabla)^i}(a)-\forall _{i+1}y^{(\Delta\nabla)^i\Delta}(a)=0,\quad \gamma_{i+1}y^{(\Delta\nabla)^i(b)+b_{i+1}y^(\Delta\nabla)'\Delta}(b)=0$$ for $0\le i\le n-1$, $f:[a,b]\times \Bbb R\to\Bbb R$ is continuous. First, we establish the existence of at least three positive solutions by using the well-known Leggett-Williams fixed point theorem. We also establish the existence of at least $2m-1$ positive solutions for arbitrary positive integer $m$.

34N05Dynamic equations on time scales or measure chains
34B15Nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
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