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Eigenvalue problems for \(p\)-Laplacian functional dynamic equations on time scales. (English) Zbl 1146.39017

Summary: This paper is concerned with the existence and nonexistence of positive solutions of the \(p\)-Laplacian functional dynamic equation on a time scale, \[ [\varphi_p(x^\Delta(t))]^\nabla+\lambda a(t)f(x(t),x(u(t)))=0,\quad t\in(0,T),\quad x_0(t)=\psi(t),\quad t\in[-\tau,0], \] \(x(0)-B_0(x^\Delta(0))=0\), \(x^\Delta(T)=0\). We show that there exists a \(\lambda^*>0\) such that the above boundary value problem has at least two, one, and no positive solutions for \(0<\lambda<\lambda^*\), \(\lambda=\lambda^*\) and \(\lambda>\lambda^*\), respectively.

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
34L05 General spectral theory of ordinary differential operators
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

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