Positive solutions for nonlinear discrete second-order boundary value problems with parameter dependence. (English) Zbl 1260.39020

The paper deals with the existence, multiplicity and nonexistence of positive solutions for the difference equation \[ \begin{aligned} &-\Delta\left[ p\left( t-1\right) \Delta u\left( t-1\right) \right]\\ &+q\left( t\right) u\left( t\right) =\lambda f\left( t,u\left( t\right) \right) ,\;t\in\mathbb{Z},\text{ }1\leq t\leq T, \end{aligned} \] subject to the boundary conditions \[ \begin{aligned} & u\left( 0\right) =u\left( T\right) ,\text{ }p\left( 0\right) \Delta\\ & u\left( 0\right) =p\left( T\right) \Delta u\left( T\right) , \end{aligned} \] where function \(f\) is continuous in the second variable, \(p,q\) are positive functions, \(\lambda\) is a positive parameter, and \(\Delta u\left( t\right) =u\left( t+1\right) -u\left( t\right) .\) For some values of \(\lambda,\) the above boundary value problem admits at least two positive solutions, for other values of \(\lambda\) it has at least one positive solutions, and for other values of \(\lambda,\) the problem admits no positive solution. The proofs employ the fixed point index theory.


39A22 Growth, boundedness, comparison of solutions to difference equations
39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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