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.

MSC:

 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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References:

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