Nontrivial solutions of inverse discrete problems with sign-changing nonlinearities
Abstract: This paper is concerned with the existence of solutions of an inverse discrete problem with sign-changing nonlinearity. This kind of problems includes, as a particular case, nth order difference equations coupled with suitable conditions on the boundary of the interval of definition. It would be valid for the case in which the related Green's function is positive on a subset of its rectangle of definition. The existence results follow from spectral theory, as an application of the Krein-Rutman theorem and by m… Show more
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In this article, we consider a discrete nonlinear third-order boundary value problem Δ 3 u ( k − 1 ) = λ a ( k ) f ( k , u ( k ) ) , k ∈ [ 1 , N − 2 ] Z , Δ 2 u ( η ) = α Δ u ( N − 1 ) , Δ u ( 0 ) = − β u ( 0 ) , u ( N ) = 0 , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{\Delta }^{3}u\left(k-1)=\lambda a\left(k)f\left(k,u\left(k)),\hspace{1em}k\in {\left[1,N-2]}_{{\mathbb{Z}}},\hspace{1.0em}\\ {\Delta }^{2}u\left(\eta )=\alpha \Delta u\left(N-1),\Delta u\left(0)=-\beta u\left(0),\hspace{1em}u\left(N)=0,\hspace{1.0em}\end{array}\right. where N > 4 N\gt 4 is an integer, λ > 0 \lambda \gt 0 is a parameter. f : [ 1 , N − 2 ] Z × [ 0 , + ∞ ) → [ 0 , + ∞ ) f:{\left[1,N-2]}_{{\mathbb{Z}}}\times \left[0,+\infty )\to \left[0,+\infty ) is continuous, a : [ 1 , N − 2 ] Z → ( 0 , + ∞ ) a:{\left[1,N-2]}_{{\mathbb{Z}}}\to \left(0,+\infty ) , α ∈ 0 , 1 N − 1 \alpha \in \left[0,\frac{1}{N-1}\right) , β ∈ 0 , 2 ( 1 − α ( N − 1 ) ) N ( 2 − α ( N − 1 ) ) \beta \in \left[0,\frac{2\left(1-\alpha \left(N-1))}{N\left(2-\alpha \left(N-1))}\right) , and η ∈ N − 2 2 + 1 , N − 2 Z \eta \in {\left[\left[\frac{N-2}{2}\right]+1,N-2\right]}_{{\mathbb{Z}}} . With the sign-changing Green’s function, we obtain not only the existence of positive solutions but also the multiplicity of positive solutions to this problem.
In this article, we consider a discrete nonlinear third-order boundary value problem Δ 3 u ( k − 1 ) = λ a ( k ) f ( k , u ( k ) ) , k ∈ [ 1 , N − 2 ] Z , Δ 2 u ( η ) = α Δ u ( N − 1 ) , Δ u ( 0 ) = − β u ( 0 ) , u ( N ) = 0 , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{\Delta }^{3}u\left(k-1)=\lambda a\left(k)f\left(k,u\left(k)),\hspace{1em}k\in {\left[1,N-2]}_{{\mathbb{Z}}},\hspace{1.0em}\\ {\Delta }^{2}u\left(\eta )=\alpha \Delta u\left(N-1),\Delta u\left(0)=-\beta u\left(0),\hspace{1em}u\left(N)=0,\hspace{1.0em}\end{array}\right. where N > 4 N\gt 4 is an integer, λ > 0 \lambda \gt 0 is a parameter. f : [ 1 , N − 2 ] Z × [ 0 , + ∞ ) → [ 0 , + ∞ ) f:{\left[1,N-2]}_{{\mathbb{Z}}}\times \left[0,+\infty )\to \left[0,+\infty ) is continuous, a : [ 1 , N − 2 ] Z → ( 0 , + ∞ ) a:{\left[1,N-2]}_{{\mathbb{Z}}}\to \left(0,+\infty ) , α ∈ 0 , 1 N − 1 \alpha \in \left[0,\frac{1}{N-1}\right) , β ∈ 0 , 2 ( 1 − α ( N − 1 ) ) N ( 2 − α ( N − 1 ) ) \beta \in \left[0,\frac{2\left(1-\alpha \left(N-1))}{N\left(2-\alpha \left(N-1))}\right) , and η ∈ N − 2 2 + 1 , N − 2 Z \eta \in {\left[\left[\frac{N-2}{2}\right]+1,N-2\right]}_{{\mathbb{Z}}} . With the sign-changing Green’s function, we obtain not only the existence of positive solutions but also the multiplicity of positive solutions to this problem.
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