Heteroclinic solutions for a difference equation involving the mean curvature operator. (English) Zbl 1539.39012
This article is concerned with generalizing the work of A. Cabada and S. Tersian [Nonlinear Anal., Real World Appl. 12, No. 4, 2429–2434 (2011; Zbl 1235.39002)] to the following discrete boundary value problem (\(n \in \mathbb{Z} \)):
\[
\left\{ \begin{array}{ll} -\Delta\left[ \phi_c\left(\Delta x(n-1)\right)\right] + \lambda q(n)x(n) + \lambda w(n,x(n)) ,\\
x(0)=\dfrac{a-b}{2}, \quad x(+\infty)=0. \end{array}\right.
\]
where \(\Delta u(n)=u(n+1)-u(n)\) is the forward difference operator, \(\lambda\) is a number, and \(q\) and \(w\) are certain real-valued functions.
The main result of the paper shows that a positive solution exists under assumptions that \(q\) is positive-valued and diverges to \(\infty\), \(w\) has certain restricted growth and form, and \(\lambda\) is sufficiently large. One example is presented which takes \(q\) to be a quadratic polynomial and \(w\) to be a quartic polynomial in \(x\) independent of \(n\).
The main result of the paper shows that a positive solution exists under assumptions that \(q\) is positive-valued and diverges to \(\infty\), \(w\) has certain restricted growth and form, and \(\lambda\) is sufficiently large. One example is presented which takes \(q\) to be a quadratic polynomial and \(w\) to be a quartic polynomial in \(x\) independent of \(n\).
Reviewer: Tom Cuchta (Huntington)
MSC:
| 39A27 | Boundary value problems for difference equations |
| 39A70 | Difference operators |
| 39A12 | Discrete version of topics in analysis |
| 53E10 | Flows related to mean curvature |
Citations:
Zbl 1235.39002References:
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