Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions

• Published in: Nonlinear analysis Vol. 67; no. 4; pp. 1236 - 1245
• Main Authors: Rachůnková, Irena, Tisdell, Christopher C.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 15-08-2007; Elsevier
• Subjects: Convergence of solutions; Discrete Dirichlet boundary value problem; Exact sciences and technology; Existence of solutions; Lower and upper solutions; Mathematical analysis; Mathematics; Non-spurious solutions; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, boundary value problems; Sciences and techniques of general use; Discrete Dirichlet boundary value problem; 34B15; 39A12; Lower and upper solutions; Convergence of solutions; Existence of solutions; Non-spurious solutions; Boundary value problem; Existence of solution; Discrete Dirichlet boundary value problem; Existence of solutions; Non-spurious solutions; Lower and upper solutions; Convergence of solutions; 39A12;34B15; Nonlinear analysis; Dirichlet problem; Spurious solution; Mathematical model; Multidisciplinary; Convergence
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2006.07.010

This paper investigates the solvability of discrete Dirichlet boundary value problems by the lower and upper solution method. Here, the second-order difference equation with a nonlinear right hand side
f
is studied and
f
(
t
,
u
,
v
)
can have a superlinear growth both in
u
and in
v
. Moreover, the growth conditions on
f
are one-sided. We compute
a priori bounds on solutions to the discrete problem and then obtain the existence of at least one solution. It is shown that solutions of the discrete problem will converge to solutions of ordinary differential equations.