Within this treatise we establish conditions for the existence of solutions to two-point, discrete, non-linear boundary value problems. We will be examining two different variations of the problem. First, we will be examining generalized discrete nonlinear
systems of the form
x(t + 1) = Ax(t) + f(x(t)), t ∈ {0, 1, ..., N – 1}
subject to
Bx(0) + Dx(N) = 0.
We demonstrate the existence of solutions to this type of problem when the associated linear, homogeneous boundary value problem has only the trivial solution, and the nonlinear element exhibits sublinear growth.
Next, we will consider scalar, two-point, nonlinear boundary value problems of the form
y(t + n) + an–1y(t + n – 1) + ··· + a0y(t) = g(y(t)),
for t ∈ {0, 1, ..., N – 1}, subject to
n∑j=1bijy(j – 1) + n∑j=1dijy(j + N – 1) = 0,
for i = 1, 2, ..., n.
In this case, we assume the associated linear homogeneous boundary value problem has a one-dimensional solution space and establish criteria that guarantee the existence of solutions by analyzing the relationship between the nonlinear element and the solution space of the associated linear boundary value problem through a projection scheme.