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On the Existence of Solutions to Discrete, Two Point, Non-linear Boundary Value Problems

Haught, Damon

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2010, Master of Science in Mathematics, Youngstown State University, Department of Mathematics and Statistics.

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

nj=1bijy(j – 1) + nj=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.

Padraic Taylor, PhD (Advisor)
David Pollack, PhD (Committee Member)
Richard Goldthwait, PhD (Committee Member)
42 p.

Recommended Citations

Citations

  • Haught, D. (2010). On the Existence of Solutions to Discrete, Two Point, Non-linear Boundary Value Problems [Master's thesis, Youngstown State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1299522079

    APA Style (7th edition)

  • Haught, Damon. On the Existence of Solutions to Discrete, Two Point, Non-linear Boundary Value Problems. 2010. Youngstown State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ysu1299522079.

    MLA Style (8th edition)

  • Haught, Damon. "On the Existence of Solutions to Discrete, Two Point, Non-linear Boundary Value Problems." Master's thesis, Youngstown State University, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1299522079

    Chicago Manual of Style (17th edition)