A mathematical theory for numerical treatment of nonlinear two-point boundary value problems. (English) Zbl 1105.34009

Yamamoto, Tetsuro; Oishi, Shin’ichi

Japan J. Ind. Appl. Math. 23, No. 1, 31-62 (2006).

The authors study the boundary value problem \[ -(p(x)u(x))'+f(x,u(x),u'(x)) = 0, \quad a \leq x \leq b, \]
\[ \alpha_0 u(a) - \alpha_1 u'(a) = \alpha, \quad \beta_0 u(b) - \beta_1 u(b) = \beta, \] where \(p(x) > 0\) on \([a,b]\), \(\alpha_0, \alpha_1, \beta_0, \beta_1 \geq 0\), \(\alpha_0+\beta_0 > 0\), \(\alpha_0 + \alpha_1 > 0\), \(\beta_0 + \beta_1 > 0\). The first result of this paper is analytic.
The authors obtain some criteria for existence and uniqueness under the assumptions of the continuity of \(f(x,u,v)\) and the uniform Lipschitz continuity in \(u\) and \(v\) on \(\mathcal R = [a,b] \times \mathbb{R} \times \mathbb{R}\) (\(p(x)\) is assumed to be continuously differentiable). The additional assumptions are \(\frac{\partial f}{\partial u} \geq 0\) and \(\left| \frac{\partial f}{\partial v} \right| \leq M\) for some \(M>0\) on \(\mathcal R\). This result is obtained using the Schauder fixed-point theorem and improves several known results.
Two more results is this paper deal with a discrete boundary value problem. Firstly, the existence and uniqueness criteria, in this case, are established based on an application of the Brouwer fixed-point theorem. Secondly, under additional assumption on \(p\) and \(f\), the discrete Sturm-Liouville boundary value problem is shown to have the second-order accuracy \(O(h^2)\), where \(h\) is the maximum step mesh size. The paper generalizes and extends several known results. It may be found useful not only for a numerical analyst but also for those interested in the fixed-point theory for difference equations.

Reviewer: Nickolai Kosmatov (Little Rock)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
39A12	Discrete version of topics in analysis
65L10	Numerical solution of boundary value problems involving ordinary differential equations

Keywords:

two-point boundary value problems; existence of solution; fixed-point theorems; finite difference methods

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