Nonlinear Analysis: Theory, Methods & Applications
Fixed point theorems in ordered Banach spaces via quasilinearization
Introduction
Let be an ordered Banach space with order cone . In the first part of this paper (Section 2) we prove abstract fixed point results for mappings based on the method of sub- and supersolution (also called lower and upper solution) combined with the idea of quasilinearization. We develop an iteration scheme for the th iterate involving and its Frechet derivative at the th iterate that allows us to approximate fixed points of in a constructive and monotone way. Moreover, under certain additional assumptions on the convergence rate can be shown to be quadratic (rapid convergence). In our first fixed point theorem (Theorem 2.1) we assume a regular order cone , i.e., all order bounded and increasing sequences of converge. However, in various applications this assumption is too strong. By imposing a compactness property on which is often met in applications, we are able to weaken the regularity of and may instead assume merely a normal order cone (see Theorem 2.2).
In the second part of this paper (Section 3) we provide two prototypes of applications of the abstract results to an ODE and a PDE problem, which clearly demonstrate that the abstract setting given here reflects in a proper way the characteristic features of what is known as the method of quasilinearization for ODE and PDE problems. As for the quasilinearization method to concrete problems for ODE and PDE we refer, e.g., to [1], [2], [3], [4].
Section snippets
The main results
Our first abstract fixed point result is as follows.
Theorem 2.1 Let be an ordered Banach space with regular order cone . Assume that satisfies the following hypotheses. There exist such that , and . The Frechet derivative exists for every , and is increasing on for all . exists and is a bounded and positive operator for all . Then, for , relationsdefine an
ODE initial value problem
In this section we consider the initial value problem
under the following hypotheses:
- (H1)
Let satisfy:
- (H2)
For some , the function is continuously differentiable.
- (i)
Let be such that
- (ii)
The derivative is increasing and Lipschitz continuous.
- (i)
Remark 3.1 From (H2) (i) it
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