Abstract
In this paper, we obtain the unique solution for the following periodic problem where is a continuous function, by constructing an auxiliary system with bounded solutions. The upper and lower solution method and an anti-maximum principle are employed to establish the monotone iterative sequences and obtain the extremal solutions for the auxiliary system. MSC:34B15, 34C25.
Highlights
1 Introduction This paper is concerned with the unique solution to the following periodic problem:
It is well known that the upper and lower solution method together with the iterative technique is a powerful tool for proving the existence results for boundary value problems
The case when the upper solution and the lower solution are in the reversed order has received some attention
Summary
This paper is concerned with the unique solution to the following periodic problem:. where g : [a, b] × R → R is a continuous function. The monotone approximation method can be used in the case the lower and upper solutions are in the reversed order β ≤ α This method works for any boundary value problem such that a uniform anti-maximum principle holds. In [ ], Zuo et al further developed the monotone method and invested the T-periodic solution of y (t) = f t, y(t), y w(t) , t ∈ R They used the monotone iterative technique with upper and lower solution in reversed order to define two sequences that converge uniformly to extremal solution of By adopting an auxiliary periodic system and an anti-maximum principle which are different from that in the reference [ ], we obtain the unique solution for the problem
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