The existence of multiple positive periodic solutions for functional differential equations☆
Introduction
Functional differential equations have been extensively studied by many authors by using the coincidence degree theory of Mawhin, the fixed point theorem in cones, Banach’s contraction mapping principle, method of lower and upper solutions, the Leggett–Williams fixed point theorem and so on. We refer the reader to [1], [2], [3], [4], [5], [6] and references cited therein.
In [2], Sun et al. studied functional differential equationunder the following assumptions: is a T-periodic function for t;, and f is a bounded function; are T-periodic function with respect to t; where is a constant. They obtained the existence of positive periodic solutions (see [2]). In [3], Zhang et al. investigated the existence of positive periodic solutions for the scalar functional differential equationunder the following assumptions: are -periodic functions, ; is -periodic with respect to the first variable. They got a few theorems about the existence of positive periodic solutions of (1.2).
These types of equation have been proposed as models for a variety physiological processes and conditions including production of blood cells, respiration, and cardiac arrhythmias.
In this paper, we obtain a few theorems of existence of positive periodic solutions for integral equationWhat is interesting is that may be singular at any point of R. Using these results, we obtain new results on the existence of multiple positive periodic solutions for functional differential Eqs. (1.1), (1.2). Our results different from those of [2], [3] and extend the results of [2], [3]. Particularly, we do not need any continuous assumption on the terms and , which is essential for the technique used in [2], [3].
In this paper, we will always suppose the following conditions are satisfied:
- (C1)
, a.e. . is a -periodic function and
- (C2)
is measurable and . For . is monotone with respect to t. For every , we have
- (C3)
maps bounded subset of into bounded subset of , f is -periodic with respect to the first variable and satisfies Caratheodory conditions;
- (C4)
is periodic, .
Section snippets
Preliminaries
Lemma 2.1 Let K be a reproducing cone in a real Banach space X and let be a compact linear operator with is the spectral radius of L. If , then there is such that . Lemma 2.2 Let X be a Banach space, P be a cone in X and be a bounded open subset in P (relative topology). Suppose that is a completely continuous operator. Then the following results hold: If there exists such that , then the fixed point index . If [7]
[8]
Fixed point index calculations
For convenience, in this paper, we make the following denotations:where .
In this section, we always suppose hold. Lemma 3.1 If , then there exists constant such that , for . Proof From , we get that there exist and such thatwhere
The existence of positive solutions
In this section we always suppose hold. Theorem 4.1 Suppose one of the following conditions holds: ; .
Then (1.3) has at least one positive periodic solution.
Proof
(1) By and Lemma 3.1, we get that there exists such that . By and Lemma 3.4, we get that there exists such that either there exists with or . Suppose (otherwise the proof is completed). By the properties of index, we can get that A has at
Positive periodic solutions for (1.1)
(1.1) equivalent to integral equation (see [2]):where . Suppose the following conditions hold:
, a.e. , is a T-periodic function;
is bounded and satisfies Caratheodory conditions, and is T-periodic function with respect to t;
maps bounded subset of into bounded subset of and satisfies Caratheodory conditions. . are T-periodic
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Cited by (0)
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This work is supported by the Natural Science Foundation of China (10875094 and 60874003) and the Natural Science Foundation of Hebei Province (08M007).