- Research Article
- Open Access
Multiple Positive Solutions for Nonlinear First-Order Impulsive Dynamic Equations on Time Scales with Parameter
© D.-B.Wang and W.Guan. 2009
Received: 13 February 2009
Accepted: 14 May 2009
Published: 22 June 2009
By using the Leggett-Williams fixed point theorem, the existence of three positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales with parameter are obtained. An example is given to illustrate the main results in this paper.
Let be a time scale, that is, is a nonempty closed subset of . Let be fixed and be points in , an interval denoting time scales interval, that is, Other types of intervals are defined similarly. Some definitions concerning time scales can be found in [1–5].
where is a time scale which has at least finitely-many right-dense points, is regressive and right-dense continuous, is given function, . The paper  obtained the existence of one solution to problem (1.2) by using the nonlinear alternative of Leray-Schauder type.
They proved the existence of one solution to the problem (1.3) by applying Schaefer's fixed point theorem and the nonlinear alternative of Leray-Schauder type.
In , Li and Shen studied the problem (1.3). Some existence results to problem (1.3) are established by using a fixed point theorem, which is due to Krasnoselskii and Zabreiko, and the Leggett-Williams fixed point theorem.
In , the first author studied the problem (1.1) when . The existence of positive solutions to the problem (1.1) was obtained by means of the well-known Guo-Krasnoselskii fixed point theorem.
Motivated by the results mentioned above, in this paper, we shall show that the problem (1.1) has at least three positive solutions for suitable by using the Leggett-Williams fixed point theorem . We note that for the case and problem (1.1) reduces to the problem studied by .
In the remainder of this section, we state the following theorem, which are crucial to our proof.
Theorem 1.1 (see ).
Since the method is similar to that of in [27, Lemma 3.1], we omit it here.
It is obvious, so we omit it here.
By [27, Lemmas 3.3 and 3.4], it is easy to see that is completely continuous.
3. Main Result
The authors express their gratitude to the anonymous referee for his/her valuable suggestions.
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