In the paper we provide sufficient conditions for the existence of positive solutions for some second-order differential equation subject to periodic boundary conditions. Our method employs a Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima. Two examples are given to illustrate the main result of the paper.
Keywords:periodic boundary value problem; positive solution; coincidence equation
In the paper we are interested in the existence of positive solutions for the periodic boundary value problem (PBVP)
where and are continuous functions. Our study is motivated by current activity of many researchers in the area of theory and applications of PVBPs; see, for example, [1-4] and references therein. In particular, in a recent paper , Chu, Fan and Torres have studied the existence of positive periodic solutions for the singular damped differential equation
by combining the properties of the Green’s function of the PBVP
with a nonlinear alternative of Leray-Schauder type (see, for example, ). It should be noted that was the key assumption used in . If , then PBVP (2) has nontrivial solutions, which means that the problem we are concerned with here, that is, PBVP (1), is at resonance. There are several methods to deal with the resonant PBVPs. For example, in , Torres studied the existence of a positive solution for the PBVP
by considering the equivalent problem
via Krasnoselskii’s theorem on cone expansion and compression. Further results in this direction can be found in  and . In  Rachůnková, Tvrdý and Vrkoč applied the method of upper and lower solutions and topological degree arguments to establish the existence of nonnegative and nonpositive solutions for the PBVP
The same PBVP was studied by Wang, Zhang and Wang in . Their existence and multiplicity results on positive solutions are based on the theory of a fixed point index for A-proper semilinear operators on cones developed by Cremins .
The goal of our paper is to provide sufficient conditions that ensure the existence of positive solutions of (1) with the function h positive on . Our general result is illustrated by two examples. The method we use in the paper is to rewrite BVP (1) as a coincidence equation , where L is a Fredholm operator of index zero and N is a nonlinear operator, and to apply the Leggett-Williams norm-type theorem for coincidences obtained by O’Regan and Zima . We would like to emphasize that the idea of results of  and , as well as these of [13-15], goes back to the celebrated Mawhin’s coincidence degree theory . For more details on this significant tool, its modifications and wide applications, we refer the reader to [17-22] and references therein.
In this paper, for the first time, the existence theorem from  is used for studying the boundary value problem with the nonlinearity f depending also on the derivative. In general, the presence of in f makes the problem much harder to handle. We point out that, to the best of our knowledge, there are only a few papers on PBVPs that discuss such a nonlinearity; we refer the reader to [15,23-25] for some results of that type. We also complement several results in the literature, for example, in [1,26] and . It is evident that the existence theorems for PBVP (1) can be established by the shift method used in , that is, one can employ the results of  to the periodic problem we study here. However, the conditions imposed on f in  are not comparable with ours.
2 Coincidence equation
For the convenience of the reader, we begin this section by providing some background on cone theory and Fredholm operators in Banach spaces.
Definition 1 A nonempty subset C, , of a real Banach space X is called a cone if C is closed, convex and
(i) for all and ,
(ii) x, implies .
Every cone induces a partial ordering in X as follows: for , we say that
The following property holds for every cone in a Banach space.
Lemma 1For every , there exists a positive number such that
for all .
Consider a linear mapping and a nonlinear operator , where X and Y are Banach spaces. If L is a Fredholm operator of index zero, that is, ImL is closed and , then there exist continuous projections and such that and (see, for example, [14,16]). Moreover, since , there exists an isomorphism . Denote by the restriction of L to . Then is an isomorphism from to ImL and its inverse
As a result, the coincidence equation is equivalent to , where
Let be a retraction, that is, a continuous mapping such that for all . Put
Let , be open bounded subsets of X with and . Assume that
1∘L is a Fredholm operator of index zero,
2∘ is continuous and bounded and is compact on every bounded subset of X,
3∘ for all and ,
4∘ρ maps subsets of into bounded subsets of C,
5∘ , where stands for the Brouwer degree,
6∘ there exists such that for , where
and is such that for every ,
7∘ and .
Under the assumptions 1∘-7∘the equation has a solution in the set .
In the next section, we use Theorem 1 to prove the existence of a positive solution for PBVP (1). For applications of Theorem 1 to nonlocal boundary value problems at resonance, we refer the reader to ,  and .
3 Periodic boundary value problem
We now provide sufficient conditions for the existence of positive solutions for PBVP (1). For convenience and ease of exposition, we make use of the following notation:
We observe that on . Moreover, we put
where M is a positive constant.
We assume that
(H1) and are continuous functions.
We also assume that there exist , , , , , and a continuous function such that
(H2) for ,
(H3) for ,
(H4) and for ,
(H5) for and ,
(H6) for ,
(H7) for and .
Theorem 2Under the assumptions (H1)-(H7), PBVP (1) has a positive solution on .
Proof Let denote the supremum norm in the space , that is, . Consider the Banach spaces with the norm , and with the norm .
We write problem (1) as a coincidence equation
with . Then
where ψ is given by (5).
Clearly, ImL is closed and with
Since , we have . Moreover, , which gives . Consequently, L is Fredholm of index zero, and the assumption 1∘ is satisfied.
Define the projections by
It is a routine matter to show that for , the inverse of is given by
with the kernel k defined by (6). Clearly, the assumption 2∘ is satisfied. For , define
Then J is an isomorphism from ImQ to KerL. Next, consider a cone
For , we have and
Obviously, and are open bounded subsets of X, and .
To verify 3∘, suppose that there exist and such that . Then on , ,
There are two cases to consider.
1. If , then there exists such that . For , we get , contrary to the assumption (H3). Similarly, if or , BCs (9) imply . Hence, which contradicts (H3) again.
2. If , then there exists such that . Observe that (H2) implies for and . Suppose that . If , we get from (8)
a contradiction. For , we have
contrary to (H5). By similar arguments, if or , BCs (9) and (H4) imply either (10) or (11). Thus, 3∘ is fulfilled.
Next, for , define (see )
Clearly, ρ is a retraction and maps subsets of into bounded subsets of C, so 4∘ holds.
To verify 5∘, it is enough to consider, for and , the mapping
Observe that if , then on and . Suppose for . Then . For , we have and in view of (H3), we get
which is a contradiction. If , then , hence
which contradicts (H2). Thus, for and . This implies
We next show that 6∘ holds. Let . Then for , we have , , and by (H6) and (H7), we obtain
This implies for , so 6∘ is satisfied.
Finally, we must check if 7∘ holds. If , then in view of (H2), we get
Moreover, for , we have from (H2) and (H7)
Thus, 7∘ is fulfilled and the assertion follows. □
We now give two examples illustrating Theorem 2. Some calculations have been made with Mathematica. In the first example, the function h is constant, while in the second and f is independent of t.
Consider the following PBVP:
Then , , , , and
Moreover, (7) with reads
and the assumptions (H2)-(H7) are met with , , , , , and . By Theorem 2, problem (12) has a positive solution.
Consider the PBVP
In this case, we have , , and
The assumptions of Theorem 2 are fulfilled with , , , , , , and .
The authors declare that they have no competing interests.
MZ and PD contributed equally to the manuscript and read and approved its final version.
Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.
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