Existence of positive solutions for a kind of periodic boundary value problem at resonance
© Zima and Drygaś; licensee Springer. 2013
Received: 20 December 2012
Accepted: 21 January 2013
Published: 11 February 2013
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.
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.
for all and ,
x, implies .
The following property holds for every cone in a Banach space.
for all .
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,
and is such that for every ,
7∘ and .
Theorem 1 
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
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 2 Under 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 .
where ψ is given by (5).
Since , we have . Moreover, , which gives . Consequently, L is Fredholm of index zero, and the assumption 1∘ is satisfied.
Obviously, and are open bounded subsets of X, and .
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)(10)
contrary to (H5). By similar arguments, if or , BCs (9) and (H4) imply either (10) or (11). Thus, 3∘ is fulfilled.
Clearly, ρ is a retraction and maps subsets of into bounded subsets of C, so 4∘ holds.
This implies for , so 6∘ is satisfied.
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.
and the assumptions (H2)-(H7) are met with , , , , , and . By Theorem 2, problem (12) has a positive solution.
The assumptions of Theorem 2 are fulfilled with , , , , , , and .
Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.
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