Abstract
In the paper we provide sufficient conditions for the existence of positive solutions for some secondorder differential equation subject to periodic boundary conditions. Our method employs a LeggettWilliams normtype theorem for coincidences due to O’Regan and Zima. Two examples are given to illustrate the main result of the paper.
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1 Introduction
In the paper we are interested in the existence of positive solutions for the periodic boundary value problem (PBVP)
where f:[0,T]\times [0,\mathrm{\infty})\times \mathbb{R}\to \mathbb{R} and h:[0,T]\to (0,\mathrm{\infty}) 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 [1], 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 LeraySchauder type (see, for example, [5]). It should be noted that a\not\equiv 0 was the key assumption used in [1]. If a\equiv 0, 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 [6], 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 [7] and [8]. In [9] 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 [10]. Their existence and multiplicity results on positive solutions are based on the theory of a fixed point index for Aproper semilinear operators on cones developed by Cremins [11].
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 [0,T]. Our general result is illustrated by two examples. The method we use in the paper is to rewrite BVP (1) as a coincidence equation Lx=Nx, where L is a Fredholm operator of index zero and N is a nonlinear operator, and to apply the LeggettWilliams normtype theorem for coincidences obtained by O’Regan and Zima [12]. We would like to emphasize that the idea of results of [11] and [12], as well as these of [13–15], goes back to the celebrated Mawhin’s coincidence degree theory [16]. 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 [12] is used for studying the boundary value problem with the nonlinearity f depending also on the derivative. In general, the presence of {x}^{\prime} 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 [27]. It is evident that the existence theorems for PBVP (1) can be established by the shift method used in [6], that is, one can employ the results of [1] to the periodic problem we study here. However, the conditions imposed on f in [1] 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, C\ne \{0\}, of a real Banach space X is called a cone if C is closed, convex and

(i)
\lambda x\in C for all x\in C and \lambda \ge 0,

(ii)
x, x\in C implies x=0.
Every cone induces a partial ordering in X as follows: for x,y\in X, we say that
The following property holds for every cone in a Banach space.
Lemma 1 [28]For every u\in C\setminus \{0\}, there exists a positive number \sigma (u) such that
for all x\in C.
Consider a linear mapping L:domL\subset X\to Y and a nonlinear operator N:X\to Y, where X and Y are Banach spaces. If L is a Fredholm operator of index zero, that is, ImL is closed and dimKerL=codimImL<\mathrm{\infty}, then there exist continuous projections P:X\to X and Q:Y\to Y such that ImP=KerL and KerQ=ImL (see, for example, [14, 16]). Moreover, since dimImQ=codimImL, there exists an isomorphism J:ImQ\to KerL. Denote by {L}_{P} the restriction of L to KerP\cap domL. Then {L}_{P} is an isomorphism from KerP\cap domL to ImL and its inverse
is defined.
As a result, the coincidence equation Lx=Nx is equivalent to x=\mathrm{\Psi}x, where
Let \rho :X\to C be a retraction, that is, a continuous mapping such that \rho (x)=x for all x\in C. Put
Let {\mathrm{\Omega}}_{1}, {\mathrm{\Omega}}_{2} be open bounded subsets of X with {\overline{\mathrm{\Omega}}}_{1}\subset {\mathrm{\Omega}}_{2} and C\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1})\ne \mathrm{\varnothing}. Assume that
1^{∘} L is a Fredholm operator of index zero,
2^{∘} QN:X\to Y is continuous and bounded and {K}_{P}(IQ)N:X\to X is compact on every bounded subset of X,
3^{∘} Lx\ne \lambda Nx for all x\in C\cap \partial {\mathrm{\Omega}}_{2}\cap domL and \lambda \in (0,1),
4^{∘} ρ maps subsets of {\overline{\mathrm{\Omega}}}_{2} into bounded subsets of C,
5^{∘} {d}_{B}([I(P+JQN)\rho ]{}_{KerL},KerL\cap {\mathrm{\Omega}}_{2},0)\ne 0, where {d}_{B} stands for the Brouwer degree,
6^{∘} there exists {u}_{0}\in C\setminus \{0\} such that \parallel x\parallel \le \sigma ({u}_{0})\parallel \mathrm{\Psi}x\parallel for x\in C({u}_{0})\cap \partial {\mathrm{\Omega}}_{1}, where
and \sigma ({u}_{0}) is such that \parallel x+{u}_{0}\parallel \ge \sigma ({u}_{0})\parallel x\parallel for every x\in C,
7^{∘} (P+JQN)\rho (\partial {\mathrm{\Omega}}_{2})\subset C and {\mathrm{\Psi}}_{\rho}({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1})\subset C.
Theorem 1 [12]
Under the assumptions 1^{∘}7^{∘} the equation Lx=Nx has a solution in the set C\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}).
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 [22], [29] and [30].
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:
and
We observe that 0<\psi (t)<\frac{1}{e(T)(1e(T))} on [0,T]. Moreover, we put
and
where M is a positive constant.
We assume that
(H1) f:[0,T]\times [0,\mathrm{\infty})\times \mathbb{R}\to \mathbb{R} and h:[0,T]\to (0,\mathrm{\infty}) are continuous functions.
We also assume that there exist R>0, 0<\alpha \le \beta, 0<M\le \frac{e(T)(1e(T)){\int}_{0}^{T}\psi (\tau )\phantom{\rule{0.2em}{0ex}}d\tau}{\alpha T}, r\in (0,R), m\in (0,1), \eta \in [0,T] and a continuous function g:[0,T]\to [0,\mathrm{\infty}) such that
(H2) f(t,x,y)>\alpha x+\beta y for (t,x,y)\in [0,T]\times [0,R]\times [R,R],
(H3) f(t,R,0)<0 for t\in [0,T],
(H4) f(0,x,R)=f(T,x,R) and f(0,x,R)=f(T,x,R) for x\in [0,R],
(H5) f(t,x,R)\le h(t)R for t\in [0,T] and x\in [0,R),
(H6) f(t,x,y)\ge g(t)(x+y) for (t,x,y)\in [0,T]\times (0,r]\times [r,r],
(H7) \frac{1}{\alpha T}\ge K(t,s)\ge 0 for t,s\in [0,T] and m{\int}_{0}^{T}K(\eta ,s)g(s)\phantom{\rule{0.2em}{0ex}}ds\ge 1.
Theorem 2 Under the assumptions (H1)(H7), PBVP (1) has a positive solution on [0,T].
Proof Let {\parallel \cdot \parallel}_{\mathrm{\infty}} denote the supremum norm in the space C[0,T], that is, {\parallel x\parallel}_{\mathrm{\infty}}={sup}_{t\in [0,T]}x(t). Consider the Banach spaces X={C}^{1}[0,T] with the norm \parallel x\parallel =max\{{\parallel x\parallel}_{\mathrm{\infty}},{\parallel {x}^{\prime}\parallel}_{\mathrm{\infty}}\}, and Y=C[0,T] with the norm {\parallel \cdot \parallel}_{\mathrm{\infty}}.
We write problem (1) as a coincidence equation
where
and
with domL=\{x\in X:{x}^{\u2033}\in C[0,T],x(0)=x(T),{x}^{\prime}(0)={x}^{\prime}(T)\}. Then
and
where ψ is given by (5).
Clearly, ImL is closed and Y={Y}_{1}+ImL with
Since {Y}_{1}\cap ImL=\{0\}, we have Y={Y}_{1}\oplus ImL. Moreover, dim{Y}_{1}=1, which gives codimImL=1. Consequently, L is Fredholm of index zero, and the assumption 1^{∘} is satisfied.
Define the projections P:X\to X by
and Q:Y\to Y by
It is a routine matter to show that for y\in ImL, the inverse {K}_{P} of {L}_{P} is given by
with the kernel k defined by (6). Clearly, the assumption 2^{∘} is satisfied. For y\in ImQ, define
Then J is an isomorphism from ImQ to KerL. Next, consider a cone
For {u}_{0}(t)\equiv 1, we have \sigma ({u}_{0})=1 and
Let
and
Obviously, {\mathrm{\Omega}}_{1} and {\mathrm{\Omega}}_{2} are open bounded subsets of X, and {\overline{\mathrm{\Omega}}}_{1}\subset {\mathrm{\Omega}}_{2}.
To verify 3^{∘}, suppose that there exist {x}_{0}\in C\cap \partial {\mathrm{\Omega}}_{2}\cap domL and {\lambda}_{0}\in (0,1) such that L{x}_{0}={\lambda}_{0}N{x}_{0}. Then x(t)\ge 0 on [0,T], \parallel {x}_{0}\parallel =R,
and
There are two cases to consider.

1.
If \parallel {x}_{0}\parallel ={\parallel {x}_{0}\parallel}_{\mathrm{\infty}}, then there exists {t}_{0}\in [0,T] such that x({t}_{0})=R. For {t}_{0}\in (0,T), we get 0\le {x}^{\u2033}({t}_{0})={\lambda}_{0}f({t}_{0},R,0), contrary to the assumption (H3). Similarly, if {t}_{0}=0 or {t}_{0}=T, BCs (9) imply {x}^{\prime}(0)={x}^{\prime}(T)=0. Hence, 0\le {x}^{\u2033}({t}_{0})={\lambda}_{0}f({t}_{0},R,0) which contradicts (H3) again.

2.
If \parallel {x}_{0}\parallel ={\parallel {x}_{0}^{\prime}\parallel}_{\mathrm{\infty}}>{\parallel {x}_{0}\parallel}_{\mathrm{\infty}}, then there exists {t}_{0}\in [0,T] such that {x}^{\prime}({t}_{0})=R. Observe that (H2) implies f(t,x,\pm R)>0 for t\in [0,T] and x\in [0,R]. Suppose that {t}_{0}\in (0,T). If {x}^{\prime}({t}_{0})=R, we get from (8)
h({t}_{0})R={\lambda}_{0}f({t}_{0},{x}_{0}({t}_{0}),R),(10)
a contradiction. For {x}^{\prime}({t}_{0})=R, we have
contrary to (H5). By similar arguments, if {t}_{0}=0 or {t}_{0}=T, BCs (9) and (H4) imply either (10) or (11). Thus, 3^{∘} is fulfilled.
Next, for x\in X, define (see [15])
Clearly, ρ is a retraction and maps subsets of {\overline{\mathrm{\Omega}}}_{2} into bounded subsets of C, so 4^{∘} holds.
To verify 5^{∘}, it is enough to consider, for x\in KerL\cap {\mathrm{\Omega}}_{2} and \lambda \in [0,1], the mapping
Observe that if x\in KerL\cap {\mathrm{\Omega}}_{2}, then x(t)=c on [0,T] and \parallel x\parallel <R. Suppose H(x,\lambda )=0 for x\in \partial {\mathrm{\Omega}}_{2}. Then c=\pm R. For c=R, we have (\rho x)(t)=x(t) and in view of (H3), we get
which is a contradiction. If c=R, then (\rho x)(t)=0, hence
which contradicts (H2). Thus, H(x,\lambda )\ne 0 for x\in \partial {\mathrm{\Omega}}_{2} and \lambda \in [0,1]. This implies
and
We next show that 6^{∘} holds. Let x\in C({u}_{0})\cap \partial {\mathrm{\Omega}}_{1}. Then for t\in [0,T], we have r\ge x(t)\ge m{\parallel x\parallel}_{\mathrm{\infty}}>0, r\ge {x}^{\prime}(t)\ge {\parallel {x}^{\prime}\parallel}_{\mathrm{\infty}}, and by (H6) and (H7), we obtain
This implies \parallel x\parallel \le \parallel \mathrm{\Psi}x\parallel for x\in C({u}_{0})\cap \partial {\mathrm{\Omega}}_{1}, so 6^{∘} is satisfied.
Finally, we must check if 7^{∘} holds. If x\in \partial {\mathrm{\Omega}}_{2}, then in view of (H2), we get
Moreover, for x\in {\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}, 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 h(t)=1/(1+t) and f is independent of t.
Example 1
Consider the following PBVP:
Then e(t)={e}^{t}, \phi (t)=1{e}^{t}, \mathrm{\Phi}(t)=t+{e}^{t}1, \psi (t)=\frac{e}{e1}, and
Moreover, (7) with M=\frac{3}{2} reads
and the assumptions (H2)(H7) are met with R=20, \alpha =\frac{2}{9}, \beta =\frac{3}{4}, r=\frac{36}{53}, m\in [\frac{12(e1)}{17+7e},1), \eta =0 and g(t)=t(1t)+1. By Theorem 2, problem (12) has a positive solution.
Example 2
Consider the PBVP
In this case, we have e(t)=\frac{1}{1+t}, \phi (t)=ln(1+t), \mathrm{\Phi}(t)=t+ln(1+t)+tln(1+t) and
The assumptions of Theorem 2 are fulfilled with M=1, R=10, \alpha =\frac{1}{3}, \beta =\frac{1}{2}, r=\frac{1}{100}, m=0.9, \eta =\frac{1}{4} and g(t)=3.
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Acknowledgements
Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.
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MZ and PD contributed equally to the manuscript and read and approved its final version.
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Zima, M., Drygaś, P. Existence of positive solutions for a kind of periodic boundary value problem at resonance. Bound Value Probl 2013, 19 (2013). https://doi.org/10.1186/16872770201319
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DOI: https://doi.org/10.1186/16872770201319