 Research
 Open access
 Published:
Existence of positive solutions for a kind of periodic boundary value problem at resonance
Boundary Value Problems volume 2013, Article number: 19 (2013)
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.
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.
References
Chu J, Fan N, Torres PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 2012, 388: 665675. 10.1016/j.jmaa.2011.09.061
Cabada A, Cid JÁ: On comparison principles for the periodic Hill’s equation. J. Lond. Math. Soc. 2012, 86: 272290. 10.1112/jlms/jds001
Graef JR, Kong L, Wang H: Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. J. Differ. Equ. 2008, 245: 11851197. 10.1016/j.jde.2008.06.012
Ma R, Xu J, Han X: Global structure of positive solutions for superlinear secondorder periodic boundary value problems. Appl. Math. Comput. 2012, 218: 59825988. 10.1016/j.amc.2011.11.079
Meehan M, O’Regan D: Existence theory for nonlinear Volterra integrodifferential and integral equations. Nonlinear Anal. 1998, 31: 317341. 10.1016/S0362546X(96)003136
Torres PJ: Existence of onesigned periodic solutions of some secondorder differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 2003, 190: 643662. 10.1016/S00220396(02)001523
Yao Q: Positive solutions of nonlinear secondorder periodic boundary value problems. Appl. Math. Lett. 2007, 20: 583590. 10.1016/j.aml.2006.08.003
Ma R, Gao C, Chen R: Existence of positive solutions of nonlinear secondorder periodic boundary value problems. Bound. Value Probl. 2010., 2010: Article ID 626054. doi:10.1155/2010/626054
Rachůnková I, Tvrdý M, Vrkoč I: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differ. Equ. 2001, 176: 445469. 10.1006/jdeq.2000.3995
Wang F, Zhang F, Wang F: The existence and multiplicity of positive solutions for secondorder periodic boundary value problem. J. Funct. Spaces Appl. 2012., 2012: Article ID 725646. doi:10.1155/2012/725646
Cremins CT: A fixed point index and existence theorems for semilinear equations in cones. Nonlinear Anal. 2001, 46: 789806. 10.1016/S0362546X(00)001449
O’Regan D, Zima M: LeggettWilliams normtype theorems for coincidences. Arch. Math. 2006, 87: 233244. 10.1007/s0001300616616
Gaines RE, Santanilla J: A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations. Rocky Mt. J. Math. 1982, 12: 669678. 10.1216/RMJ1982124669
Santanilla J: Some coincidence theorems in wedges, cones, and convex sets. J. Math. Anal. Appl. 1985, 105: 357371. 10.1016/0022247X(85)900538
Santanilla J: Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations. J. Math. Anal. Appl. 1987, 126: 397408. 10.1016/0022247X(87)900497
Mawhin J: Equivalence theorems for nonlinear operator equations and coincidence degree theory for mappings in locally convex topological vector spaces. J. Differ. Equ. 1972, 12: 610636. 10.1016/00220396(72)900289
Gaines RE, Mawhin J Lect. Notes Math. 568. In Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin; 1977.
Webb JRL: Solutions of semilinear equations in cones and wedges. IIV. In World Congress of Nonlinear Analysts ’92 (Tampa, FL 1992). de Gruyter, Berlin; 1996:137147.
Feng W, Webb JRL: Solvability of threepoint boundary value problems at resonance. Nonlinear Anal. 1997, 30: 32273238. 10.1016/S0362546X(96)001186
Liu B: Solvability of multipoint boundary value problems at resonance. IV. Appl. Math. Comput. 2003, 143: 275299. 10.1016/S00963003(02)003612
Kosmatov N: Multipoint boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 2008, 68: 21582171. 10.1016/j.na.2007.01.038
Franco D, Infante G, Zima M: Second order nonlocal boundary value problems at resonance. Math. Nachr. 2011, 284: 875884. 10.1002/mana.200810841
Cabada A, Pouso R:Existence result for the problem {(\varphi ({u}^{\prime}))}^{\prime}=f(t,u,{u}^{\prime}) with periodic and Neumann boundary conditions. Nonlinear Anal. 1997, 30: 17331742. 10.1016/S0362546X(97)002496
Sȩdziwy S: Nonlinear periodic boundary value problem for a second order ordinary differential equation. Nonlinear Anal. 1998, 32: 881890. 10.1016/S0362546X(97)005336
Kiguradze I, Staněk S: On periodic boundary value problem for the equation {u}^{\u2033}=f(t,u,{u}^{\prime}) with onesided growth restrictions on f . Nonlinear Anal. 2002, 48: 10651075. 10.1016/S0362546X(00)002352
Torres PJ: Existence and stability of periodic solutions of a Duffing equation by using a new maximum principle. Mediterr. J. Math. 2004, 1: 479486. 10.1007/s0000900400253
Cheng Z, Ren J: Harmonic and subharmonic solutions for superlinear damped Duffing equation. Nonlinear Anal., Real World Appl. 2013, 14: 11551170. 10.1016/j.nonrwa.2012.09.007
Petryshyn WV: On the solvability of x\in Tx+\lambda Fx in quasinormal cones with T and F k set contractive. Nonlinear Anal. 1981, 5: 585591. 10.1016/0362546X(81)90105X
Infante G, Zima M: Positive solutions of multipoint boundary value problems at resonance. Nonlinear Anal. 2008, 69: 24582465. 10.1016/j.na.2007.08.024
Zhang HE, Sun JP: Positive solutions of thirdorder nonlocal boundary value problems at resonance. Bound. Value Probl. 2012., 2012: Article ID 102
Acknowledgements
Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MZ and PD contributed equally to the manuscript and read and approved its final version.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/16872770201319