- Open Access
Existence of positive solutions for a kind of periodic boundary value problem at resonance
Boundary Value Problemsvolume 2013, Article number: 19 (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.
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
for all and ,
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 1 For 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 .
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
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 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 .
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.
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.
If , then there exists such that . Observe that (H2) implies for and . Suppose that . If , we get from (8)(10)
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 .
Chu J, Fan N, Torres PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 2012, 388: 665-675. 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: 272-290. 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: 1185-1197. 10.1016/j.jde.2008.06.012
Ma R, Xu J, Han X: Global structure of positive solutions for superlinear second-order periodic boundary value problems. Appl. Math. Comput. 2012, 218: 5982-5988. 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: 317-341. 10.1016/S0362-546X(96)00313-6
Torres PJ: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 2003, 190: 643-662. 10.1016/S0022-0396(02)00152-3
Yao Q: Positive solutions of nonlinear second-order periodic boundary value problems. Appl. Math. Lett. 2007, 20: 583-590. 10.1016/j.aml.2006.08.003
Ma R, Gao C, Chen R: Existence of positive solutions of nonlinear second-order 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: 445-469. 10.1006/jdeq.2000.3995
Wang F, Zhang F, Wang F: The existence and multiplicity of positive solutions for second-order 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: 789-806. 10.1016/S0362-546X(00)00144-9
O’Regan D, Zima M: Leggett-Williams norm-type theorems for coincidences. Arch. Math. 2006, 87: 233-244. 10.1007/s00013-006-1661-6
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: 669-678. 10.1216/RMJ-1982-12-4-669
Santanilla J: Some coincidence theorems in wedges, cones, and convex sets. J. Math. Anal. Appl. 1985, 105: 357-371. 10.1016/0022-247X(85)90053-8
Santanilla J: Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations. J. Math. Anal. Appl. 1987, 126: 397-408. 10.1016/0022-247X(87)90049-7
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: 610-636. 10.1016/0022-0396(72)90028-9
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. I-IV. In World Congress of Nonlinear Analysts ’92 (Tampa, FL 1992). de Gruyter, Berlin; 1996:137-147.
Feng W, Webb JRL: Solvability of three-point boundary value problems at resonance. Nonlinear Anal. 1997, 30: 3227-3238. 10.1016/S0362-546X(96)00118-6
Liu B: Solvability of multi-point boundary value problems at resonance. IV. Appl. Math. Comput. 2003, 143: 275-299. 10.1016/S0096-3003(02)00361-2
Kosmatov N: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal. 2008, 68: 2158-2171. 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: 875-884. 10.1002/mana.200810841
Cabada A, Pouso R:Existence result for the problem with periodic and Neumann boundary conditions. Nonlinear Anal. 1997, 30: 1733-1742. 10.1016/S0362-546X(97)00249-6
Sȩdziwy S: Nonlinear periodic boundary value problem for a second order ordinary differential equation. Nonlinear Anal. 1998, 32: 881-890. 10.1016/S0362-546X(97)00533-6
Kiguradze I, Staněk S: On periodic boundary value problem for the equation with one-sided growth restrictions on f . Nonlinear Anal. 2002, 48: 1065-1075. 10.1016/S0362-546X(00)00235-2
Torres PJ: Existence and stability of periodic solutions of a Duffing equation by using a new maximum principle. Mediterr. J. Math. 2004, 1: 479-486. 10.1007/s00009-004-0025-3
Cheng Z, Ren J: Harmonic and subharmonic solutions for superlinear damped Duffing equation. Nonlinear Anal., Real World Appl. 2013, 14: 1155-1170. 10.1016/j.nonrwa.2012.09.007
Petryshyn WV: On the solvability of in quasinormal cones with T and F k -set contractive. Nonlinear Anal. 1981, 5: 585-591. 10.1016/0362-546X(81)90105-X
Infante G, Zima M: Positive solutions of multi-point boundary value problems at resonance. Nonlinear Anal. 2008, 69: 2458-2465. 10.1016/j.na.2007.08.024
Zhang HE, Sun JP: Positive solutions of third-order nonlocal boundary value problems at resonance. Bound. Value Probl. 2012., 2012: Article ID 102
Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.
The authors declare that they have no competing interests.
MZ and PD contributed equally to the manuscript and read and approved its final version.