Positive solutions of periodic boundary value problems for the second-order differential equation with a parameter

In this paper, we investigate the existence of positive solutions for a class of singular second-order differential equations with periodic boundary conditions. By using the fixed point theory in cones, the explicit range for λ is derived such that for any λ lying in this interval, the existence of at least one positive solution to the boundary value problem is guaranteed.


Introduction
Reaction-diffusion problems often arise in physics, chemistry, biology, economics, and various engineering fields. A class of reaction-diffusion equations includes several known evolution equations. For equation (), if traveling wave satisfies w(x, t) = W (x -Ct) with speed C, then equation () can be converted to a second-order ordinary differential equation With appropriate boundary value conditions, the existence of positive solution of equation () is significant and helpful. The Liebau phenomenon, which is in honor of the physician Liebauh's pioneering work, is the occurrence of valveless pumping through the application of a periodic force at a place which lies asymmetric with respect to system configuration. Propst [] made use of differential equations to model a periodically forced flow through different pipe-tank configurations. In one pipe-one tank configuration, ignore the singularity in the corresponding differential equation model, namely According to the physical meaning of the involved parameters, assume a ≥ , b > , c > , and e is continuous and T-periodic on [, +∞). In [], Cid et al. applied the substitution u = x μ , μ =  b+ , and then transformed the singular periodic boundary value problem () to the regular problem where Based on the lower and upper solution technique, the existence and asymptotic stability of positive solutions for () are obtained.
In this paper, we discuss the positive solutions of the following periodic boundary value problem (PBVP): is a continuous function and f (t, u) may be singular at t = , t = T and u = . In recent years, the existence of solutions for differential equations has been widely studied by many scholars in the mathematical sense (see [-] and the references therein). In [, ], through the use of Guo-Krasnosel'skii's fixed point theorem, the existence and multiplicity of positive solutions for the following periodic boundary value problem were established.
In [], the author researched PBVP () by using an L p -anti-maximum principle and obtained the existence results in order to overcome the difficulties of the symbol of Green's functions for the corresponding linear periodic problem: Motivated by the above works, we consider PBVP (). In (), if f (t, x(t)) = m  x(t)+s(t)x βr(t)x α , then () is a special case of (). Compared with [, ], in which the existence and multiplicity of positive solutions for () () are considered, we not only obtain the existence of positive solutions for (), but also increase the parameter λ and get the explicit range of λ by using the fixed point theory in cones. Therefore, our article contains, promotes, and improves the previous results to a certain extent.

Preliminaries and lemmas
In this section, we present some notations and lemmas that will be used in the proof of our main results.
has an integral representation where G(t, s) is the related Green's function.
Then G(t, s) has the following properties: where ξ is defined as Lemma .. It is easy to see that K is a positive and normal cone in X. For any  < r < R < +∞, let K r,R = {x ∈ K : r ≤ x ≤ R}. In this paper, we always assume that the following conditions hold.
Under assumptions (H  )-(H  ), for any n ∈ N, N is a natural number set, we define a nonlinear integral operator A n : K → X by where f n (t, u) = f (t, (u +  n )). Obviously, the existence of solutions to () is equivalent to the existence of solutions in K for the operator equation A n x = x defined by (). In this paper, the proof of the main theorem is based on the fixed point theory in cones. We list the following lemmas which are needed in our study.
Let A : K r,R → K be a completely continuous operator. If the following conditions are satisfied: Remark . If () and () are satisfied for x ∈ ∂K r and x ∈ ∂K R , respectively, then Lemma . is still true.
then A has at least one fixed point in K ∩ (  \  ).

Main results
Theorem . Assume that (H  )-(H  ) hold, then A n : K → K is a completely continuous operator for any fixed n ∈ N.
Proof Let λ >  and n ∈ N be fixed. For any x ∈ K and t ∈ [, T], by Lemma ., we have This implies that (A n x)(t) ≥ ξ A n x , therefore A n (K) ⊂ K . By a standard argument, under assumptions (H  )-(H  ), we know that A n : K → K is well defined. Next, for any positive integers n, m ∈ N, we define an operator A n,m : K → X by In a similar discussion, A n,m : K → X is well defined and A n,m (K) ⊆ K . In what follows, we will prove that A n,m : K → K is completely continuous for each m ≥ . Firstly, we show that , |f n (t, x υ (t))f n (t, x(t))| →  as υ → +∞. Using the Lebesgue dominated convergence theorem, we have So, A n,m : K → C[, T] is continuous for any natural numbers n, m. Then A n,m : K → K is continuous for any natural numbers n, m.
Let D ⊂ K be any bounded set, then for any x ∈ D, we have x ≤ r, and then  < ξ r ≤ x(t) ≤ r for any t ∈ [, T]. By (H  )-(H  ), for any x ∈ D, we have In order to show that A n,m is a compact operator, we only need to show that A n,m D is equicontinuous. For any ε > , by the continuity of G(t, s) Then, for any x ∈ D, for any t  , which means that A n,m D is equicontinuous. By the Arzela-Ascoli theorem, A n,m D is a relatively compact set and so A n,m : K → K is a completely continuous operator. Finally, we show that A n : K → K is a completely continuous operator. For any t ∈ [, T] and x ∈ S = {x ∈ K, x ≤ }, by (), (), we have Therefore, by A n,m : K → K is a completely continuous operator, we get that A n : K → K is a completely continuous operator. Then there exists λ >  such that PBVP () has at least one positive solution for any λ ∈ (, λ).
Proof Choose r  > , let For any x ∈ ∂K r  , t ∈ [, T], by the definition of · , we have For any λ ∈ (, λ), we have

G(t, s)f n s, x(s) ds
Thus, On the other hand, by the inequality in (H  ), choose l >  such that λlξ r  b a G(s, s) ds > , then there exists N * >  such that For any x ∈ ∂K r  , μ > , n ∈ N, we will show Hence, we conclude that This implies that r  > r  , which is a contradiction. This yields that () holds. It follows from the above discussion, (), (), Lemma . and Theorem . that, for any n ∈ N, λ ∈ (, λ), A n has a fixed point x n ∈ K r  \ K r  .
Let {x n } ∞ n= be the sequence of solutions of PBVP (). It is easy to see that they are uniformly bounded. From x n ∈ K r  \ K r  , we know that For any ε > , by the continuity of G(t, s) on [, T] × [, T], there exists δ  >  such that for any t  , t  , s ∈ [, T], |t t  | < δ  , we have

Thus,
A n x ≤ x for any x ∈ ∂K R  .
(   ) It follows from the above discussion, (), (), Lemma . and Theorem . that, for n ∈ N, λ ∈ (λ, +∞), A n has a fixed point x n ∈ K R  \ K R  satisfying R  ≤ x n ≤ R  . The rest of the proof is similar to Theorem .. That is the proof of Theorem ..
Corollary . The conclusion of Theorem . is valid if (H  ) is replaced by the following: Remark . From the proof of Theorems . and ., we can obtain the main results under the condition that the function f (t, u) not only has singularity on t but also has singularity on u, and we use the approximation method to overcome the difficulty caused by singularity.
Remark . In this paper, we can get the positive solution of PBVP () when the parameter λ is sufficiently large and small; concretely, we can choose λ ∈ (, ) and λ ∈ (, +∞). What is more, the solution x in PBVP () satisfies x(t) >  for any t ∈ [, T].

Examples
Consider the PBVP So all the conditions of Theorem . are satisfied. By Theorem ., PBVP () has at least one positive solution provided λ is small enough.