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Positive periodic solutions of second-order difference equations with weak singularities
Advances in Difference Equations volume 2012, Article number: 90 (2012)
Abstract
We study the existence of positive periodic solutions of the second-order difference equation
via Schauder's fixed point theorem, where a, c : ℤ → ℝ+ are T -periodic functions, f ∈ C(ℤ × (0, ∞), ℝ) is T -periodic with respect to t and singular at u = 0.
Mathematics Subject Classifications: 34B15.
1 Introduction and the main results
Let ℤ denote the integer set, for a, b ∈ ℤ with a < b, [a, b]ℤ : = {a, a + 1,..., b} and ℝ+ : = [0; ∞). In this article, we are concerned with the existence of positive periodic solutions of the second-order difference equation
where a, c : ℤ → ℝ+ are T-periodic functions, f ∈ C(ℤ × (0, ∞), ℝ) is T-periodic with respect to t and singular at u = 0.
Positive periodic solutions of second-order difference equations have been studied by many authors, see [1–6]. However, in these therein, the nonlinearities are nonsingular, what would happen if the nonlinearity term is singular? It is of interest to note here that singular boundary value problems in the continuous case have been studied in great detail in the literature [7–20]. In 1987, Lazer and Solimini [7] firstly investigated the existence of the positive periodic solutions of the problem
where c ∈ C(ℝ, ℝ) is T-periodic. They proved that for λ ≥ 1 (called strong force condition in a terminology first introduced by Gordon [8, 9]), a necessary and sufficient condition for the existence of a positive periodic solution of (1.2) is that the mean value of c is negative,
Moreover, if 0 < λ < 1 (weak force condition) they found examples of functions c with negative mean values and such that periodic solutions do not exist. Subsequently, many authors studied the existence of positive solutions of the problem
where a ∈ L1(ℝ/T ℤ; ℝ+), c ∈ L1(R/T ℤ; ℝ), f ∈ Car(ℝ/T ℤ × (0, ∞), ℝ) and is singular at u = 0, see [7–20]. The first existence result with weak force condition appears in Rachunková et al. [16]. Since then, the Equation (1.3) with f has weak singularities has been studied by several authors, see Torres [17, 18], Franco and Webb [19], Chu and Li [20].
Recently, Torres [18] showed how a weak singularity can play an important role if Schauder's fixed point theorem is chosen in the proof of the existence of positive periodic solution for (1.3). For convenience, for a given function ξ ∈ L∞[0, T], we denote the essential supremum and infimum of ξ by ξ* and ξ*, respectively. We write ξ ≻ 0 if ξ ≥ 0 for a.e. t ∈ [0, T] and it is positive in a set of positive measure. Under the assumption
(H1) The linear equation u"+ a(t)u = 0 is nonresonant and the corresponding Green's function
Torres showed the following three results
Theorem A. [[18], Theorem 1] Let (H1) hold and define
Assume that
(H2) there exist b∈ L1(0, T) with b ≻ 0 and λ > 0 such that
If γ * > 0, then there exists a positive T-periodic solution of (1.3).
Theorem B. [[18], Theorem 2] Let (H1) hold. Assume that
(H3) there exist two functions b, with b, and a constant λ ∈ (0, 1) such that
If γ* = 0. Then (1.3) has a positive T-periodic solution.
Theorem C. [[18], Theorem 4] Let (H1) and (H3) hold. Let
If γ* ≤ 0 and
Then (1.3) has a positive T-periodic solution.
However, the discrete analogue of (1.3) has received almost no attention. In this article, we will discuss in detail the singular discrete problem (1.1) with our goal being to fill the above stated gap in the literature. For other results on the existence of positive solution for the other singular discrete boundary value problem, see [21–24] and their references. From now on, for a given function ξ ∈ l∞(0, ∞), we denote the essential supremum and infimum of ξ by ξ* and ξ*, respectively. We write ξ ≻ 0 if ξ ≥ 0 for t [0, T ]ℤ and it is positive in a set of positive measure.
Assume that
(A1) The linear equation Δ2u(t - 1)+ a(t)u(t) = 0 is nonresonant and the corresponding Green's function
(A2) There exist b, e : [1, T]ℤ → ℝ+ with b, e ≻ 0, α, β ∈ (0, ∞), m ≤ 1 ≤ M, such that
and
(A3) There exist b1, b2, e : [1, T ]ℤ → ℝ+ with b1, b2, e ≻ 0, α, β, μ, v ∈ (0, 1), such that
and
To prove the main results, we will use the following notations.
Our main results are the following
Theorem 1.1. Let (A1) and (A2) hold. If γ* > 0. Then (1.1) has a positive T-periodic solution.
Theorem 1.2. Let (A1) and (A3) hold. If γ* = 0. Then (1.1) has a positive T-periodic solution.
Theorem 1.3. Let (A1) and (A3) hold. Assume that
If γ* ≤ 0 and
Then (1.1) has a positive T-periodic solution.
Remark 1.1. Let us consider the function
where ε, η > 0. Obviously, f0 satisfies (A2) with M = m = 1, b(t) = e(t) ≡ 1. However, it is fail to satisfy (H2) since it can not be bounded by a single function for any γ ∈ (0, ∞) and any h ≻ 0. □
Remark 1.2. If ε, η ∈ (0, 1), then the function f0 defined by (1.6) satisfies (A3) with ν = μ = η, α = β = ε, and b1(t) ≡ b2(t) ≡ e(t) ≡ 1. However, it is fail to satisfy (H3). □
2 Proof of Theorem 1.1
Let
under the norm. Then (X, || · ||) is a Banach space.
A T-periodic solution of (1.1) is just a fixed point of the completely continuous map A : X → X defined as
By Schauder's fixed point theorem, the proof is finished if we prove that A maps the closed convex set defined as
into itself, where R > r > 0 are positive constants to be fixed properly.
For given u∈ K, let us denote
Given u ∈ K, by the nonnegative sign of G and f, we have
Let
Then, it follows from the continuity of f that Λ < ∞, and consequently, for every u ∈ K,
Therefore, A(K) ⊂ K if r = γ* and , and the proof is finished. □
3 Proof of Theorem 1.2
We follow the same strategy and notations as in the proof of Theorem 1.1. Define a closed convex set
By a direct application of Schauder's fixed point theorem, the proof is finished if we prove that A maps the closed convex set K into itself, where R and r are positive constants to be fixed properly and they should satisfy R > r > 0 and R > 1.
For given u ∈ K, let us denote
Then for given u ∈ K, by the nonnegative sign of G and f, it follows that
On the other hand, for every u∈ K,
Thus Au ∈ K if r, R are chosen so that
Note that B1* , E* > 0 and taking , it is sufficient to find R > 1 such that
and these inequalities hold for R big enough because σ < 1 and ν < 1. □
Remark 3.1. It is worth remarking that Theorem 1.2 is also valid for the special case that c(t) ≡ 0, which implies that γ* = 0. □
4 Proof of Theorem 1.3
Define a closed convex set
By a direct application of Schauder's fixed point theorem, the proof is finished if we prove that A maps the closed convex set K into itself, where R and r are positive constants to be fixed properly and they should satisfy R > 1 > r > 0.
Recall that δ = max{ν, β} and r < 1, for given u ∈ K,
where J i (i = 1, 2) is defined as in Section 3 and .
On the other hand, since σ = max {μ, α} and R > 1, for every u∈ K,
In this case, to prove that A(K) ⊂ K it is sufficient to find r < R with 0 < r < 1 < R such that
If we fix , then the first inequality holds if r verifies
or equivalently,
The function f(r) possesses a minimum in . Taking r = r0, (1.4) implies that r < 1. Then the first inequality in (4.1) holds if γ* ≥ f (r0), which is just condition (1.5). The second inequality in (4.1) holds directly by the choice of R, and it would remain to prove that . To the end, it follows from (1.4) that
This completes the proof. □
Remark 4.1. Note that the condition (1.4), which is stated as
is crucial to guarantee that R > 1 > r0, and in the proof of Theorem 1.3 we require R > 1 > r0 because the exponents in inequalities of (A3) is different. However, in the special case that
if we define ω (t): = max{b2(t), e(t)}, t ∈ [0, T]ℤ, then the condition (1.4) is needn't because R > r0 can be easily verified by
□
Example 4.1. Let us consider the second order periodic boundary value problem
where
and is a constant.
It is easy to check that (4.2) is equivalent to the operator equation
here
Clearly, G(t, s) > 0 for all (t, s) ∈ 0[4]ℤ × 0[4]ℤ.
Let
Then
and
Thus, the condition (A3) is satisfied. By simple computations, we get
and . So the condition (1.4) is satisfied. Moreover,
and so
Finally, since , it follows that
Consequently, Theorem 1.3 yields that (4.2) has a positive solution. □
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Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11061030), NSFC (No. 11126296), the Fundamental Research Funds for the Gansu Universities.
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RM completed the main study, carried out the results of this article and drafted the manuscript. YL checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
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Ma, R., Lu, Y. Positive periodic solutions of second-order difference equations with weak singularities. Adv Differ Equ 2012, 90 (2012). https://doi.org/10.1186/1687-1847-2012-90
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DOI: https://doi.org/10.1186/1687-1847-2012-90
Keywords
- positive periodic solutions
- difference equations
- Schauder's fixed point theorem
- weak singularities.