Electronic Journal of Qualitative Theory of Differential Equations Proc. 7th Coll. QTDE, 2004, No. 19 1-8; http://www.math.u-szeged.hu/ejqtde/

Construction of non-constant lower and upper functions for impulsive periodic problems

Irena Rachůnková^{∗} and Milan Tvrdý^{†}

Summary. We present conditions ensuring the existence of piecewise linear lower and upper functions for the nonlinear impulsive periodic boundary value problem u′′ = f(t, u, u′), u(t_{i}+) = J_{i}(u(t_{i})), u′(t_{i}+) = M_{i}(u′(t_{i})), i = 1, 2, …, m, u(0) = u(T), u′(0) = u′(T). This together with the existence principles which we proved in [5]–[7] allows us to prove new existence criteria, see Theorems 3.1 and 3.2. Mathematics Subject Classification 2000. 34B37, 34B15, 34C25 Keywords. Second order nonlinear ordinary differential equation, impulse, periodic solution, lower function, upper function.

1 . Introduction

This paper deals with the impulsive periodic boundary value problem

(1.1)  u′′ =f(t, u, u′),
(1.2)  u(t_{i}+) = J_{i}(u(t_{i})), u′(t_{i}+) = M_{i}(u′(t_{i})), i = 1, 2, …, m,
(1.3)  u(0) = u(T), u′(0) = u′(T),

where

(1.4)  \{ \begin{matrix}0 < t_{1} < ⋯ < t_{m} < T < ∞, \\ f satisfies the Carathéodory conditions on [0, T] × ℝ^{2}, \\ J_{i} and M_{i}, i = 1, 2, …, m, are continuous functions on ℝ. \end{matrix}

There are several papers providing the existence results for such problems in terms of lower and upper functions, see e.g. [1]–[4], [8] and our papers [5]–[7]. However, up to now, only Proposition 1.3 in [6] gives conditions ensuring the existence of nonconstant (in particular, piecewise constant) lower and upper functions. The main

∗Supported by the grant No. 201/01/1451 of the Grant Agency of the Czech Republic and by the Council of Czech Government J14/98:153100011

†Supported by the grant No. 201/01/1199 of the Grant Agency of the Czech Republic This paper is in final form and no version of it will be submitted for publications elsewhere.

goal of this paper is to find conditions for f, J_{i}, M_{i} giving piecewise linear lower and upper functions for (1.1)–(1.3). This together with the existence principles which we proved in [5]–[7] allows us to present new existence criteria.

Throughout the paper we keep the following notation and conventions: For J ⊂ ℝ, ℂ(J) is the set of real valued functions which are continuous on J, ℂ^{1}(J) is the set of functions having continuous first derivatives on J, 𝕃(J) is the set of functions Lebesgue integrable on J and 𝕃_{∞}(J) is the set of functions essentially bounded on J. For u ∈ 𝕃_{∞}[0, T], we denote ∥u∥_{∞} = sup ess_{t∈[0,T]} |u(t)|. Further, D = {t_{1}, t_{2}, …, t_{m}}, t_{0}=0, t_{m+1}=T and ℂ_{D}^{1}[0, T] is the set of functions u: [0, T] ↦ ℝ of the form

u(t) = \{ \begin{matrix}u_{[0]}(t) & if t ∈ [0, t_{1}], \\ u_{[1]}(t) & if t ∈ (t_{1}, t_{2}], \\ … & … \\ u_{[m]}(t) & if t ∈ (t_{m}, T], \end{matrix}

where u_{[i]} ∈ ℂ^{1}[t_{i}, t_{i+1}] for i = 0, 1, …, m. Moreover, 𝔸ℂ_{D}^{1}[0, T] stands for the set of functions u ∈ ℂ_{D}^{1}[0, T] having first derivatives absolutely continuous on each subinterval (t_{i}, t_{i+1}), i = 0, 1, …, m. For u ∈ ℂ_{D}^{1}[0, T] and i = 1, 2, …, m + 1 we define u′(t_{i}) = u′(t_{i}−) = lim_{t→t_{i}−} u′(t) and u′(0) = u′(0+) = lim_{t→0+} u′(t).

1.1 Definition. A solution of the problem (1.1)–(1.3) is a function u ∈ 𝔸ℂ_{D}^{1}[0, T] which satisfies the conditions (1.2) and (1.3) and for a.e. t ∈ [0, T] fulfils the equation (1.1).

1.2 Definition. A function σ_{1} ∈ 𝔸ℂ_{D}^{1}[0, T] is called a lower function of the problem (1.1)–(1.3) if

σ′′_{1}(t) ≥ f(t, σ_{1}(t), σ′_{1}(t)) for a.e. t ∈ [0, T], (1.5)

σ_{1}(t_{i}+) = J_{i}(σ_{1}(t_{i})), σ′_{1}(t_{i}+) ≥ M_{i}(σ′_{1}(t_{i})), i = 1, 2, …, m, (1.6)

σ_{1}(0) = σ_{1}(T), σ′_{1}(0) ≥ σ′_{1}(T). (1.7)

A function σ_{2} ∈ 𝔸ℂ_{D}^{1}[0, T] is an upper function of (1.1)–(1.3) if it satisfies

σ′′_{2}(t) ≤ f(t, σ_{2}(t), σ′_{2}(t)) for a.e. t ∈ [0, T], (1.8)

σ_{2}(t_{i}+) = J_{i}(σ_{2}(t_{i})), σ′_{2}(t_{i}+) ≤ M_{i}(σ′_{2}(t_{i})), i = 1, 2, …, m, (1.9)

σ_{2}(0) = σ_{2}(T), σ′_{2}(0) ≤ σ′_{2}(T). (1.10)

2 . Construction of nonconstant lower and upper functions

2.1 Theorem. Assume (1.4) and

(2.1) lim_{x→∞}(J_{i}(x) − x) = c_{i} ∈ ℝ for i = 1, 2, …, m.

Denote

(2.2)  c = −\frac{1}{T} ∑_{i=1}^{m} c_{i}

and suppose that there are A ∈ ℝ, δ > 0 and ν ∈ {1, 2} such that (2.3) (−1)^{ν} f(t, x, y) ≥ 0 for a.e. t ∈ [0, T] and all x ≥ A, y ∈ [c − δ, c + δ] and (−1)^{ν} (M_{i}(y) − y) ≥ 0 for y ∈ [c − δ, c + δ], i = 1, 2, …, m. (2.4)

Then for each Ã ∈ [A, ∞) there exist k̃ ∈ (c − δ, c + δ) and σ_{ν} ∈ 𝔸ℂ_{D}^{1}[0, T] such that σ_{ν}(t) ≥ Ã, σ′_{ν}(t) = k̃ for t ∈ [0, T] and, for ν = 1 (ν = 2), σ_{ν} is a lower (upper) function of (1.1)−(1.3). Proof. Let ν = 2 and Ã ∈ [A, ∞).

• Step 1. For a, k ∈ ℝ, x ∈ ℝ and i = 1, 2, …, m, define

φ(t, a, k) = \{ \begin{matrix}a + k t & if t ∈ [0, t_{1}], \\ J_{i}(φ(t_{i}, a, k)) + k (t − t_{i}) & if t ∈ (t_{i}, t_{i+1}], i = 1, 2, …, m, \end{matrix}

and

ε_{i}(x) = J_{i}(x) − x − c_{i}.

By virtue of (2.1), there are constants A_{i} > Ã, i = 1, 2, …, m, such that

(2.5)  |ε_{i}(x)| < \frac{δT}{m} for all x ≥ A_{i} and i = 1, 2, …, m.

Furthermore, we have

(2.6)  φ(t, a, k) = \{ \begin{matrix}a + k t if t ∈ [0, t_{1}], \\ a + k t + ∑_{j=1}^{i} (c_{j} + ε_{j}(φ(t_{j}, a, k))) \\ if t ∈ (t_{i}, t_{i+1}], i = 1, 2, …, m. \end{matrix}

Put

ã = max_{j=1,...,m} A_{j} + 2 ( ∑_{j=1}^{m} |c_{j}| + δ T )

and suppose that k ∈ [c − δ, c + δ]. Then

φ(t_{1}, ã, k) = ã + k t_{1} ≥ A_{1} + (1 − \frac{t_{1}}{T} ) ∑_{j=1}^{m} |c_{j}| ≥ A_{1}.

Furthermore, by (2.5), we have |ε_{1}(φ(t_{1}, ã, k))| ≤ \frac{δT}{m} . Consequently, in view of (2.6), we get

φ(t_{1}+, ã, k) = ã + k t_{1} + c_{1} + ε_{1}(φ(t_{1}, ã, k))

≥ A_{2} + |c_{1}| + δ T + (1 − \frac{t_{1}}{T}) ∑_{j=1}^{m} |c_{j}| − |c_{1}| − \frac{δT}{m} ≥ A_{2}.

Now, let 1 < i ≤ m and let

(2.7)  φ(t_{j}+, ã, k) ≥ A_{j+1} and φ(t_{j}, ã, k) ≥ A_{j}

for each j = 1, 2, …, i − 1. Then, by (2.5), we have

|ε_{j}(φ(t_{j}, ã, k))| ≤ \frac{δT}{m} for each j = 1, 2, …, i − 1.

Hence, using (2.6) and (2.5), we get

φ(t_{i}, ã, k) ≥ A_{i} + ∑_{j=1}^{m} (1 − \frac{t_{i}}{T} ) |c_{j}| + ∑_{j=i}^{m} |c_{j}| + (1 − \frac{i − 1}{m}) δ T ≥ A_{i}

and |ε_{i}(φ(t_{i}, ã, k))| ≤ \frac{δT}{m} . In view of (2.6), we have

φ(t_{i}+, ã, k) = ã + k t_{i} + ∑_{j=1}^{i}(c_{j} + ε_{j}(φ(t_{j}, ã, k)))

≥ A_{i+1} + ∑_{j=1}^{m} (1 − \frac{t_{i}}{T} ) |c_{j}| + (1 − \frac{i}{m}) δ T ≥ A_{i+1}

which means that (2.7) is true for any j ∈ {1, 2, …, m}. Similarly we can show that φ(T, ã, k) ≥ A_{m}. Thus

(2.8)  \{ \begin{matrix}φ(t, ã, k) ≥ Ã and |ε_{i}(φ(t_{i}, ã, k))| < \frac{δT}{m} \\ for all t ∈ [0, T], k ∈ [c − δ, c + δ] and i = 1, 2, …, m. \end{matrix}

• Step 2. We will prove that there is k̃ ∈ (c − δ, c + δ) such that

(2.9)  φ(0, ã, k̃) = φ(T, ã, k̃).

By virtue of (2.6) and (2.8), ∑_{i=1}^{m} |ε_{i}(φ(t_{i}, ã, k)| < δ T and

φ(T, ã, k) − φ(0, ã, k) = T (k − c + \frac{1}{T} ∑_{i=1}^{m} ε_{i}(φ(t_{i}, ã, k)))

for each k ∈ [c − δ, c + δ]. In particular, φ(T, ã, c − δ) − φ(0, ã, c − δ) < 0 and φ(T, ã, c+δ)−φ(0, ã, c+δ) > 0. Since φ is continuous, the existence of k̃ ∈ (c−δ, c+δ) satisfying (2.9) follows.

• Step 3. Define σ_{2}(t) = φ(t, ã, k̃) for t ∈ [0, T]. Then σ′_{2}(t) = k̃ for t ∈ [0, T] and σ_{2}(t_{i}+) = J_{i}(σ_{2}(t_{i})) for i = 1, 2, …, m. By (2.4) we have k̃ ≤ M_{i}(k̃), i.e. σ_{2} satisfies (1.9). Moreover, by (2.8) and (2.9), we have σ_{2}(t) ≥ Ã on [0, T] and σ_{2}(0) = σ_{2}(T) and so (1.10) is true. Finally, by (2.3), σ_{2} fulfils (1.8), i.e. σ_{2} is an upper function for (1.1)–(1.3).

The case ν = 1 can be treated analogously.

Theorem 2.1 gives piecewise linear lower and upper functions which are bounded below. Now we will show conditions guaranteeing the existence of lower or upper functions bounded above. This is the contents of the next theorem. Its proof is similar to that of Theorem 2.1.

2.2 Theorem. Assume (1.4). Further, let d_{i} ∈ ℝ, i = 1, 2, …, m, d, B ∈ ℝ, η > 0 and ν ∈ {1, 2} be such that

lim_{x→−∞}(J_{i}(x) − x) = d_{i} ∈ ℝ for i = 1, 2, …, m, d = −\frac{1}{T} ∑_{i=1}^{m} d_{i},

(−1)^{ν} f(t, x, y) ≥ 0 for a.e. t ∈ [0, T] and all x ≤ B, y ∈ [d − η, d + η],

(−1)^{ν} (M_{i}(y) − y) ≥ 0 for y ∈ [d − η, d + η], i = 1, 2, …, m,

Then for each B̃ ≤ B there exist k̃ ∈ (d − η, d + η) and σ_{ν} ∈ 𝔸ℂ_{D}^{1}[0, T] such that σ_{ν}(t) ≤ B̃, σ′_{ν}(t) = k̃ for t ∈ [0, T] and, for ν = 1 (ν = 2), σ_{ν} is a lower (upper) function of (1.1)−(1.3).

2.3 Remark. Let (1.4) hold. Assume that c_{i}, d_{i} ∈ ℝ and A ∈ (0, ∞) are such that

J_{i}(x) = \{ \begin{matrix}x + c_{i} for x ≥ A \\ x + d_{i} for x ≤ −A, i = 1, 2, …, m, \end{matrix}

\{ \begin{matrix}f(t, x, c) ≥ 0 for a.e. t ∈ [0, T] and all x ≥ A, \\ f(t, x, d) ≤ 0 for a.e. t ∈ [0, T] and all x ≤ −A, \end{matrix} (2.10)

and M_{i}(c) ≥ c, M_{i}(d) ≤ d, i = 1, 2, …, m. (2.11)

Let Ã ≥ A. Then, according to the proof of Theorem 2.1, we can see that

σ_{2}(t) = \{ \begin{matrix}Ã + ∑_{j=1}^{m} |c_{j}| + c t & for t ∈ [0, t_{1}), \\ Ã + ∑_{j=1}^{m} |c_{j}| + c t + ∑_{j=1}^{i} c_{j} & for t ∈ (t_{i}, t_{i+1}], i = 1, 2, …, m \end{matrix}

and

σ_{1}(t) = \{ \begin{matrix}−Ã − ∑_{j=1}^{m} |d_{j}| + d t & for t ∈ [0, t_{1}), \\ −Ã − ∑_{j=1}^{m} |d_{j}| + d t + ∑_{j=1}^{i} d_{j} & for t ∈ (t_{i}, t_{i+1}], i = 1, 2, …, m \end{matrix}

are respectively upper and lower functions of (1.1)–(1.3) satisfying

Ã ≤ σ_{2}(t) ≤ Ã + 2 ∑_{j=1}^{m} |c_{j}| and − Ã − 2 ∑_{j=1}^{m} |d_{j}| ≤ σ_{1}(t) ≤ −Ã for t ∈ [0, T].

If all inequalities in (2.10) and (2.11) are reversed, then σ_{2} becomes a lower function and σ_{1} an upper function.

3 . New existence criteria

Our main results are Theorems 3.1 and 3.2 which provides new existence criteria for the problem (1.1)–(1.3).

3.1 Theorem. Let the assumptions of Theorem 2.1 be satisfied for ν = 2 and let the assumptions of Theorem 2.2 be satisfied for ν = 1. Assume that J_{i} are increasing on ℝ and M_{i} are nondecreasing on ℝ for i = 1, 2, …, m. Finally, let for each compact interval K ⊂ ℝ there exist h_{K} ∈ 𝕃[0, T] and ω_{K} ∈ ℂ([1, ∞)) such that h_{K} ≥ 0 on [0, T], ω_{K} > 0 on [1, ∞), ∫_{1}^{∞} ds/ω_{K}(s) = ∞ and |f(t, x, y)| ≤ ω_{K}(|y|) (|y| + h_{K}(t)) for a.e. t ∈ [0, T] and all x ∈ K, |y| > 1. Then the problem (1.1)−(1.3) has a solution.

Proof. By Theorem 2.1, for each Ã ≥ A, there is an upper function σ_{2} of (1.1)–(1.3) such that σ_{2} ≥ Ã on [0, T]. By Theorem 2.2, for each B̃ ≤ B, there is a lower function σ_{1} of (1.1)–(1.3) such that σ_{1} ≤ B̃ on [0, T]. Choose Ã, B̃ in such a way that B̃ ≤ Ã. Hence σ_{1} ≤ σ_{2} on [0, T] and all the assumptions of [5, Theorem 3.1] are satisfied. Therefore (1.1)–(1.3) has a solution.

3.2 Theorem. Let the assumptions of Theorem 2.1 be satisfied for ν = 1 and let the assumptions of Theorem 2.2 be satisfied for ν = 2. Assume that J_{i} are increasing on ℝ and M_{i} are nondecreasing on ℝ for i = 1, 2, …, m. Finally, let there exist h ∈ 𝕃[0, T] such that |f(t, x, y)| ≤ h(t) for a.e. t ∈ [0, T] and all x, y ∈ ℝ. Then the problem (1.1)−(1.3) has a solution.

Proof. By Theorems 2.1 and 2.2 there are a lower function σ_{1} and an upper function σ_{2} of (1.1)–(1.3). The existence of a solution to (1.1)–(1.3) follows by [7, Theorem 3.1].

3.3 Example. Let k ∈ ℕ ∪ {0}, γ ∈ (0, ∞), p_{j} ∈ 𝕃_{∞}[0, T], j = 0, 1, …, 2k, p_{2k+1} ∈ 𝕃[0, T], p_{2k+1} ≥ γ a.e. on [0, T], q_{1}, q_{2} ∈ 𝕃_{∞}[0, T], c_{i} ∈ (0, ∞), α_{i} ∈ [0, 1) and β_{i} ∈ ℝ, i = 1, 2, …, m. Consider the problem (1.1)–(1.3), where

f(t, x, y) = ∑_{j=0}^{2k+1} p_{j}(t) x^{j} +q_{1}(t) y+q_{2}(t) y^{2}, J_{i}(x) = x+ \frac{2c_{i}}{π} arctan x, M_{i}(y) = α_{i} y+β_{i}

for a.e. t ∈ [0, T], all x, y ∈ ℝ and i = 1, 2, …, m. Let c be given by (2.2) and let β_{i} ∈ (c (1 − α_{i}), −c (1 − α_{i})), i = 1, 2, …, m. Then the conditions of Theorem 2.1 are satisfied for ν = 2 and the conditions of Theorem 2.2 are satisfied for ν = 1 and d_{i} = −c_{i}, i = 1, 2, …, m. Since c_{i} > 0 and α_{i} ≥ 0, the functions J_{i} are increasing and M_{i} are nondecreasing on ℝ for i = 1, 2, …, m. Choose an arbitrary compact interval K ⊂ ℝ and denote κ_{1} = max_{x∈K} (|x|^{2k+1}) and κ_{2} = ∑_{j=1}^{2k} (∥p_{j}∥_{∞} max_{x∈K} |x|^{j}). Then |f(t, x, y)| ≤ ω(|y|) (|y| + h(t)) for a.e. t ∈ [0, T] and all x ∈ K, |y| > 1, where ω(s) = 1 + ∥q_{1}∥_{∞} + ∥q_{2}∥_{∞} s and h(t) = κ_{1} |p_{2k+1}(t)| + κ_{2}. Thus, the existence of a solution to (1.1)–(1.3) follows by means of Theorem 3.1.

3.4 Example. Let γ ∈ (0, ∞), p ∈ 𝕃_{∞}[0, T], q_{1} ∈ 𝕃[0, T], q_{2} ∈ 𝕃_{∞}[0, T], q_{1} ≥ 0, q_{2} ≥ 0 and q_{1} + q_{2} ≥ γ a.e. on [0, T], φ ∈ ℂ(ℝ), lim_{|x|→∞} φ(x) = 0. Consider the problem (1.1)–(1.3), where

f(t, x, y) = p(t) φ(x) + q_{1}(t) y + q_{2}(t) y^{2} sgn y for a.e. t ∈ [0, T] and all x, y ∈ ℝ

and J_{i}(x) and M_{i}(y), i = 1, 2, …, m, are given as in Example 3.3, but with c_{i} ∈ (−π/2, 0) and α_{i} ∈ (1, ∞), i = 1, 2, …, m. Let c be given by (2.2) and let β_{i} ∈ (c (1 − α_{i}), c (α_{i} − 1)) for i = 1, 2, …, m. We have c > 0, M_{i}(c) > c, M_{i}(−c) < −c, f(t, x, c) > 0 and f(t, x, −c) < 0 for a.e. t ∈ [0, T] and all x ∈ ℝ with |x| sufficiently large. Thus, the assumptions of Theorem 2.1 are satisfied for ν = 2 and the assumptions of Theorem 2.2 are satisfied for ν = 1 and d_{i} = −c_{i}, i = 1, 2, …, m. Furthermore, J_{i} are increasing and M_{i} are nondecreasing on ℝ for i = 1, 2, …, m. Since φ is bounded on ℝ, we can find ω ∈ ℂ([1, ∞)) such that ω > 0 on [1, ∞), ∫_{1}^{∞} ds/ω(s) = ∞ and |f(t, x, y)| ≤ ω(|y|)|y| for a.e. t ∈ [0, T] and all x ∈ ℝ, |y| > 1. Thus, by Theorem 3.1, the given problem has a solution.

3.5 Example. Let γ ∈ (0, ∞), p ∈ 𝕃_{∞}[0, T], q ∈ 𝕃[0, T], q ≥ γ a.e. on [0, T], φ ∈ ℂ(ℝ), lim_{|x|→∞} φ(x) = 0. Consider the problem (1.1)–(1.3), where

f(t, x, y) = p(t) φ(x) + \frac{q(t) y}{1 + y^{2}} for a.e. t ∈ [0, T] and all x, y ∈ ℝ

and J_{i}(x) and M_{i}(y), i = 1, 2, …, m, are given as in Example 3.4, but with c_{i} ∈ (0, ∞). We can see that the assumptions of Theorem 2.1 are satisfied for ν = 1 and the assumptions of Theorem 2.2 are satisfied for ν = 2 and d_{i} = −c_{i}, i = 1, 2, …, m. As in the previous examples, J_{i} are increasing and M_{i} are nondecreasing on ℝ for i = 1, 2, …, m. Moreover, since the functions φ and \frac{y}{y^{2}+1} are bounded on ℝ, we can find h ∈ 𝕃[0, T] such that |f(t, x, y)| ≤ h(t) for a.e. t ∈ [0, T] and all x, y ∈ ℝ. Thus, by Theorem 3.2, our problem has a solution.

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Irena Rachůnková, Department of Mathematics, Palacký University, 779 00 OLO-MOUC, Tomkova 40, Czech Republic (e-mail: rachunko@inf.upol.cz)

Milan Tvrdý, Mathematical Institute, Academy of Sciences of the Czech Republic, 115 67 PRAHA 1, Žitná 25, Czech Republic (e-mail: tvrdy@math.cas.cz)

(Received September 30, 2003)