On periodic solutions of second–order diﬀerential equations with attractive–repulsive singularities

(1)

On periodic solutions

of second–order differential equations with attractive–repulsive singularities

Robert Hakl, Pedro J. Torres

(2)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

ω, λ, µ > 0 g, h ∈ L`[0, ω]; R+´ f ∈ L`[0, ω]; R´

Positive solution u : [0, ω] → ]0, +∞[

u, u⁰are absolutely continuous and satisfy (2), abbr. u ∈ AC¹_ω`[0, ω]; R´ u satisfies (1)

(3)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

ω, λ, µ > 0

g, h ∈ L`[0, ω]; R+´ f ∈ L`[0, ω]; R´

(4)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

ω, λ, µ > 0 g, h ∈ L`[0, ω]; R+´

f ∈ L`[0, ω]; R´

(5)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

ω, λ, µ > 0 g, h ∈ L`[0, ω]; R+´ f ∈ L`[0, ω]; R´

(6)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

ω, λ, µ > 0 g, h ∈ L`[0, ω]; R+´ f ∈ L`[0, ω]; R´

(7)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

ω, λ, µ > 0 g, h ∈ L`[0, ω]; R+´ f ∈ L`[0, ω]; R´

u, u⁰ are absolutely continuous and satisfy (2), abbr. u ∈ AC_ω¹`[0, ω]; R´

u satisfies (1)

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u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

ω, λ, µ > 0 g, h ∈ L`[0, ω]; R+´ f ∈ L`[0, ω]; R´

u, u⁰ are absolutely continuous and satisfy (2), abbr. u ∈ AC_ω¹`[0, ω]; R´

u satisfies (1)

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Special cases of (1)

u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u⁰⁰(t) = − h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (4) u⁰⁰(t) = g(t)

u^µ(t)+ f (t) for a. e. t ∈ [0, ω], (5) Attractive singularity

x→0lim₊q(t, x) = −∞ for a. e. t ∈ [0, ω]

Repulsive singularity

x→0lim₊q(t, x) = +∞ for a. e. t ∈ [0, ω]

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u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u⁰⁰(t) = − h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (4)

u⁰⁰(t) = g(t)

x→0lim₊q(t, x) = −∞ for a. e. t ∈ [0, ω]

x→0lim₊q(t, x) = +∞ for a. e. t ∈ [0, ω]

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u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u⁰⁰(t) = − h(t)

u^µ(t)+ f (t) for a. e. t ∈ [0, ω], (5)

Attractive singularity

x→0lim₊q(t, x) = −∞ for a. e. t ∈ [0, ω]

x→0lim₊q(t, x) = +∞ for a. e. t ∈ [0, ω]

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u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u⁰⁰(t) = − h(t)

x→0lim₊q(t, x) = −∞ for a. e. t ∈ [0, ω]

x→0lim₊q(t, x) = +∞ for a. e. t ∈ [0, ω]

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u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u⁰⁰(t) = − h(t)

x→0lim₊q(t, x) = −∞ for a. e. t ∈ [0, ω]

x→0lim₊q(t, x) = +∞ for a. e. t ∈ [0, ω]

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A. C. Lazer, S. Solimini (1987)

u⁰⁰= − 1

u^λ+ f (t); u(0) = u(ω), u⁰(0) = u⁰(ω) (6)

u⁰⁰= 1

u^µ+ f (t); u(0) = u(ω), u⁰(0) = u⁰(ω) (7) Theorem A. f continuous; (6) has a positive solution iffRω

0 f (s)ds > 0. Theorem B. f ∈ L`[0, ω]; R´, µ ≥ 1; (7) has a positive solution iffRω

0 f (s)ds < 0. I. Rach˚unkov´a, M. Tvrd´y, I. Vrkoˇc (2001)

Theorem C. f ∈ L`[0, ω]; R´, µ < 1; (7) has a positive solution ifRω

0 f (s)ds < 0 and

ess inf˘f (t) : t ∈ [0, ω]¯ > −

„ π² ω²µ

«_1+µ^µ (1 + µ).

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u⁰⁰= − 1

u^λ+ f (t); u(0) = u(ω), u⁰(0) = u⁰(ω) (6)

u⁰⁰= 1

0 f (s)ds < 0 and

ess inf˘f (t) : t ∈ [0, ω]¯ > −

„ π² ω²µ

«_1+µ^µ (1 + µ).

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u⁰⁰= − 1

u^λ+ f (t); u(0) = u(ω), u⁰(0) = u⁰(ω) (6)

u⁰⁰= 1

u^µ+ f (t); u(0) = u(ω), u⁰(0) = u⁰(ω) (7)

Theorem A. f continuous; (6) has a positive solution iffRω

0 f (s)ds < 0 and

ess inf˘f (t) : t ∈ [0, ω]¯ > −

„ π² ω²µ

«_1+µ^µ (1 + µ).

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u⁰⁰= − 1

u^λ+ f (t); u(0) = u(ω), u⁰(0) = u⁰(ω) (6)

u⁰⁰= 1

0 f (s)ds > 0.

Theorem B. f ∈ L`[0, ω]; R´, µ ≥ 1; (7) has a positive solution iffRω

0 f (s)ds < 0 and

ess inf˘f (t) : t ∈ [0, ω]¯ > −

„ π² ω²µ

«_1+µ^µ (1 + µ).

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u⁰⁰= − 1

u^λ+ f (t); u(0) = u(ω), u⁰(0) = u⁰(ω) (6)

u⁰⁰= 1

0 f (s)ds > 0.

0 f (s)ds < 0.

I. Rach˚unkov´a, M. Tvrd´y, I. Vrkoˇc (2001)

0 f (s)ds < 0 and

ess inf˘f (t) : t ∈ [0, ω]¯ > −

„ π² ω²µ

«_1+µ^µ (1 + µ).

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u⁰⁰= − 1

u^λ+ f (t); u(0) = u(ω), u⁰(0) = u⁰(ω) (6)

u⁰⁰= 1

0 f (s)ds > 0.

0 f (s)ds < 0.

0 f (s)ds < 0 and

ess inf˘f (t) : t ∈ [0, ω]¯ > −

„ π² ω²µ

«_1+µ^µ (1 + µ).

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u⁰⁰= − 1

u^λ+ f (t); u(0) = u(ω), u⁰(0) = u⁰(ω) (6)

u⁰⁰= 1

0 f (s)ds > 0.

0 f (s)ds < 0.

0 f (s)ds < 0 and

ess inf˘f (t) : t ∈ [0, ω]¯ > −

„ π² ω²µ

«_1+µ^µ (1 + µ).

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u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

Notation

For the sake of brevity we will use the following notation:

G = Z ω

0

g(s)ds, H = Z ω

0

h(s)ds,

F = Zω

0

f (s)ds, F+= Z ω

0

[f (s)]+ds, F−= Z ω

0

[f (s)]−ds.

Note that F = F+− F−.

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u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Theorem

Let H > 0, F > 0, functions w, σ ∈ AC_ω¹`[0, ω]; R´ be such that the equalities w⁰⁰(t) = Hg(t) − Gh(t) for a. e. t ∈ [0, ω], σ⁰⁰(t) = −F

Hh(t) + f (t) for a. e. t ∈ [0, ω], are fulfilled, and let there exist x0∈ ]0, +∞[ such that

x0`w(t) − m_w´ + σ(t) − m_σ≤

„ H

x0GH + F

«1/λ

−

„ 1 x0H

«1/µ

for t ∈ [0, ω],

where

mw= min˘w(t) : t ∈ [0, ω]¯, mσ= min˘σ(t) : t ∈ [0, ω]¯. Then the equation (1), (2) has at least one positive solution.

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u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Theorem

Let H > 0, F > 0, functions w, σ ∈ AC_ω¹`[0, ω]; R´ be such that the equalities w⁰⁰(t) = Hg(t) − Gh(t) for a. e. t ∈ [0, ω], σ⁰⁰(t) = −F

Hh(t) + f (t) for a. e. t ∈ [0, ω], are fulfilled, and let there exist x0∈ ]0, +∞[ such that

x0`w(t) − m_w´ + σ(t) − m_σ≤

„ H

x0GH + F

«1/λ

−

„ 1 x0H

«1/µ

for t ∈ [0, ω],

where

mw= min˘w(t) : t ∈ [0, ω]¯, mσ= min˘σ(t) : t ∈ [0, ω]¯.

Then the equation (1), (2) has at least one positive solution.

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u⁰⁰(t) = − h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (4)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let H > 0, F > 0, σ ∈ AC_ω¹`[0, ω]; R´ be such that

σ⁰⁰(t) = −F

Hh(t) + f (t) for a. e. t ∈ [0, ω], and let

`M_σ− m_σ´λ

F < H, where

Mσ= max˘σ(t) : t ∈ [0, ω]¯, mσ= min˘σ(t) : t ∈ [0, ω]¯. Then the problem (4), (2) has at least one positive solution.

Corollary

Let H > 0, F > 0, and let

“ω 4F+

”λ

F ≤ H. Then the problem (4), (2) has at least one positive solution.

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u⁰⁰(t) = − h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (4)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

σ⁰⁰(t) = −F

`M_σ− mσ

´λ

F < H, where

Mσ= max˘σ(t) : t ∈ [0, ω]¯, mσ= min˘σ(t) : t ∈ [0, ω]¯.

Then the problem (4), (2) has at least one positive solution.

Corollary

“ω 4F+

”λ

F ≤ H. Then the problem (4), (2) has at least one positive solution.

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u⁰⁰(t) = − h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (4)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

σ⁰⁰(t) = −F

`M_σ− mσ

´λ

F < H, where

Mσ= max˘σ(t) : t ∈ [0, ω]¯, mσ= min˘σ(t) : t ∈ [0, ω]¯.

Corollary

“ω 4F+

”λ

F ≤ H.

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u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Lemma

Let there exist positive functions α, β ∈ AC_ω¹`[0, ω]; R´ such that

α⁰⁰(t) ≥ g(t) α^µ(t)− h(t)

α^λ(t)+ f (t) for a. e. t ∈ [0, ω], (8) β⁰⁰(t) ≤ g(t)

β^µ(t)− h(t)

β^λ(t)+ f (t) for a. e. t ∈ [0, ω], (9) α(t) ≤ β(t) for t ∈ [0, ω].

Then there exists at least one positive solution to (1), (2).

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u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Lemma

α⁰⁰(t) ≥ g(t) α^µ(t)− h(t)

α^λ(t)+ f (t) for a. e. t ∈ [0, ω], (8) β⁰⁰(t) ≤ g(t)

β^µ(t)− h(t)

β^λ(t)+ f (t) for a. e. t ∈ [0, ω], (9) α(t) ≤ β(t) for t ∈ [0, ω].

Then there exists at least one positive solution to (1), (2).

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w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t), x0`w(t) − mw´ + σ(t) − mσ≤

„ H

x0GH + F

«1/λ

−

„ 1 x0H

«1/µ

Proof.

α(t) =

„ 1 x0H

«1/µ

+ x0`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

α⁰⁰(t) = x0Hg(t) −

„ x0G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

for t ∈ [0, ω].

α⁰⁰(t) ≥ g(t) α^µ(t)− h(t)

α^λ(t)+ f (t) for a. e. t ∈ [0, ω].

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w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t), x0`w(t) − mw´ + σ(t) − mσ≤

„ H

x0GH + F

«1/λ

−

„ 1 x0H

«1/µ

Proof.

α(t) =

„ 1 x0H

«1/µ

+ x0`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

α⁰⁰(t) = x0Hg(t) −

„ x0G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

for t ∈ [0, ω].

α⁰⁰(t) ≥ g(t) α^µ(t)− h(t)

α^λ(t)+ f (t) for a. e. t ∈ [0, ω].

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w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t), x0`w(t) − mw´ + σ(t) − mσ≤

„ H

x0GH + F

«1/λ

−

„ 1 x0H

«1/µ

Proof.

α(t) =

„ 1 x0H

«1/µ

+ x0`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

α⁰⁰(t) = x0Hg(t) −

„ x0G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

for t ∈ [0, ω].

α⁰⁰(t) ≥ g(t) α^µ(t)− h(t)

α^λ(t)+ f (t) for a. e. t ∈ [0, ω].

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w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t), x0`w(t) − mw´ + σ(t) − mσ≤

„ H

x0GH + F

«1/λ

−

„ 1 x0H

«1/µ

Proof.

α(t) =

„ 1 x0H

«1/µ

+ x0`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

α⁰⁰(t) = x0Hg(t) −

„ x0G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

for t ∈ [0, ω].

α⁰⁰(t) ≥ g(t) α^µ(t)− h(t)

α^λ(t)+ f (t) for a. e. t ∈ [0, ω].

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w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t), x0`w(t) − mw´ + σ(t) − mσ≤

„ H

x0GH + F

«1/λ

−

„ 1 x0H

«1/µ

Proof.

α(t) =

„ 1 x0H

«1/µ

+ x0`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

α⁰⁰(t) = x0Hg(t) −

„ x0G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

for t ∈ [0, ω].

α⁰⁰(t) ≥ g(t) α^µ(t)− h(t)

α^λ(t)+ f (t) for a. e. t ∈ [0, ω].

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w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t),

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

.

Proof.

x1∈ ]0, x0] : x1`w(t) − m_w´ + σ(t) − m_σ≤

„ 1 x1H

«1/µ

−

„ H

x1GH + F

«1/λ

for t ∈ [0, ω]

β(t) =

„ H

x1GH + F

«1/λ

+ x1`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

β⁰⁰(t) = x1Hg(t) −

„ x1G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ H

x1GH + F

«1/λ

≤ β(t) ≤

„ 1 x1H

«1/µ

for t ∈ [0, ω].

β⁰⁰(t) ≤ g(t) β^µ(t)− h(t)

β^λ(t)+ f (t) for a. e. t ∈ [0, ω]. α(t) ≤ β(t) for t ∈ [0, ω].

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w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t),

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

.

Proof.

x1∈ ]0, x0] : x1`w(t) − m_w´ + σ(t) − m_σ≤

„ 1 x1H

«1/µ

−

„ H

x1GH + F

«1/λ

for t ∈ [0, ω]

β(t) =

„ H

x1GH + F

«1/λ

+ x1`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

β⁰⁰(t) = x1Hg(t) −

„ x1G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ H

x1GH + F

«1/λ

≤ β(t) ≤

„ 1 x1H

«1/µ

for t ∈ [0, ω].

β⁰⁰(t) ≤ g(t) β^µ(t)− h(t)

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w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t),

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

.

Proof.

x1∈ ]0, x0] : x1`w(t) − m_w´ + σ(t) − m_σ≤

„ 1 x1H

«1/µ

−

„ H

x1GH + F

«1/λ

for t ∈ [0, ω]

β(t) =

„ H

x1GH + F

«1/λ

+ x1`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

β⁰⁰(t) = x1Hg(t) −

„ x1G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ H

x1GH + F

«1/λ

≤ β(t) ≤

„ 1 x1H

«1/µ

for t ∈ [0, ω].

β⁰⁰(t) ≤ g(t) β^µ(t)− h(t)

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w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t),

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

.

Proof.

x1∈ ]0, x0] : x1`w(t) − m_w´ + σ(t) − m_σ≤

„ 1 x1H

«1/µ

−

„ H

x1GH + F

«1/λ

for t ∈ [0, ω]

β(t) =

„ H

x1GH + F

«1/λ

+ x1`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

β⁰⁰(t) = x1Hg(t) −

„ x1G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ H

x1GH + F

«1/λ

≤ β(t) ≤

„ 1 x1H

«1/µ

for t ∈ [0, ω].

β⁰⁰(t) ≤ g(t) β^µ(t)− h(t)

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w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t),

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

.

Proof.

x1∈ ]0, x0] : x1`w(t) − m_w´ + σ(t) − m_σ≤

„ 1 x1H

«1/µ

−

„ H

x1GH + F

«1/λ

for t ∈ [0, ω]

β(t) =

„ H

x1GH + F

«1/λ

+ x1`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

β⁰⁰(t) = x1Hg(t) −

„ x1G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ H

x1GH + F

«1/λ

≤ β(t) ≤

„ 1 x1H

«1/µ

for t ∈ [0, ω].

β⁰⁰(t) ≤ g(t) β^µ(t)− h(t)

(39)

w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t),

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

.

Proof.

x1∈ ]0, x0] : x1`w(t) − m_w´ + σ(t) − m_σ≤

„ 1 x1H

«1/µ

−

„ H

x1GH + F

«1/λ

for t ∈ [0, ω]

β(t) =

„ H

x1GH + F

«1/λ

+ x1`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

β⁰⁰(t) = x1Hg(t) −

„ x1G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ H

x1GH + F

«1/λ

≤ β(t) ≤

„ 1 x1H

«1/µ

for t ∈ [0, ω].

β⁰⁰(t) ≤ g(t) β^µ(t)− h(t)

β^λ(t)+ f (t) for a. e. t ∈ [0, ω].

α(t) ≤ β(t) for t ∈ [0, ω].

(40)

w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = −F

Hh(t) + f (t),

„ 1 x0H

«1/µ

≤ α(t) ≤

„ H

x0GH + F

«1/λ

.

Proof.

x1∈ ]0, x0] : x1`w(t) − m_w´ + σ(t) − m_σ≤

„ 1 x1H

«1/µ

−

„ H

x1GH + F

«1/λ

for t ∈ [0, ω]

β(t) =

„ H

x1GH + F

«1/λ

+ x1`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω].

β⁰⁰(t) = x1Hg(t) −

„ x1G +F

H

«

h(t) + f (t) for a. e. t ∈ [0, ω],

„ H

x1GH + F

«1/λ

≤ β(t) ≤

„ 1 x1H

«1/µ

for t ∈ [0, ω].

β⁰⁰(t) ≤ g(t) β^µ(t)− h(t)

β^λ(t)+ f (t) for a. e. t ∈ [0, ω].

α(t) ≤ β(t) for t ∈ [0, ω].

(41)

Definition

A function ϕ ∈ L`[0, ω]; R+´ is said to verify the property (P ) if the implication u ∈ AC_ω¹`[0, ω]; R´

u⁰⁰(t) + ϕ(t)u(t) ≥ 0 for a. e. t ∈ [0, ω]

)

=⇒ u(t) ≥ 0 for t ∈ [0, ω]

holds.

(42)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Theorem

Let G > 0, F < 0, functions w, σ ∈ AC_ω¹`[0, ω]; R´ be such that the equalities

w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = |F |

Gg(t) + f (t) for a. e. t ∈ [0, ω] are fulfilled, and let there exist x0∈ ]0, +∞[ such that

x0`w(t) − m_w´ + σ(t) − m_σ≤

„ G

x0GH + |F |

«1/µ

−

„ 1 x0G

«1/λ

for t ∈ [0, ω].

Moreover, let us define

β(t) =

„ 1 x0G

«1/λ

+ x0`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω]

and assume that ϕ(t) =_β^µg(t)_1+µ_(t) verifies the property (P ). Then the problem (1), (2) has at least one positive solution.

(43)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Theorem

Let G > 0, F < 0, functions w, σ ∈ AC_ω¹`[0, ω]; R´ be such that the equalities

w⁰⁰(t) = Hg(t) − Gh(t), σ⁰⁰(t) = |F |

Gg(t) + f (t) for a. e. t ∈ [0, ω]

are fulfilled, and let there exist x0∈ ]0, +∞[ such that

x0`w(t) − m_w´ + σ(t) − m_σ≤

„ G

x0GH + |F |

«1/µ

−

„ 1 x0G

«1/λ

for t ∈ [0, ω].

Moreover, let us define

β(t) =

„ 1 x0G

«1/λ

+ x0`w(t) − m_w´ + σ(t) − m_σ for t ∈ [0, ω]

(44)

u⁰⁰(t) = g(t)

u^µ(t)+ f (t) for a. e. t ∈ [0, ω], (5)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let G > 0, F < 0, σ ∈ AC_ω¹`[0, ω]; R´ be such that

σ⁰⁰(t) =|F |

Gg(t) + f (t) for a. e. t ∈ [0, ω], and let

`Mσ− mσ´µ

|F | < G. Let, moreover, either

µ|F |

1+µ µ g(t)

`G^1/µ− |F |^1/µ`Mσ− σ(t)´´1+µ≤“π ω

”2

for a. e. t ∈ [0, ω],

or

µ|F |

1+µ µ

Z ω 0

g(s)

`G^1/µ− |F |^1/µ`Mσ− σ(s)´´1+µds ≤ 4 ω. Then the problem (5), (2) has at least one positive solution.

(45)

u⁰⁰(t) = g(t)

u^µ(t)+ f (t) for a. e. t ∈ [0, ω], (5)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let G > 0, F < 0, σ ∈ AC_ω¹`[0, ω]; R´ be such that

σ⁰⁰(t) =|F |

Gg(t) + f (t) for a. e. t ∈ [0, ω], and let

`Mσ− mσ´µ

|F | < G.

Let, moreover, either µ|F |

1+µ µ g(t)

`G^1/µ− |F |^1/µ`Mσ− σ(t)´´1+µ≤“π ω

”2

for a. e. t ∈ [0, ω],

or

µ|F |^1+µ^µ Z ω

0

g(s)

`G^1/µ− |F |^1/µ`Mσ− σ(s)´´1+µds ≤ 4 ω. Then the problem (5), (2) has at least one positive solution.

(46)

u⁰⁰(t) = g(t)

u^µ(t)+ f (t) for a. e. t ∈ [0, ω], (5)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let G > 0 and F < 0. Let, moreover,

“ω 4µG”_1+µ¹

|F |^1/µ+ω

4F−|F |^1/µ≤ G^1/µ. Then the problem (5), (2) has at least one positive solution.

(47)

u⁰⁰(t) = g(t)

u^µ(t)+ f (t) for a. e. t ∈ [0, ω], (5)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let G > 0 and F < 0. Let, moreover,

“ω 4µG”_1+µ¹

|F |^1/µ+ω

4F−|F |^1/µ≤ G^1/µ. Then the problem (5), (2) has at least one positive solution.

(48)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Lemma

α⁰⁰(t) ≥ g(t) α^µ(t)− h(t)

α^λ(t)+ f (t) for a. e. t ∈ [0, ω], β⁰⁰(t) ≤ g(t)

β^µ(t)− h(t)

β^λ(t)+ f (t) for a. e. t ∈ [0, ω], β(t) ≤ α(t) for t ∈ [0, ω].

Let, moreover, there exists ϕ ∈ L`[0, ω]; R₊´ with the property (P ) and such that g(t)

u^µ(t)− h(t) u^λ(t)−

„ g(t) v^µ(t)− h(t)

v^λ(t)

«

≤ ϕ(t)`v(t) − u(t)´ for a. e. t ∈ [0, ω],

whenever β(t) ≤ u(t) ≤ v(t) ≤ α(t) for t ∈ [0, ω]. Then there exists at least one positive solution to (1), (2).

(49)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Lemma

α⁰⁰(t) ≥ g(t) α^µ(t)− h(t)

α^λ(t)+ f (t) for a. e. t ∈ [0, ω], β⁰⁰(t) ≤ g(t)

β^µ(t)− h(t)

β^λ(t)+ f (t) for a. e. t ∈ [0, ω], β(t) ≤ α(t) for t ∈ [0, ω].

Let, moreover, there exists ϕ ∈ L`[0, ω]; R₊´ with the property (P ) and such that g(t)

„ g(t) v^µ(t)− h(t)

v^λ(t)

«

≤ ϕ(t)`v(t) − u(t)´ for a. e. t ∈ [0, ω],

whenever β(t) ≤ u(t) ≤ v(t) ≤ α(t) for t ∈ [0, ω]. Then there exists at least one positive solution to (1), (2).

(50)

ϕ(t) = µg(t)

β^1+µ(t) for t ∈ [0, ω], g(t)

„ g(t) v^µ(t)− h(t)

v^λ(t)

«

≤ ϕ(t)`v(t) − u(t)´ for a. e. t ∈ [0, ω],

whenever β(t) ≤ u(t) ≤ v(t) ≤ α(t) for t ∈ [0, ω],

Proof.

ψ(y) = µ β^1+µy + 1

y^µ is nondecreasing for y ≥ β.

g(t)

„ µ

β^1+µ(t)u(t) + 1 u^µ(t)

«

− h(t) u^λ(t)≤ g(t)

„ µ

β^1+µ(t)v(t) + 1 v^µ(t)

«

− h(t)

v^λ(t) for t ∈ [0, ω] whenever β(t) ≤ u(t) ≤ v(t) for t ∈ [0, ω].

g(t) u^µ(t)− h(t)

u^λ(t)−

„ g(t) v^µ(t)− h(t)

v^λ(t)

«

≤ µg(t)

β^1+µ(t)(v(t) − u(t)).

(51)

ϕ(t) = µg(t)

β^1+µ(t) for t ∈ [0, ω], g(t)

„ g(t) v^µ(t)− h(t)

v^λ(t)

«

≤ ϕ(t)`v(t) − u(t)´ for a. e. t ∈ [0, ω],

whenever β(t) ≤ u(t) ≤ v(t) ≤ α(t) for t ∈ [0, ω], Proof.

ψ(y) = µ β^1+µy + 1

g(t)

„ µ

β^1+µ(t)u(t) + 1 u^µ(t)

«

− h(t) u^λ(t)≤ g(t)

„ µ

β^1+µ(t)v(t) + 1 v^µ(t)

«

− h(t)

v^λ(t) for t ∈ [0, ω] whenever β(t) ≤ u(t) ≤ v(t) for t ∈ [0, ω].

u^λ(t)−

„ g(t) v^µ(t)− h(t)

v^λ(t)

«

≤ µg(t)

β^1+µ(t)(v(t) − u(t)).

(52)

ϕ(t) = µg(t)

β^1+µ(t) for t ∈ [0, ω], g(t)

„ g(t) v^µ(t)− h(t)

v^λ(t)

«

≤ ϕ(t)`v(t) − u(t)´ for a. e. t ∈ [0, ω],

ψ(y) = µ β^1+µy + 1

g(t)

„ µ

β^1+µ(t)u(t) + 1 u^µ(t)

«

− h(t) u^λ(t)≤ g(t)

„ µ

β^1+µ(t)v(t) + 1 v^µ(t)

«

− h(t)

v^λ(t) for t ∈ [0, ω]

whenever β(t) ≤ u(t) ≤ v(t) for t ∈ [0, ω].

u^λ(t)−

„ g(t) v^µ(t)− h(t)

v^λ(t)

«

≤ µg(t)

β^1+µ(t)(v(t) − u(t)).

(53)

ϕ(t) = µg(t)

β^1+µ(t) for t ∈ [0, ω], g(t)

„ g(t) v^µ(t)− h(t)

v^λ(t)

«

≤ ϕ(t)`v(t) − u(t)´ for a. e. t ∈ [0, ω],

ψ(y) = µ β^1+µy + 1

g(t)

„ µ

β^1+µ(t)u(t) + 1 u^µ(t)

«

− h(t) u^λ(t)≤ g(t)

„ µ

β^1+µ(t)v(t) + 1 v^µ(t)

«

− h(t)

v^λ(t) for t ∈ [0, ω]

whenever β(t) ≤ u(t) ≤ v(t) for t ∈ [0, ω].

u^λ(t)−

„ g(t) v^µ(t)− h(t)

v^λ(t)

«

≤ µg(t)

β^1+µ(t)(v(t) − u(t)).

(54)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Theorem

Let λ > µ, H > 0, G > 0, F = 0, functions w, σ ∈ AC¹_ω`[0, ω]; R´ be such that w⁰⁰(t) = Hg(t) − Gh(t) for a. e. t ∈ [0, ω],

σ⁰⁰(t) = f (t) for a. e. t ∈ [0, ω], and let there exist x0∈ ]0, +∞[ such that

x0`w(t) − m_w´ + σ(t) − m_σ≤

„ 1 x0G

«1/λ

−

„ 1 x0H

«1/µ

for t ∈ [0, ω],

where

mw= min˘w(t) : t ∈ [0, ω]¯, mσ= min˘σ(t) : t ∈ [0, ω]¯. Then the problem (1), (2) has at least one positive solution.

(55)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Theorem

Let λ > µ, H > 0, G > 0, F = 0, functions w, σ ∈ AC_ω¹`[0, ω]; R´ be such that w⁰⁰(t) = Hg(t) − Gh(t) for a. e. t ∈ [0, ω],

x0`w(t) − m_w´ + σ(t) − m_σ≤

„ 1 x0G

«1/λ

−

„ 1 x0H

«1/µ

for t ∈ [0, ω],

where

mw= min˘w(t) : t ∈ [0, ω]¯, mσ= min˘σ(t) : t ∈ [0, ω]¯.

(56)

u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let λ > µ, H > 0, G > 0, and let w ∈ AC_ω¹`[0, ω]; R´ be such that w⁰⁰(t) = Hg(t) − Gh(t) for a. e. t ∈ [0, ω] is fulfilled. Let, moreover,

Mw− mw≤H^λ−µ^1+λ G^λ−µ^1+µ

„ (1 + λ)µ (1 + µ)λ

«^(1+λ)µ

λ−µ λ − µ (1 + µ)λ, where

Mw= max˘w(t) : t ∈ [0, ω]¯, mw= min˘w(t) : t ∈ [0, ω]¯. Then the problem (3), (2) has at least one positive solution.

(57)

u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let λ > µ, H > 0, G > 0, and let w ∈ AC_ω¹`[0, ω]; R´ be such that w⁰⁰(t) = Hg(t) − Gh(t) for a. e. t ∈ [0, ω]

is fulfilled. Let, moreover,

Mw− mw≤H^λ−µ^1+λ G^1+µ^λ−µ

„ (1 + λ)µ (1 + µ)λ

«^(1+λ)µ_λ−µ λ − µ (1 + µ)λ, where

Mw= max˘w(t) : t ∈ [0, ω]¯, mw= min˘w(t) : t ∈ [0, ω]¯.

(58)

u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let λ > µ, H > 0, and G > 0. Let, moreover, G^1+λ

H^1+µ ≤„ 4 ω

«λ−µ„ (1 + λ)µ (1 + µ)λ

«(1+λ)µ„ λ − µ (1 + µ)λ

«λ−µ

.

(59)

u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let λ > µ, H > 0, and G > 0. Let, moreover, G^1+λ

H^1+µ ≤„ 4 ω

«λ−µ„ (1 + λ)µ (1 + µ)λ

«(1+λ)µ„ λ − µ (1 + µ)λ

«λ−µ

.

(60)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Theorem

Let µ > λ, H > 0, G > 0, F = 0, functions w, σ ∈ AC¹_ω`[0, ω]; R´ be such that w⁰⁰(t) = Hg(t) − Gh(t) for a. e. t ∈ [0, ω],

x0`w(t) − mw´ + σ(t) − mσ≤

„ 1 x0H

«1/µ

−

„ 1 x0G

«1/λ

for t ∈ [0, ω].

Moreover, assume that ϕ(t) = ^µg(t)

β^1+µ(t) verifies the property (P ), where β is given by

β(t) =

„ 1 x0G

«1/λ

+ x0`w(t) − mw´ + σ(t) − mσ for t ∈ [0, ω].

(61)

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Theorem

Let µ > λ, H > 0, G > 0, F = 0, functions w, σ ∈ AC_ω¹`[0, ω]; R´ be such that w⁰⁰(t) = Hg(t) − Gh(t) for a. e. t ∈ [0, ω],

x0`w(t) − mw´ + σ(t) − mσ≤

„ 1 x0H

«1/µ

−

„ 1 x0G

«1/λ

for t ∈ [0, ω].

Moreover, assume that ϕ(t) = ^µg(t)

β^1+µ(t) verifies the property (P ), where β is given by

β(t) =

„ 1 x0G

«1/λ

+ x0`w(t) − mw´ + σ(t) − mσ for t ∈ [0, ω].

(62)

u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let µ > λ, H > 0, G > 0, w ∈ AC_ω¹`[0, ω]; R´ be such that

w⁰⁰(t) = Hg(t) − Gh(t) for a. e. t ∈ [0, ω] is fulfilled, and let

Mw− mw≤ G^µ−λ^1+µ H

1+λ µ−λ

„ (1 + µ)λ (1 + λ)µ

«^(1+µ)λ_µ−λ µ − λ (1 + λ)µ. Moreover, let us define

β(t) =„ (1 + µ)λ (1 + λ)µ

«_µ−λ^µ „ G H

«_µ−λ¹

+„ (1 + λ)µ (1 + µ)λ

«_µ−λ^λµ H^µ−λ^λ G

µ µ−λ

`w(t) − mw´

for t ∈ [0, ω],

(63)

u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let µ > λ, H > 0, G > 0, w ∈ AC_ω¹`[0, ω]; R´ be such that

w⁰⁰(t) = Hg(t) − Gh(t) for a. e. t ∈ [0, ω]

is fulfilled, and let

Mw− mw≤ G^1+µ^µ−λ H

1+λ µ−λ

„ (1 + µ)λ (1 + λ)µ

«^(1+µ)λ_µ−λ µ − λ (1 + λ)µ. Moreover, let us define

β(t) =„ (1 + µ)λ (1 + λ)µ

«_µ−λ^µ „ G H

«_µ−λ¹

+„ (1 + λ)µ (1 + µ)λ

«_µ−λ^λµ H^µ−λ^λ G

µ µ−λ

`w(t) − mw´

for t ∈ [0, ω],

(64)

u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let µ > λ, H > 0, and G > 0. Let, moreover,

H^1+µ G^1+λ ≤„ 4

ω

«µ−λ„ (1 + µ)λ (1 + λ)µ

«(1+µ)λ„ µ − λ (1 + λ)µ

«µ−λ

×

× min (

1,„ 1 + λ µ − λ

«µ−λ„ (1 + µ)λ (1 + λ)µ

«(µ−λ)(1+µ)) .

Then the problem (3), (2) has at least one solution.

(65)

u⁰⁰(t) = g(t) u^µ(t) − h(t)

u^λ(t) for a. e. t ∈ [0, ω], (3)

u(0) = u(ω), u⁰(0) = u⁰(ω) (2)

Corollary

Let µ > λ, H > 0, and G > 0. Let, moreover,

H^1+µ G^1+λ ≤„ 4

ω

«µ−λ„ (1 + µ)λ (1 + λ)µ

«(1+µ)λ„ µ − λ (1 + λ)µ

«µ−λ

×

× min (

1,„ 1 + λ µ − λ

«µ−λ„ (1 + µ)λ (1 + λ)µ

«(µ−λ)(1+µ)) .

Then the problem (3), (2) has at least one solution.

(66)

Open problem:

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u^λ(t)+ f (t) for a. e. t ∈ [0, ω], (1) λ = µ and F = 0.

J. L. Bravo, P. J. Torres (preprint)

u⁰⁰=a(t)

u³ ; u(0) = u(ω), u⁰(0) = u⁰(ω) (8)

a(t) =

(a+ for t ∈ [0, t0[

−a− for t ∈ [t0, ω] with a+, a−> 0.

Theorem. (8) has a positive solution iffRω

0 a(s)ds < 0.

(67)

Open problem:

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u⁰⁰=a(t)

u³ ; u(0) = u(ω), u⁰(0) = u⁰(ω) (8)

a(t) =

(a+ for t ∈ [0, t0[

−a− for t ∈ [t0, ω]

with a+, a−> 0.

0 a(s)ds < 0.

(68)

Open problem:

u⁰⁰(t) = g(t) u^µ(t)− h(t)

u⁰⁰=a(t)

u³ ; u(0) = u(ω), u⁰(0) = u⁰(ω) (8)

a(t) =

(a+ for t ∈ [0, t0[

−a− for t ∈ [t0, ω]

with a+, a−> 0.

0 a(s)ds < 0.