Periodic solutions to the Liénard type equations with phase attractive singularities

Robert Hakl; Manuel Zamora

doi:10.1186/1687-2770-2013-47

Research
Open Access

Periodic solutions to the Liénard type equations with phase attractive singularities

Robert Hakl¹Email author and
Manuel Zamora²

Boundary Value Problems20132013:47

https://doi.org/10.1186/1687-2770-2013-47

Received: 10 December 2012

Accepted: 18 February 2013

Published: 6 March 2013

Abstract

Sufficient conditions are established guaranteeing the existence of a positive ω-periodic solution to the equation

u^{″} + f (u) u^{'} + g (u) = h (t, u),

where $f, g : (0, + \infty) \to R$ are continuous functions with possible singularities at zero and $h : [0, ω] \times R \to R$ is a Carathéodory function. The results obtained are rewritten for the equation of the type

u^{″} + \frac{c u^{'}}{u^{μ}} + \frac{g_{1}}{u^{ν}} - \frac{g_{2}}{u^{γ}} = h_{0} (t) u^{δ},

where $g_{1}$ , $g_{2}$ , δ are non-negative constants, c, μ, ν, γ are real numbers, and $h_{0} \in L ([0, ω]; R)$ . The last equation also covers the so-called Rayleigh-Plesset equation, frequently used in fluid mechanics to model the bubble dynamics in liquid. In the paper, the case when $ν > γ$ , i.e., the case which covers the attractive singularity of the function g, is studied. The results obtained assure that there exists a positive ω-periodic solution to the above-mentioned equation if the power μ or ν is sufficiently large.

MSC:34C25, 34B16, 34B18, 76N15.

Keywords

Rayleigh-Plesset equation
singular equation
periodic solution
upper and lower function

1 Introduction

The topic of singular boundary value problems has been of substantial and rapidly growing interest for many scientists and engineers. The importance of such investigation is emphasized by the fact that numerical simulations of solutions to such problems usually break down near singular points.

On the other hand, problems of this type arise frequently in applied science. Namely, in fluid mechanics, since 1917 the physicists have used the Rayleigh equation,

ρ [R \ddot{R} + \frac{3}{2} {\dot{R}}^{2}] = p (R) - p_{\infty},

to model the bubble dynamics in liquid, where

R (t)

is the ratio of the bubble at the time t, ρ is the liquid density,

p_{\infty}

is the pressure in the liquid at a large distance from the bubble, and

p (R)

is the pressure in the liquid at the bubble boundary. In 1949, Plesset proposed to use a more exact equation involving the surface-tension constant S and the coefficient of the liquid viscosity

μ_{*}

, which was finally improved by adding a term with polytropic coefficient k in 1977, nowadays known as a Rayleigh-Plesset equation (see [1])

ρ [R \ddot{R} + \frac{3}{2} {\dot{R}}^{2}] = [P_{v} - p_{\infty} (t)] + P_{g_{0}} {(\frac{R_{0}}{R})}^{3 k} - \frac{2 S}{R} - \frac{4 μ_{*} \dot{R}}{R} .

The transformation

R = u^{\frac{2}{5}}

in the previous equation leads to the equation

\ddot{u} = \frac{5 [P_{v} - p_{\infty} (t)]}{2 ρ} u^{\frac{1}{5}} + (\frac{5 P_{g_{0}} R_{0}^{3 k}}{2 ρ}) \frac{1}{u^{\frac{6 k - 1}{5}}} - \frac{5 S}{u^{\frac{1}{5}}} - 4 μ_{*} \frac{\dot{u}}{u^{\frac{4}{5}}} .

Consequently, the class of equations

u^{″} (t) + \frac{c u^{'} (t)}{u^{μ} (t)} + \frac{g_{1}}{u^{ν} (t)} - \frac{g_{2}}{u^{γ} (t)} = h_{0} (t) u^{δ} (t) for a.e. t \in [0, ω],

(1.1)

with non-negative constants

g_{1}

,

g_{2}

, δ, real numbers c, μ, ν, γ, and

h_{0} \in L ([0, ω]; R)

, plays an important role in fluid mechanics. Therefore, the equation

u^{″} (t) + f (u (t)) u^{'} (t) + g (u (t)) = h (t, u (t)) for a.e. t \in [0, ω],

(1.2)

subjected to the periodic conditions

u (0) = u (ω), u^{'} (0) = u^{'} (ω),

(1.3)

is investigated in the presented paper. Here, $f, g : (0, + \infty) \to R$ are continuous, having possible singularities at zero, and $h : [0, ω] \times R \to R$ is a Carathéodory function, i.e., $h (\cdot, x) : [0, ω] \to R$ is measurable for all $x \in R$ , $h (t, \cdot) : R \to R$ is continuous for a.e. $t \in [0, ω]$ , and for every $r > 0$ , there exists a non-negative function $q_{r} \in L ([0, ω]; R)$ such that $| h (t, x) | \leq q_{r} (t)$ for a.e. $t \in [0, ω]$ and all $| x | \leq r$ . By a solution to (1.2), (1.3) we understand a function $u : [0, ω] \to R$ which is positive, absolutely continuous together with its first derivative, satisfies (1.2) almost everywhere on $[0, ω]$ , and verifies (1.3). In spite of the fact that the problem (1.2), (1.3) was investigated by many mathematicians (see, e.g., [2–27]), most of the mentioned works deal with the repulsive case and/or when f has no singularity. However, the physical model, covered by equation (1.1), justifies considering the types of equations with a singular friction-like term.

A particular case of (1.1) is the equation

u^{″} + \frac{1}{u^{ν}} = h_{0} (t)

(1.4)

studied by Lazer and Solimini. Their results were published in 1987 (see [11]) and they proved, among others, that (1.4), (1.3) has at least one solution if and only if ${\bar{h}}_{0} > 0$ , provided $h_{0}$ is bounded. Recently, we have proved (see [28]) that this result cannot be extended to the case when $h_{0}$ is a general integrable (and so unbounded) function unless some additional conditions are introduced. In particular, (1.4), (1.3) is solvable for any $h_{0} \in L ([0, ω]; R)$ with ${\bar{h}}_{0} > 0$ if $ν \geq 1$ ; and, moreover, for any $ν \in (0, 1)$ , there exists a function $h_{0} \in L ([0, ω]; R)$ with ${\bar{h}}_{0} > 0$ such that (1.4), (1.3) has no solution. At this point, we would like to emphasize the important fact that the condition $ν \geq 1$ can be weakened if (1.4) is generalized to equation (1.1), see Remark 2.2 below.

The structure of the paper is as follows. After the introduction and basic notation, we recall the definition of lower and upper functions to the problem (1.2), (1.3), and we formulate the classical theorem on the existence of a solution to (1.2), (1.3) in the case when there exists a couple of well-ordered lower and upper functions. In Section 2, we establish our main results and their consequences. Sections 3 and 4 are devoted to auxiliary propositions and proofs of the main results, respectively.

For convenience, we finish the introduction with a list of notations which are used throughout the paper:

ℕ is the set of all natural numbers, ℝ is the set of all real numbers, $R^{+} = (0, + \infty)$ , ${[x]}_{+} = max {x, 0}$ , ${[x]}_{-} = max {- x, 0}$ .

C ([0, ω]; R)

is the Banach space of continuous functions

u : [0, ω] \to R

with the norm

{∥ u ∥}_{\infty} = max {| u (t) | : t \in [0, ω]} .

$C (R^{+}; R)$ , resp. $C (R^{+}; R^{+})$ , is the set of continuous functions $u : R^{+} \to R$ , resp. $u : R^{+} \to R^{+}$ .

$C^{1} (R^{+}; R^{+})$ is the set of functions $u : R^{+} \to R^{+}$ which are continuous together with their first derivative.

$A C^{1} ([0, ω]; R)$ is a set of all functions $u : [0, ω] \to R$ such that u and $u^{'}$ are absolutely continuous.

L ([0, ω]; R)

is the Banach space of the Lebesgue integrable functions

p : [0, ω] \to R

endowed with the norm

{∥ p ∥}_{1} = \int_{0}^{ω} | p (s) | d s .

For a given

p \in L ([0, ω]; R)

, its mean value is defined by

\bar{p} = \frac{1}{ω} \int_{0}^{ω} p (s) d s .

Given

φ, ψ \in L ([0, ω]; R)

, then

\begin{matrix} Φ_{+} = \int_{0}^{ω} {[φ (s)]}_{+} d s, Φ_{-} = \int_{0}^{ω} {[φ (s)]}_{-} d s, \\ Ψ_{+} = \int_{0}^{ω} {[ψ (s)]}_{+} d s, Ψ_{-} = \int_{0}^{ω} {[ψ (s)]}_{-} d s . \end{matrix}

The following definitions of lower and upper functions are suitable for us. For more general definitions, one can see, e.g., [[12], Definition 8.2].

Definition 1.1 A function

α \in A C^{1} ([0, ω]; R)

is called a lower function to the problem (1.2), (1.3) if

α (t) > 0

for every

t \in [0, ω]

and

\begin{matrix} α^{″} (t) + f (α (t)) α^{'} (t) + g (α (t)) \geq h (t, α (t)) for a.e. t \in [0, ω], \\ α (0) = α (ω), α^{'} (0) \geq α^{'} (ω) . \end{matrix}

Definition 1.2 A function

β \in A C^{1} ([0, ω]; R)

is called an upper function to the problem (1.2), (1.3) if

β (t) > 0

for every

t \in [0, ω]

and

\begin{matrix} β^{″} (t) + f (β (t)) β^{'} (t) + g (β (t)) \leq h (t, β (t)) for a.e. t \in [0, ω], \\ β (0) = β (ω), β^{'} (0) \leq β^{'} (ω) . \end{matrix}

The following theorem is well known in the theory of differential equations (see, e.g., [[12], Theorem 8.12]).

Theorem 1.1 Let α and β be lower and upper functions to the problem (1.2), (1.3) such that

α (t) \leq β (t) for t \in [0, ω] .

(1.5)

Then there exists a solution u to the problem (1.2), (1.3) such that

α (t) \leq u (t) \leq β (t) for t \in [0, ω] .

2 Main results

Theorem 2.1 Let

ρ_{0} \in C^{1} (R^{+}; R^{+})

and

ρ_{1} \in C (R^{+}; R^{+})

be non-decreasing functions,

h_{0}, h_{1} \in L ([0, ω]; R)

, and

x_{0} > 0

be such that

h_{1} (t) ρ_{1} (x) \leq h (t, x) \leq h_{0} (t) ρ_{0} (x) for a.e. t \in [0, ω], x \geq x_{0},

(2.1)

and let there exist

c_{0}, c_{1} \in R

such that

\frac{g (x)}{ρ_{0} (x)} \leq c_{0} < {\bar{h}}_{0} for x \geq x_{0},

(2.2)

\frac{g (x)}{ρ_{1} (x)} \leq c_{1} \leq {\bar{h}}_{1} for x \geq x_{0} .

(2.3)

Let, moreover, there exist

λ \in [0, 1]

such that

\int_{0}^{1} \frac{d s}{ρ_{0}^{λ} (s)} < + \infty,

(2.4)

lim_{x \to 0_{+}} \frac{g (x)}{ρ_{0}^{λ} (x)} = + \infty,

(2.5)

and let either

\int_{0}^{1} (\frac{{[f (s)]}_{+}}{ρ_{0}^{λ} (s)} + \frac{{[g (s)]}_{+}}{ρ_{0}^{2 λ} (s)}) d s = + \infty, \int_{0}^{1} \frac{{[f (s)]}_{-}}{ρ_{0}^{λ} (s)} d s < + \infty

(2.6)

or

\int_{0}^{1} (\frac{{[f (s)]}_{-}}{ρ_{0}^{λ} (s)} + \frac{{[g (s)]}_{+}}{ρ_{0}^{2 λ} (s)}) d s = + \infty, \int_{0}^{1} \frac{{[f (s)]}_{+}}{ρ_{0}^{λ} (s)} d s < + \infty .

(2.7)

Furthermore, let us suppose that

ρ_{0}

fulfills at least one of the following conditions:

(a)

there exists a sequence ${y_{n}}_{n = 1}^{+ \infty}$ of positive numbers such that
$lim_{n \to + \infty} y_{n} = + \infty, lim_{n \to + \infty} \frac{ρ_{0}^{1 - λ} (y_{n})}{σ (y_{n})} = 0,$

(2.8)

and there exist

ε_{0} > 0

,

ε_{1} \in (0, ε_{0}]

, and

n_{0} \in N

such that

\frac{ρ_{0}^{1 - λ} ((1 + ε_{0}) y_{n})}{ρ_{0}^{1 - λ} (y_{n})} Φ_{-} \leq Φ_{+} - ε_{0} for n \geq n_{0},

(2.9)

(1 + ε_{1}) σ (y_{n}) \leq σ ((1 + ε_{0}) y_{n}) for n \geq n_{0},

(2.10)

where

φ (t) = h_{0} (t) - c_{0}

for almost every

t \in [0, ω]

and

σ (x) = \int_{0}^{x} \frac{d s}{ρ_{0}^{λ} (s)};

(2.11)

(b)

the function $\frac{ρ_{0}^{1 - λ} (x)}{σ (x)}$ is non-increasing and
$\frac{ω}{4} Φ_{+} Φ_{-} \frac{ρ_{0}^{1 - λ} (x_{0})}{σ (x_{0})} < Φ_{+} - Φ_{-},$

(2.12)

where $φ (t) = h_{0} (t) - c_{0}$ for almost every $t \in [0, ω]$ and σ is given by (2.11).

Besides, let us suppose that

ρ_{1}

fulfills at least one of the following conditions:

(c)

there exists a sequence ${z_{n}}_{n = 1}^{+ \infty}$ of positive numbers such that
$lim_{n \to + \infty} z_{n} = + \infty, lim_{n \to + \infty} \frac{ρ_{1} (z_{n})}{z_{n}} = 0,$

and there exist

ε_{2} > 0

and

n_{1} \in N

such that

\frac{ρ_{1} (z_{n} (1 + ε_{2}))}{ρ_{1} (z_{n})} Ψ_{-} \leq Ψ_{+} for n \geq n_{1},

where

ψ (t) = h_{1} (t) - c_{1}

for almost every

t \in [0, ω]

;

(d)

the function $\frac{ρ_{1} (x)}{x}$ is non-increasing and
$\frac{ω}{4} Ψ_{+} Ψ_{-} \frac{ρ_{1} (x_{0})}{x_{0}} \leq Ψ_{+} - Ψ_{-},$

where $ψ (t) = h_{1} (t) - c_{1}$ for almost every $t \in [0, ω]$ .

Then there exists at least one solution to the problem (1.2), (1.3).

Remark 2.1 Note that there exists a suitable

ε_{1}

such that (2.10) holds, e.g., if

\underset{x \to + \infty}{lim sup} \frac{ρ_{0}^{λ} ((1 + ε_{0}) x)}{ρ_{0}^{λ} (x)} < 1 + ε_{0} .

For equation (1.1), from Theorem 2.1 we get the following assertion.

Corollary 2.1 Let

g_{1} > 0

,

g_{2} \geq 0

,

0 \leq δ < 1

,

ν > γ

,

ν + δ > 0

and

either (μ + δ) sgn | c | \geq 1 or ν + 2 δ \geq 1 .

If

{\bar{h}}_{0} > - lim_{x \to + \infty} \frac{g_{2}}{x^{γ + δ}},

then (1.1), (1.3) has at least one solution.

Remark 2.2 In [28], it is proved, among others, that the equation

u^{″} + \frac{1}{u^{ν}} = h_{0} (t),

(2.13)

with $h_{0} \in L ([0, ω]; R)$ and ${\bar{h}}_{0} > 0$ , has a positive ω-periodic solution if $ν \geq 1$ . Moreover, there is also an example introduced showing that for any $ν \in (0, 1)$ , there exists $h_{0} \in L ([0, ω]; R)$ with ${\bar{h}}_{0} > 0$ such that (2.13), (1.3) has no positive solution.

Corollary 2.1 says that if a friction-like term or sub-linear term are added to (2.13), the condition

ν \geq 1

can be weakened. For example,

u^{″} + \frac{u^{'}}{u^{μ}} + \frac{1}{u^{ν}} = h_{0} (t)

has a positive solution satisfying (1.3) for any

ν > 0

if

μ \geq 1

, provided

{\bar{h}}_{0} > 0

. Also, the equation

u^{″} + \frac{1}{u^{ν}} = h_{0} (t) u^{δ}

subjected to the boundary conditions (1.3) is solvable for any $ν > 0$ if $δ \in [1 / 2, 1)$ , provided ${\bar{h}}_{0} > 0$ .

Example 2.1 As it was mentioned in the introduction, the particular case of (1.1) is the so-called Rayleigh-Plesset equation frequently used in fluid mechanics. This equation has the following form:

u^{″} + \frac{c u^{'}}{u^{\frac{4}{5}}} + \frac{g_{1}}{u^{\frac{1}{5}}} - \frac{g_{2}}{u^{γ}} = h_{0} (t) u^{\frac{1}{5}},

(2.14)

where $h_{0} \in L ([0, ω]; R)$ , c, $g_{1}$ , $g_{2}$ are positive constants and $γ \in R$ (see [9, 10]).

The results dealing with the existence of positive ω-periodic solutions of (2.14) were established in [10] provided

h_{0}

is bounded from above (see [[10], Theorems 4.4, 4.6, 4.7]). However, Corollary 2.1 says that in the case when

γ < 1 / 5

, the problem (2.14), (1.3) is solvable if one of the following items is fulfilled:

1.

$γ > - 1 / 5$ and ${\bar{h}}_{0} > 0$ ;
2.

$γ = - 1 / 5$ and ${\bar{h}}_{0} > - g_{2}$ ;
3.

$γ < - 1 / 5$ .

Thus, Corollary 2.1 assures that the boundedness of $h_{0}$ is not necessary.

Corollary 2.2 Let

g_{1} > 0

,

g_{2} \geq 0

,

ν > γ

,

ν + 1 > 0

. Let, moreover, either

g^{*} = - \infty

or

{\bar{h}}_{0} > g^{*} > - \infty, \frac{ω}{4} Φ_{+} Φ_{-} < Φ_{+} - Φ_{-},

(2.15)

where

φ (t) = h_{0} (t) - g^{*}

for almost every

t \in [0, ω]

, and

g^{*} = - lim_{x \to + \infty} \frac{g_{2}}{x^{γ + 1}} .

Then the problem (1.1), (1.3) with $δ = 1$ has at least one solution.

Remark 2.3 According to [29] and Theorem 1.1, it can be easily verified that the problem

u^{″} + \frac{g_{1}}{u^{ν}} = h_{0} (t) u; u (0) = u (ω), u^{'} (0) = u^{'} (ω),

(2.16)

with $g_{1} > 0$ and $ν > 0$ , has a positive solution if and only if the inclusion $L [0, - h_{0}] \in V^{-}$ holds (see notation in [29]).

Indeed, according to [[29], Definition 1.1], the inclusion

L [0, - h_{0}] \in V^{-}

implies the existence of a positive solution v to the problem

v^{″} = h_{0} (t) v - g_{1}; u (0) = u (ω), u^{'} (0) = u^{'} (ω) .

Therefore there exist constants

x > 0

and

y > 0

such that

x^{1 + ν} \leq v (t) \leq y^{1 + ν}

for

t \in [0, ω]

. By setting

α (t) \overset{def}{=} \frac{v (t)}{y^{ν}}, β (t) \overset{def}{=} \frac{v (t)}{x^{ν}} for t \in [0, ω],

one can easily realize that α and β are, respectively, lower and upper functions to (2.16) satisfying (1.5). Now, the existence of a positive solution to (2.16) follows from Theorem 1.1.

On the other hand, the existence of a positive solution to (2.16) implies the inclusion $L [0, - h_{0}] \in V^{-}$ (see [[29], Theorem 2.1]).

However, one of the optimal effective conditions guaranteeing such an inclusion is

h_{0} ≢ 0

and

\frac{ω}{4} \int_{0}^{ω} {[h_{0} (s)]}_{+} d s \int_{0}^{ω} {[h_{0} (s)]}_{-} d s \leq \int_{0}^{ω} {[h_{0} (s)]}_{+} d s - \int_{0}^{ω} {[h_{0} (s)]}_{-} d s

(see [[29], Corollary 2.5]). Therefore, the condition (2.15) is natural in a certain sense.

When the right-hand side of equation (1.3) does not depend on u, i.e., when

h (t, x) \equiv h_{0} (t)

, then (1.3) has the form

u^{″} (t) + f (u (t)) u^{'} (t) + g (u (t)) = h_{0} (t) for a.e. t \in [0, ω] .

(2.17)

From Theorem 2.1, for equation (2.17) we get the following assertion.

Corollary 2.3 Let there exist

x_{0} > 0

and

c_{0} \in R

such that

g (x) \leq c_{0} < {\bar{h}}_{0} for x \geq x_{0}

and let

lim_{x \to 0_{+}} g (x) = + \infty .

Let, moreover, either

\int_{0}^{1} ({[f (s)]}_{+} + {[g (s)]}_{+}) d s = + \infty, \int_{0}^{1} {[f (s)]}_{-} d s < + \infty

or

\int_{0}^{1} ({[f (s)]}_{-} + {[g (s)]}_{+}) d s = + \infty, \int_{0}^{1} {[f (s)]}_{+} d s < + \infty .

Then there exists at least one solution to the problem (2.17), (1.3).

In the following result, the assumptions do not depend on the friction-like term. On the other hand, a certain smallness of oscillation of the primitive to $h_{0}$ is supposed. Clearly, Theorems 2.1 and 2.2 are independent.

Theorem 2.2 Let

ρ_{0} \in C^{1} (R^{+}; R^{+})

and

ρ_{1} \in C (R^{+}; R^{+})

be non-decreasing functions,

h_{0}, h_{1} \in L ([0, ω]; R)

, and

0 < x_{0} \leq x_{1} < + \infty

be such that

h (t, x) \leq h_{0} (t) ρ_{0} (x) for a.e. t \in [0, ω], 0 < x \leq x_{0},

(2.18)

h (t, x) \geq h_{1} (t) ρ_{1} (x) for a.e. t \in [0, ω], x \geq x_{1} .

(2.19)

Let, moreover,

\frac{ω}{8} {∥ h_{0} - {\bar{h}}_{0} ∥}_{1} < \int_{0}^{x_{0}} \frac{d s}{ρ_{0} (s)} < + \infty,

(2.20)

\frac{g (x)}{ρ_{0} (x)} \geq {\bar{h}}_{0} for 0 < x \leq x_{0},

(2.21)

and let there exist

c_{1} \in R

such that

\frac{g (x)}{ρ_{1} (x)} \leq c_{1} \leq {\bar{h}}_{1} for x \geq x_{1} .

(2.22)

Besides, let us suppose that $ρ_{1}$ fulfills at least one of the conditions (c) or (d) of Theorem 2.1. Then there exists at least one solution to the problem (1.2), (1.3).

In the particular case, when equation (1.2) has the form (1.1), the following assertion immediately follows from Theorem 2.2.

Corollary 2.4 Let

0 \leq δ < 1

, and let

0 < x_{0} \leq x_{1} < + \infty

be such that

(1 - δ) \frac{ω}{8} {∥ h_{0} - {\bar{h}}_{0} ∥}_{1} < x_{0}^{1 - δ} .

Let, moreover,

\begin{matrix} \frac{g_{1}}{x^{ν + δ}} - \frac{g_{2}}{x^{γ + δ}} \geq {\bar{h}}_{0} if 0 < x \leq x_{0}, \\ \frac{g_{1}}{x^{ν + δ}} - \frac{g_{2}}{x^{γ + δ}} \leq {\bar{h}}_{0} if x \geq x_{1} . \end{matrix}

Then the problem (1.1), (1.3) has at least one solution.

Remark 2.4 The consequence of Theorem 2.2 for the problem (2.17), (1.3) coincides with the result obtained in [[10], Theorem 3.6].

3 Auxiliary propositions

In what follows, we will show the existence of a solution to the equation

u^{″} (t) + f (u (t)) u^{'} (t) + g (u (t)) = h_{0} (t) ρ_{0} (u (t)) for a.e. t \in [0, ω],

(3.1)

satisfying the boundary conditions (1.3). Here,

ρ_{0} \in C (R^{+}; R^{+})

is a non-decreasing function,

h_{0} \in L ([0, ω]; R)

, and

f, g \in C (R^{+}; R)

. Together with (3.1), for every

k \in N

, consider the auxiliary equation

u^{″} (t) + f (u (t)) u^{'} (t) + g (u (t)) = h_{0 k} (t) ρ_{0} (u (t)) for a.e. t \in [0, ω],

(3.2)

where

h_{0 k} (t) = {\begin{matrix} k & if h_{0} (t) > k, \\ h_{0} (t) & if h_{0} (t) \leq k \end{matrix} for a.e. t \in [0, ω], k \in N .

(3.3)

Obviously,

h_{0 k} (t) \leq h_{0 m} (t) \leq h_{0} (t) for a.e. t \in [0, ω], k \leq m,

(3.4)

and

lim_{k \to + \infty} {\bar{h}}_{0 k} = {\bar{h}}_{0} .

(3.5)

The following three results can be found in [10].

Lemma 3.1 (see [[10], Corollary 2.17])

Let

x_{0} > 0

and

c \in R

be such that

\frac{g (x)}{ρ_{0} (x)} \leq c \leq {\bar{h}}_{0} for x \geq x_{0} .

(3.6)

Let, moreover, there exist a sequence

{y_{n}}_{n = 1}^{+ \infty}

of positive numbers such that

lim_{n \to + \infty} y_{n} = + \infty, lim_{n \to + \infty} \frac{ρ_{0} (y_{n})}{y_{n}} = 0,

(3.7)

and let there exist

ε > 0

and

n_{0} \in N

such that

\frac{ρ_{0} ((1 + ε) y_{n})}{ρ_{0} (y_{n})} Φ_{-} \leq Φ_{+} for n \geq n_{0},

where

φ (t) = h_{0} (t) - c

for almost every

t \in [0, ω]

. Then there exists an upper function β to the problem (3.1), (1.3) satisfying

β (t) \geq x_{0} for t \in [0, ω] .

(3.8)

Lemma 3.2 (see [[10], Corollary 2.18])

Let

x_{0} > 0

and

c \in R

be such that (3.6) holds. If

\frac{ρ_{0} (x)}{x}

is a non-increasing function such that

\frac{ω}{4} Φ_{+} Φ_{-} \frac{ρ_{0} (x_{0})}{x_{0}} \leq Φ_{+} - Φ_{-},

where $φ (t) = h_{0} (t) - c$ for almost every $t \in [0, ω]$ , then there exists an upper function β to the problem (3.1), (1.3) satisfying (3.8).

Lemma 3.3 (see [[10], Corollary 2.11])

Let

x_{0} > \frac{ω}{8} {∥ h_{0} - {\bar{h}}_{0} ∥}_{1}

be such that

g (x) \geq {\bar{h}}_{0} for 0 < x \leq x_{0} .

Then there exists a lower function α to the problem (2.17), (1.3) with

0 < α (t) \leq x_{0} for t \in [0, ω] .

(3.9)

Lemma 3.4 Let

x_{0} > 0

and

c_{0} \in R

be such that (2.2) holds. Let, moreover, there exist a sequence

{y_{n}}_{n = 1}^{+ \infty}

of positive numbers such that (3.7) is fulfilled, and let there exist

ε > 0

and

n_{0} \in N

such that

\frac{ρ_{0} ((1 + ε) y_{n})}{ρ_{0} (y_{n})} Φ_{-} \leq Φ_{+} - ε for n \geq n_{0},

(3.10)

where $φ (t) = h_{0} (t) - c_{0}$ for almost every $t \in [0, ω]$ . Then there exist $k_{0} \in N$ and an upper function β to the problems (3.2), (1.3) for $k \geq k_{0}$ satisfying (3.8).

Proof

Put

φ_{k} (t) = h_{0 k} (t) - c_{0} for a.e. t \in [0, ω],

(3.11)

Φ_{k +} = \int_{0}^{ω} {[φ_{k} (s)]}_{+} d s, Φ_{k -} = \int_{0}^{ω} {[φ_{k} (s)]}_{-} d s .

(3.12)

Then, obviously, in view of (3.3), we have

lim_{k \to + \infty} Φ_{k +} = Φ_{+}, lim_{k \to + \infty} Φ_{k -} = Φ_{-}

(3.13)

and, consequently, on account of (2.2), (3.5), (3.10), and (3.13), there exists

k_{0} \in N

such that

\frac{g (x)}{ρ_{0} (x)} \leq c_{0} \leq {\bar{h}}_{0 k_{0}} \leq {\bar{h}}_{0} for x \geq x_{0},

(3.14)

\frac{ρ_{0} ((1 + ε) y_{n})}{ρ_{0} (y_{n})} Φ_{k_{0} -} \leq Φ_{k_{0} +} for n \geq n_{0} .

(3.15)

Therefore, according to Lemma 3.1, there exists an upper function β to (3.2), (1.3) with $k = k_{0}$ satisfying (3.8). Obviously, in view of (3.4) and the non-negativity of $ρ_{0}$ , it follows that β is also an upper function to (3.2), (1.3) for $k \geq k_{0}$ . □

Lemma 3.5 Let

x_{0} > 0

and

c_{0} \in R

be such that (2.2) holds. If

\frac{ρ_{0} (x)}{x}

is a non-increasing function such that

\frac{ω}{4} Φ_{+} Φ_{-} \frac{ρ_{0} (x_{0})}{x_{0}} < Φ_{+} - Φ_{-},

(3.16)

where $φ (t) = h_{0} (t) - c_{0}$ for almost every $t \in [0, ω]$ , then there exist $k_{0} \in N$ and an upper function β to the problems (3.2), (1.3) for $k \geq k_{0}$ satisfying (3.8).

Proof Define

φ_{k}

,

Φ_{k +}

, and

Φ_{k -}

by (3.11) and (3.12). Then, obviously, in view of (3.3), we have that (3.13) holds and, consequently, on account of (2.2), (3.5), (3.13), and (3.16), there exists

k_{0} \in N

such that (3.14) is valid and

\frac{ω}{4} Φ_{k_{0} +} Φ_{k_{0} -} \frac{ρ_{0} (x_{0})}{x_{0}} \leq Φ_{k_{0} +} - Φ_{k_{0} -} .

(3.17)

Therefore, according to Lemma 3.2, there exists an upper function β to (3.2), (1.3) with $k = k_{0}$ satisfying (3.8). Obviously, in view of (3.4) and the non-negativity of $ρ_{0}$ , it follows that β is also an upper function to (3.2), (1.3) for $k \geq k_{0}$ . □

Lemma 3.6 Let

\underset{x \to 0 +}{lim inf} g (x) > - \infty,

(3.18)

and let either

\int_{0}^{1} {[f (s)]}_{+} d s < + \infty

(3.19)

or

\int_{0}^{1} {[f (s)]}_{-} d s < + \infty .

(3.20)

Then, for every

K > 0

, there exists a constant

K_{1} > 0

such that for any

k \in N

and any positive solution u of (3.2), (1.3) with

{∥ u ∥}_{\infty} \leq K,

(3.21)

we have the estimate

{∥ u^{'} ∥}_{\infty} \leq K_{1} .

(3.22)

Proof Assume that (3.20) is fulfilled. Let u be a positive solution to (3.2), (1.3) satisfying (3.21). Then there exist

t_{0}, t_{1} \in [0, ω]

such that

u (t_{0}) = min {u (t) : t \in [0, ω]}, u (t_{1}) = max {u (t) : t \in [0, ω]} .

(3.23)

Define the operator ϑ of ω-periodic prolongation by

ϑ (v) (t) = {\begin{matrix} v (t) & if t \in [0, ω], \\ v (t - ω) & if t \in (ω, 2 ω] . \end{matrix}

(3.24)

Then, obviously, from (3.2) and (1.3) it follows that

\begin{aligned} ϑ {(u)}^{″} (t) + f (ϑ (u) (t)) ϑ {(u)}^{'} (t) + g (ϑ (u) (t)) \\ = ϑ (h_{0 k}) (t) ρ_{0} (ϑ (u) (t)) for a.e. t \in [0, 2 ω] . \end{aligned}

(3.25)

The integration of (3.25) from

t_{0}

to t, on account of (3.23), yields

\begin{aligned} ϑ {(u)}^{'} (t) = & - \int_{t_{0}}^{t} f (ϑ (u) (s)) ϑ {(u)}^{'} (s) d s - \int_{t_{0}}^{t} g (ϑ (u) (s)) d s \\ + \int_{t_{0}}^{t} ϑ (h_{0 k}) (s) ρ_{0} (ϑ (u) (s)) d s for t \in [t_{0}, t_{0} + ω] . \end{aligned}

(3.26)

From (3.21), (3.23), and (3.24) it follows that

0 < ϑ (u) (t_{0}) \leq ϑ (u) (t) \leq K for t \in [t_{0}, t_{0} + ω] .

(3.27)

Put

μ = sup {{[g (s)]}_{-} : s \in (0, K]} .

(3.28)

According to (3.18), we have

0 \leq μ < + \infty .

(3.29)

Thus, using (3.3), (3.20), (3.21), and (3.27)-(3.29) in (3.26), we arrive at

ϑ {(u)}^{'} (t) \leq \int_{0}^{K} {[f (s)]}_{-} d s + ω μ + {∥ h_{0} ∥}_{1} ρ_{0} (K) for t \in [t_{0}, t_{0} + ω] .

(3.30)

Put

K_{1} = \int_{0}^{K} {[f (s)]}_{-} d s + ω μ + {∥ h_{0} ∥}_{1} ρ_{0} (K) .

Then, on account of (3.24) and (3.30), we have

u^{'} (t) \leq K_{1} for t \in [0, ω] .

(3.31)

On the other hand, the integration of (3.25) from t to

t_{1} + ω

, with respect to (3.23), results in

\begin{aligned} ϑ {(u)}^{'} (t) = & \int_{t}^{t_{1} + ω} f (ϑ (u) (s)) ϑ {(u)}^{'} (s) d s + \int_{t}^{t_{1} + ω} g (ϑ (u) (s)) d s \\ - \int_{t}^{t_{1} + ω} ϑ (h_{0 k}) (s) ρ_{0} (ϑ (u) (s)) d s for t \in [t_{1}, t_{1} + ω] . \end{aligned}

(3.32)

Now, using (3.3), (3.20), (3.21), and (3.27)-(3.29) in (3.32), we obtain

- ϑ {(u)}^{'} (t) \leq K_{1} for t \in [t_{1}, t_{1} + ω] .

(3.33)

Therefore, in view of (3.24), from (3.33) we get

- u^{'} (t) \leq K_{1} for t \in [0, ω] .

(3.34)

Consequently, (3.31) and (3.34) result in (3.22).

Now, suppose that (3.19) is fulfilled. Put

v (t) = u (ω - t) for t \in [0, ω] .

(3.35)

Then, according to (3.2), we have

v^{″} (t) - f (v (t)) v^{'} (t) + g (v (t)) = {\tilde{h}}_{0 k} (t) ρ_{0} (v (t)) for a.e. t \in [0, ω],

(3.36)

where

{\tilde{h}}_{0 k} (t) = h_{0 k} (ω - t) for a.e. t \in [0, ω] .

Analogously to the above-proved, using (3.19) instead of (3.20), we obtain

{∥ v^{'} ∥}_{\infty} \leq K_{1}

(3.37)

with

K_{1} = \int_{0}^{K} {[f (s)]}_{+} d s + ω μ + {∥ h_{0} ∥}_{1} ρ_{0} (K) .

Thus, (3.35) and (3.37) yield (3.22). □

Lemma 3.7 Let

lim_{x \to 0_{+}} g (x) = + \infty

(3.38)

and let either

\int_{0}^{1} ({[f (s)]}_{+} + {[g (s)]}_{+}) d s = + \infty, \int_{0}^{1} {[f (s)]}_{-} d s < + \infty

(3.39)

or

\int_{0}^{1} ({[f (s)]}_{-} + {[g (s)]}_{+}) d s = + \infty, \int_{0}^{1} {[f (s)]}_{+} d s < + \infty .

(3.40)

Then, for every

K > 0

, there exists a constant

a > 0

such that for any

k \in N

and any positive solution u of (3.2), (1.3) satisfying (3.21), we have the estimate

a \leq u (t) for t \in [0, ω] .

(3.41)

Proof Let u be a positive solution to (3.2), (1.3) satisfying (3.21). Thus, the integration of (3.2) from 0 to ω, in view of (1.3) and (3.4), yields

\int_{0}^{ω} g (u (s)) d s \leq {∥ h_{0} ∥}_{1} ρ_{0} (K) .

(3.42)

On the other hand, (3.38) implies the existence of

x_{0} \in (0, + \infty)

such that

g (x) > \frac{{∥ h_{0} ∥}_{1} ρ_{0} (K)}{ω} \geq 0 for x \in (0, x_{0}) .

(3.43)

Let

t_{m} \in [0, ω]

be such that

u (t_{m}) = min {u (t) : t \in [0, ω]} .

(3.44)

Obviously, either

u (t_{m}) \geq x_{0}

or

u (t_{m}) < x_{0} .

(3.45)

Obviously, it is sufficient to show the estimate (3.41) is valid just in the case when (3.45) is fulfilled. Let, therefore, (3.45) hold.

If

u (t) < x_{0}

for

t \in [0, ω]

, then applying (3.43) in (3.42) we obtain a contradiction. Thus, there exist points

t_{1}, t_{2} \in (t_{m}, t_{m} + ω)

such that

ϑ (u) (t) < x_{0} for t \in [t_{m}, t_{1}), ϑ (u) (t_{1}) = x_{0},

(3.46)

ϑ (u) (t) < x_{0} for t \in (t_{2}, t_{m} + ω], ϑ (u) (t_{2}) = x_{0},

(3.47)

where ϑ is the operator defined by (3.24). Obviously, (3.25) holds.

Assume that (3.39) holds. Then, according to Lemma 3.6, there exists

K_{1} > 0

such that (3.22) holds. The integration of (3.25) from

t_{m}

to

t_{1}

, in view of (3.4), (3.21), (3.22), (3.39), (3.43), (3.44), and (3.46), results in

\begin{aligned} ϑ {(u)}^{'} (t_{1}) + \int_{ϑ (u) (t_{m})}^{x_{0}} {[f (s)]}_{+} d s + \frac{1}{K_{1}} \int_{ϑ (u) (t_{m})}^{x_{0}} {[g (s)]}_{+} d s \\ \leq \int_{0}^{x_{0}} {[f (s)]}_{-} d s + {∥ h_{0} ∥}_{1} ρ_{0} (K) . \end{aligned}

(3.48)

Note that in view of (3.46), we have

ϑ {(u)}^{'} (t_{1}) \geq 0

. Consequently, from (3.48) we obtain

\int_{ϑ (u) (t_{m})}^{x_{0}} ({[f (s)]}_{+} + {[g (s)]}_{+}) d s \leq K_{2},

(3.49)

where

K_{2} = (K_{1} + 1) (\int_{0}^{x_{0}} {[f (s)]}_{-} d s + {∥ h_{0} ∥}_{1} ρ_{0} (K)) .

Note that $K_{2}$ does not depend on k. Therefore, if we apply (3.39) in (3.49), it can be easily seen, with respect to (3.44), that there exists a constant $a > 0$ such that (3.41) holds.

If (3.40) holds, we integrate (3.25) from

t_{2}

to

t_{m} + ω

and apply similar steps as above, just using (3.47) instead of (3.46). Finally, we arrive at

\int_{ϑ (u) (t_{m} + ω)}^{x_{0}} ({[f (s)]}_{-} + {[g (s)]}_{+}) d s \leq K_{2}

with

K_{2} = (K_{1} + 1) (\int_{0}^{x_{0}} {[f (s)]}_{+} d s + {∥ h_{0} ∥}_{1} ρ_{0} (K)) .

Therefore, also in this case, there exists a constant $a > 0$ such that (3.41) holds. □

Lemma 3.8 Let $x_{0} > 0$ and $c_{0} \in R$ be such that (2.2) holds. Let, moreover, (3.38) be fulfilled, and let either (3.39) or (3.40) be valid. Let, in addition, there exist a sequence ${y_{n}}_{n = 1}^{+ \infty}$ of positive numbers such that (3.7) holds, and let there exist $ε > 0$ and $n_{0} \in N$ such that (3.10) is fulfilled, where $φ (t) = h_{0} (t) - c_{0}$ for almost every $t \in [0, ω]$ . Then there exists a positive solution u to (3.1), (1.3).

Proof According to Lemma 3.4, there exist

k_{0} \in N

and an upper function β to the problems (3.2), (1.3) for

k \geq k_{0}

satisfying (3.8). On the other hand, in view of (3.4) and (3.38), there exists

x_{k} \in (0, x_{0}]

for

k \geq k_{0}

such that

g (x_{k}) \geq h_{0 k} (t) ρ_{0} (x_{k}) for a.e. t \in [0, ω] .

Thus, if we put

α_{k} (t) = x_{k}

for

t \in [0, ω]

, according to Theorem 1.1, there exists a solution

u_{k}

to (3.2), (1.3) for

k \geq k_{0}

satisfying

0 < α_{k} (t) \leq u_{k} (t) \leq β (t) for t \in [0, ω] .

(3.50)

Moreover, according to Lemmas 3.6 and 3.7, in view of (3.50), there exist constants

K > 0

,

K_{1} > 0

, and

a > 0

, not depending on k, such that

{∥ u_{k} ∥}_{\infty} \leq K, {∥ u_{k}^{'} ∥}_{\infty} \leq K_{1} for k \geq k_{0},

(3.51)

a \leq u_{k} (t) for t \in [0, ω], k \geq k_{0},

(3.52)

| u_{k}^{″} (t) | \leq f_{0} K_{1} + g_{0} + | h_{0} (t) | ρ_{0} (K) for a.e. t \in [0, ω], k \geq k_{0},

(3.53)

where

f_{0} = max {| f (x) | : x \in [a, K]}, g_{0} = max {| g (x) | : x \in [a, K]} .

Therefore, according to the Arzelà-Ascoli theorem, there exist

u_{0} \in C ([0, ω]; R)

and

v_{0} \in C ([0, ω]; R)

such that

lim_{k \to + \infty} {∥ u_{k} - u_{0} ∥}_{\infty} = 0, lim_{k \to + \infty} {∥ u_{k}^{'} - v_{0} ∥}_{\infty} = 0 .

(3.54)

Moreover, since $u_{k}$ are solutions to (3.2), (1.3), in view of (3.3), (3.52), and (3.54), we have $u_{0} \in A C^{1} ([0, ω]; R)$ , $u_{0}^{'} \equiv v_{0}$ , and $u_{0}$ is a positive solution to (3.1), (1.3). □

The following assertion can be proved analogously to Lemma 3.8, just Lemma 3.5 is used instead of Lemma 3.4.

Lemma 3.9 Let $x_{0} > 0$ and $c_{0} \in R$ be such that (2.2) holds. Let, moreover, (3.38) be fulfilled, and let either (3.39) or (3.40) be valid. Let, in addition, $\frac{ρ_{0} (x)}{x}$ be a non-increasing function and let (3.16) be fulfilled, where $φ (t) = h_{0} (t) - c_{0}$ for almost every $t \in [0, ω]$ . Then there exists a positive solution u to (3.1), (1.3).

Lemma 3.10 Let $ρ_{0} \in C^{1} (R^{+}; R^{+})$ be non-decreasing, $x_{0} > 0$ , and $c_{0} \in R$ be such that (2.2) holds. Let, moreover, there exist $λ \in [0, 1]$ such that (2.4) and (2.5) are valid, and let either (2.6) or (2.7) be fulfilled. Let, in addition, there exist a sequence ${y_{n}}_{n = 1}^{+ \infty}$ of positive numbers such that (2.8) holds and let there exist $ε_{0} > 0$ , $ε_{1} \in (0, ε_{0}]$ , and $n_{0} \in N$ such that (2.9) and (2.10) are fulfilled, where $φ (t) = h_{0} (t) - c_{0}$ for almost every $t \in [0, ω]$ and σ is given by (2.11). Then there exists a lower function α to the problem (3.1), (1.3).

Proof Because

ρ_{0}

is a positive function, from (2.4) and (2.11) we obtain that σ is a positive increasing function. Therefore, there exists an inverse function

σ^{- 1}

to σ which is also increasing. Moreover, in view of (2.4) and (2.11), it follows that

\begin{aligned} lim_{x \to 0_{+}} σ (x) = 0, lim_{x \to 0_{+}} σ^{- 1} (x) = 0, \\ lim_{x \to + \infty} σ (x) = + \infty, lim_{x \to + \infty} σ^{- 1} (x) = + \infty . \end{aligned}

(3.55)

Consider the auxiliary equation

\begin{aligned} u^{″} (t) + f (σ^{- 1} (u (t))) u^{'} (t) + \frac{g (σ^{- 1} (u (t)))}{ρ_{0}^{λ} (σ^{- 1} (u (t)))} \\ = h_{0} (t) ρ_{0}^{1 - λ} (σ^{- 1} (u (t))) for a.e. t \in [0, ω] . \end{aligned}

(3.56)

Put

z = σ (x)

,

z_{0} = σ (x_{0})

. Then from (2.2) we get

\frac{g (σ^{- 1} (z))}{ρ_{0} (σ^{- 1} (z))} \leq c_{0} < {\bar{h}}_{0} for z \geq z_{0}

(3.57)

and, in view of (3.55), from (2.5) we have

lim_{z \to 0_{+}} \frac{g (σ^{- 1} (z))}{ρ_{0}^{λ} (σ^{- 1} (z))} = + \infty .

(3.58)

Furthermore, the substitution

r = σ (s)

in (2.6), resp (2.7), with respect to (2.11) yields

\int_{0}^{1} ({[f (σ^{- 1} (r))]}_{+} + \frac{{[g (σ^{- 1} (r))]}_{+}}{ρ_{0}^{λ} (σ^{- 1} (r))}) d r = + \infty, \int_{0}^{1} {[f (σ^{- 1} (r))]}_{-} d r < + \infty,

(3.59)

resp.

\int_{0}^{1} ({[f (σ^{- 1} (r))]}_{-} + \frac{{[g (σ^{- 1} (r))]}_{+}}{ρ_{0}^{λ} (σ^{- 1} (r))}) d r = + \infty, \int_{0}^{1} {[f (σ^{- 1} (r))]}_{+} d r < + \infty .

(3.60)

Moreover, put

z_{n} = σ (y_{n})

for

n \in N

. Then from (2.8), in view of (3.55), we get

lim_{n \to + \infty} z_{n} = + \infty, lim_{n \to + \infty} \frac{ρ_{0}^{1 - λ} (σ^{- 1} (z_{n}))}{z_{n}} = 0 .

(3.61)

Finally, (2.10) results in

σ^{- 1} ((1 + ε_{1}) z_{n}) \leq (1 + ε_{0}) y_{n} for n \geq n_{0},

and so, since

ρ_{0}

is a non-decreasing function, from (2.9) we obtain

\frac{ρ_{0}^{1 - λ} (σ^{- 1} ((1 + ε_{1}) z_{n}))}{ρ_{0}^{1 - λ} (σ^{- 1} (z_{n}))} Φ_{-} \leq Φ_{+} - ε_{1} for n \geq n_{0} .

(3.62)

Therefore, applying Lemma 3.8, according to (3.57)-(3.62), there exists a positive solution u to the problem (3.56), (1.3).

Now, we put

α (t) = σ^{- 1} (u (t))

for

t \in [0, ω]

, i.e., in view of (2.11),

u (t) = \int_{0}^{α (t)} \frac{d s}{ρ_{0}^{λ} (s)} for t \in [0, ω] .

Obviously,

α \in A C^{1} ([0, ω]; R)

is a positive function and

\begin{matrix} u^{'} (t) = \frac{α^{'} (t)}{ρ_{0}^{λ} (α (t))} for t \in [0, ω], \\ u^{″} (t) = \frac{α^{″} (t)}{ρ_{0}^{λ} (α (t))} - \frac{λ α^{' 2} (t) ρ_{0}^{'} (α (t))}{ρ_{0}^{1 + λ} (α (t))} \leq \frac{α^{″} (t)}{ρ_{0}^{λ} (α (t))} for a.e. t \in [0, ω] . \end{matrix}

Thus, it can be easily seen that α is a lower function to the problem (3.1), (1.3). □

Analogously to the proof of Lemma 3.10, one can prove the following assertion applying Lemma 3.9 instead of Lemma 3.8.

Lemma 3.11 Let $ρ_{0} \in C^{1} (R^{+}; R^{+})$ be non-decreasing, $x_{0} > 0$ , and $c_{0} \in R$ be such that (2.2) holds. Let, moreover, there exist $λ \in [0, 1]$ such that (2.4) and (2.5) are valid, and let either (2.6) or (2.7) be fulfilled. Let, in addition, $\frac{ρ_{0}^{1 - λ} (x)}{σ (x)}$ be a non-increasing function and let (2.12) be fulfilled, where $φ (t) = h_{0} (t) - c_{0}$ for almost every $t \in [0, ω]$ and σ is given by (2.11). Then there exists a lower function α to the problem (3.1), (1.3).

4 Proofs of the main results

Proof of Theorem 2.1 According to Lemmas 3.1, 3.2, 3.10, and 3.11, the conditions of the theorem guarantee a well-ordered couple of lower and upper functions, therefore the result is a direct consequence of Theorem 1.1. □

Proof of Corollary 2.1 It follows from Theorem 2.1 with

h_{1} \equiv h_{0}

,

ρ_{0} (x) = ρ_{1} (x) = x^{δ}

,

λ = 1

, and

c_{0} = c_{1}

such that

{\bar{h}}_{0} > c_{0} > - lim_{x \to + \infty} \frac{g_{2}}{x^{γ + δ}} .

Then items (a) and (c) of Theorem 2.1 are fulfilled. □

Proof of Corollary 2.2 It follows from Theorem 2.1 with $h_{1} \equiv h_{0}$ , $ρ_{0} (x) = ρ_{1} (x) = x$ , and $λ < 1$ such that $ν + λ > 0$ , $ν + 2 λ \geq 1$ . Then items (b) and (d) of Theorem 2.1 are fulfilled. □

Proof of Corollary 2.3 It immediately follows from Theorem 2.1 with $h_{1} \equiv h_{0}$ , $ρ_{i} (x) \equiv 1$ ( $i = 0, 1$ ). □

Proof of Theorem 2.2

Put

σ (x) = \int_{0}^{x} \frac{d s}{ρ_{0} (s)} for x \geq 0 .

(4.1)

Because $ρ_{0}$ is a positive function, from (2.20) and (4.1) we obtain that σ is an increasing function. Therefore, there exists an inverse function $σ^{- 1}$ to σ which is also increasing.

Consider the auxiliary equation

u^{″} (t) + f (σ^{- 1} (u (t))) u^{'} (t) + \frac{g (σ^{- 1} (u (t)))}{ρ_{0} (σ^{- 1} (u (t)))} = h_{0} (t) for a.e. t \in [0, ω] .

(4.2)

Put

z = σ (x)

,

z_{0} = σ (x_{0})

. Then from (2.20) and (2.21), in view of (4.1), we get

\begin{matrix} \frac{ω}{8} {∥ h_{0} - {\bar{h}}_{0} ∥}_{1} < z_{0}, \\ \frac{g (σ^{- 1} (z))}{ρ_{0} (σ^{- 1} (z))} \geq {\bar{h}}_{0} for 0 < z \leq z_{0} . \end{matrix}

Therefore, according to Lemma 3.3, there exists a lower function w to the problem (4.2), (1.3) satisfying

0 < w (t) \leq σ (x_{0}) for t \in [0, ω] .

(4.3)

Now, we put

α = σ^{- 1} (w (t))

for

t \in [0, ω]

, i.e., in view of (4.1),

w (t) = \int_{0}^{α (t)} \frac{d s}{ρ_{0} (s)} for t \in [0, ω] .

Obviously, with respect to (4.3),

α \in A C^{1} ([0, ω]; R)

is a positive function satisfying (3.9), and

\begin{matrix} w^{'} (t) = \frac{α^{'} (t)}{ρ_{0} (α (t))} for t \in [0, ω], \\ w^{″} (t) = \frac{α^{″} (t)}{ρ_{0} (α (t))} - \frac{α^{' 2} (t) ρ_{0}^{'} (α (t))}{ρ_{0}^{2} (α (t))} \leq \frac{α^{″} (t)}{ρ_{0} (α (t))} for a.e. t \in [0, ω] . \end{matrix}

Thus, on account of (2.18), (3.9), and (4.2), it can be easily seen that α is a lower function to the problem (1.2), (1.3).

The existence of an upper function β to (1.2), (1.3) satisfying

β (t) \geq x_{1} for t \in [0, ω]

(4.4)

follows from (2.19) and Lemma 3.1, resp. 3.2.

Obviously, in view of (3.9) and (4.4), we have that (1.5) holds. Thus the theorem follows from Theorem 1.1. □

Proof of Corollary 2.4 It follows from Theorem 2.2 with $h_{1} \equiv h_{0}$ , $ρ_{0} (x) = ρ_{1} (x) = x^{δ}$ , and $c_{1} = {\bar{h}}_{0}$ . □

Declarations

Acknowledgements

The first author was supported by RVO: 67985840; the second author was supported by Ministerio de Educación y Ciencia, Spain, project MTM2011-23652.

Authors’ Affiliations

(1)

Institute of Mathematics, Academy of Sciences of the Czech Republic, Brno, Czech Republic

(2)

Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, Granada, Spain

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