30
L. H. F. de Andrade et al.
In the following lemma, we find an equivalence for the inclusion of the Musielak–
Orlicz classes and in the next proposition we use this lemma to find an equivalence
for the occurrence of Condition 2.2.
Lemma 2.2 [11, Theorem 8.4] Let and be finite-value Musielak–Orlicz functions. Then, the inclusion ˜
L
⊂ ˜
L
is satisfied if and only if there exist α > 0 and a
non-negative function f ∈ ˜
L
such that
αα(t, u) ≤ (t, u), for all u > f (t).
Proposition 2.4 A measurable function u 0 satisfies Condition 2.2 if and only if for
some measurable function c : T → R such that ϕ(c) is μ-integrable, we can find
constants λ, α > 0 and a non-negative function f ∈ ˜
L
c such that
αα c (t, u) ≤ c−λu 0 (t, u),
for all u > f (t).
(2.9)
Proof Suppose that u 0 satisfies Condition 2.2. Let c : T → R be any measurable
function such that
T ϕ(c)dμ < ∞. As u is a measurable function with
T ϕ(c −
λu 0 + u)dμ < ∞ then
T
ϕ(c + u)dμ =
T
ϕ(c − λu 0 + u + λu 0 )dμ < ∞.
This result implies ˜
L
c−λu 0 ⊂ ˜
L
c and inequality (2.9) follows from Lemma 2.2.
Now suppose that inequality (2.9) is satisfied. By Lemma 2.2 we have ˜
L
c−λu 0 ⊂
˜
L
c . Therefore, u ∈ ˜
L
c implies u + λu 0 ∈ ˜
L
c−λu 0 ⊂ ˜
L
c . Or, equivalently, if u is
a measurable function such that ϕ(c + u) is μ-integrable, then ϕ(c + u + λu 0 ) is
μ-integrable. As a result, we conclude that
T ϕ(c + u + λu 0 )dμ < ∞ for all λ >
0. Let c : T → R be any measurable function satisfying
T ϕ( c)dμ < ∞. Denote
A = { c > c}. Thus, for each λ > 0, it follows that
T
ϕ( c + λu 0 )dμ =
T
ϕ(c + ( c − c) + λu 0 )dμ ≤
T
ϕ(c + ( c − c)χ A + λu 0 )dμ < ∞,
which shows that u 0 is stated under Condition 2.2.
From Proposition 2.4 we have that Condition 2.2 is not satisfied if, and only
if, there exists a measurable function u : T → R such that
T ϕ(c + u)dμ = ∞
but
T ϕ(c + u − λu 0 )dμ < ∞ for some λ > 0. For our result we make use of the
following lemmas.
Lemma 2.3 [11, Lemma 8.3] Consider a non-atomic and σ -finite measure μ . If
{u n } is a sequence of finite-value, non-negative, measurable functions, and {α n } is a
sequence of positive, real numbers, such that
L. H. F. de Andrade et al.
In the following lemma, we find an equivalence for the inclusion of the Musielak–
Orlicz classes and in the next proposition we use this lemma to find an equivalence
for the occurrence of Condition 2.2.
Lemma 2.2 [11, Theorem 8.4] Let and be finite-value Musielak–Orlicz functions. Then, the inclusion ˜
L
⊂ ˜
L
is satisfied if and only if there exist α > 0 and a
non-negative function f ∈ ˜
L
such that
αα(t, u) ≤ (t, u), for all u > f (t).
Proposition 2.4 A measurable function u 0 satisfies Condition 2.2 if and only if for
some measurable function c : T → R such that ϕ(c) is μ-integrable, we can find
constants λ, α > 0 and a non-negative function f ∈ ˜
L
c such that
αα c (t, u) ≤ c−λu 0 (t, u),
for all u > f (t).
(2.9)
Proof Suppose that u 0 satisfies Condition 2.2. Let c : T → R be any measurable
function such that
T ϕ(c)dμ < ∞. As u is a measurable function with
T ϕ(c −
λu 0 + u)dμ < ∞ then
T
ϕ(c + u)dμ =
T
ϕ(c − λu 0 + u + λu 0 )dμ < ∞.
This result implies ˜
L
c−λu 0 ⊂ ˜
L
c and inequality (2.9) follows from Lemma 2.2.
Now suppose that inequality (2.9) is satisfied. By Lemma 2.2 we have ˜
L
c−λu 0 ⊂
˜
L
c . Therefore, u ∈ ˜
L
c implies u + λu 0 ∈ ˜
L
c−λu 0 ⊂ ˜
L
c . Or, equivalently, if u is
a measurable function such that ϕ(c + u) is μ-integrable, then ϕ(c + u + λu 0 ) is
μ-integrable. As a result, we conclude that
T ϕ(c + u + λu 0 )dμ < ∞ for all λ >
0. Let c : T → R be any measurable function satisfying
T ϕ( c)dμ < ∞. Denote
A = { c > c}. Thus, for each λ > 0, it follows that
T
ϕ( c + λu 0 )dμ =
T
ϕ(c + ( c − c) + λu 0 )dμ ≤
T
ϕ(c + ( c − c)χ A + λu 0 )dμ < ∞,
which shows that u 0 is stated under Condition 2.2.
From Proposition 2.4 we have that Condition 2.2 is not satisfied if, and only
if, there exists a measurable function u : T → R such that
T ϕ(c + u)dμ = ∞
but
T ϕ(c + u − λu 0 )dμ < ∞ for some λ > 0. For our result we make use of the
following lemmas.
Lemma 2.3 [11, Lemma 8.3] Consider a non-atomic and σ -finite measure μ . If
{u n } is a sequence of finite-value, non-negative, measurable functions, and {α n } is a
sequence of positive, real numbers, such that