Note on Some Nonlinear Integral Inequalities and Applications to Differential Equations

Theorem 1.1 (see [3].)If u(t) and f(t) are non-nonnegative continuous functions on [0, ∞[ satisfying

for some constant c ≥ 0, then

Lemma 2.1. For x ∈ ℝ₊,   y ∈ ℝ₊,   1/p + 1/q = 1, one has

Lemma 2.2 (see [1].)Let b(t) and f(t) be continuous functions for t ≥ α, let v(t) be a differentiable function for t ≥ α and suppose

Then for t ≥ α,

Theorem 2.3. Let u, a, b, h_i  (i = 1, …, n) be real-valued nonnegative continuous functions and there exists a series of positive real numbers p₁, p₂, …, p_n and u(t) satisfy the following integral inequality,

for t ∈ ℝ₊ then for p ≥ p^⋆ = max {p_i, i = 1, …, n}.

Proof. Define a function v(t) by

then v(0) = 0 and (2.4) can be written as By (2.7) and Lemma 2.1, we get Differentiating (2.6), we get Using (2.8) and (2.9), it yields where By Lemma 2.2, we have Using (2.7) and (2.12), we get

This achieves the proof of the theorem.

Remark 2.4. if we take n = 2,   p ≥ 1,   p₁ = p,   and  p₂ = 1, then the inequality established in Theorem 2.3 become the inequality given in [4,  Theorem 1(a₁)].

Theorem 2.5. Suppose that the hypothesis of Theorem 2.3 holds. Assume that the function (a(t) + p^⋆/p)/b(t) is nondecreasing and

for t ∈ ℝ₊ then for p_⋆ = min {p_i, i = 1, …, n} ≤ p < p^⋆ and $$where

Proof. For p_⋆ ≤ p < p^⋆

By (2.7) and the fact that (p^⋆/p) > 1, one gets:

Differentiating (2.6) and using (2.17), we obtain then Since the function (a(t) + p^⋆/p)/b(t) is nondecreasing, for 0 ≤ t ≤ τ then, where Consequently For τ = t, we can see that then the function M(t) can be estimated as Let Now we estimate the expression $$ by using (2.24) to get Remarking that we integrate (2.27) from 0 to t to get replacing L(t) by its value in (2.28), we obtain then Using (2.7), (2.23), and (2.30) we have, This achieves the proof of the theorem.

Theorem 2.6. Suppose that the hypothesis of Theorem 2.3 holds. Assume that the function (a(t) + p_⋆/p)/b(t), is nondecreasing and

for t ∈ ℝ₊ then for p < p_⋆ and $$,

Proof. for p < p_⋆ = min {p_i, i = 1, …, n}.

Using (2.7), the fact that the function (a(t) + p_⋆/p)/b(t) is nondecreasing, and p_⋆/p≻1, we have

Differentiating (2.6) and using (2.35), we obtain then from the proof of Theorem 2.5, we get the required inequality in (2.33).

Remark 2.7. if we take n = 2, then the inequalities established in Theorems 2.5 and 2.6 become the inequalities given in [5, Theorem 1.2].

Theorem 2.8. Suppose that the hypothesis of Theorem 2.3 holds and moreover the function b(t) is decreasing. Let c be a real valued nonnegative continuous and nondecreasing function for t ∈ ℝ₊. If

then

• (1)

   

for p ≥ p^⋆, where

• (2)

   

for p_⋆≤p < p^⋆ and$$where

• (3)

   

for and $$where

Proof. Since c(t) is a nonnegative, continuous, and nondecreasing function, for t∈ℝ₊, from (2.38) we observe that

we put then, we have

Then a direct application of the inequalities established in Theorems 2.3, 2.5, and 2.6 gives the required results.

Theorem 2.9. Suppose that the hypothesis of Theorem 2.3 holds. Assume that the functions (a(t) + p^⋆/p)/b(t), (a(t) + p_⋆/p)/b(t) are nondecreasing and let k(t, s) and its derivative partial ∂/∂sk(t, s) be real-valued nonnegative continuous functions, for 0 ≤ s ≤ t ≤ ∞. If

then

• (1)

   

where for p ≥ p^⋆.

• (2)

   

For $$

• (3)

   

For p < p_⋆ and $$, where

Proof. Let

 (1) for p ≥ p^⋆.

Differentiating  (2.57) we get

Using (2.8) and (2.58) and the fact that v(t) is nondecreasing, we obtain, for 0 ≤ s ≤ t, then

By Lemma 2.2, we have

where Finally using (2.61) in (2.7), we get the required inequality.

• (2)

  For p_⋆≤p < p^⋆. Using (2.17) and (2.58), we get

  $$ ()

For 0 ≤ s ≤ t ≤ τ, and the fact that (a(t) + p^⋆/p)/b(t)    is nondecreasing, we have then, where From the proof of Theorem 2.5, we get the required inequality.

• (3)

  Using (2.35) and (2.58), we get

taking account the fact that (a(t)+(p_⋆/p))/b(t) is nondecreasing and from the proof of Theorem 2.6, we get the required inequality.

Remark 2.10. if we take n = 2,  p ≥ 1,  p₁ = p,  p₂ = 1, then the inequality established in Theorem 2.9 (part 1) becomes the inequality given in [4, Theorem 1(a₃)].

Theorem 3.1. Assume that u(t) and f(t) are non-nonnegative continuous functions on [0, ∞[ and c ≥ 0 is a constant. If k(t, s) is defined as in Theorem 2.9, then

implies where where p ≠ 0, 0 ≤ q ≤ p and 0 ≤ r ≤ p.

Proof. Define a function z(t) by the right side of (3.1) then

Define a function v(t) by Then and v(t) is nondecreasing for t ∈ ℝ₊.

Then,

its follow from (3.4), (3.6), and (3.7) that then (3.8) can be written as where by Lemma 2.2 we obtain from (3.6) and (3.11), it follows that integrating (3.12), we obtain but then the result required is found.

Example 4.1. We consider a nonlinear differential equation

Assume that p, p_i  (i = 1, …, n) ≥ 0, are fixed real numbers,    u₀ is a real constant, and H : ℝ₊ × ℝ₊ → ℝ₊, h_i, l : ℝ₊ → ℝ₊ are continuous functions.

Example 4.2. Consider the following initial value problem:

where f(t) and k(t, s) are as defined in Theorem 3.1, and p ≠ 0,  0 ≤ q ≤ p and 0 ≤ r ≤ p,  and  c is a constant.

Theorem 4.3. Assume u(t) is a solution of (4.4), then

where A^*(t) and B^*(t) are defined in (3.10).

Proof. The solution u(t) of (4.4) satisfies the following equivalent equation:

It follows from (4.6) that

Using Theorem 3.1, we obtain (4.5).