Abstract
In this article we discuss some new generalized nonlinear Gronwall-Bellman-Type integral inequalities with two variables, which include a non-constant term outside the integrals. We use our result to deal with the estimate on the solutions of partial differential equations with the initial and boundary conditions.
Mathematics Subject Classification 2000: 26D10; 26D15; 26D20; 34A40.
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1 Introduction
Various generalizations of Gronwall inequality [1, 2] are fundamental tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of differential equations, integral equations, and differential-integral equations. There are a lot of articles investigating its generalizations such as [3–23]. Recently, Pachpatte [19] provided the explicit estimations of following integral inequalities:
and
where c is a constant. Cheung [7] investigated the inequality
Agarwal et al. [3] obtained the explicit bounds to the solutions of the following retarded integral inequalities:
where c is a constant. Chen et al. [6] discussed the following inequalities:
where c is a constant.
In this article, motivated mainly by the works of Agarwal et al. [3] and Chen et al. [6], Cheung [7], Pachpatte [19], we discuss more general forms of following integral inequalities:
for (x, y) ∈ [x0, x1) × [y0, y1), where a(x, y), b(x, y) are nonnegative and nondecreasing functions in each variable. In inequalities (1.1)-(1.3), we generalized the constant c in [1, 5] to the function a(x,y), the function u(x) in [1] to the u(x,y) with two variables.
2 Main result
Throughout this article, x0, x1, y0, y1 ∈ ℝ are given numbers. I := [x0,x1), J := [y0,y1), Δ:= [x0,x1) × [y0,y1), ℝ+ := [0,∞). Consider (1.1)-(1.3), and suppose that
(H1) ψ ∈ C(ℝ+, ℝ+) is a strictly increasing function with ψ(0) = 0 and ψ(t) → ∞ as t → ∞;
(H2) a, b: Δ → (0, ∞) are nondecreasing in each variable;
(H3) w, ϕ, ϕ1, ϕ2 ∈ C(ℝ+,ℝ+) are nondecreasing with w(0) > 0, ϕ(r) > 0, ϕ1(r) > 0 and ϕ2(r) > 0 for r > 0;
(H4) α i ∈ C1(I,I) and β i ∈ C1(J,J) are nondecreasing such that α i (x) ≤ x, α i (x0) = x0, β i (y) ≤ y and β i (y0) = y0, i = 1, 2,..., n;
(H5) f i , g i ∈ C(Δ,ℝ+), i = 1,2,...,n.
Theorem 1. Suppose that (H1-H5) hold and u(x,y) is a nonnegative and continuous function on Δ satisfying (1.1). Then we have
for all (x,y) ∈ [x0,X1) × [y0,Y1), where
ψ-1, W-1 and Φ-1 denote the inverse function of ψ, W and Φ, respectively, and (X1,Y1) ∈ Δ is arbitrarily given on the boundary of the planar region
Proof. From assumption H2 and the inequality (1.1), we have
for all (x,y) ∈ [x0,X] × [y0,y1), where x0 ≤ X ≤ X1 is chosen arbitrarily. Define a function η(x, y) by the right-hand side of (2.7). Clearly, η(x, y) is a positive and nondecreasing function in each variable, η(x0,y) = a(X,y) > 0. Then, (2.7) is equivalent to
for all (x,y) ∈ [x0,X] × [y0,y1). By the fact that α i (x) ≤ x for x ∈ [x0,x1), β i (y) ≤ y for y ∈ [y0,y1),i = 1,2,...,n, and the monotonicity of w,ψ-1,η, we have for all (x,y) ∈ [x0,X] × [y0,y1),
From (2.9), we get
for all (x,y) ∈ [x0,X] × [y0,y1). Integrating (2.10) from x0 to x, by the definition of W in (2.2), we get for all (x,y) ∈ [x0,X] × [y0,y1),
where
Now, define a function Γ(x,y) by the right-hand side of (2.11). Clearly, Γ(x,y) is a positive and nondecreasing function in each variable, Γ(x0,y) = c(X, y) > 0. then, (2.11) is equivalent to
for all (x,y) ∈ [x0,X] × [y0,Y1), where Y1 is defined in (2.6). By the fact that α i (x) ≤ x for x ∈ [x0,x1), β i (y) ≤ y for y ∈ [y0,y1), i = 1, 2,...,n, and the monotonicity of ϕ, ψ-1, W-1, Γ, we have for all (x,y) ∈ [x0,X] × [y0,Y1),
From (2.14), we have for all (x,y) ∈ [x0,X] × [y0,Y1),
Integrating (2.15) from x0 to x, by the definition of Φ in (2.3), we get
for all (x,y) ∈ [x0,X] × [y0,Y1). From (2.12) and (2.16), we find
for all (x, y) ∈ [x0, X] × [y0, Y1). From (2.8), (2.13), and (2.17), we get
for all (x, y) ∈ [x0,X] × [y0,Y1). Let x = X, from (2.18), we observe that
for all (X, y) ∈ [x0, X1) × [y0, Y1), where X1 is defined by (2.6). Since X ∈ [x0, X1) is arbitrary, from (2.19), we get the required estimations
for all (x,y) ∈ [x0,X1) × [y0,Y1). Theorem 1 is proved.
Remark that Theorem 1 generalizes Theorem 2.1 in [3].
Theorem 2. Suppose that (H1-H5) hold and u(x,y) is a nonnegative and continuous function on Δ satisfying (1.2). Then
(i) if ϕ1(u) ≥ ϕ2(log(u)), we have
for all (x,y) ∈ [x0,X2) × [y0,Y2),
(ii) if ϕ1(u) < ϕ2(log(u)), we have
for all (x,y) ∈ [x0,X3) × [y0,Y3), where W is defined by (2.2) in Theorem 1,
j = 1, 2, ψ-1, W-1, and denote the inverse function of ψ, W, Ψ1 and Ψ2, respectively, (X2,Y2) is arbitrarily given on the boundary of the planar region
and (X3,Y3) is arbitrarily given on the boundary of the planar region
Proof. From the inequality (1.2), we have
for all (x,y) ∈ [x0,X] × [y0,y1), where x0 ≤ X ≤ X2 is chosen arbitrarily. Let Ξ(x,y) denote the right-hand side of (2.25), which is a positive and nondecreasing function in each variable, Ξ(x0,y) = a(X,y). Then, (2.25) is equivalent to u(x,y) ≤ ψ-1(Ξ(x,y)). By the fact that α i (x) ≤ x for x ∈ [x0, x1), β i (y) ≤ y for y ∈ [y0, y1), i = 1, 2,..., n, and the monotonicity of w,ψ-1,Ξ, we have for all (x,y) ∈ [x0,X] × [y0,y1),
for all (x,y) ∈ [x0,X] × [y0,y1). From (2.26), we have
for all (x,y) ∈ [x0,X] × [y0,y1). Integrating (2.27) from x0 to x, by the definition of W in (2.2), we get
for all (x,y) ∈ [x0,X] × [y0,y1).
When ϕ1(u) ≥ ϕ2(log(u)), from the inequality (2.28), we have
for all (x,y) ∈ [x0,X] × [y0,y1). Now, define a function Θ(x,y) by the right-hand side of (2.29). Clearly, Θ(x,y) is a positive and nondecreasing function in each variable, Θ(x0,y) = W(a(X,y)) > 0. Then, (2.29) is equivalent to
where Y2 is defined by (2.23). Differentiating Θ(x,y) in x for any fixed y ∈ [y0,Y2), we have
for all (x,y) ∈ [x0,X] × [y0,Y2). From (2.31), we have
for all (x,y) ∈ [x0,X] × [y0,Y2). Integrating (2.32) from x0 to x, by the definition of Ψ1 in (2.22), we obtain
From (2.30) and (2.33), we conclude
for all (x,y) ∈ [x0,X] × [y0,Y2). Let x = X, from (2.34), we get
Since X ∈ [x0,X2) is arbitrary, from the inequality (2.35), we obtain the required inequality in (2.20).
When ϕ1(u) ≤ ϕ2(log(u)), from the inequality (2.28), we have
for all (x,y) ∈ [x0,X] × [y0,y1), where x0 ≤ X ≤ X3. Similarly to the above process from (2.29) to (2.35), from (2.36), we obtain
Since X ∈ [x0,X3) is arbitrary, where X3 is defined by (2.24), from the inequality (2.37), we obtain the required inequality in (2.21).
Theorem 3. Suppose that (H1-H5) hold and that L, satisfy
for s, t, u, v ∈ ℝ+ with u > v ≥ 0. If u(x,y) is a nonnegative and continuous function on Δ satisfying (1.3), then we have
for all (x,y) ∈ [x0,X4) × [y0,Y4), where W is defined by (2.2),
ψ-1,W-1 and denote the inverse function of ψ, W and Ψ3, respectively, and (X4,Y4) ∈ Δ is arbitrarily given on the boundary of the planar region
Proof. From the inequality (1.3), we have
for all (x,y) ∈ [x0,X] × [y0,y1), where x0 ≤ X ≤ X4 is chosen arbitrarily. Let P(x,y) denote the right-hand side of (2.42), which is a positive and nondecreasing function in each variable, P(x0,y) = a(X,y). Similarly to the process in the proof of Theorem 2 from (2.25) to (2.28), we obtain
From the inequality (2.38) and (2.43), we get
for all (x, y) ∈ [x0,X] × [y0,y1). Similarly to the process in the proof of Theorem 2 from (2.29) to (2.35), we obtain
where Ψ3 is defined by (2.40). Since X ∈ [x0,X4) is arbitrary, where X4 is defined by (2.41), from the inequality (2.44), we obtain the required inequality in (2.39).
3 Applications to BVP
In this section we use our result to study certain properties of solution of the following boundary value problem (simply called BVP):
for x ∈ I,y ∈ J, where x0,y0,x1,y1 ∈ ℝ+ are constants, I := [x0,x1), J := [y0,y1), F ∈ C(I × J × ℝn,ℝ), ψ: ℝ → ℝ is strictly increasing on ℝ+ with ψ(0) = 0, |ψ(r)| = ψ(|r|) > 0, for |r| > 0 and ψ(t) → ∞ as t → ∞; functions α i ∈ C1(I,I);β i ∈ C1(J,J),i = 1,2,...,n are nondecreasing such that α i (x) ≤ x, β i (y) ≤ y,α i (x0) = x0, β i (y0) = y0; |a1| ∈ C1(I,ℝ+), |a2| ∈ C1(J,ℝ+) are both nondecreasing. Though this equation is similar to the equation discussed in Section 3 in [3], our results are more general than the results obtained in [3].
We first give an estimate for solutions of the BVP (3.1) so as to obtain a condition for boundedness.
Corollary 1. Consider BVP (3.1) and suppose that F ∈ C(I × J × ℝn,ℝ) satisfies
where f i ,g i ∈ C(I × J,ℝ+) and w,ϕ ∈ C(ℝ+,ℝ+) are nondecreasing such that w(u) > 0,ϕ(u) > 0 for u > 0. Then all solutions z(x,y) of BVP (3.1) have the estimate
for all (x,y) ∈ [x0,X1) × [y0,Y1), where
for all (x,y) ∈ [x0,X1) × [y0,Y1), where functions W, W-1, Φ, Φ-1 and real numbers X1, Y1 are given as in Theorem 1.
Proof. The equivalent integral equation of BVP (3.1) is
By (3.2) and (3.4), we get that
where a change of variables s1 = α i (s), t1 = β i (t),i = 1,2,...,n are made. Clearly, the inequality (3.5) is in the form of (1.1). Thus the estimate (3.3) of the solution z(x,y) in this corollary is obtained immediately by our Theorem 1.
Our Corollary 1 actually gives a condition of boundedness for solutions. Concretely, if
on [x0,X1) × [y0,Y1), then every solution z(x,y) of BVP (3.1) is bounded on [x0,X1) × [y0,Y1).
Next, we discuss the uniqueness of solutions for BVP (3.1).
Corollary 2. Consider BVP (3.1) and suppose that F ∈ C(I × J × ℝn,ℝ) satisfies
for all (x,y) ∈ I × J and u i , v i ∈ ℝ, i = 1, 2,..., n, where I = [x0, x1], J = [y0, y1] are two finite intervals, and f i ∈ C(I × J, ℝ+),i = 1,2,...,n. Then BVP (3.1) has at most one solution on I × J.
Proof. Assume that both z(x,y) and are solutions of BVP (3.1). From the equivalent integral Equations (3.4) and (3.6), we have
for all (x,y) ∈ I × J, where changes of variables s1 = α i (s), t1 = β i (t) are made, ε > 0 is an arbitrary small number. Clearly, the inequality (3.7) is in the form of (1.1). Suitably applying our Theorem 1 to (3.7), we get an estimate of the form (2.1) for all (x,y) ∈ I × J,
Letting ε → 0+, since is finite on finite intervals I and J, ψ is a strictly increasing function, from (3.8), we conclude that , implying that for all (x,y) ∈ I × J. The uniqueness is proved.
Remark Suppose that F ∈ C(I × J × ℝn,ℝ) in BVP (3.1) satisfies
By using Theorem 2, we can give an estimate for solutions of the BVP (3.1).
Suppose that F ∈ C(I × J × ℝn,ℝ) in BVP (3.1) satisfies
By using Theorem 3, we can give an estimate for solutions of the BVP (3.1) too.
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Acknowledgements
The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions. This research was supported by the National Natural Science Foundation of China (Project No. 11161018), Guangxi Natural Science Foundation (Project No. 0991265), the Key Project of Hechi University (2009YAZ-N001) and the Key Discipline of Applied Mathematics of Hechi University of China(200725).
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Wang, WS. Some generalized nonlinear retarded integral inequalities with applications. J Inequal Appl 2012, 31 (2012). https://doi.org/10.1186/1029-242X-2012-31
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DOI: https://doi.org/10.1186/1029-242X-2012-31