Abstract
The main objective of this paper is to establish some new retarded nonlinear sum-difference inequalities with two independent variables, which provide explicit bounds on unknown functions. These inequalities given here can be used as handy tools in the study of boundary value problems in partial difference equations.
2000 Mathematics Subject Classification: 26D10; 26D15; 26D20.
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1 Introduction
Being important tools in the study of differential, integral, and integro-differential equations, various generalizations of Gronwall inequality [1, 2] and their applications have attracted great interests of many mathematicians (cf. [3–16], and the references cited therein). Recently, Agarwal et al. [3] studied the inequality
Cheung [17] investigated the inequality
Agarwal et al. [18] obtained explicit bounds to the solutions of the following retarded integral inequalities:
where c is a constant, and Chen et al. [19] did the same for the following inequalities:
where c is a constant.
Along with the development of the theory of integral inequalities and the theory of difference equations, more attentions are drawn to some discrete versions of Gronwall type inequalities (e.g., [20–22] for some early works). Some recent works can be found, e.g., in [6, 23–25] and some references therein. Found in [26], the unknown function u in the fundamental form of sum-difference inequality
can be estimated by . In [6], the inequality of two variables
was discussed, and the result was generalized in [23] to the inequality
In this paper, motivated mainly by the works of Cheung [17, 23], Agarwal et al. [3, 18], and Chen et al. [19], we shall discuss upper bounds of the function u(m, n) satisfying one of the following general sum-difference inequalities
for (m, n) ∈ [m 0, m 1) ∩ ℕ+ × [n 0, n 1) ∩ ℕ+, where a(m, n), b(m, n) are nonnegative and nonde-creasing functions in each variable. Inequalities (1.1), (1.2), and (1.3) are the discrete versions of Agarwal et al. [18] and Chen et al. [19]. They not only generalized the forms with one variable into the ones with two variables but also extended the constant 'c' out of integral into a function 'a(m, n)'. These inequalities will play an important part in the study on difference equation. To illustrate the action of their inequalities, we also gave an example of boundary value problem in partial difference equation.
2 Main result
Throughout this paper, k, m 0, m 1, n 0, n 1 are fixed natural numbers. ℕ+ := {1, 2, 3, . . .}, I := [m 0, m 1] ∩ ℕ+, I m := [m 0, m] ∩ ℕ+, J := [n 0, n 1] ∩ ℕ+, J n := [n 0, n] ∩ ℕ+, ℝ+ := [0, ∞). For functions s(m), z(m, n), m, n ∈ ℕ, their first-order (forward) differences are defined by Δs(m) = s(m + 1) - s(m), Δ1 z(m, n) = z(m + 1, n) - z(m, n) and Δ2 z(m, n) = z(m, n + 1) - z(m, n). Obviously, the linear difference equation Δx(m) = b(m) with initial condition x(m 0) = 0 has solution . For convenience, in the sequel, we define . We make the following assumptions:
(H 1) ψ ∈ C(ℝ+, ℝ+) is strictly increasing with ψ(0) = 0 and ψ (t) → ∞ as t → ∞;
(H 2) a, b : I × J → (0, ∞) are nondecreasing in each variable;
(H 3) w, φ, φ 1, φ 2 ∈ C(ℝ+,ℝ+) are nondecreasing with w(0) > 0, φ(r) > 0, φ 1(r) > 0 and φ 2(r) > 0 for r > 0;
(H 4) α i : I → I and β i : J → J are nondecreasing with α i (m) ≤ m and β i (n) ≤ n, i = 1, 2, . . . , k;
(H 5) f i , g i : I × J → ℝ+, i = 1, 2, . . . , k.
Theorem 1. Suppose (H 1 - H 5) hold and u(m, n) is a nonnegative function on I × J satisfying (1.1). Then, we have
for all , where
and (M 1, N 1) ∈ I × J is arbitrarily chosen such that
Proof. From assumption (H 2) and the inequality (1.1), we have
for all (m, n) ∈ I M × J, where m 0 ≤ M ≤ M 1 is a natural number chosen arbitrarily. Define a function η(m, n) by the right-hand side of (2.6). Clearly, η(m, n) is positive and nondecreasing in each variable, with η(m 0, n) = a(M, n) > 0. Hence (2.6) is equivalent to
for all (m, n) ∈ I M × J. By (H 4) and the monotonicity of w, ψ -1 and η, we have, for all (m, n) ∈ I M × J,
On the other hand, by the monotonicity of w and ψ -1,
From (2.8) and (2.9), we have
for (m, n), (m + 1, n) ∈ I M × J. Keeping n fixed and substituting m with s in (2.10), and then summing up both sides over s from m 0 to m - 1, we get
for (m, n) ∈ I M × J, where
Now, define a function Γ(m, n) by the right-hand side of (2.11). Clearly, Γ(m, n) is positive and nondecreasing in each variable, with Γ(m 0, n) = c(M, n) > 0. Hence (2.11) is equivalent to
for all , where N 1 is defined in (2.5). By (H 4) and the monotonicity of φ, ψ -1, W -1 and Γ , we have, for all ,
On the other hand, by the monotonicity of φ, ψ -1, and W -1, we have
From (2.14) and (2.15), we obtain
for . Keeping n fixed and substituting m with s in (2.16), and then summing up both sides over s from m 0 to m - 1, we get
for . From (2.12) and (2.17), we have
From (2.7), (2.13), and (2.18), we get
for . Let m = M, from (2.20), we observe that
for all , where M 1 is defined by (2.5). Since is arbitrary, from (2.21), we get the required estimate
for all . Theorem 1 is proved.
Theorem 2. Suppose (H 1 - H 5) hold and u(m, n) is a nonnegative function on I × J satisfying (1.2). Then
(i) if φ 1(u) ≥ φ 2(log u), we have
for all ,
(ii) if φ 1(u) ≤ φ 2(log u), we have
for all , where
j = 1, 2; (M 2, N 2) is arbitrarily given on the boundary of the planar region
and (M 3, N 3) is arbitrarily given on the boundary of the planar region
Proof. (i) When φ 1(u) ≥ φ 2(log u), from inequality (1.2), we have
for all (m, n) ∈ I M × J, where m 0 ≤ M ≤ M 2 is chosen arbitrarily. Let Ξ(m, n) denote the right-hand side of (2.27), which is a positive and nondecreasing function in each variable with Ξ (m 0, n) = a(M, n). Hence (2.27) is equivalent to
By (H 4) and the monotonicity of w, ψ -1, and Ξ, we have, for all (m, n) ∈ I M × J,
for all (m, n) ∈ I M × J. Similar to the process from (2.9) to (2.11), we obtain
for all (m, n) ∈ I M × J. Now, define a function Θ(m, n) by the right-hand side of (2.30). Clearly, Θ(m, n) is positive and nondecreasing in each variable, with Θ(m 0, n) = W (a(M, n)) > 0. Thus, (2.30) is equivalent to
where N 2 is defined by (2.25). Similar to the process from (2.14) to (2.18), we obtain
for all . From (2.28), (2.31), and (2.32), we conclude that
for all . Let m = M , from (2.33), we get
Since is arbitrary, from inequality (2.34), we obtain the required inequality in (2.22).
(ii) When φ 1(u) ≤ φ 2(log u), similar to the process from (2.27) to (2.30), from inequality (1.2), we have
for all , where M 3 is defined in (2.26). Similar to the process from (2.30) to (2.34), we obtain
Since is arbitrary, from inequality (2.36), we obtain the required inequality in (2.23).
Theorem 3. Suppose (H 1 - H 5) hold and that L, satisfy
for s, t, u, v ∈ ℝ+ with u > v ≥ 0. If u(m, n) is a nonnegative function on I × J satisfying (1.3) then we have
for all , where W is defined by (2.2),
and (M 4, N 4) ∈ I × J is arbitrarily given on the boundary of the planar region
Proof. From inequality (1.3), we have
for all (m, n) ∈ I M × J, where m 0 ≤ M ≤ M 4 is chosen arbitrarily. Let P (m, n) denote the right-hand side of (2.41), which is a positive and nondecreasing function in each variable, with P(m 0, n) = a(M, n). Similar to the process in the proof of Theorem 2 from (2.27) to (2.30), we obtain
for all (m, n) ∈ I M × J. From inequality (2.37) and (2.42), we get
for all (m, n) ∈ I M × J. Similar to the process in the proof of Theorem 2 from (2.30) to (2.34), we obtain
Since is arbitrary, where M 4 is defined in (2.40), from inequality (2.43), we obtain the required inequality in (2.38).
3 Applications to BVP
In this section, we use our result to study certain properties of the solutions of the following boundary value problem (BVP):
for m ∈ I, n ∈ J, where m 0, n 0, m 1, n 1 ∈ ℝ+ are constants, I := [m 0, m 1] ∩ ℕ+, J := [n 0, n 1] ∩ ℕ+, F : I × J × ℝ k → ℝ, ψ : ℝ → ℝ is strictly increasing on ℝ+ with ψ(0) = 0, |ψ(r)| = ψ(|r|), and ψ(t) → ∞ as t → ∞; functions α i : I → I and β i : J → J are nondecreasing such that α i (m) ≤ m and β i (n) ≤ n, i = 1, 2, . . . , k; |a 1| : I → ℝ+, |a 2| : J → ℝ+ are both nondecreasing.
We give an upper bound estimate for solutions of BVP (3.1).
Corollary 1. Consider BVP (3.1) and suppose that F satisfies
where f i , g i : I × J → ℝ+ and w, φ ∈ C 0(ℝ+, ℝ+) are nondecreasing with w(u) > 0, φ(u) > 0 for u > 0. Then, all solutions z(m, n) of BVP (3.1) satisfy
for all , where
for all , with W, W -1, Φ, Φ-1 and M 1, N 1 as given in Theorem 1.
Proof. BVP (3.1) is equivalent to
By (3.2) and (3.5), we get
Clearly, inequality (3.6) is in the form of (1.1). Thus the estimate (3.3) of the solution z(m, n) follows immediately from Theorem 1.
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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 National Natural Science Foundation of China(Project No. 11161018), Guangxi Natural Science Foundation(Project No. 0991265), and the Research Grants Council of the Hong Kong SAR, Project No. HKU7016/07P.
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Wang, WS., Li, Z. & Cheung, WS. Some new nonlinear retarded sum-difference inequalities with applications. Adv Differ Equ 2011, 41 (2011). https://doi.org/10.1186/1687-1847-2011-41
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DOI: https://doi.org/10.1186/1687-1847-2011-41