A Generalization of Linear and Nonlinear Retarded Integral Inequalities in Two Independent Variables.

1. Introduction

With the development of science and technology, various inequalities have been paid more and more attention, and the generalization of inequalities has become one of the important research directions in modern mathematics. The integral inequality, which has integrals of unknown functions, is an important type of inequality. For nonlinear differential equations derived from the natural science and engineering technology, especially from various branches of mathematics, it is difficult or impossible to obtain explicit solutions in most cases. Therefore, it is of great significance to get the bounds of the solutions to those nonlinear differential equations. Integral inequalities just can provide the bounds of the solutions to the nonlinear differential equations and integral equations. Hence, integral inequalities are used to serve as handy tools in the study of the qualitative properties of solutions to differential and integral equations, such as existence, uniqueness, boundedness, oscillation, stability, and invariant manifold. For example, these inequalities have been widely employed to investigate the stability of switched systems which can be applied to modeling many engineering system problems in real world, such as traffic control, automobile engine control, switching power converters, and multiagent consensus [1-5]. For some related contributions on various classes of integral inequalities, we refer the reader to [1-20] and the references cited therein.

For convenience, throughout this paper, R represents the set of real numbers, [R.sub.+] = [0, [infinity]), and C(A, B) signifies the class of all continuous functions defined on set A with range in the set B.

In what follows, we provide some background details that motivated our study. One of the most famous and widespread integral inequalities in the study of differential and integral equations is Gronwall-Bellman-type inequality [6-8], which can be described as follows.

Theorem 1. Let u and f be nonnegative continuous functions on an interval [a, b] satisfying

u(t) [less than or equal to] c + [[integral].sup.t.sub.a] f(s) u(s) ds, t [member of] [a, b] (1)

for some constant c [greater than or equal to] 0. Then

u(t) [less than or equal to] c exp ([[integral].sup.t.sub.a] f(s) u(s) ds), t [member of] [a, b] (2)

In recent years, many scholars have done a lot of researches and generalization of the above integral inequality, which make the integral inequalities develop continually and the application fields expand gradually. Pachpatte [9, 10] investigated the inequality

[mathematical expression not reproducible] (3)

and the retarded inequality

[mathematical expression not reproducible], (4)

where [alpha] [member of] [C.sup.1](I, I) is nondecreasing with [alpha](t) [less than or equal to] t on I = [0, T) and [u.sub.1] and [u.sub.2] are constants. Abdeldaim and El-Deeb  generalized  and analyzed the following retarded linear and nonlinear inequalities:

[mathematical expression not reproducible], (5)

respectively. Tian et al.  introduced the retarded inequalities in two independent variables as follows.

Theorem 2 (see [16, Theorem 1]). Let u, f, and g [member of] C([R.sub.+] x [R.sub.+], [R.sub.+]), a(x) > 0, b(y) > 0, a'(x) [greater than or equal to] 0, b'(y) [greater than or equal to] 0, and [alpha], [beta] [member of] [C.sup.1]([R.sub.+], [R.sub.+]) be nondecreasing with [alpha](x) [less than or equal to] x and [beta](y) [less than or equal to] y on [R.sub.+]. Moreover, let [phi] [member of] [C.sup.1]([R.sub.+], [R.sub.+]) bean increasing function with [phi]([infinity]) = [infinity] and let [phi](x) > 0 on (0, [infinity]), [psi] [member of] [C.sup.1]([R.sub.+], [R.sub.+]) be a nondecreasing function with [psi](x) > 0 on (0, [infinity]). If

[mathematical expression not reproducible], (6)

then, for 0 [less than or equal to] x < [[xi].sub.1], 0 [less than or equal to] y < [[eta].sub.1],

[mathematical expression not reproducible], (7)

where

[mathematical expression not reproducible], (8)

[[OMEGA].sup.-1], [[phi].sup.-1], and [G.sup.-1] are the inverses of [OMEGA], [phi], and G, respectively; ([[xi].sub.1], [[eta].sub.1]) [member of] [R.sub.+] x [R.sub.+] is chosen so that

[mathematical expression not reproducible] (9)

with dom(*) denoting the function domain.

Theorem 3 (see [16, Corollary 1]). Assume that u, f, g, a, b, [alpha], and [beta] are defined as in Theorem 2. Let [phi](u) = [u.sup.p] and [psi](u) = [u.sup.q-1] in Theorem 2, where p [greater than or equal to] q > 1 are positive constants. If

[mathematical expression not reproducible] (10)

then, for all (x, y) [member of] [R.sub.+] x [R.sub.+],

[mathematical expression not reproducible], (11)

where

[mathematical expression not reproducible]. (12)

Motivated by the recent contributions of Abdeldaim and El-Deeb , Zhang and Meng , and Tian et al. , our principal goal is to extend the inequalities with one variable in  to those with two variables which include Theorems 2 and 3 as special cases.

The rest of the work is organized as follows. A useful lemma that plays a fundamental role in the proofs of the main theorems is presented in Section 2. In Section 3, we propose our main theorems and corollary on several new types of linear and nonlinear retarded integral inequalities in two independent variables. An illustrative example is given to indicate the usefulness of these inequalities in Section 4, which is followed by a short conclusion in Section 5.

2. Lemma

The subsequent lemma is helpful in proving our main theorems.

Lemma 4. Assume that u, f, and g [member of] C([R.sub.+] x [R.sub.+], [R.sub.+]) and [phi] [member of] C([R.sub.+], [R.sub.+]) is an increasing function with [phi]([infinity]) = [infinity] and [psi] [member of] C([R.sub.+], [R.sub.+]) is a nondecreasing function. Suppose that c is a nonnegative constant and [alpha], [beta] [member of] [C.sup.1] ([R.sub.+], [R.sub.+]) are nondecreasing with [alpha](x) [less than or equal to] x, [beta](y) [less than or equal to] y, [alpha](0) = 0, and [beta](0) = 0 on [R.sub.+]. If

[mathematical expression not reproducible] (13)

then, for 0 [less than or equal to] x < [xi], 0 [less than or equal to] y < [eta],

[mathematical expression not reproducible], (14)

where

[mathematical expression not reproducible]; (15)

[[phi].sup.-1] and [G.sup.-1] are the inverses of [phi] and G, respectively; ([xi], [eta]) [member of] [R.sub.+] x [R.sub.+] is chosen so that

[mathematical expression not reproducible]. (16)

Proof. Define the nondecreasing positive function z by

[mathematical expression not reproducible], (17)

where [epsilon] is an arbitrary small positive number. Utilizing inequality (13) and the monotonicity of [[phi].sup.-1], we get

u(x, y) [less than or equal to] [[phi].sup.-1] (z(x, y)). (18)

Differentiating (17) with respect to x and combining (18) and the monotonicities of [[phi].sup.-1], z, and [psi], we conclude that

[mathematical expression not reproducible]. (19)

On account of [psi][[phi].sup.-1] (z(x, y))] [greater than or equal to] [psi][[[phi].sup.-1] (c+[epsilon])] > 0, we deduce that

[mathematical expression not reproducible]. (20)

Integrating the latter inequality on [0, x] and letting [epsilon] [right arrow] 0, we have

[mathematical expression not reproducible] (21)

owing to (15). By virtue of (16), (18), and the last inequality, we obtain inequality (14). The proof is complete.

Remark 5. Assume that [mathematical expression not reproducible]. Then G([infinity]) = [infinity] and (14) is valid on [R.sub.+] x [R.sub.+]; that is, one can select [xi] = [infinity] and [eta] = [infinity].

3. Main Results

The following are the main results of this paper.

Theorem 6. Let u, a, f, g, and h [member of] C([R.sub.+] x [R.sub.+], [R.sub.+]) and let [alpha], [beta] [member of] [C.sup.1]([R.sub.+], [R.sub.+]) be nondecreasing with [alpha](x) [less than or equal to] x, [beta](y) [less than or equal to] y, [alpha](0) = 0, and [beta](0) = 0 on [R.sub.+]. If the inequality

[mathematical expression not reproducible] (22)

holds, for all (x, y) [member of] [R.sub.+] x [R.sub.+], then

[mathematical expression not reproducible]. (23)

Proof. Letting

[mathematical expression not reproducible]. (24)

then z(0, y) = z(x, 0) = 0 and

u (x, y) [less than or equal to] [alpha] (x, y) + z (x, y). (25)

Our assumptions on f, u, h, g, [alpha], and [beta] indicate that z is a positive function which is nondecreasing with respect to each of the two variables. Differentiating z with respect to x and using (25), we arrive at

[mathematical expression not reproducible]. (26)

By virtue of the monotonicity of z, we get

[mathematical expression not reproducible]. (27)

Multiplying the latter inequality by [mathematical expression not reproducible] yields

[mathematical expression not reproducible]. (28)

Integrating this inequality on [0, x], we deduce that

[mathematical expression not reproducible]. (29)

Combining (25) with (29), we get inequality (23). This completes the proof.

Theorem 7. Let u, a, f, g, and h [member of] C([R.sub.+] x [R.sub.+], [R.sub.+]), a(x, y) > 0, [a.sub.x] [greater than or equal to] 0, [a.sub.y] [greater than or equal to] 0, and [alpha], [beta] [member of] [C.sup.1] ([R.sub.+], [R.sub.+]) be nondecreasing with [alpha](x) [less than or equal to] x, [beta](y) [less than or equal to] y, [alpha](0) = 0, and [beta](0) = 0 on [R.sub.+]. Moreover, let [gamma] and [psi] [member of] [C.sup.1]([R.sub.+], [R.sub.+]) be nondecreasing function with [gamma] > 0 and [psi] > 0 on (0, [infinity]). If

[mathematical expression not reproducible], (30)

then, for 0 [less than or equal to] x < [xi], 0 [less than or equal to] y < [eta],

[mathematical expression not reproducible], (31)

where

[mathematical expression not reproducible], (32)

[mathematical expression not reproducible], (33)

[mathematical expression not reproducible]. (34)

[[OMEGA].sup.-1] and [G.sup.-1] are the inverses of [OMEGA] and G, respectively; ([xi], [eta]) [member of] [R.sub.+] x [R.sub.+] is chosen so that

[mathematical expression not reproducible] (35)

for 0 [less than or equal to] x < [xi], 0 [less than or equal to] y < [eta].

Proof. Define the nondecreasing function z by

[mathematical expression not reproducible]. (36)

Then

u (x, y) [less than or equal to] a (x, y) + z (x, y). (37)

Differentiating (36) and using (37) and the monotonicity of [gamma], we obtain

[mathematical expression not reproducible]. (38)

Let [T.sub.1] [less than or equal to] [xi] and [T.sub.2] [less than or equal to] [eta] be arbitrary numbers. Utilizing (38) and the monotonicities of a, z, and y, we get that, for 0 [less than or equal to] x < [T.sub.1] and 0 [less than or equal to] y < [T.sub.2],

[mathematical expression not reproducible]. (39)

For a([T.sub.1], [T.sub.2]) > 0 and [gamma][a([T.sub.1], [T.sub.2]) + z(x, y)] > 0,

[mathematical expression not reproducible]. (40)

+ h(a(x),[beta](y))+f(a(x),[beta](y))

From another point of view,

[mathematical expression not reproducible]. (41)

It follows from (40) and (41) that

[mathematical expression not reproducible]. (42)

Integrating the above inequality on [0, y] with respect to the second variable and taking [z.sub.x](x, 0) = 0 into account, we have

[mathematical expression not reproducible]. (43)

From (33), the latter relation gives

[mathematical expression not reproducible]. (44)

Integrating the last inequality over [0, x], we get

[mathematical expression not reproducible], (45)

where P is defined as in (32). Combining (37) and the monotonicity of a and employing Lemma 4, we obtain

[mathematical expression not reproducible], (46)

where G is defined as in (34). Taking x = [T.sub.1] and y = [T.sub.2], we conclude that

[mathematical expression not reproducible]. (47)

As [T.sub.1] [less than or equal to] [xi] and [T.sub.2] [less than or equal to] [eta] are arbitrary, we get the desired inequality (31). The proof is complete.

Theorem 8. Assume that u, a, f, g, h, [alpha], [beta], [gamma], and [psi] are defined as in Theorem 7. Moreover, let [phi] [member of] [C.sup.1]([R.sub.+], [R.sub.+]) be increasing function with f(x) = x and <p(x) > 0 on (0, x). If

[mathematical expression not reproducible], (48)

then, for 0 [less than or equal to] x < [xi], 0 [less than or equal to] y [less than or equal to] [eta],

[mathematical expression not reproducible], (49)

where

[mathematical expression not reproducible]; (50)

[[phi].sup.-1], [[OMEGA].sup.-1], and [G.sup.-1] are the inverses of [phi], [OMEGA], and G, respectively; ([xi], [eta]) [member of] [R.sub.+] x [R.sub.+] is chosen so that

[mathematical expression not reproducible] (51)

for 0 [less than or equal to] x < [xi], 0 [less than or equal to] y < [eta].

Proof. Define function z by (36). Then

u (x, y) [less than or equal to] [[phi].sup.-1] [a (x, y) + z (x, y)]. (52)

The rest of the proof is similar to that of Theorem 7 and hence is omitted.

Remark 9. Letting a(x, y) = a(x) + b(y), [gamma](u(x, y)) = u(x, y), and g(x, y) = 0 in Theorem 8, Theorem 8 turns out to be Theorem 2. Therefore, the inequality established in Theorem 8 generalizes that of [16, Theorem 1].

If [phi](u) = [u.sup.p], [gamma](u) = [u.sup.q], and [psi](u) = [u.sup.n] in Theorem 8, where p [greater than or equal to] q + n > l, and p, q, and n are positive constants, then we have the following corollary.

Corollary 10. Assume that u, a, f, g, h, [alpha], and [beta] are defined as in Theorem 8. If

[mathematical expression not reproducible], (53)

then, for all (x, y) [member of] [R.sub.+] x [R.sub.+],

[mathematical expression not reproducible], (54)

where

[mathematical expression not reproducible]. (55)

Proof. Assume that p > q + n and let [phi](u) = [u.sup.p], [gamma](u) = [u.sup.q], and [psi](u) = [u.sup.n]. Then we have [[phi].sup.-1](u) = [u.sup.1/p], and so

[mathematical expression not reproducible], (56)

where [lambda] and [theta] are defined in (55). Using Theorem 8, one can easily obtain

[mathematical expression not reproducible]. (57)

When p = q + n,

[mathematical expression not reproducible], (58)

where [lambda] and [theta] are the same as in (55). By Theorem 8, similar discussions can give

[mathematical expression not reproducible]. (59)

This completes the proof.

Remark 11. Letting a(x, y) = a(x) + b(y), q = 1, n = q-1, and g(x, y) = 0, Corollary 10 reduces to Theorem 3. Hence, the inequality established in Corollary 10 includes the result of [16, Corollary 1].

4. Example

Example 1. Consider the integral equation

[mathematical expression not reproducible], (60)

where k : [R.sub.+] x [R.sub.+] [right arrow] R and F : [R.sub.+] x [R.sub.+] x R x R [right arrow] R are continuous functions, [alpha], [beta] [member of] [C.sup.1]([R.sub.+], [R.sub.+]) is nondecreasing with [alpha](x) [less than or equal to] x, [beta](y) [less than or equal to] y, [alpha](0) = 0, and [beta](0) = 0 on [R.sub.+], and p [greater than or equal to] 4 is a constant. Suppose that

[mathematical expression not reproducible], (61)

where a, f, h, and g are defined as in Corollary 10. Combining (60)-(61) yields

[mathematical expression not reproducible]. (62)

Exploiting Corollary 10, we obtain an explicit bound to the solutions of (60):

[mathematical expression not reproducible], (63)

where [lambda] and [theta] are defined as in Corollary 10.

5. Conclusions

This paper investigates some new types of linear and nonlinear retarded integral inequalities in two independent variables. Several theorems and a corollary of these inequalities are obtained based on some analysis techniques, such as amplification method, differential, and integration. An illustrative example is studied to demonstrate the effectiveness of the new results.

https://doi.org/10.1155/2017/5129051

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants 61304130 and 51277116).

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Ying Jiang, Guojing Xing, and Chenghui Zhang

School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

Correspondence should be addressed to Guojing Xing; xgjsdu@sdu.edu.cn

Received 26 August 2017; Accepted 5 December 2017; Published 25 December 2017

Academic Editor: Eric R. Kaufmann
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