Integrodifferential Inequalities Arising in the Theory of Differential Equations

Abstract

The goal of this paper is to achieve some new results related to integrodifferential inequalities of one independent variable which can be applied as a study of qualitative and quantitative properties of solutions of some nonlinear integral equations.

1. Introduction

Integral and integrodifferential inequalities play a significant role in recent years by many authors [1–11], which provide an explicit bounds on the solutions of a class of differential and integral equations.

Lemma 1. Pachpatte (1995) studied the following useful integral inequality: Let , , and be nonnegative continuous functions defined on and and be positive constants. Iffor , wherefor , thenfor all .

2. Main Results

Here by using Lemma 1, we establish some new results in the form of integrodifferential inequalities instead of integral inequality.

Theorem 2. Let , , , and be nonnegative real valued continuous functions defined for . Let and be positive constants. Iffor all , thenwhere .
, , and are defined as in (2), (3), and (4), respectively, for all .

Proof. Define a function by the right-hand side of (6), such thatwhereFrom (6) and (8), we getBy differentiating (8) and using the fact thatwe observeorLetDifferentiating (14) with respect to , we getwhereBy substituting (14) and (15) in (13), we haveInequality (17) implies the estimation for and by using (16), we observe thatwhere and are defined as in (3) and (4) and by applying (11) and (14) it is noticed thatwhere is defined as in (2). This completes the proof.

Theorem 3. Let , , , , , and be defined as in Theorem 2 for . Iffor all , thenwhere andfor all .

Proof. Define a function by the right-hand side of (20), such thatwhereFrom (20) and (24), we getBy differentiating (24) and since is monotone nondecreasing function for and using the fact thatwe observe thatorLetBy repeating the same steps from (14)–(18) in (29) with suitable modifications, the estimation for impliesFrom (27) and (30) in (31), we getfor all , where , , and are defined as in (3), (22), and (23), respectively. This completes the proof.

Theorem 4. Let , , , , , and be defined as in Theorem 2 for . Iffor all , thenwhere and .for all .

Proof. Define a function by the right-hand side of (33), such thatwhereFrom (33) and (37), we getorBy differentiating (37) and since is monotone nondecreasing function for , we observe thatorLetBy repeating the same steps from (14)–(18) in (42) with suitable modifications, the estimation for impliesFrom (40) and (43) in (44), we getfor all , where , , and are defined as in (3), (35), and (36), respectively. This completes the proof.

Theorem 5. Let , , , , , and be defined as in Theorem 2 for . Iffor all , thenwhere and for all .

Proof. Define a function by the right-hand side of (46), such thatwhereFrom (46) and (50), we getorBy differentiating (50) and since is monotone nondecreasing function for , we observe thatorLetBy repeating the same steps from (14)–(18) in (55) with suitable modifications, the estimation for impliesFrom (53) and (56) in (57), we getfor all , where , , and are defined as in (3), (48), and (49), respectively. This completes the proof.

Theorem 6. Let , , , , , and be defined as in Theorem 2 for . Iffor all , thenwhere , , and .for all .

Proof. Define a function by the right-hand side of (59), such thatwhereFrom (59) and (62), we getorBy differentiating (62) and since is monotone nondecreasing function for , we observe thatorLetBy repeating the same steps from (14)–(18) in (67) with suitable modifications, the estimation for impliesFrom (65) and (68) in (69), we getfor all , where and are defined as in (3) and (61), respectively. This completes the proof.

3. Application

As an application, the explicit bounds of some of the integral inequalities can be found by some examples.

Example 1. Let us consider the explicit bound on the solution of the nonlinear integrodifferential equationwhere is a nonnegative real valued continuous function and every solution of of (71) exists for .
By using the application of Theorem 4 to (71), we observe thatwhereTherefore the right-hand side of (74) provides the bound of the solution of (75) of known quantitiesfor .

Example 2. Let us consider the nonlinear integrodifferential equation of the formwhere is a nonnegative real valued continuous function and every solution of of (77) exists for .
By using the application of Theorem 5 to (77), we observe thatwhereTherefore the right-hand side of (80) provides the bound of the solution of (77) of known quantitiesfor .

Example 3. Now let us consider the boundedness and asymptotic behaviour of the solutions of nonlinear Volterra integrodifferential inequality of the form is nonnegative real valued continuous function defined on and , , , are real valued continuous function defined on .
We assume that every solution of in (81) exists on , and and are defined as in Theorem 6. Define the following hypotheses on the function of (81) asAlsofor all , . , , , are nonnegative real constants and , are nonnegative real valued continuous function defined on .

Proof. For the boundedness of the solution of nonlinear integrodifferential equation (83), let us suppose that the hypotheses (84), (85), and (86) are satisfied and let be a solution of (83); then we observe thatReplacing by and by and applying the same proof with some modifications of Theorem 6 in (88) and with and being the same as defined in Theorem 6, we noticed that every solution of of (88) that exists on is bounded; that is,For the asymptotic behaviour of the solution of nonlinear integrodifferential equation (83), assume the following hypothesesAlsoare satisfied. Let be a solution of (83); thenLet be for in (92) and by applying the same proof with some changes of Theorem 6 in (92), we getTherefore the solution of (83) is asymptotically stable.

Competing Interests

There is no conflict of interests regarding the publication of this paper.