On a Generalized Retarded Integral Inequality with Two Variables
© W.-S. Wang and C.-X. Shen. 2008
Received: 16 November 2007
Accepted: 22 April 2008
Published: 5 May 2008
This paper improves Pachpatte's results on linear integral inequalities with two variables, and gives an estimation for a general form of nonlinear integral inequality with two variables. This paper does not require monotonicity of known functions. The result of this paper can be applied to discuss on boundedness and uniqueness for a integrodifferential equation.
for boundedness and uniqueness of solutions.
for all . Obviously, appears linearly in (1.1), but in our (1.3) it is generalized to nonlinear terms: and . Our strategy is to monotonize functions s with other two nondecreasing ones such that one has stronger monotonicity than the other. We apply our estimation to an integrodifferential equation, which looks similar to (1.2) but includes delays, and give boundedness and uniqueness of solutions.
2. Main Result
Throughout this paper, are given numbers. Let and . Consider inequality (1.3), where we suppose that is strictly increasing such that , and are nondecreasing, such that and , , and are given, and are functions satisfying and for all .
Furthermore, let which is also nondecreasing in for each fixed , and and satisfies .
Here denotes the domain of a function.
for all .
for all , where is defined by (2.8).
which proves the claimed (2.12).
for all . This proves (2.6).
for all . Letting , we obtain (2.6) because of continuity of in and continuity of , and . This completes the proof.
for all , where , and are supposed to be as in Theorem 2.1; , , , and are all continuous functions such that . Obviously, the estimation obtained in  cannot be applied to (3.2).
and , and are defined as in Theorem 2.1.
on , then every solution of (3.2) is bounded on .
Next, we give the condition of the uniqueness of solutions for (3.2).
on . Then, (3.2) has at most one solution on , where are defined as in Theorem 2.1.
This work is supported by the Scientific Research Fund of Guangxi Provincial Education Department (no. 200707MS112), the Natural Science Foundation (no. 2006N001), and the Applied Mathematics Key Discipline Foundation of Hechi College of China.
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