Nonlinear Retarded Integral Inequalities with Two Variables and Applications
© Wu-Sheng Wang et al. 2010
Received: 8 June 2010
Accepted: 6 October 2010
Published: 11 October 2010
We consider some new nonlinear retarded integral inequalities with two variables, which extend the results in the work of W.-S. Wang (2007), and the one in the work of Y.-H. kim (2009). These inequalities include not only a nonconstant term outside the integrals but also more than one distinct nonlinear integrals without assumption of monotonicity. Finally, we give some applications to the boundary value problem of a partial differential equation for boundedness and uniqueness.
The purpose of the present paper is to establish some new nonlinear retarded integral inequalities of Gronwall-Bellman type with two variables. We can demonstrate that inequalities (1.4), (1.5), and (1.7), considered in [9, 12, 16], respectively, can also be solved with our results. We also apply our results to study the boundedness and uniqueness of the solutions of the boundary value problem of a partial differential equation.
2. Main Result
Throughout this paper, denotes the set of real numbers, and are given numbers. , , are the subsets of and . For any , let denote the subset of . denotes the set of continuous differentiable functions of into .
Our inequality (2.1) not only includes a nonconstant term outside the integrals but also more than one distinct nonlinear integral without assumption of monotonicity. When , and , our inequality (2.1) reduces to (1.7) studied in . When , and our inequality (2.1) reduces to (1.5) studied in .
3. Proofs and Remarks
Proof of Theorem 2.1.
The proofs of Corollary 2.2, 2.3, 2.5 and Theorem 2.4 are similar to the argument in the proofs of Theorem 2.1 with appropriate modification. We omit the details here.
When and , Corollary 2.2 reduces to Theorem in .
When and , Corollary 2.3 reduces to Theorem of Wang .
When , Theorem 2.4 reduces to Theorem of Kim .
for all , where is defined as in Section 2, , are defined in , is a continuous and strictly increasing odd function on , satisfying , for , , , , and are nondecreasing continuous functions, and the ratio is also nondecreasing, and for .
In the following corollary, we firstly apply our result to discuss boundedness on the solution of problem (4.1).
Next, we discuss the uniqueness of the solutions of (4.1).
for and , where is defined as in Section 2, is a constant, , , are continuous nondecreasing with the nondecreasing ratio such that for all , and , , and is a strictly increasing odd function satisfying for all . Then, (4.1) has at most one solution on
The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions. This work is supported by the Natural Science Foundation of Guangxi Autonomous Region (0991265), the Scientfic Research Foundation of the Education Department of Guangxi Autonomous Region (200707MS112), the Key Discipline of Applied Mathematics (200725) and the Key Project (2009YAZ-N001) of Hechi University of China.
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