On Lyapunov-Type Inequalities for Two-Dimensional Nonlinear Partial Systems
© Lian-Ying Chen et al. 2010
Received: 27 February 2010
Accepted: 23 June 2010
Published: 11 July 2010
We establish a new Laypunov-type inequality for two nonlinear systems of partial differential equations and the discrete analogue is also established. As application, boundness of the two-dimensional Emden-Fowler-type equation is proved.
In this paper, we obtain new Lyapunov-type inequalities for the two-dimensional nonlinear system and discrete nonlinear system, respectively.
2. The Lyapunov-Type Integral Inequality for the Two-Dimensional Nonlinear System
The proof is complete.
This is just a new Lyapunov-type inequality which was given by Tiryaki et al. .
3. The Lyapunov-Type Discrete Inequality for the Two-Dimensional Nonlinear System
where denotes the forward difference operator for that is, and denotes the forward difference operator for that is, We shall assume the existence of nontrivial solution of the system (3.1), and furthermore, (3.1) satisfies the following assumptions ( ), ( ), and ( ):
This completes the proof.
This is just a new Lyapunov-type inequality which was given by Ünal et al. .
4. An application
This completes the proof.
This research is supported by National Natural Sciences Foundation of China (10971205). It is also partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P) and an HKU Seed Grant for Basic Research.
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