Lyapunov-type Inequalities For Higher-order Nonlinear Systems

In this paper,we mainly consider the Wirtinger-type inequalities and Lyapunov-type inequalities.In chapter 1,we introduce the importance of inequalities.In chapter 2,on the basis of[4]and[5],we defined a new differential operator,by using Picone-type differential identity,we can obtain some new generalized integral inequalities of the Wirtinger type.In chapter 3,we established the new Lyapunov-type inequalities for the following ⁽nonlinear system（3.1.8）,where p,q>1,1/p+1/q=1,A,B,C:R→R^n×nand B^T（t）=B（t）,C^T（t）=C（t）.We will establish some sufficient conditions such that system（3.1.8）has no solution（x（t）,y（t））satisfying condition（?）.Moreover,these conditions are actually Lyapunov-type inequalities.We can establish some criteria for the non-existence of homoclinc solutions of system（3.1.8）by using these necessary conditions.In chapter 4,we set up the new Lyapunov-type inequalities for the following nonlinear difference system（?）（4.1.5）,where p,q>1,1/p+1/q=1,x,y are k×1 vectors,A,B,C are k×k matrices and B^T（n）=B（n）,C^T（n）=C（n）.Similarity,we will establish some sufficient conditions such that system（4.1.5）has no solution（x（n）,y（n））satisfying condition （?）.Hence,we can establish some criteria for the non-existence of homoclinc solutions of system（4.1.5）by using these necessary conditions.