Several integral inequalities and their applications in nonlinear differential systems

In this paper, several new integral inequalities are presented, which are effective in dealing with the integro-differential inequalities whose variable exponents are greater than one. Compared with existing integral inequalities, those proposed here can be applied to more complicated differential equations. The notions of uniform Lipschitz stability are generalized and the relations between these notions are analyzed. Several sufficient conditions about uniform Lipschitz asymptotic stability of nonlinear systems are established by the proposed integral inequalities. These sufficiently conditions can be similarly generalized to linearly perturbed differential systems that appear in the literature. Finally, an example of uniform Lipschitz asymptotic stability of nonlinear differential systems is shown.