Generalization of integral inequalities and (c₁, c₁) stability of neutral differential equations with time-varying delays

A uniform stability analysis is developed for a type of neutral delays differential equations which depend on more general nonlinear integral inequalities. Many original investigations and results are obtained. Firstly, generations of two integral nonlinear inequalities are presented, which are very effective in dealing with the complicated integro-differential inequalities whose variable exponents are greater than zero. Compared with existed integral inequalities, those proposed here can be applied to more complicated differential equations, such as time-varying delay neutral differential equations. Secondly, the notions of (ω, Ω) uniform stable and (ω, Ω) uniform asymptotically stable, especially (c₁, c₁) uniform stable and (c₁, c₁) uniform asymptotically stable, are presented. Sufficient conditions on about (c₁, c₁) uniform stable and (c₁, c₁) uniform asymptotically stable of time-varying delay neutral differential equations are established by the proposed integral inequalities. Finally, a complex numerical example is presented to illustrate the main results effectively. The above work allows to provide further applications on the proposed stability analysis and control system design for different nonlinear systems.