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About differential inequalities for nonlocal boundary value problems with impulsive delay equations

Alexander Domoshnitsky, Irina Volinsky (2015)

Mathematica Bohemica

We propose results about sign-constancy of Green's functions to impulsive nonlocal boundary value problems in a form of theorems about differential inequalities. One of the ideas of our approach is to construct Green's functions of boundary value problems for simple auxiliary differential equations with impulses. Careful analysis of these Green's functions allows us to get conclusions about the sign-constancy of Green's functions to given functional differential boundary value problems, using the...

Asymptotic behaviour of a two-dimensional differential system with a nonconstant delay under the conditions of instability

Josef Kalas, Josef Rebenda (2011)

Mathematica Bohemica

We present several results dealing with the asymptotic behaviour of a real two-dimensional system x ' ( t ) = 𝖠 ( t ) x ( t ) + k = 1 m 𝖡 k ( t ) x ( θ k ( t ) ) + h ( t , x ( t ) , x ( θ 1 ( t ) ) , , x ( θ m ( t ) ) ) with bounded nonconstant delays t - θ k ( t ) 0 satisfying lim t θ k ( t ) = , under the assumption of instability. Here 𝖠 , 𝖡 k and h are supposed to be matrix functions and a vector function, respectively. The conditions for the instable properties of solutions together with the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the real system considered to one equation with...

Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay

Josef Rebenda (2009)

Archivum Mathematicum

In this article, stability and asymptotic properties of solutions of a real two-dimensional system x ' ( t ) = 𝐀 ( t ) x ( t ) + 𝐁 ( t ) x ( τ ( t ) ) + 𝐡 ( t , x ( t ) , x ( τ ( t ) ) ) are studied, where 𝐀 , 𝐁 are matrix functions, 𝐡 is a vector function and τ ( t ) t is a nonconstant delay which is absolutely continuous and satisfies lim t τ ( t ) = . Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented.

Boundedness and stability in third order nonlinear differential equations with multiple deviating arguments

Moussadek Remili, Lynda D. Oudjedi (2016)

Archivum Mathematicum

In this paper, we establish some new sufficient conditions which guarantee the stability and boundedness of solutions of certain nonlinear and non autonomous differential equations of third order with delay. By defining appropriate Lyapunov function, we obtain some new results on the subject. By this work, we extend and improve some stability and boundedness results in the literature.

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