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Equivalence and reduction of delay-differential systems

Mohamed Boudellioua (2007)

International Journal of Applied Mathematics and Computer Science

A new direct method is presented which reduces a given high-order representation of a control system with delays to a first-order form that is encountered in the study of neutral delay-differential systems. Using the polynomial system description (PMD) setting due to Rosenbrock, it is shown that the transformation connecting the original PMD with the first-order form is Fuhrmann's strict system equivalence. This type of system equivalence leaves the transfer function and other relevant structural...

Existence and controllability results for semilinear neutral functional differential inclusions with nonlocal conditions

S.K. Ntouyas, D. O'Regan (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, we prove existence and controllability results for first and second order semilinear neutral functional differential inclusions with finite or infinite delay in Banach spaces, with nonlocal conditions. Our theory makes use of analytic semigroups and fractional powers of closed operators, integrated semigroups and cosine families.

Existence and Stability of Periodic Solutions for Nonlinear Neutral Differential Equations with Variable Delay Using Fixed Point Technique

Mouataz Billah MESMOULI, Abdelouaheb Ardjouni, Ahcene Djoudi (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Our paper deals with the following nonlinear neutral differential equation with variable delay d d t D u t ( t ) = p ( t ) - a ( t ) u ( t ) - a ( t ) g ( u ( t - τ ( t ) ) ) - h ( u ( t ) , u ( t - τ ( t ) ) ) . By using Krasnoselskii’s fixed point theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness. A sufficient condition is established for the positivity of the above equation. Stability results of this equation are analyzed. Our results extend and complement some results obtained in the work [Yuan, Y., Guo, Z.: On the existence and stability of...

Existence for nonoscillatory solutions of higher order nonlinear neutral differential equations

Yong Zhou, Bing Gen Zhang, Y. Q. Huang (2005)

Czechoslovak Mathematical Journal

Consider the forced higher-order nonlinear neutral functional differential equation d n d t n [ x ( t ) + C ( t ) x ( t - τ ) ] + i = 1 m Q i ( t ) f i ( x ( t - σ i ) ) = g ( t ) , t t 0 , where n , m 1 are integers, τ , σ i + = [ 0 , ) , C , Q i , g C ( [ t 0 , ) , ) , f i C ( , ) , ( i = 1 , 2 , , m ) . Some sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general Q i ( t ) ( i = 1 ...

Existence of nonnegative periodic solutions in neutral integro-differential equations with functional delay

Imene Soulahia, Abdelouaheb Ardjouni, Ahcene Djoudi (2015)

Commentationes Mathematicae Universitatis Carolinae

The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay x ' ( t ) = - t - τ ( t ) t a ( t , s ) g ( x ( s ) ) d s + d d t Q ( t , x ( t - τ ( t ) ) ) + G ( t , x ( t ) , x ( t - τ ( t ) ) ) . We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for τ , g , a , Q and G to show that this sum of mappings fits into the framework of a modification of Krasnoselskii’s...

Existence of nonoscillatory and oscillatory solutions of neutral differential equations with positive and negative coefficients

John R. Graef, Bo Yang, Bing Gen Zhang (1999)

Mathematica Bohemica

In this paper, we study the existence of oscillatory and nonoscillatory solutions of neutral differential equations of the form x ( t ) - c x ( t - r ) P ( t ) x ( t - θ ) - Q ( t ) x ( t - δ ) =0 where c > 0 , r > 0 , θ > δ 0 are constants, and P , Q C ( + , + ) . We obtain some sufficient and some necessary conditions for the existence of bounded and unbounded positive solutions, as well as some sufficient conditions for the existence of bounded and unbounded oscillatory solutions.

Existence of periodic solutions for first-order totally nonlinear neutral differential equations with variable delay

Abdelouaheb Ardjouni, Ahcène Djoudi (2014)

Commentationes Mathematicae Universitatis Carolinae

We use a modification of Krasnoselskii’s fixed point theorem due to Burton (see [Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem, Nonlinear Stud. 9 (2002), 181–190], Theorem 3) to show that the totally nonlinear neutral differential equation with variable delay x ' ( t ) = - a ( t ) h ( x ( t ) ) + c ( t ) x ' ( t - g ( t ) ) Q ' ( x ( t - g ( t ) ) ) + G ( t , x ( t ) , x ( t - g ( t ) ) ) , has a periodic solution. We invert this equation to construct a fixed point mapping expressed as a sum of two mappings such that one is compact and the other is a large contraction. We show that the mapping fits...

Existence of Periodic Solutions for Nonlinear Neutral Dynamic Equations with Functional Delay on a Time Scale

Abdelouaheb Ardjouni, Ahcène Djoudi (2013)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Let 𝕋 be a periodic time scale. The purpose of this paper is to use a modification of Krasnoselskii’s fixed point theorem due to Burton to prove the existence of periodic solutions on time scale of the nonlinear dynamic equation with variable delay x t = - a t h x σ t + c ( t ) x ˜ t - r t + G t , x t , x t - r t , t 𝕋 , where f is the -derivative on 𝕋 and f ˜ is the -derivative on ( i d - r ) ( 𝕋 ) . We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a compact map. We show...

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