Emil Minchev

Monotone iterative methods for impulsive hyperbolic differential-functional equations

Theorems on impulsive hyperbolic differential-functional inequalities are considered. Comparison results and a uniqueness criterion are obtained. A method of approximation of the solutions of impulsive hyperbolic differential-functional equations by means of solutions of the associated linear problems is established. The difference between the exact and the approximate solutions is estimated.

Forced oscillation of solutions of certain hyperbolic equations of neutral type

Some oscillation properties of solutions of certain hyperbolic equations of neutral type are studied and sufficient conditions for the oscillation of the solutions of a boundary value problem in a cylindrical domain are given.

Oscillation of the solutions of impulsive parabolic equations

Oscillation properties of the solutions of impulsive parabolic equations are investigated via the method of differential inequalities.

Estimates of solutions of impulsive parabolic equations and applications to the population dynamics

A theorem on estimates of solutions of impulsive parabolic equations by means of solutions of impulsive ordinary differential equations is proved. An application to the population dynamics is given.

Oscillation of solutions of impulsive nonlinear parabolic differential-difference equations

Sufficient conditions for oscillation of the solutions of impulsive nonlinear parabolic differential-difference equations are obtained.

On first order impulsive partial differential inequalities

Abstract This paper deals with the Cauchy problem for nonlinear impulsive partial differential equations of first order. Theorems on impulsive differential inequalities are obtained. Comparison results implying uniqueness criteria are proved.

Comparison principles for impulsive hyperbolic equations of first order

Strong and weak comparison principles for impulsive hyperbolic equations of first order are proved. Uniqueness criterion is obtained.

Difference methods for impulsive differential-functional equations

Abstract We consider a class of difference methods for initial value problems for first-order impulsive partial differential-functional equations. We give sufficient conditions for the convergence of a sequence of approximate solutions under the assumptions that the right-hand sides satisfy the nonlinear estimates of Perron type with respect to the functional argument. The proof of the stability of difference methods is based on a general theorem on the error estimate of approximate solutions for difference-functional equations of Volterra type with an unknown function of several variables.

Periodic boundary value problem for impulsive hyperbolic partial differential equations of first order

Abstract The paper deals with the periodic boundary value problem for impulsive hyperbolic equations of first order. A comparison result for impulsive differential inequalities is obtained. This result is applied to get a uniqueness criterion for the solutions of impulsive hyperbolic equations of first order.

The finite difference method for first order impulsive partial differential-functional equations

We consider initial boundary value problems for first order impulsive partial differential-functional equations. We give sufficient conditions for the convergence of a general class of one step difference methods. We assume that given functions satisfy the non-linear estimates of the Perron type with respect to the functional argument. The proof of stability is based on a theorem on difference functional inequalities generated by an impulsive differential-functional problem. It is an essential assumption in our consideration that given functions satisfy the Volterra condition. We give a numerical example.ZusammenfassungWir betrachten Anfangs- und Randwertprobleme von partiellen Impuls-Differential-Funktionalgleichungen erster Ordnung. Wir leiten hinreichende Bedingungen für die Konvergenz einer allgemeinen Klasse von Einschrittmethoden her. Wir nehmen weiters an, daß die gegebenen Funktionen bezüglich ihrer Funktionalargumente einer nichtlinearen Abschätzung vom Perron-Typus genügen. Der Stabilitätsbeweis basiert auf einem Theorem über Differenzen-Funktionalgleichungen, die von einem Impuls-Differential-Funktionalproblem stammen. Grundsätzlich nehmen wir an, daß die betrachteten Funktionen die Volterra-Bedingung erfüllen. Ein numerisches Beispiel wird angeführt.