On the boundedness and periodicity of solutions of certain third-order non-linear differential equations

Summary This paper extends a number of known results on the boundedness of solutions and the existence of periodic solutions to the more general equation1.1(I). Entrata in Redazione il 24 dicembre 1968.     

Boundedness and periodicity of solutions of a certain system of third-order non-linear differential equations

Summary In this paper a number of known results on the boundedness of solutions and the existence of periodic solutions of the scalar equation 1.1(2) are extended to the vector equation 1.1(1).     

Ultimate boundedness of impulsive fractional differential equations

This article investigates nonlinear impulsive Caputo fractional differential equations. Utilizing Lyapunov functions, Laplace transforms of fractional derivatives and boundedness of Mittag-Leffler functions, several sufficient conditions are derived to ensure the global ultimate boundedness and the exponential stability of the systems. An example is given to explain the obtained results.     

Ultimate boundedness of impulsive fractional delay differential equations

This paper concerns with the ultimate boundedness problem for impulsive fractional delay differential equations. Based on the impulsive fractional differential inequality, the boundedness of Mittag-Leffler functions, and the successful construction of suitable Lyapunov functionals, some algebraic criteria are derived for testing the global ultimate boundedness of the equations, and the estimations of the global attractive sets are provided as well. One example is also given to show the effectiveness of the obtained theoretical results.     

A note on the boundedness and the stability of solutions of certain third-order differential equations

Summary This paper establishes conditions under which solutions of equations of the form (1.1) are bounded or tend to zero as t → ∞. Entrata in Redazione il 14 aprile 1971.     

On the boundedness of solutions of third-order delay differential equations

We establish certain sufficient conditions guaranteeing the boundedness of all solutions of some third-order delay differential equations.     

Asymptotic and exponential stability of certain third-order non-linear delayed differential equations: Frequency domain method

We use the frequency domain method to prove that the zero solution of certain third order nonlinear delayed differential equations is asymptotically stable, (when there is no forcing term). We also prove the existence of a bounded solution which is exponentially stable, (when there is a bounded forcing term). The situation for which the non-linear term is delayed is also proved.     

Stability and boundedness of a kind of third-order delay differential system

The paper studies equation (1.1) in two cases:

• •(i)p ≡ 0,
• •(ii)p ≠ 0.

In Case (i), the asymptotic stability of the solution x = 0 is studied; in Case (ii), the uniform boundedness and uniform ultimate boundedness of all solutions of (1.1) are proved.     

On a system of non-linear differential equations

Summary E. Kasner(1925) considered a system of non-linear differential equations in three variables. The authors of this paper have extended this system to n variables by means of a linear transformation with complex coefficients. The formulas obtained for the solutions of the differential equations give a simplification of Kasner’s solutions. Entrata in Redazione il 26 agosto 1968.     

Ultimate boundedness and periodicity for some partial functional differential equations with infinite delay

In this work we study the existence of periodic solutions for some partial functional differential equation with infinite delay. We assume that the linear part is not necessarily densely defined and satisfies the known Hille-Yosida condition. Firstly, we give some estimates of the solutions. Secondly, we prove that the Poincaré map is condensing which allows us to prove the existence of periodic solutions when the solutions are ultimately bounded.     

Partial differential equations associated with certain non-linear algorithms

Zusammenfassung In der letzten Zeit sind eine Anzahl nicht-linearer Algorithmen untersucht worden, die in der Analysis und numerischen Mathematik von besonderer Wichtigkeit sind.Diese Algorithmen betreffen Grössen, die in einem zweidimensionalen Schema angeordnet werden können, und verknüpfen vier Grössen miteinander, die an den Eckpunkten eines Rhombus in diesem Schema vorkommen. Wenn die Dimensionen dieses Rhombus unendlich klein gemacht werden, dann führen die algorithmischen Beziehungen zu Systemen von zwei simultanen partiellen Differentialgleichungen erster Ordnung.In dieser Arbeit werden einige allgemeinere Ergebnisse hergeleitet, die sich auf solche partielle Differentialgleichungssysteme beziehen; die Ergebnisse werden an Hand spezieller Beispiele erläutert. Der -Algorithmus, derq — d Algorithmus und verschiedene andere Algorithmen werden im Detail untersucht. Diese Algorithmen führen auf die partielle Differentialgleichung der Padéschen Fläche, auf die korrespondierende Kettenbruchkoeffizientenfläche und auf eine Anzahl weiterer partieller Differentialgleichungen.

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Communication MR 65 of the Computation Department of the Mathematical Centre, Amsterdam.     

On certain non-linear differential equations in complex domains

We describe meromorphic solutions of certain non-linear (ordinary and partial) differential equations in complex domains. Received: 13 November 2007