Viability for a class of semilinear differential equations of retarded type

Let X be a Banach space, A ： D（A） X → X the generator of a compact C0- semigroup S（t） ： X → X, t ≥ 0, D a locally closed subset in X, and f ： （a, b） × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u＇（t） = Au（t） ＋ f（t, u（t - q））, t ∈ [to, to ＋ T], with initial condition uto = φ ∈C（[-q, 0]; X）, is the tangency condition lim infh10 h^-1d（S（h）v（O）＋hf（t, v（-q））; D） = 0 for almost every t ∈ （a, b） and every v ∈ C（[-q, 0]; X） with v（0）, v（-q）∈ D.     

Invariant manifolds for singularly perturbed linear functional differential equations

The existence of “slow” and “fast” manifolds, and of invariant manifolds approaching the manifold of orbits of the degenerate system, is discussed for singularly perturbed systems of linear retarded functional differential equations (FDE). It is shown that these manifolds exist only in very degenerate situations and, consequently, the geometry of the flow of singularly perturbed ordinary differential equations does not generalize to FDEs.     

Invariant manifolds of singularly perturbed ordinary differential equations

For singularly perturbed systems of ODE's satisfying a certain stability assumption the existence of an asymptotically stable (unstable) invariant manifold is proved. This invariant manifold is -close to the so-called reduced manifold of such a system. As an illustrative example a 3-dimensional autonomous system describing a model in biochemistry is considered.

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Zusammenfassung Für singulär gestörte Differentialgleichungssysteme, die einer gewissen Stabilitätsbedingung genügen, wird die Existenz einer asymptotisch stabilen (instabilen) invarianten Mannigfaltigkeit nachgewiesen, die in einer -Umgebung der sogenannten reduzierten Mannigfaltigkeit eines solchen Systems liegt. Als Anschauungsbeispiel wird ein dreidimensionales autonomes System betrachtet, welches ein biochemisches Modell beschreibt.

     

The invariance of limit sets for retarded differential equations

Résumé On démontre la semi-invariance ou la quasi-invariance des ensembles limites pour une équation différentielle à retard, sans supposer l'unicité ni la prolongeabilité des solutions, et sans supposer non plus que la solution engendrant l'ensemble limite soit contenue dans un ensemble compact ou fermé.

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A Dario Graffi, pour son 70-ème anniversaire

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Entrata in Redazione il 7 maggio 1975.     

Regularity properties of perturbed Hille-Yosida operators and retarded differential equations

In this work we give sufficient conditions on the semigroup (T ₀ (t)) _tgeq0 , generated by the part of a Hille—Yosida operator in the closure of its domain, in order that certain perturbations preserve some regularity properties of (T ₀ (t)) _tgeq0 , like the norm continuity, compactness and differentiability. An application of our abstract results to retarded differential equations is given.     

Complete guiding sets for a class of almost-periodic differential equations

A functional differential equation admitting a complete guiding set always has an a priori bounded solution. The introduction of this concept and the idea of reducibility allow the authors to apply a well-known method for the autonomous case to a non-autonomous problem. A previous result of Ortega and Tineo is extended to a non-hyperbolic semilinear functional differential equation with almost-periodic coefficients in its lineal part. The non-hyperbolicity condition is established in terms of the Sacker-Sell spectrum of the associated linear system.     

Invariant stabilization of a certain class of functional differential equations

Consider the system

Integral sets of a class of differential equations with impulses

The problem of the existence and stability of an invariant set of a class of systems of differential equations with impulses is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 994–997, July, 1990.     

Attracting and invariant sets for a class of impulsive functional differential equations

In this article, a class of nonlinear and nonautonomous functional differential systems with impulsive effects is considered. By developing a delay differential inequality, we obtain the attracting set and invariant set of the impulsive system. An example is given to illustrate the theory.     

Invariant manifolds for differential equations

In this paper we discuss the concept ‘generalized exponential dichotomy’ and give the existence ofC ^k invariant manifolds for abstract nonautonomous differential equations in Banach or Hilbert spaces. Also we give a classification of invariant manifolds and an estimate of the locality radius of invariant manifolds.     

Quantum resurgence for a class of Airy type differential equations

In this article, we investigate the resurgent properties of divergent WKB solutions of a class of Airy type perturbed differential equations. In particular, we extend and propose a new proof of a reduction theorem, due to Aoki et al. [T. Aoki, T. Kawai, Y. Takei, The Bender-Wu analysis and the Voros theory, in: Special Functions, ICM-90 Satell. Conf. Proc., Okayama, 1990, Springer, Tokyo, 1991, pp. 1-29], near a simple turning point, in the framework of exact WKB analysis. Our scheme of proof is based on a Laplace-integral representation derived from an existence theorem of holomorphic solutions for a singular linear partial differential equation.     

On a new almost periodic type solution of a class of singularly perturbed differential equations with piecewise constant argument

A new class of ergodic sequences, pseudo almost periodic sequence, is introduced, and the existence of pseudo almost periodic sequence to difference equation is studied. Based on these, we investigate the existence of pseudo almost periodic solutions for a nonautonomous, singularly perturbed differential equations with piecewise constant argument.     

Periodic solutions for perturbed differential equations

A variant of the Alekseev variation of constants integral equation is obtained relating the solutions of systems of the form $\text{x}\text{?}\text{= f(t, x, λ)}$ and $\text{y}\text{?}\text{= f(t, y, ψ(t, y)) + g(t, y)}$. For the case when f, g, and ψ have period P in t several theorems are given for the existence of periodic solutions extending known results when f is linear in x and does not depend on the parameter m-vector λ. Comparison with an older technique gives hypotheses where the method above is advantageous for establishing periodic solutions. An example is given for constructing limit cycles of autonomous second-order systems.     

Global exponential stability of a class of retarded impulsive differential equations with applications

This paper studies the dynamics of a class of retarded impulsive differential equations (IDE), which generalizes the delayed cellular neural networks (DCNN), delayed bidirectional associative memory (BAM) neural networks and some population growth models. Some sufficient criteria are obtained for the existence and global exponential stability of a unique equilibrium. When the impulsive jumps are absent, our results reduce to its corresponding results for the non-impulsive systems. The approaches are based on Banach’s fixed point theorem, matrix theory and its spectral theory. Due to this method, our results generalize and improve many previous known results such as [3], [5], [6], [9], [17], [18], [23], [32], [38], [43], [51], [52]. Some examples are also included to illustrate the feasibility and effectiveness of the results obtained.     

Stability for retarded functional differential equations

It is known that retarded functional differential equations can be regarded as Banach-space-valued generalized ordinary differential equations (GODEs). In this paper, some stability concepts for retarded functional differential equations are introduced and they are discussed using known stability results for GODEs. Then the equivalence of the different concepts of stabilities considered here are proved and converse Lyapunov theorems for a very wide class of retarded functional differential equations are obtained by means of the correspondence of this class of equations with GODEs. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 107–126, January, 2008.     

Averaging for retarded functional differential equations

We consider retarded functional differential equations in the setting of Kurzweil-Henstock integrable functions and we state an averaging result for these equations. Our result generalizes previous ones.