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Convergence to equilibria in a differential equation with small delay

Mihály Pituk (2002)

Mathematica Bohemica

Consider the delay differential equation x ˙ ( t ) = g ( x ( t ) , x ( t - r ) ) , ( 1 ) where r > 0 is a constant and g 2 is Lipschitzian. It is shown that if  r is small, then the solutions of (1) have the same convergence properties as the solutions of the ordinary differential equation obtained from (1) by ignoring the delay.

Differentiability of perturbed semigroups and delay semigroups

Charles J. K. Batty (2007)

Banach Center Publications

Suppose that A generates a C₀-semigroup T on a Banach space X. In 1953 R. S. Phillips showed that, for each bounded operator B on X, the perturbation A+B of A generates a C₀-semigroup on X, and he considered whether certain classes of semigroups are stable under such perturbations. This study was extended in 1968 by A. Pazy who identified a condition on the resolvent of A which is sufficient for the perturbed semigroups to be immediately differentiable. However, M. Renardy showed in 1995 that immediate...

Functional differential inequalities with unbounded delay

Z. Kamont, S. Kozieł (2006)

Annales Polonici Mathematici

Classical solutions of functional partial differential inequalities with initial boundary conditions are estimated by maximal solutions of suitable problems for ordinary functional differential equations. Uniqueness of solutions and continuous dependence on given functions are obtained as applications of the comparison result. A theorem on weak functional differential inequalities generated by mixed problems is proved. Our method is based on an axiomatic approach to equations with unbounded delay....

New interval oscillation criteria for second-order functional differential equations with nonlinear damping

Süleyman Öǧrekçi (2015)

Open Mathematics

This paper concerns the oscillation problem of second-order nonlinear damped ODE with functional terms.We give some new interval oscillation criteria which is not only based on constructing a lower solution of a Riccati type equation but also based on constructing an upper solution for corresponding Riccati type equation. We use a recently developed pointwise comparison principle between those lower and upper solutions to obtain our results. Some illustrative examples are also provided to demonstrate...

On a theorem of Haimo regarding concave mappings

Martin Chuaqui, Peter Duren, Brad Osgood (2011)

Annales UMCS, Mathematica

A relatively simple proof is given for Haimo's theorem that a meromorphic function with suitably controlled Schwarzian derivative is a concave mapping. More easily verified conditions are found to imply Haimo's criterion, which is now shown to be sharp. It is proved that Haimo's functions map the unit disk onto the outside of an asymptotically conformal Jordan curve, thus ruling out the presence of corners.

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