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A criterion for convergence of solutions of homogeneous delay linear differential equations

Josef Diblík (1999)

Annales Polonici Mathematici

The linear homogeneous differential equation with variable delays ( t ) = j = 1 n α j ( t ) [ y ( t ) - y ( t - τ j ( t ) ) ] is considered, where α j C ( I , ͞ ͞ ) , I = [t₀,∞), ℝ⁺ = (0,∞), j = 1 n α j ( t ) > 0 on I, τ j C ( I , ) , the functions t - τ j ( t ) , j=1,...,n, are increasing and the delays τ j are bounded. A criterion and some sufficient conditions for convergence of all solutions of this equation are proved. The related problem of nonconvergence is also discussed. Some comparisons to known results are given.

An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces

C. S. Barroso, M. A. M. Marrocos, M. P. Rebouças (2013)

Studia Mathematica

We establish some results that concern the Cauchy-Peano problem in Banach spaces. We first prove that a Banach space contains a nontrivial separable quotient iff its dual admits a weak*-transfinite Schauder frame. We then use this to recover some previous results on quotient spaces. In particular, by applying a recent result of Hájek-Johanis, we find a new perspective for proving the failure of the weak form of Peano's theorem in general Banach spaces. Next, we study a kind of algebraic genericity...

Approximate solutions for integrodifferential equations of the neutral type

B. G. Pachpatte (2010)

Commentationes Mathematicae Universitatis Carolinae

The main objective of the present paper is to study the approximate solutions for integrodifferential equations of the neutral type with given initial condition. A variant of a certain fundamental integral inequality with explicit estimate is used to establish the results. The discrete analogues of the main results are also given.

Approximation of solutions of a difference-differential equation

B. G. Pachpatte (2010)

Archivum Mathematicum

In the present paper we study the approximate solutions of a certain difference-differential equation under the given initial conditions. The well known Gronwall-Bellman integral inequality is used to establish the results. Applications to a Volterra type difference-integral equation are also given.

Functional differential equations

Tadeusz Jankowski (2002)

Czechoslovak Mathematical Journal

The method of quasilinearization is a well-known technique for obtaining approximate solutions of nonlinear differential equations. In this paper we apply this technique to functional differential problems. It is shown that linear iterations converge to the unique solution and this convergence is superlinear.

On an elasto-dynamic evolution equation with non dead load and friction

Oanh Chau (2006)

Applications of Mathematics

In this paper, we are interested in the dynamic evolution of an elastic body, acted by resistance forces depending also on the displacements. We put the mechanical problem into an abstract functional framework, involving a second order nonlinear evolution equation with initial conditions. After specifying convenient hypotheses on the data, we prove an existence and uniqueness result. The proof is based on Faedo-Galerkin method.

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