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Appendix to the paper "Osgood type conditions for an mth order differential equation"

100%

Szufla S.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2001

tom 21

nr 1

149-153

Osgood type conditions for an m th-order differential equation

100%

Szufla S.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1998

tom 18

nr 1-2

45-55

We present a new theorem on the differential inequality $u^{(m)} ≤ w(u)$. Next, we apply this result to obtain existence theorems for the equation $x^{(m)} = f(t,x)$.

A verified method for solving piecewise smooth initial value problems

80%

Auer E. , Kiel S. , Rauh A.

International Journal of Applied Mathematics and Computer Science

2013

tom 23

nr 4

731-747

In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VAL E NC IA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.

Existence of mild solutions on infinite intervals to first order initial value problems for a class of differential inclusions in banach spaces

60%

Benchohra M.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1999

tom 19

nr 1-2

111-121

In this paper we investigate the existence of mild solutions on an unbounded real interval to first order initial value problems for a class of differential inclusions in Banach spaces. We shall make use of a theorem of Ma, which is an extension to multivalued maps on locally convex topological spaces of Schaefer's theorem.