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On an integral transform by R. S. Phillips

100%
Bjon S.
Open Mathematics
|
2010
|
tom 8
|
nr 1
98-113
EN
The properties of a transformation $$ f \mapsto \tilde f_h $$ by R.S. Phillips, which transforms an exponentially bounded C 0-semigroup of operators T(t) to a Yosida approximation depending on h, are studied. The set of exponentially bounded, continuous functions f: [0, ∞[→ E with values in a sequentially complete L c-embedded space E is closed under the transformation. It is shown that $$ (\tilde f_h )\widetilde{_k } = \tilde f_{h + k} $$ for certain complex h and k, and that $$ f(t) = \lim _{h \to 0^ + } \tilde f_h (t) $$, where the limit is uniform in t on compact subsets of the positive real line. If f is Hölder-continuous at 0, then the limit is uniform on compact subsets of the non-negative real line. Inversion formulas for this transformation as well as for the Laplace transformation are derived. Transforms of certain semigroups of non-linear operators on a subset X of an L c-embedded space are studied through the C 0-semigroups, which they define by duality on a space of functions on X.
2
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Optimal time and space regularity for solutions of degenerate differential equations

100%
Favaron A.
Open Mathematics
|
2009
|
tom 7
|
nr 2
249-271
EN
We derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.
3
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The norm convergence of a Magnus expansion method

100%
Bátkai A., Sikolya E.
Open Mathematics
|
2012
|
tom 10
|
nr 1
150-158
EN
We consider numerical approximation to the solution of non-autonomous evolution equations. The order of convergence of the simplest possible Magnus method is investigated.
4
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On the nonlocal Cauchy problem for semilinear fractional order evolution equations

84%
Wang J., Zhou Y., Fečkan M.
Open Mathematics
|
2014
|
tom 12
|
nr 6
911-922
EN
In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.
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