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A remark on local fractional calculus and ordinary derivatives

100%
Almeida R., Guzowska M., Odzijewicz T.
Open Mathematics
|
2016
|
tom 14
|
nr 1
1122-1124
EN
In this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.
2
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Oscillation of second-order linear delay differential equations

76%
Ohriska J.
Open Mathematics
|
2008
|
tom 6
|
nr 3
439-452
EN
The aim of this paper is to derive sufficient conditions for the linear delay differential equation (r(t)y′(t))′ + p(t)y(τ(t)) = 0 to be oscillatory by using a generalization of the Lagrange mean-value theorem, the Riccati differential inequality and the Sturm comparison theorem.
3
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A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic

64%
de Amo E., Carrillo M. D., Fernández-Sánchez J.
Open Mathematics
|
2016
|
tom 14
|
nr 1
1039-1052
EN
We study a parameterized family of singular functions which appears in a paper by H. Okamoto and M. Wunsch (2007). Various properties are revisited from the viewpoint of fractal geometry and probabilistic techniques. Hausdorff dimensions are calculated for several sets related to these functions, and new properties close to fractal analysis and strong negations are explored.
4
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Filippov Lemma for certain second order differential inclusions

64%
Bartuzel G., Fryszkowski A.
Open Mathematics
|
2012
|
tom 10
|
nr 6
1944-1952
EN
In the paper we give an analogue of the Filippov Lemma for the second order differential inclusions with the initial conditions y(0) = 0, y′(0) = 0, where the matrix A ∈ ℝd×d and multifunction is Lipschitz continuous in y with a t-independent constant l. The main result is the following: Assume that F is measurable in t and integrably bounded. Let y 0 ∈ W 2,1 be an arbitrary function fulfilling the above initial conditions and such that where p 0 ∈ L 1[0, 1]. Then there exists a solution y ∈ W 2,1 to the above differential inclusions such that a.e. in [0, 1], .
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