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1

Results in Q-measure

Jisha C. R.

Mathematica Applicanda

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2022

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Vol. 50, no. 2

183--215

EN

This paper introduces the notion of a generalized measure for a sequence of functions with oscillation and concentration effects. This measure is constructed by averaging the sequence of Borel measurable functions using singular or regular perturbations. In this way, the generalized limits of such sequences are conceptualized by enlarging the space of functions to measure spaces. It is a modification of the Young measure. This modified measure was termed a Q-measure. It can be difficult to determine the Young measure for a broad function. The Q-measure can be easily calculated for particular functions. This is one of the advantages of this study. As an application of the measure, we can define another weaker type of Monotone convergence theorem, the Lebesgue-dominated convergent theorem. A notion of average for underlying sequences to define the Q-measure is given, as also its application in signal analysis and atmospheric sciences.

PL

W tym artykule autor wprowadza nową miarę, którą nazywa miarą Q, reprezentującą słabą∗ granicę barycentrum ciągu funkcji borelowskich. Omawia niektóre wyniki zwi¡zane z tą miarą, co jest pomocne przy wyznaczaniu miary Q dla poszczególnych typów funkcji. Ponadto omówiono zastosowanie koncepcji średniej w analizie sygnałów i naukach o atmosferze.

2

Q-functional applications

Jisha C. R.

Mathematica Applicanda

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2022

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Vol. 50, no. 2

217--233

EN

The Q-measure indicate a weak∗ limit of the barycenter of a sequence of Borel measurable functions. In this paper, we will look only at Q-functional. Q-functional is defined by Q- measure, it is useful in the field of optimization. Computational results for Q-functional are presented and compared with Young functional. The obtained analytical results demonstrate relative error in Q-functional is lesser compared to Young functional.

PL

Miara Q wyznacza słabą∗ granicę barycentrum ciągu funkcji borelowskich. W tym artykule przyjrzymy się tylko funkcjonałom Q. Q-funkcjonalność jest definiowana przez miarę Q i jest przydatna w zastosowaniu do zadań optymalizacji. Przedstawiono wyniki obliczeń dla funkcjonału Q i porównano je z funkcjonałem Younga. Otrzymane wyniki analityczne pokazują, że błąd względny w funkcjonale Q jest mniejszy w porównaniu z funkcjonałem Younga.

3

On strongly quasilinear elliptic systems with weak monotonicity

Azroul Elhoussine, Balaadich Farah

Journal of Applied Analysis

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2021

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Vol. 27, nr 1

153--162

EN

In this paper, we prove existence results in the setting of Sobolev spaces for a strongly quasilinear elliptic system by means of Young measures and mild monotonicity assumptions.

4

Three boundary value problems for second order differential inclusions in Banach spaces

Azzam D., Castaing C., Thibault L.

Control and Cybernetics

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2002

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Vol. 31, no 3

659-693

EN

The paper studies, in the context of Banach spaces, the problem of three boundary conditions for both second order differential inclusions and second order ordinary differential equations. The results are obtained in several new settings of Sobolev-type spaces involving Bochner and Pettis integrals. Some classes of second order multivalued evolution equations associated with m-accretive operators are also considered. Applications to some control problems are provided with the help of narrow convergence for Young measures.