Minimization of Quadratic Functionals Through Γ-Hilbert Space

Yıl 2022, Cilt: 19 Sayı: 1, 22 - 28, 01.05.2022

Sahın Injamamul Islam , Nırmal Sarkar , Ashoke Das

Öz

In this article we introduce the Gateaux differential and Frechet differential in Γ-Hilbert
space. We show the examples and related theorems in this space. We have noticed that
two differentials mentioned above will be equal for certain condition. Also, we discuss
the relative extremum and the stationary point of a functional in Γ-Hilbert space. We
already investigated the characteristics of both bounded and unbounded operators of
Γ-Hilbert space. Now, by using previous concept we elaborate optimization problems
and extremum of quadratic functionals in Γ-Hilbert space. Here we observe that how
the function of the solution of a operator equation minimizes the quadratic functionals.
Finally we describe the Minimization of quadratic functionals and its related theorem
via Γ-Hilbert space.

Anahtar Kelimeler

Γ-Hilbert space, Gateaux Γ-derivative, Frechet Γ-derivative, Relative extremum, Stationary point, Quadratic functionals.

Kaynakça

• T. E. Aman and D. K. Bhattacharya, "Γ-Hilbert Space and linear quadratic control problem," Revista de la Academia Canaria de Ciencias, vol. 15, no. 1-2, pp. 107-114, 2004.
• A. Gosh, A. Das and T. E. Aman, "Representation Theorem on Γ-Hilbert Space," International Journal of Mathematics Trends and Technology, vol. 52, no. 9, pp. 608-615, 2017.
• S. Islam and A. Das, "On Some bounded Operators and their characterizations in Γ-Hilbert Space," Cumhuriyet Science Journal, vol. 41, no. 4, pp. 854-861, 2020.
• A. Das, A. Ghosh and T. E. Aman, "Calculas on Γ-Hilbert Space," Journal of Interdisciplinary Cycle Research. vol. 12, no. 7, pp. 254-268, 2020.