Browsing by Author "Toufik, HERAIZ: Supervisor"

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    On Spectral Theory of Ordinary Differential Operator Of Fractional Order
    (Mohamed Boudiaf University of M'sila, 2024-06-10) Lachache, Qamar; Toufik, HERAIZ: Supervisor
    Fractional analysis is a branch of mathematical analysis which studies the possibility of defining noninteger powers of derivative and integrating operators. we define some useful functions such as Gamma function, Beta function and MittagLeffler function. These functions play a very important role in the theory of fractional order differential calculus. we will present the definition of a fractional derivative and then we study the three most popular approaches to fractional derivatives: the Grünwald-Letnikov, Riemann-Liouville and Caputo approaches as well as their properties. Finally, we study some examples of fractional derivatives. Keywords: Fractional derivation, Grünwald-Letnikov, Riemann-Liouville, Caputo. and operator theory to study the spectral properties (such as eigenvalues and eigenfunctions) of fractional differential operators.
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    ON THE WEYL AND BROWDER THEOREMS OF BOUNDED LINEAR OPERATOR
    (Mohamed Boudiaf University of M'sila, 2024-06-10) Bouguerra, Malika; Toufik, HERAIZ: Supervisor
    The Weyl and Browder theorems stand as pivotal landmarks in the realm of functional analysis, offering profound insights into the spectral behavior and structural properties of bounded linear operators. Pietro Aiena's seminal contributions have significantly enriched our understanding of these theorems, unraveling their intricate connections and extending their applicability to various classes of operators. In this research endeavor, we embark on a comprehensive exploration of Aiena's work, delving deep into the nuanced implications and far-reaching consequences of the Weyl and Browder theorems. Through a meticulous examination of Aiena's results, we elucidate the underlying principles governing the spectral decomposition, essential spectra, and related spectral properties of bounded linear operators. Moreover, we investigate the interplay between these theorems and other fundamental concepts in functional analysis, such as compact operators, spectral mapping theorems, and perturbation theory. By synthesizing Aiena's insights with contemporary developments in the field, we aim to provide a unified framework for understanding the spectral theory of bounded linear operators, with implications for diverse areas including operator theory, mathematical physics, and dynamical systems. Our research not only consolidates and extends Aiena's seminal contributions but also sheds light on new avenues for theoretical exploration and practical applications in functional analysis and its myriad interdisciplinary ramifications