Sectorial Operators Research Articles

Complex Powers of Fractional Sectorial Operators and Quasilinear Equations with Riemann–Liouville Derivatives

Complex Powers of Fractional Sectorial Operators and Quasilinear Equations with Riemann–Liouville Derivatives

Approximate controllability of Hilfer fractional neutral stochastic systems of the Sobolev type by using almost sectorial operators

<abstract><p>The main aim of this work is to conduct an analysis of the approximate controllability of Hilfer fractional (HF) neutral stochastic differential systems under the condition of an almost sectorial operator with delay. The theoretical ideas linked to stochastic analysis, fractional calculus and semigroup theory, along with the fixed-point technique, are utilized to establish the key results of this article. More precisely, the main theorem of this study is devoted to proving the fact that the relevant linear system is approximately controllable. Finally, to help this research be as comprehensive as possible, we provide a theoretical application and filter system.</p></abstract>

Strong solution for singularly nonautonomous evolution equation with almost sectorial operators

<p style='text-indent:20px;'>In this paper we consider the singularly nonautonomous evolution problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t +A(t) u = f(t), \mbox{ } \tau<t<\tau+T; \quad u(\tau) = u_0 \in X, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>associated with a family of uniformly almost sectorial linear operators <inline-formula><tex-math id="M1">\begin{document}$ A(t):D\subset X \rightarrow X $\end{document}</tex-math></inline-formula>, that is, a family for which a sector of the complex plane is contained in the resolvent of <inline-formula><tex-math id="M2">\begin{document}$ -A(t) $\end{document}</tex-math></inline-formula> and satisfies <inline-formula><tex-math id="M3">\begin{document}$ \|(\lambda+A(t))^{-1}\|_{\mathcal{L} (X)} \leq \frac{C}{|\lambda|^{\alpha}} $\end{document}</tex-math></inline-formula>, for some <inline-formula><tex-math id="M4">\begin{document}$ \alpha \in (0, 1) $\end{document}</tex-math></inline-formula>, uniformly in <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula>. Under a proper condition on the value of <inline-formula><tex-math id="M6">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> we prove that the linear process associated to the family <inline-formula><tex-math id="M7">\begin{document}$ A(t) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula>, is strongly differentiable and that the singularly nonautonomous problem has a strong solution. An example of a singularly nonautonomous reaction-diffusion equation in a domain with a handle illustrates the abstracts results obtained.</p>

The e-positive mild solutions for impulsive evolution fractional differential equations with sectorial operator

The e-positive mild solutions for impulsive evolution fractional differential equations with sectorial operator

Introducing and solving generalized Black–Scholes PDEs through the use of functional calculus

We introduce some families of generalized Black–Scholes equations which involve the Riemann–Liouville and Weyl space-fractional derivatives. We prove that these generalized Black–Scholes equations are well-posed in (L^1-L^infty )-interpolation spaces. More precisely, we show that the elliptic-type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial-like operators and sectorial operators in an abstract setting. Such a connection extends the scaling property of sectorial operators to a wider family of both operators and the functions involved.

A New Result Concerning Nonlocal Controllability of Hilfer Fractional Stochastic Differential Equations via almost Sectorial Operators

This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors establish new criteria for the mild existence of solutions and derive fundamental characteristics of the nonlocal controllability of a system. In addition, researchers offer theoretical and real-world examples to demonstrate the effectiveness and suitability of our suggested solutions.

Analysis on the Controllability of Hilfer Fractional Neutral Differential Equations with Almost Sectorial Operators and Infinite Delay via Measure of Noncompactness

Analysis on the Controllability of Hilfer Fractional Neutral Differential Equations with Almost Sectorial Operators and Infinite Delay via Measure of Noncompactness

INFINITE INTERVAL PROBLEMS FOR HILFER FRACTIONAL EVOLUTION EQUATIONS WITH ALMOST SECTORIAL OPERATORS

We investigate the Cauchy problem on an infinite interval for the fractional evolution equation with Hilfer fractional derivative, which is a generalization of both Riemann–Liouville and Caputo fractional derivatives. Our methods are based on the generalized Ascoli–Arzelà theorem, Schauder’s fixed point theorem, the Wright function and Kuratowski’s measure of noncompactness. We obtain sufficient conditions of the existence for global mild solutions and attractive solutions when the semigroup associated with an almost sectorial operator is compact as well as noncompact. Two examples are provided to illustrate the results.

Existence of Hilfer Fractional Stochastic Differential Equations with Nonlocal Conditions and Delay via Almost Sectorial Operators

In this article, we examine the existence of Hilfer fractional (HF) stochastic differential systems with nonlocal conditions and delay via almost sectorial operators. The major methods depend on the semigroup of operators method and the Mo¨nch fixed-point technique via the measure of noncompactness, and the fundamental theory of fractional calculus. Finally, to clarify our key points, we provide an application.

Nonlinear Inverse Problems for Fractional Differential Equations with Sectorial Operators

Nonlinear Inverse Problems for Fractional Differential Equations with Sectorial Operators

The Lp-regularity properties of nonlocal wave equations and applications

In this paper, the Cauchy problem for linear and nonlinear wave equations is studied. The equation involves abstract operator [Formula: see text] in a Banach space [Formula: see text] and convolution terms. Here, assuming enough smoothness on the initial data and on coefficients, the existence, uniqueness and regularity properties of local and global solutions are established in terms of fractional powers of a given sectorial operator [Formula: see text]. We obtain the regularity properties of a wide class of wave equations by choosing the space [Formula: see text] and the operator [Formula: see text] which appear in the field of physics.

Discussion on the Approximate Controllability of Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators

This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost sectorial operators, fractional differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove the result. We start by emphasizing the existence of a mild solution and demonstrate the approximate controllability of the fractional system. In addition, an example is presented to demonstrate the principle.

On the c-Entropy of L-Systems with Schrödinger Operator

We study L-systems whose main operators are extensions of one-dimensional half-line Schrödinger operators with deficiency indices (1, 1), the Schrödinger L-systems. Introducing new concepts of an c-entropy and dissipation coefficient for an L-system we discuss the following dual problems: describe Schrödinger L-systems (1) with a given c-entropy and minimal dissipation coefficient, and (2) with a given dissipation coefficient and maximal c-entropy. Also, we analyze in detail the dual c-entropy problems for Schrödinger L-systems with sectorial and extremal main operators.

Existence and controllability of fractional evolution inclusions with impulse and sectorial operator

Many evolutionary operations fromdiverse fields of engineering and physical sciences go through abrupt modifications of state at specific moments of time among periods of non-stop evolution. These operations are more conveniently modeled via impulsive differential equations and inclusions. In this work, firstly we address the existence of mild solutions for nonlocal fractional impulsive semilinear differential inclusions related to Caputo derivative in Banach spaces when the linear part is sectorial. Secondly, we determine the enough, conditions for the controllability of the studied control problem. We apply effectively fixed point theorems, contraction mapping, multivalued analysis and fractional calculus. Moreover, we enhance our results by introducing an illustrative examples.

Existence of Mild Solutions for Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators

This manuscript focuses on the existence of a mild solution Hilfer fractional neutral integro-differential inclusion with almost sectorial operator. By applying the facts related to fractional calculus, semigroup, and Martelli’s fixed point theorem, we prove the primary results. In addition, the application is provided to demonstrate how the major results might be applied.

New Outcomes Regarding the Existence of Hilfer Fractional Stochastic Differential Systems via Almost Sectorial Operators

In this paper, we focus on the existence of Hilfer fractional stochastic differential systems via almost sectorial operators. The main results are obtained by using the concepts and ideas from fractional calculus, multivalued maps, semigroup theory, sectorial operators, and the fixed-point technique. We start by confirming the existence of the mild solution by using Dhage’s fixed-point theorem. Finally, an example is provided to demonstrate the considered Hilferr fractional stochastic differential systems theory.

Global solution to non-self-adjoint stochastic Volterra equation

In this paper, we establish the existence and uniqueness of the mild solution for stochastic Volterra equation with a non-self-adjoint operator. The specific Volterra equation that we consider is a generalization of the fractional differential equation. To obtain the mild solution for the case of multiplicative problem, the resolvent property of the linear perturbation of a sectorial operator will be considered.

Optimal controls for nonlocal Cauchy problems of multi-term fractional evolution equations

Abstract This paper is mainly concerned with a controlled multi-term fractional evolution equation in Banach spaces. Firstly, we give formula of its mild solutions and show the existence result for the problem via $\omega $-sectorial operator technique. Secondly, we establish the Lagrange optimal control and time optimal control for the system invoked by the nonlocal Cauchy problems of multi-term fractional evolution equation by properties of resolvent operators.

Hilfer Fractional Neutral Stochastic Volterra Integro-Differential Inclusions via Almost Sectorial Operators

In our paper, we mainly concentrate on the existence of Hilfer fractional neutral stochastic Volterra integro-differential inclusions with almost sectorial operators. The facts related to fractional calculus, stochastic analysis theory, and the fixed point theorem for multivalued maps are used to prove the result. In addition, an illustration of the principle is provided.

On Hilfer-Type Fractional Impulsive Differential Equations

Using the Schauder fixed point theorem, we prove the existence of impulsive fractional differential equations using Hilfer fractional derivative and nearly sectorial operators in this paper. We’ve gone over the two scenarios where the related semigroup is compact and noncompact for this purpose. We also go over an example to back up the main points.