Nonlocal Conditions Research Articles

An analysis on the approximate controllability results for Caputo fractional hemivariational inequalities of order 1 < r < 2 using sectorial operators

In this paper, we investigate the effect of hemivariational inequalities on the approximate controllability of Caputo fractional differential systems. The main results of this study are tested by using multivalued maps, sectorial operators of type (P, η, r, γ ), fractional calculus, and the fixed point theorem. Initially, we introduce the idea of mild solution for fractional hemivariational inequalities. Next, the approximate controllability results of semilinear control problems were then established. Moreover, we will move on to the system involving nonlocal conditions. Finally, an example is provided in support of the main results we acquired.

ALTERNATING DIRECTION IMPLICIT METHOD FOR POISSON EQUATION WITH INTEGRAL CONDITIONS

In this paper, we investigate the convergence of the Peaceman-Rachford Alternating Direction Implicit method for the system of difference equations, approximating the two-dimensional elliptic equations in rectangular domain with nonlocal integral conditions. The main goal of the paper is the analysis of spectrum structure of difference eigenvalue problem with nonlocal conditions. The convergence of iterative method is proved in the case when the system of eigenvectors is complete. The main results are generalized for the system of difference equations, approximating the differential problem with truncation error O(h4).

Approximate controllability for some integrodifferential evolution equations with nonlocal conditions

The main objective of this work is to investigate the existence of mild solutions and approximate controllability for some integrodifferential evolution equations with nonlocal conditions. Assuming that the linear part is exactly null and approximately controllable, using resolvent operator theory, we provide our main results. An example is given to illustrate the basic results of this work.

A class of stochastic nonlocal evolution equations with nonlocal initial conditions

In this paper, we consider the existence and uniqueness of mild solutions for stochastic nonlocal evolution equations with Lévy diffusion operator and nonlocal initial conditions. Based on the continuity of the Lévy semigroup and the technique of the measure of noncompactness, we establish the local existence of mild solutions in [Formula: see text] under some weaker growth conditions. Moreover, we obtain the existence of mild solutions on any finite interval by using the general growth conditions on the nonlinear. Finally, the global existence and uniqueness of mild solutions follow from the additional Lipschitz conditions on nonlinear terms.

An Outlook on Hybrid Fractional Modeling of a Heat Controller with Multi-Valued Feedback Control

In this study, we extend the investigations of fractional-order models of thermostats and guarantee the solvability of hybrid Caputo fractional models for heat controllers, satisfying some nonlocal hybrid multi-valued conditions with multi-valued feedback control, which involves the Chandrasekhar kernel, by using hybrid Dhage’s fixed point theorem. A part of this study is dedicated to transforming this problem into an equivalent integral representation and then proving some existence results to achieve our aims. Furthermore, the continuous dependence of the unique solution on the control variable and on the set of selections will be discussed. Moreover, we provide an illustration to support our results.

Fractional partial differential variational inequality

In this present paper, we introduce and study a dynamical systems involving fractional derivative operator and nonlocal condition, which is constituted of a fractional evolution equation and a time-dependent variational inequality, and is named as fractional partial differential variational inequality (FPDVI, for short). By employing the estimates involving the one-and two-parameter Mittag-Leffler functions, fixed-point theory for set-value mappings, and non-compactness measure theory, we develop a general framework to establish the existence of smooth solutions to (FPDVI).

Nonlocal partial fractional evolution equations with state dependent delay

In this work, we propose sufficient conditions guaranteeing an existence result of mild solutions by using the nonlinear Leray-Schauder alternative in Banach spaces combined with the semigroup theory for the class of Caputo partial semilinear fractional evolution equations with finite state-dependent delay and nonlocal conditions.

Rothe’s Fixed Point Theorem and the Approximate Controllability of Semilinear Heat Equation with Impulses, Delays, and Nonlocal Conditions

Rothe’s Fixed Point Theorem and the Approximate Controllability of Semilinear Heat Equation with Impulses, Delays, and Nonlocal Conditions

An analysis on the approximate controllability of neutral impulsive stochastic integrodifferential inclusions via resolvent operators

This article focuses on the approximate controllability of impulsive neutral stochastic integrodifferential inclusions in Hilbert spaces. We used resolvent operators, fixed point approaches, and semigroup theory to achieve the article's main results. First, we focus on the existence of approximate controllability, and we develop the existence results with nonlocal conditions. At last, an application is provided to illustrate the concept.

Existence of sign-changing solutions for a third-order boundary value problem with nonlocal conditions of integral type

We prove the existence of at least one sign-changing solution for a third-order nonlocal boundary value problem by applying Leray-Schauder Continuation Principle. To illustrate the applicability of the obtained results, we consider an example.

Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay

Abstract In this work, we investigate the existence and uniqueness of solutions for retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay. To achieve this goal, we select an appropriate phase space that satisfies the axiomatic theory developed by Hale and Kato for studying differential equations with infinite delay. Then, we reformulate the problem of existence of solutions as the problem of finding fixed points of an operator. To this end, we apply the Karakostas fixed point theorem, which is an extension of the well-known Krasnoselskii fixed point theorem. Under certain conditions, we establish the uniqueness of solutions for our problem. Finally, we analyze the prolongation of solutions and demonstrate that, given certain conditions, these solutions are globally defined. Our research has important implications for the study of delayed dynamical systems, including models of population dynamics, physiological processes, and engineering systems.

Simultaneous inversion for a fractional order and a time source term in a time-fractional diffusion-wave equation

Abstract In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg–Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm.

Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven.

Results on approximate controllability for fractional stochastic delay differential systems of order r ∈ (1,2)

In this paper, we deal with the approximate controllability of fractional stochastic delay differential inclusions of order [Formula: see text]. By using fractional calculus, stochastic analysis, the theory of cosine family and Dhage fixed point techniques, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear fractional stochastic system. In particular, the results are established with the assumption that the associated linear part of the system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result.

Study of Non-Linear Impulsive Neutral Fuzzy Delay Differential Equations with Non-Local Conditions

This manuscript aims to investigate the existence and uniqueness of fuzzy mild solutions for non-local impulsive neutral functional differential equations of both first and second order, incorporating finite delay. Furthermore, the study explores the properties of fuzzy set-valued mappings of a real variable, where these mappings exhibit characteristics such as normality, convexity, upper semi-continuity, and compact support. The application of the Banach fixed-point theorem is employed to derive the results. The research extensively employs fundamental concepts from fuzzy set theory, functional analysis, and the Hausdorff metric. Additionally, an illustrative example is provided to exemplify the practical implementation of the proposed concept.

Efficient results on Hilfer pantograph model with nonlocal integral condition

The optimistic plan of this work reveals the solvability conditions of Hilfer fractional pantograph integrodifferential equations of type 0≤q2≤1 and order 0<q1<1. The presence of an integral solution and its uniqueness have been discussed with the ultimate fixed point techniques. At last, the observed results are substantiated by the illustration.

Mild solution in the α-norm for some partial integrodifferential equations involving a nonlocal condition

Abstract In this work, we investigate the existence of mild solutions in the α \alpha -norm for a class of nonlocal integrodifferential equations. We employ the theory of resolvent operators introduced by R. Grimmer. The Leray-Schauder alternative is the principal working tool for our analysis.

Mild solutions and controllability of fractional evolution inclusions of Clarke's subdifferential type with nonlocal conditions in Hilbert spaces

In this work, our aim is to ensure the existence of mild solutions and study the controllability of fractional evolution inclusions of Clarke's subdifferential type of order q∈(1,2) with nonlocal conditions in the setting of Hilbert spaces. Using fixed point techniques, fractional calculus, multivalued maps, cosine and sine function operators, we discuss the existence of mild solutions for the considered system. Moreover, under suitable conditions, we investigate the controllability of the proposed system. Finally, we present an illustrated example.

Modeling the Chromosphere and Transition Region of Planet-hosting Star GJ 436

Abstract Ahead of upcoming space missions intending to conduct observations of low-mass stars in the ultraviolet (UV) spectral region it becomes imperative to simultaneously conduct atmospheric modeling from the UV to the visible (VIS) and near-infrared (NIR). Investigations on extended spectral regions will help to improve the overall understanding of the diversity of spectral lines arising from very different atmospheric temperature regions. Here we investigate atmosphere models with a chromosphere and transition region for the M2.5V star GJ 436, which hosts a close-in Hot Neptune. The atmosphere models are guided by observed spectral features from the UV to the VIS/NIR originating in the chromosphere and transition region of GJ 436. High-resolution observations from the Hubble Space Telescope and Calar Alto high-Resolution search for M dwarfs with Exo-earths with Near-infrared and optical Echelle Spectrographs (CARMENES) are used to obtain an appropriate model spectrum for the investigated M dwarf. We use a large set of atomic species considered in nonlocal thermodynamic equilibrium conditions within our PHOENIX model computations to approximate the physics within the low-density atmospheric regions. In order to obtain an overall match for the nonsimultaneous observations, it is necessary to apply a linear combination of two model spectra, where one of them better reproduces the UV lines while the other better represents the lines from the VIS/NIR range. This is needed to adequately handle different activity states across the observations.

Hilfer fractional neutral stochastic integro‐differential evolution hemivariational inequalities and optimal controls

The focus of this study is a control system with a nonlocal state that is driven by a neutral stochastic integro‐differential evolution hemivariational inequality known as Hilfer fractional (HF). In order to demonstrate effective requirements for the existence of mild solutions to the relevant control system, we first make use of the features of the extended Clarke subdifferential and a fixed point theorem for multivalued functions. We next prove that there are optimal state‐control sets that are induced by HF neutral stochastic integro‐differential evolution hemivariational inequalities with a nonlocal condition by using limited Lagrange optimal structures. The distinctiveness of the control system's solutions is not taken into consideration while developing optimal control outcomes. At last, an example is employed to go over the major findings.