scholarly journals Controllability of Semilinear Multi-Valued Differential Inclusions with Non-Instantaneous Impulses of Order α ∈ (1,2) without Compactness

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 566
Author(s):  
Zainab Alsheekhhussain ◽  
Ahmed Gamal Ibrahim
Keyword(s):  
Differential Equation ◽  
Differential Inclusions ◽  
Linear Part ◽  
Previous Literature ◽  
Cosine Family ◽  
Semi Group ◽  
Strongly Continuous Cosine Family

Herein, we investigated the controllability of a semilinear multi-valued differential equation with non-instantaneous impulses of order α∈(1,2), where the linear part is a strongly continuous cosine family without compactness. We did not assume any compactness conditions on either the semi-group, the multi-valued function, or the inverse of the controllability operator, which is different from the previous literature.

scholarly journals Discrete Nonlinear Schrödinger Systems for Periodic Media with Nonlocal Nonlinearity: The Case of Nematic Liquid Crystals

2021 ◽  
Vol 11 (10) ◽  
pp. 4420
Author(s):  
Panayotis Panayotaros
Keyword(s):  
Differential Equation ◽  
Infinite System ◽  
Linear Part ◽  
Optical Power ◽  
Explicit Construction ◽  
Translation Invariance ◽  
Nonlocal Nonlinearity ◽  
Wannier Functions ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger Systems

We study properties of an infinite system of discrete nonlinear Schrödinger equations that is equivalent to a coupled Schrödinger-elliptic differential equation with periodic coefficients. The differential equation was derived as a model for laser beam propagation in optical waveguide arrays in a nematic liquid crystal substrate and can be relevant to related systems with nonlocal nonlinearities. The infinite system is obtained by expanding the relevant physical quantities in a Wannier function basis associated to a periodic Schrödinger operator appearing in the problem. We show that the model can describe stable beams, and we estimate the optical power at different length scales. The main result of the paper is the Hamiltonian structure of the infinite system, assuming that the Wannier functions are real. We also give an explicit construction of real Wannier functions, and examine translation invariance properties of the linear part of the system in the Wannier basis.

Nonlocal fractional semilinear differential inclusions with noninstantaneous impulses and of order α ∈ (1, 2)

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
JinRong Wang ◽  
Ahmed G. Ibrahim ◽  
Donal O’Regan ◽  
Adel A. Elmandouh
Keyword(s):  
Differential Inclusions ◽  
Mild Solutions ◽  
Multivalued Function ◽  
Linear Operators ◽  
Solution Set ◽  
Cosine Family ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Noninstantaneous Impulses

AbstractIn this paper, we establish the existence of mild solutions for nonlocal fractional semilinear differential inclusions with noninstantaneous impulses of order α ∈ (1,2) and generated by a cosine family of bounded linear operators. Moreover, we show the compactness of the solution set. We consider both the case when the values of the multivalued function are convex and nonconvex. Examples are given to illustrate the theory.

scholarly journals On controllability for a nondensely defined fractional differential equation with a deviated argument

2019 ◽  
Vol 13 (4) ◽  
pp. 407-413
Author(s):  
A. Raheem ◽  
M. Kumar
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Measure Of Noncompactness ◽  
Cosine Family ◽  
Abstract Result ◽  
Deviated Argument ◽  
Approximately Controllable ◽  
Fractional Differential

Abstract This article deals with a fractional differential equation with a deviated argument defined on a nondense set. A fixed-point theorem and the concept of measure of noncompactness are used to prove the existence of a mild solution. Furthermore, by using the compactness of associated cosine family, we proved that system is approximately controllable and obtains an optimal control which minimizes the performance index. To illustrate the abstract result, we included an example.

Existence for Second Order Differential Inclusions on ℝ+ Governed by Monotone Operators

2014 ◽  
Vol 14 (3) ◽  
Author(s):  
Gheorghe Moroşanu
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Differential Inclusions ◽  
Real Hilbert Space ◽  
Monotone Operators ◽  
Second Order

AbstractConsider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u″(t) + q(t)u′(t) ∈ Au(t) + f (t) for a.a. t ∈ ℝ

scholarly journals Existence results for an impulsive neutral integro-differential equations in Banach spaces

2019 ◽  
Vol 27 (3) ◽  
pp. 231-257
Author(s):  
Venkatesh Usha ◽  
Dumitru Baleanu ◽  
Mani Mallika Arjunan
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Group Theory ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Banach Spaces ◽  
Mild Solution ◽  
Existence Results ◽  
Semi Group ◽  
Schaefer Fixed Point Theorem

AbstractIn this manuscript we investigate the existence of mild solution for a abstract impulsive neutral integro-differential equation by using semi-group theory and Krasnoselskii-Schaefer fixed point theorem in different approach. At last, an example is also provided to illustrate the obtained results.

scholarly journals A Convolution Type Nonlinear Integro-Differential Equation with a Variable Coefficient and an Inhomogeneity in the Linear Part

2020 ◽  
pp. 16-27
Author(s):  
С.Н. Асхабов
Keyword(s):  
Differential Equation ◽  
Variable Coefficient ◽  
Linear Part ◽  
Convolution Type ◽  
Integro Differential Equation

Изучается вольтерровское интегро-дифференциальное уравнение типа свертки со степенной нелинейностью, переменным коэффициентом $a(x)$ и неоднородностью $f(x)$ в линейной части, которое тесно связано с соответствующим нелинейным интегральным уравнением, возникающим при исследовании инфильтрации жидкости из цилиндрического резервуара в изотропную однородную пористую среду, при описании процесса распространения ударных волн в трубах, наполненных газом, при решении задачи о нагревании полубесконечного тела при нелинейном теплопередаточном процессе, в моделях популяционной генетики и других. Важно отметить, что в связи с указанными и другими приложениями особый интерес представляют непрерывные положительные при $x>0$ решения интегрального уравнения. На основе полученных точных нижней и верхней априорных оценок решения интегрального уравнения мы строим весовое полное метрическое пространство $P_b$, инвариантное относительно нелинейного интегрального оператора свертки, порожденного этим уравнением, и, применяя метод весовых метрик (аналог метода Белицкого), доказываем глобальную теорему о существовании и единственности решения изучаемого нелинейного интегро-дифференциального уравнения как в пространстве $P_b$, так и во всем классе $Q_0^1$ непрерывно дифференцируемых положительных при $x>0$ функций. Показано, что решение может быть найдено в пространстве $P_b$ методом последовательных приближений пикаровского типа. Для последовательных приближений получены оценки скорости их сходимости к точному решению в терминах весовой метрики пространства~$P_b$. В частности, при $f(x)=0$ из этой теоремы вытекает, что соответствующее однородное нелинейное интегро-дифференциальное уравнение, в отличие от линейного случая, имеет нетривиальное решение. Приведены также примеры, иллюстрирующие полученные результаты.

Exact and trajectory controllability of second order nonlinear differential equations with deviated argument

2017 ◽  
Vol 26 (2) ◽  
pp. 181-191
Author(s):  
M. MUSLIM ◽  
AVADHESH KUMAR ◽  
RAVI P. AGARWAL
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Exact Controllability ◽  
Second Order ◽  
Fixed Point Method ◽  
Linear Operators ◽  
Cosine Family ◽  
Bounded Linear ◽  
Deviated Argument ◽  
Strongly Continuous Cosine Family

In this manuscript, we consider a control system governed by a second order nonlinear differential equations with deviated argument in a Hilbert space X. We used the strongly continuous cosine family of bounded linear operators and fixed point method to study the exact and trajectory controllability. Also, we study the exact controllability of the nonlocal control problem. Finally, we give an example to illustrate the application of these results.

scholarly journals On generalized boundary value problems for a class of fractional differential inclusions

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Irene Benedetti ◽  
Valeri Obukhovskii ◽  
Valentina Taddei
Keyword(s):  
Differential Inclusions ◽  
Nonlinear Term ◽  
Reflexive Banach Space ◽  
Mild Solutions ◽  
Linear Part ◽  
Fractional Differential Inclusions ◽  
Local Conditions ◽  
Non Local ◽  
Fractional Differential ◽  
Theoretical Results

AbstractWe prove existence of mild solutions to a class of semilinear fractional differential inclusions with non local conditions in a reflexive Banach space. We are able to avoid any kind of compactness assumptions both on the nonlinear term and on the semigroup generated by the linear part. We apply the obtained theoretical results to two diffusion models described by parabolic partial integro-differential inclusions.

scholarly journals COMPOSITION METHODS FOR THE SIMULATION OF ARRAYS OF CHUA'S CIRCUITS

1999 ◽  
Vol 09 (04) ◽  
pp. 723-733
Author(s):  
YVES MOREAU ◽  
JOOS VANDEWALLE
Keyword(s):  
Differential Equation ◽  
Differential Geometry ◽  
Linear Part ◽  
Fixed Time ◽  
Time Step ◽  
Nonlinear Part ◽  
Composition Methods ◽  
Algebra Theory ◽  
Simple Integration ◽  
Integration Rules

Composition methods are methods for the integration of ordinary differential equations arising from differential geometry, or more precisely, Lie algebra theory. We apply them here to the simulation of arrays of Chua's circuits. In these methods, we split the vector field of the array of Chua's circuits into its linear part and its nonlinear part. We then solve the elementary differential equation for each part separately — which is easy since the equations for the nonlinear part are all decoupled — and recombine these contributions into a sequence of compositions. This splitting gives rise to simple integration rules for arrays of Chua's circuits, which we compare to more classical approaches: fixed time-step explicit Euler and adaptive fourth-order Runge–Kutta.

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