scholarly journals Approximate controllability of nonlocal impulsive neutral integro-differential equations with finite delay

2021 ◽  
Author(s):  
Kamal Jeet ◽  
Dwijendra Pandey
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Operator Theory ◽  
Resolvent Operator ◽  
Semigroup Theory ◽  
Continuous Functions ◽  
Lipschitz Continuous ◽  
Finite Delay ◽  
First Case ◽  
The Resolvent Operator

In this paper, we apply the resolvent operator theory and an approximating technique to derive the existence and controllability results for nonlocal impulsive neutral integro-differential equations with finite delay in a Hilbert space. To establish the results, we take the impulsive functions as a continuous function only, and we assume that the nonlocal initial condition is Lipschitz continuous function in the first case and continuous functions only in the second case. The main tools applied in our analysis are semigroup theory, the resolvent operator theory, an approximating technique, and fixed point theorems. Finally, we illustrate the main results with the help of two examples.

scholarly journals On the invertibility of solutions of first order linear homogeneous differential equations in Banach algebras

2019 ◽  
Vol 21 (4) ◽  
pp. 430-442
Author(s):  
Oleg E. Galkin ◽  
Svetlana Y. Galkina
Keyword(s):  
Differential Equations ◽  
Banach Algebras ◽  
Foundations Of Quantum Mechanics ◽  
Continuous Functions ◽  
Heisenberg Equation ◽  
Complex Numbers ◽  
Bounded Operators ◽  
First Order ◽  
First Case ◽  
Families Of Operators

This work is devoted to the study of some properties of linear homogeneous differential equations of the first order in Banach algebras. It is found (for some types of Banach algebras), at what right-hand side of such an equation, from the invertibility of the initial condition it follows the invertibility of its solution at any given time. Associative Banach algebras over the field of real or complex numbers are considered. The right parts of the studied equations have the form [F(t)](x(t)), where {F(t)} is a family of bounded operators on the algebra, continuous with respect to t∈R. The problem is to find all continuous families of bounded operators on algebra, preserving the invertibility of elements from it, for a given Banach algebra. In the proposed article, this problem is solved for only three cases. In the first case, the algebra consists of all square matrices of a given order. For this algebra, it is shown that all continuous families of operators, preserving the invertibility of elements from the algebra at zero must be of the form [F(t)](y)=a(t)⋅y+y⋅b(t), where the families {a(t)} and {b(t)} are also continuous. In the second case, the algebra consists of all continuous functions on the segment. For this case, it is shown that all families of operators, preserving the invertibility of elements from the algebra at any time must be of the form [F(t)](y)=a(t)⋅y, where the family {a(t)} is also continuous. The third case concerns those Banach algebras in which all nonzero elements are invertible. For example, the algebra of complex numbers and the algebra of quaternions have this property. In this case, any continuous families of bounded operators preserves the invertibility of the elements from the algebra at any time. The proposed study is in contact with the research of the foundations of quantum mechanics. The dynamics of quantum observables is described by the Heisenberg equation. The obtained results are an indirect argument in favor of the fact, that the known form of the Heisenberg equation is the only correct one.

On the Uniqueness and Boundedness of Solutions of Hyperbolic Differential Equations

1962 ◽  
Vol 58 (4) ◽  
pp. 583-587 ◽  
Author(s):  
V. Lakshmikantham
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Boundedness Of Solutions ◽  
Initial Value ◽  
Lipschitz Continuous ◽  
Almost Everywhere ◽  
Uniformly Lipschitz ◽  
Image Position ◽  
Hyperbolic Differential Equations ◽  
Characteristic Initial Value Problem

Consider a characteristic initial value problem of partial differential equationswhere the functions E (x) and F (y) are real valued, uniformly Lipschitz continuous on 0 ≤ x ≤ a, 0 ≤ y ≤ b, respectively. Suppose f (x, y, u, p, q) is a real-valued and continuous function defined on 0 ≤ ≤ b. By a solution of (1), we mean a real-valued continuous function u (x, y), having partial derivatives ux (x, y), uy (x, y) and ux, y (x, y) in the domain 0 ≤ x ≤ a, 0 ≤ y ≤ b almost everywhere.

scholarly journals Linear differential equations with unbounded delays and a forcing term

2004 ◽  
Vol 2004 (4) ◽  
pp. 337-345 ◽  
Author(s):  
Jan Čermák ◽  
Petr Kundrát
Keyword(s):  
Differential Equation ◽  
Functional Equation ◽  
Continuous Function ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Functional Equations ◽  
Continuous Functions ◽  
Positive Continuous Function ◽  
All Solutions ◽  
Linear Differential

The paper discusses the asymptotic behaviour of all solutions of the differential equationy˙(t)=−a(t)y(t)+∑i=1nbi(t)y(τi(t))+f(t),t∈I=[t0,∞), with a positive continuous functiona, continuous functionsbi,f, andncontinuously differentiable unbounded lags. We establish conditions under which any solutionyof this equation can be estimated by means of a solution of an auxiliary functional equation with one unbounded lag. Moreover, some related questions concerning functional equations are discussed as well.

THE EFFECTS OF THE RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL ON THE BOX DIMENSION OF FRACTAL GRAPHS OF HÖLDER CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050052
Author(s):  
JUNRU WU
Keyword(s):  
Continuous Function ◽  
Fractional Integral ◽  
Closed Interval ◽  
Open Interval ◽  
Continuous Functions ◽  
Box Dimension ◽  
Hölder Continuous ◽  
Positive Order ◽  
Lipschitz Continuous ◽  
Hölder Continuous Functions

In this paper, the linearity of the dimensional-decrease effect of the Riemann–Liouville fractional integral is mainly explored. It is proved that if the Box dimension of the graph of an [Formula: see text]-Hölder continuous function is greater than one and the positive order [Formula: see text] of the Riemann–Liouville fractional integral satisfies [Formula: see text], the upper Box dimension of the Riemann–Liouville fractional integral of the graph of this function will not be greater than [Formula: see text]. Furthermore, it is proved that the Riemann–Liouville fractional integral of a Lipschitz continuous function defined on a closed interval is continuously differentiable on the corresponding open interval.

ON BOUNDED WEAK AND PSEUDO-SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS HAVING TRICHOTOMY WITH AND WITHOUT DELAY IN BANACH SPACES

2010 ◽  
Vol 07 (03) ◽  
pp. 357-366
Author(s):  
ADEL MAHMOUD GOMAA
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Continuous Functions ◽  
Linear Operators ◽  
Absolutely Continuous ◽  
Integrable Operator ◽  
Weakly Sequentially Continuous ◽  
Bochner Integrable Function

In the present work we consider E is a Banach space, E* is its dual space and L(E) is the space of continuous linear operators from E to itself. A function x: ℝ → E is said to be a pseudo-solution of the equation [Formula: see text] where A:ℝ → L(E) is strongly measurable and Bochner integrable function on every finite subinterval of ℝ with f:ℝ × E → E is only assumed to be weakly weakly sequentially continuous or Pettis-integrable and the linear equation [Formula: see text] has a trichotomy with constants α ≥ 1 and σ > 0, if x is absolutely continuous function and for each x* ∈ E* there exists a negligible set ℵx* such that for each t ∉ ℵx*, then we have [Formula: see text] We give an existence theorem for bounded weak and pseudo-solutions of the nonlinear differential equations [Formula: see text] Let T, r, d > 0, Br = {x > E: ‖x‖ ≤ r} and CE([-d,0]) be the Banach space of continuous functions from [-d,0] into E. Finally we prove an existence result for the differential equation with delay [Formula: see text] where fd : [a,b] × CE([-d,0]) → E is weakly weakly sequentially continuous function, [Formula: see text] is strongly measurable and Bochner integrable operator on [a,b] and θtx(s) = x(t + s) for all s ∈ [-d,0].

scholarly journals Control of systems with Holder continuous functions in the distributed delays

2014 ◽  
Vol 30 (1) ◽  
pp. 123-128
Author(s):  
NASSER-EDDINE TATAR ◽  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Continuous Functions ◽  
Distributed Delays ◽  
Exponential Stabilization ◽  
Activation Functions ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Hölder Continuous Functions ◽  
Doubly Nonlinear

An exponential stabilization result is proved for a doubly nonlinear distributed delays system of ordinary differential equations. The problem involves non-Lipschitz continuous distributed delays of non-Lipschitz continuous ”activation” functions. This extends similar previous works where the distributed delays as well as the activation functions were assumed to be Lipschitz continuous.

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