Some results of Watson, Plancherel type integral transforms related to the Hartley, Fourier convolutions and applications

Symmetric Form ◽

Original Function ◽

Type Integral ◽

Necessary And Sufficient ◽

Sequence Of Functions

In this work, we study the Watson-type integral transforms for the convolutions related to the Hartley and Fourier transformations. We establish necessary and sufficient conditions for these operators to be unitary in the L 2 (R) space and get their inverse represented in the conjugate symmetric form. Furthermore, we also formulated the Plancherel-type theorem for the aforementioned operators and prove a sequence of functions that converge to the original function in the defined L 2 (R) norm. Next, we study the boundedness of the operators (T k ). Besides, showing the obtained results, we demonstrate how to use it to solve the class of integro-differential equations of Barbashin type, the differential equations, and the system of differential equations. And there are numerical examples given to illustrate these.

On the solvability of the antiperiodic boundary value problem for systems of linear generalized differential equations

Georgian Mathematical Journal ◽

10.1515/gmj-2017-0015 ◽

2017 ◽

Vol 24 (2) ◽

pp. 169-184

Author(s):

Malkhaz Ashordia

Keyword(s):

Differential Equations ◽

Boundary Value Problem ◽

Spectral Type ◽

Unique Solvability ◽

Boundary Value ◽

Type Theorem ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Generalized Differential ◽

AbstractAn antiperiodic boundary value problem is considered for systems of linear generalized differential equations. A Green-type theorem on the unique solvability of the problem and representation of its solution are established. Effective necessary and sufficient conditions (of spectral type) are given for the unique solvability of the problem as well.

Lagrangian descriptions of dissipative systems: a review

Mathematics and Mechanics of Solids ◽

10.1177/1081286520971834 ◽

2020 ◽

pp. 108128652097183

Author(s):

Alberto Maria Bersani ◽

Paolo Caressa

Keyword(s):

Differential Equations ◽

Equations Of Motion ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Dissipative Systems ◽

Lagrangian Formalism ◽

Lagrangian Description ◽

Helmholtz Conditions ◽

In this paper, we review classical and recent results on the Lagrangian description of dissipative systems. After having recalled Rayleigh extension of Lagrangian formalism to equations of motion with dissipative forces, we describe Helmholtz conditions, which represent necessary and sufficient conditions for the existence of a Lagrangian function for a system of differential equations. These conditions are presented in different formalisms, some of them published in the last decades. In particular, we state the necessary and sufficient conditions in terms of multiplier factors, discussing the conditions for the existence of equivalent Lagrangians for the same system of differential equations. Some examples are discussed, to show the application of the techniques described in the theorems stated in this paper.

H-function with complex parameters I: existence

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201005142 ◽

2001 ◽

Vol 25 (9) ◽

pp. 571-586

Author(s):

Fadhel A. Al-Musallam ◽

Vu Kim Tuan

Keyword(s):

Sufficient Conditions ◽

Integral Transforms ◽

Necessary And Sufficient Conditions ◽

Special Cases ◽

Type Integral ◽

AnH-function with complex parameters is defined by a Mellin-Barnes type integral. Necessary and sufficient conditions under which the integral defining theH-function converges absolutely are established. Some properties, special cases, and an application to integral transforms are given.

Minimizing of the quadratic functional on Hopfield networks

Electronic journal of qualitative theory of differential equations ◽

10.14232/ejqtde.2021.1.92 ◽

2021 ◽

pp. 1-20

Author(s):

Oleksandr Boichuk ◽

Dmytro Bihun ◽

Victor Feruk ◽

Oleksandr Pokutnyi

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Sufficient Conditions ◽

Quadratic Functional ◽

Problem Solution ◽

Linear Boundary ◽

Hopfield Networks ◽

Necessary And Sufficient ◽

The Given

In this paper, we consider the continuous Hopfield model with a weak interaction of network neurons. This model is described by a system of differential equations with linear boundary conditions. Also, we consider the questions of finding necessary and sufficient conditions of solvability and constructive construction of solutions of the given problem, which turn into solutions of the linear generating problem, as the parameter $\varepsilon$ tends to zero. An iterative algorithm for finding solutions has been constructed. The problem of finding the extremum of the target functions on the given problem solution is considered. To minimize a functional, an accelerated method of conjugate gradients is used. Results are illustrated with examples for the case of three neurons.

Convergence of Solutions of Third Order Differential Equations*

Canadian Mathematical Bulletin ◽

10.4153/cmb-1969-101-0 ◽

1969 ◽

Vol 12 (6) ◽

pp. 779-792 ◽

Cited By ~ 2

Author(s):

K. E. Swick

Keyword(s):

Differential Equations ◽

Sufficient Conditions ◽

Higher Order ◽

Similar Technique ◽

Third Order ◽

Convergence Of Solutions ◽

The Matrix ◽

Necessary And Sufficient ◽

Direct Use

Consider a system of differential equations . Solutions of this system are said to be convergent if, given any pair of solutions x(t), y(t), x(t) - y(t) → 0 as t → ∞. In this case the system is also said to be extremely stable.In [6] a technique was developed which yielded the convergence of solutions of the forced Lienard equation. Here a similar technique i s applied to forced third order equations. A critical step in [6] was to show that a certain matrix was negative definite. This could be done directly in [6] since the matrix was only 2 × 2. With third and higher order equations, direct use of necessary and sufficient conditions is not feasible since the computations become unwieldy.

Solution of Some Impulsive Differential Equations via Coupled Fixed Point

Symmetry ◽

10.3390/sym13030501 ◽

2021 ◽

Vol 13 (3) ◽

pp. 501

Author(s):

Ahmed Boudaoui ◽

Khadidja Mebarki ◽

Wasfi Shatanawi ◽

Kamaleldin Abodayeh

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Fixed Points ◽

Metric Space ◽

Fixed Point Theorems ◽

Coupled Fixed Point ◽

Sufficient Conditions ◽

Impulsive Differential Equations ◽

Coupled Fixed Points

In this article, we employ the notion of coupled fixed points on a complete b-metric space endowed with a graph to give sufficient conditions to guarantee a solution of system of differential equations with impulse effects. We derive recisely some new coupled fixed point theorems under some conditions and then apply our results to achieve our goal.

Second-order impulsive differential systems with mixed and several delays

Advances in Difference Equations ◽

10.1186/s13662-021-03474-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Shyam Sundar Santra ◽

Apurba Ghosh ◽

Omar Bazighifan ◽

Khaled Mohamed Khedher ◽

Taher A. Nofal

Keyword(s):

Differential Equations ◽

Sufficient Conditions ◽

Impulsive Differential Equations ◽

Oscillation Theory ◽

Second Order ◽

Neutral Delay ◽

Differential Systems ◽

Impulsive Differential Systems ◽

Several Delays ◽

AbstractIn this work, we present new necessary and sufficient conditions for the oscillation of a class of second-order neutral delay impulsive differential equations. Our oscillation results complement, simplify and improve recent results on oscillation theory of this type of nonlinear neutral impulsive differential equations that appear in the literature. An example is provided to illustrate the value of the main results.