Douglas R. Anderson

Verified

Douglas verified their affiliation via an institutional email.

Learn more

Verified

Douglas verified their affiliation via an institutional email.

Learn more

• Ph. D.
• Professor (Full) at Concordia College

The purpose of this work is to show that the Khalil and Katagampoula conformable derivatives are equivalent to the simple change of variables x → x α /α, where α is the order of the derivative operator, when applied to differential functions. Although this means no "new mathematics" is obtained by working with these derivatives, it is a second purp...

In this study, the Ulam stability of quantum equations on time scales that alternate between two quanta is considered. We show that linear equations of first order with constant coefficient or of Euler type are Ulam stable across large regions of the complex plane, and give the best Ulam constants for those regions. We also show, however, that line...

In this paper, using the comparison theorem on time scales and some elementary algebraic inequalities, we introduce a new class of generalizations of Pachpatte-type inequalities and their extensions on time scales, which provide explicit bounds on unknown functions. Some applications of the main results for specific nonlinear dynamic equations are...

This study investigates the conditional Hyers–Ulam stability of a first-order nonlinear h-difference equation, specifically a discrete logistic model. Identifying bounds on both the relative size of the perturbation and the initial population size is an important issue for nonlinear Hyers–Ulam stability analysis. Utilizing a novel approach, we deri...

We apply a new definition of periodicity on isolated time scales introduced by Bohner, Mesquita, and Streipert to the study of Ulam stability. If the graininess (step size) of an isolated time scale is bounded by a finite constant, then the linear \(\omega \)-periodic dynamic equations are Ulam stable if and only if the exponential function has mod...

We decompose an operator associated with a right-focal boundary value problem, whose fixed points are solutions of the boundary value problem, into multiple fixed point problems. We provide conditions for the original boundary value problem to have a solution that can be found by iteration using the decomposition.

An associated Riccati equation is used to study the Ulam stability of non-autonomous linear differential equations that model the damped linear oscillator. In particular, the best (minimal) Ulam constants for these equations are derived. These robust results apply to equations with solutions that blow up in finite time and to equations with solutio...

This study deals with the Ulam stability of nonautonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two‐dimensional nonautonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are appli...

Due to the restrictive growth and/or monotonicity requirements inherent in their employment, classical iterative fixed-point theorems are rarely used to approximate solutions to an integral operator with Green’s function kernel whose fixed points are solutions of a boundary value problem. In this paper, we show how one can decompose a fixed-point p...

In this paper, we introduce certain operators 𝑇^𝑚_𝜆 and 𝑇^𝑚_𝜑 on weighted Hardy spaces 𝐻^2_𝛽, and in the following, we investigate their boundedness on 𝐻^2_𝛽. After that, we prove the Hyers–Ulam stability for certain operators on weighted Hardy spaces 𝐻^2_𝛽. Moreover, we show under what conditions these concepts are stable and unstable by using som...

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential...

In this paper, we improve the averaging theory on both finite and infinite time intervals for discrete fractional-order systems with impulses. By employing new techniques, generalized impulsive discrete fractional-order Gronwall inequality is introduced. In addition, the closeness of solutions for the discrete fractional-order systems with impulses...

We establish the Hyers–Ulam stability of a second-order linear Hill-type h-difference equation with a periodic coefficient. Using results from first-order h-difference equations with periodic coefficient of arbitrary order, both homogeneous and non-homogeneous, we also establish a Hyers–Ulam stability constant. Several interesting examples are prov...

We establish the Hyers–Ulam stability of a second-order finite difference scheme using a diamond-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document...

Converting nonlinear boundary value problems to fixed point problems of an integral operator with a Green's function kernel is a common technique to find or approximate solutions of boundary value problems. It is often difficult to apply Banach's Theorem since it is challenging to find an initial estimate with a contractive constant less than one....

This study uses an associated Riccati equation to study the Ulam stability of non-autonomous linear differential vector equations that model the damped linear oscillator. In particular, the best (minimal) Ulam constants for these non-autonomous linear differential vector equations are derived. These robust results apply to vector equations with sol...

This study deals with the Ulam stability of non-autonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two-dimensional non-autonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are app...

First, we show that first-order linear q-difference equations with 1-periodic or 2-periodic coefficients are not Ulam stable. Second, we establish the best (minimum) constant for modified Ulam stability of first-order linear q-difference equations with a 1-periodic coefficient. Third, we establish the best (minimum) constant for modified Ulam stabi...

We establish the Hyers-Ulam stability of a second-order linear Hill-type $h$-difference equation with a periodic coefficient. Using results from first-order $h$-difference equations with periodic coefficient of arbitrary order, both homogeneous and non-homogeneous, we also establish a Hyers-Ulam stability constant. Several interesting examples are...

We establish the Lyapunov stability of the equilibrium (trivial) solution for the discrete diamond–alpha difference operator, using the imaginary diamond–alpha ellipse. This unifies and extends equilibrium analysis for first-order forward (Delta) and backward (nabla) difference equations with a constant complex coefficient. We prove that for coeffi...

In this paper, we prove some new dynamic inequalities of Opial type involving higher-order derivatives of two functions, with two different weights on time scales. From these inequalities, we will derive some special cases and give an improvement of some versions of recent results.

In this paper, we establish a new class of dynamic inequalities of Hardy’s type which generalize and improve some recent results given in the literature. More precisely, we prove some new Hardy-type inequalities involving many functions on time scales. Some new discrete inequalities are deduced in seeking applications.

In this paper, we prove some new characterizations of weighted functions for dynamic inequalities of Hardy's type involving monotonic functions on a time scale T in different spaces L p (T) and L q (T) when 0 < p < q < ∞ and p ≤ 1. The main results will be proved by employing the reverse Hölder inequality, integration by parts, and the Fubini theor...

We explore the Hyers–Ulam stability of perturbations for a homogeneous linear differential system with 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\en...

We explore the Hyers-Ulam stability of perturbations for a homogeneous linear differential system with $2\times 2$ constant coefficient matrix. New necessary and sufficient conditions for the linear system to be Hyers-Ulam stable are proven, and for the first time, the best (minimal) Hyers-Ulam constant for systems is found in some cases. Several e...

In this paper, we prove some new characterizations of two weighted functions u and v in norm inequalities of Hardy’s type, in the context of dynamic inequalities on time scales \({\mathbb {T}}\). These norm inequalities studied the boundedness of the operator of Hardy’s type between the weighted spaces \( L_{v}^{p}({\mathbb {T}})\) and \(L_{u}^{q}(...

In this paper, we study discrete fractional order singular systems with multiple time-varying delays. By use of new techniques, some useful fractional order difference inequalities are given. Then, conditions based on matrix inequalities and discrete fractional Lyapunov direct method have been obtained for the asymptotic stability of such systems....

An integrating factor is used to convert a conjugate boundary value problem to a fixed-point problem. We conclude with an application illustrating the ease of use in finding an upper solution to a family of boundary value problems that one can apply iteration to in order to solve when the nonlinear term is monotonic.

We establish the Ulam stability of a first-order linear nonautonomous quantum equation with Cayley parameter in terms of the behavior of the nonautonomous coefficient function. We also provide details for some cases of Ulam instability.

We investigate the Hyers—Ulam stability (HUS) of certain second-order linear constant coefficient dynamic equations on time scales, building on recent results for first-order constant coefficient time-scale equations. In particular, for the case where the roots of the characteristic equation are non-zero real numbers that are positively regressive...

The main purpose of this study is to clarify the Hyers–Ulam stability (HUS) for the Cayley quantum equation. In addition, the result obtained for all parameters is applied to HUS of h -difference equations with a specific variable coefficient using a new transformation.

In this paper, we prove some new characterizations of weighted functions for dynamic inequalities of Hardy’s type involving monotonic functions on a time scale \(\mathbb {T}\) in different spaces \(L^{p}(\mathbb {T})\) and \(L^{q}( \mathbb {T})\) when \(0<p<q<\infty \) and \(p\le 1\). The main results will be proved by employing the reverse Hölder...

A new definition of a multivalued logarithm on time scales is introduced for delta-differentiable functions that never vanish. This new logarithm arises naturally from the definition of the cylinder transformation that is also the wellspring of the definition of exponential functions on time scales. This definition will lead to a logarithm function...

We introduce and study the Hyers–Ulam stability (HUS) of a Cayley quantum (q-difference) equation of first order, where the constant coefficient is allowed to range over the complex numbers. In particular, if this coefficient is non-zero, then the quantum equation has Hyers–Ulam stability for certain values of the Cayley parameter, and we establish...

Consider a time scale consisting of a discrete core with uniform step size, augmented with a continuous-interval periphery. On this time scale, we determine the best constants for the Hyers–Ulam stability of a first-order dynamic equation with complex constant coefficient, based on the placement of the complex coefficient in the complex plane, with...

In this book excerpt, the conformable Laplace transform on time scales using a conformable proportional derivative is explored. In particular, after defining the conformable Laplace transform and listing some of its properties, we discuss the decay of the conformable exponential function on time scales, the convergence of this transform, and then a...

In this book excerpt, linear second-order conformable dynamic equations on time scales using a conformable proportional derivative are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary conditions. Defining a Wronskian, we establish a Lagrange identity and Abel's formula. Sev...

Using an integrating factor, a second order boundary value problem is transformed into a fixed point problem. We provide growth conditions for the existence of a fixed point to the associated operator for this transformation and conclude that the index of the operator applying the standard Green’s function approach is zero; this does not guarantee...

This study deals with the Hyers–Ulam stability (HUS) for the first-order linear difference equations with two alternating step sizes, where the coefficient is allowed to be complex valued. In particular, it turns out that the best HUS constant can be determined by finding an explicit solution to the corresponding inhomogeneous linear equation. Spec...

In this paper, we give a new method to show the monotonicity results for a function f satisfying (a ∇ h)(t) ≤ 0 (or (a ∇ h, *)(t) ≤ 0) with ∈ (0,1], which has never been solved in the existing literatures. In addition, we give an example to illustrate one of our main results. KEYWORDS discrete fractional calculus, monotonicity, nabla fractional h-d...

We investigate the Hyers-Ulam stability (HUS) of a certain first-order linear complex constant coefficient dynamic equation on the time scale P α,h , which has continuous intervals of length α > 0 followed by discrete jumps of length h > 0. In particular, we establish results in the case of this specific time scale, for coefficient values in the co...

In this chapter excerpt from our book Conformable Dynamic Equations on Time Scales (2020 CRC Press), linear conformable dynamic inequalities on time scales using a conformable proportional derivative are explored. In particular, conformable versions of Gronwall, Gamidov, Rodrigues, Pachpatte, and some integral inequalities on time scales are establ...

We establish the best (minimum) constant for Ulam stability of first-order linear h-difference equations with a periodic coefficient. First, we show Ulam stability and find the Ulam stability constant for a first-order linear equation with a period-two coefficient, and give several examples. In the last section we prove Ulam stability for a periodi...

Many applications using discrete dynamics employ either q-difference equations or h-difference equations. In this work, we introduce and study the Hyers–Ulam stability (HUS) of a quantum (q-difference) equation of Euler type. In particular, we show a direct connection between quantum equations of Euler type and h-difference equations of constant st...

We introduce and study the Hyers--Ulam stability (HUS) of a Cayley quantum ($q$-difference) equation of first order, where the constant coefficient is allowed to range over the complex numbers. In particular, if this coefficient is non-zero, then the quantum equation has Hyers--Ulam stability for certain values of the Cayley parameter, and we estab...

In this paper, the Hyers-Ulam stability and generalized Hyers-Ulam stability of sequential fractional order h-difference equations are investigated using the open mapping theorem and the direct method, respectively. Finally, we give an example to illustrate one of our main results.

We establish the best (minimum) constant for Ulam stability of first-order linear $h$-difference equations with a periodic coefficient. First, we show Ulam stability and find the Ulam stability constant for a first-order linear equation with a period-two coefficient, and give several examples. In the last section we prove Ulam stability for a perio...

A Cayley h-difference equation uses the forward-difference operator with step size h, and then some proportion of the function value and the advanced-function value. For such an equation with a complex constant coefficient, we establish that the equation exhibits instability along a certain circle, but is Hyers-Ulam stable inside and outside that c...

A new definition of a multi-valued logarithm on time scales is introduced for delta-differentiable functions that never vanish. This new logarithm arises naturally from the definition of the cylinder transformation that is also at the heart of the definition of exponential functions on time scales. This definition will lead to a logarithm function...

In this paper, we decompose an operator as the sum of an increasing operator and a decreasing operator, with a condition that the change in the increasing component is greater than the change in the decreasing component on a given set, which allows us to use a monotonic iterative method to find a fixed point for the operator. We illustrate the meth...

We establish the Hyers–Ulam stability (HUS) of a certain first-order linear constant coefficient discrete diamond-alpha derivative equation. In particular, for each parameter value we determine whether the equation has HUS, and if so whether there exists a minimum HUS constant.

We introduce the imaginary diamond-alpha ellipse, which unifies and extends the left Hilger imaginary circle (forward, Delta case) and the right Hilger imaginary circle (backward, nabla case), for the discrete diamond-alpha derivative with constant step size. We then establish the Hyers-Ulam stability (HUS) of the first-order linear complex constan...

The standard methods of applying iterative techniques do not apply when the nonlinear term is neither monotonic (corresponding to an increasing or decreasing operator) nor Lipschitz (corresponding to a condensing operator). However, by applying the Layered Compression-Expansion Theorem in conjunction with an alternative inversion technique, we show...

First, we prove that the best (minimum) constant for Hyers–Ulam stability of first-order linear h-difference equations with a complex constant coefficient is the reciprocal of the absolute value of the Hilger real part of that coefficient; if the coefficient lies on the Hilger imaginary circle, then the equation is unstable in the Hyers–Ulam sense....

In this paper we will convert a fixed point problem of the form x = R(x)S(x) to a fixed point system of the form r = R(rs) and s = S(rs) so we can apply compression-expansion fixed point arguments on our operators R and S instead of on the product of the operators RS.

The Hyers-Ulam stability (HUS) of a certain first-order proportional nabla difference equation with a sign-alternating coefficient is established. For those parameter values for which HUS holds, an HUS constant is found, and in special cases it is shown that this is the minimal such constant possible. A 2-cycle solution and a 4-cycle solution are s...

We establish the Hyers-Ulam stability (HUS) of a certain first-order linear constant coefficient discrete diamond-alpha derivative equation. In particular, for each parameter value we determine whether the equation has HUS, and if so whether there exists a minimum HUS constant.

We clarify the Hyers–Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, in the case of a specific time scale with two alternating step sizes, where the exponential function changes sign. In particular, in the case of HUS, we discuss the HUS constant, and whether a minimal HUS constant can be fo...

We clarify the Hyers-Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, in the case of a specific time scale with two alternating step sizes, where the exponential function changes sign. In particular, in the case of HUS, we discuss the HUS constant, and whether a minimal HUS constant can be fo...

We establish the Hyers-Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, which include the continuous (step size zero) and the discrete (step size constant and nonzero) dynamic equations as important special cases. In particular, for certain parameter values in relation to the graininess of th...

We clarify the Hyers--Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, in the case of a specific time scale with two alternating step sizes, where the exponential function changes sign. In particular, in the case of HUS, we discuss the HUS constant, and whether a minimal constant can be found...

In this paper we show how one can replace expansion-compression conditions in the Layered Compression-Expansion Fixed Point Theorem by Leggett-Williams conditions. We then show how we can stack Leggett-Williams conditions on any of the expansioncompression conditions of a fixed point theorem. We remark on the utility of using Leggett-Williams condi...

In this paper we show how one can use suitable k-contractive conditions to prove that iterates converge to a fixed point guaranteed by a compression-expansion fixed point theorem of functional type, even though the operator is not known to be invariant on the underlying set.

The layered compression-expansion fixed point theorem is an alternative approach to the Krasnoselskii fixed point theorem for perturbed operators. The layered compression-expansion fixed point theorem is used to verify the existence of a fixed point to an operator of the form T = R + S (sum of operators) by verifying the existence of a fixed point...

This paper presents a generalization of the Functional Omitted Ray Fixed Point Theorem which provides a means to put conditions on the underlying set in our fixed point arguments. We conclude with an application to show how the range of the operator can be used to define the underlying set A.

In this study, even order self-adjoint differential equations incorporating recently introduced proportional derivatives, and their associated self-adjoint boundary conditions, are discussed. Using quasi derivatives, a Lagrange bracket and bilinear functional are used to obtain a Lagrange identity and Green's formula; this also leads to the class...

In this study, linear second-order matrix equations using a proportional derivative are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary conditions. We also introduce a generalized Wronskian and establish a Lagrange identity and Abel's formula. Two reduction-of-order theore...

Using a differential operator modeled after a proportional-derivative controller (PD controller), linear second-order differential equations are shown to be formally self adjoint with respect to a certain inner product and the associated self-adjoint boundary conditions. Defining a Wronskian, we establish a Lagrange identity and Abel's formula. Sev...

We derive Taylor’s theorem using a variation of constants formula for conformable fractional derivatives. This is then employed to extend some recent and classical integral inequalities to the conformable fractional calculus, including the inequalities of Steffensen, Chebyshev, Hermite–Hadamard, Ostrowski, and Grüss.

In this paper we show how the interplay between a function and its derivative can be used to define a comparison functional when applying the functional compression-expansion fixed point theorem.

In this paper we show how the interplay between a function and its derivative can be used to define a comparison functional when applying the functional compression-expansion fixed point theorem.

In this article we use an interval of functional type as the underlying set in our compression-expansion fixed point theorem argument which can be used to exploit properties of the operator to improve conditions that will guarantee the existence of a fixed point in applications. An example is provided to demonstrate how intervals of functional type...

We investigate two types of first-order, two-point boundary value problems (BVPs). Firstly, we study BVPs that involve nonlinear difference equations (the “discrete” BVP); and secondly, we study BVPs involving nonlinear ordinary differential equations (the “continuous” BVP). We formulate some sufficient conditions under which the discrete BVP will...

In this study, linear second-order conformable differential equations using a proportional derivative are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary conditions. Defining a Wronskian, we establish a Lagrange identity and Abel's formula. Several reduction-of-order theor...

We use a newly introduced conformable derivative to formulate several boundary value problems with three or four conformable derivatives, including those with conjugate, right-focal, and Lidstone conditions. With the conformable differential equation and boundary conditions established, we find the corresponding Green's functions and prove their po...

In this article we use an interval of functional type as the underlying set in our compression-expansion fixed point theorem argument which can be used to exploit properties of the operator to improve conditions that will guarantee the existence of a fixed point in applications. An example is provided to demonstrate how intervals of functional type...

We discuss the method of undetermined coe¢ cients for fractional di¤erential equations , where we use the (local) conformable fractional derivative presented in [1]. The concept of fractional polynomials, fractional exponentials and fractional trigonometric functions is introduced. A method similar to the case of ordinary di¤erential equations is e...

We discuss the method of undetermined coefficients for fractional differential equations, where we use the (local) conformable fractional derivative presented in KHKS . The concept of fractional polynomials, fractional exponentials and fractional trigonometric functions is introduced. A method similar to the case of ordinary differential equations...

Motivated by a proportional-derivative (PD) controller, a more precise definition of a conformable derivative is introduced and explored. Results include basic conformable derivative and integral rules, Taylor's theorem, reduction of order, variation of parameters, complete characterization of solutions for constant coefficient and Cauchy-Euler typ...

Katugampola [e-print arXiv:1410.6535] recently introduced a limit based fractional derivative, Dα (referred to in this work as the Katugampola fractional derivative) that maintains many of the familiar properties of standard derivatives such as the product, quotient, and chain rules. Typically, fractional derivatives are handled using an integral r...

Using the new conformable fractional derivative, which differs from the Riemann-Liouville and Caputo fractional derivatives, we reformulate the second-order conjugate boundary value problem in this new setting. Utilizing the corresponding positive fractional Green's function, we apply a functional compression-expansion fixed point theorem to prove...

We establish the Hyers-Ulam stability of certain first-order linear differential equations where the coefficients are allowed to vanish. We then extend these results to higher-order linear differential equations with vanishing coefficients that can be written with these first-order factors.

This paper presents a corollary of the omitted ray fixed point theorem with an example that utilizes a non-standard existence of solutions argument, in conjunction with the mean value theorem, to prove the existence of a solution to a conjugate boundary value problem.

Motivated by a proportional-derivative (PD) controller, a more precise definition of a conformable derivative is introduced and explored. Results include basic conformable derivative and integral rules, Taylor's theorem, reduction of order, variation of parameters, complete characterization of solutions for constant coefficient and Cauchy-Euler typ...