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Estimating the fraction of dementia cases in a population attributable to a risk factor or combination of risk factors (the population attributable fraction (PAF)) informs the design and choice of dementia risk-reduction activities. It is directly relevant to dementia prevention policy and practice. Current methods employed widely in the dementia literature to combine PAFs for multiple dementia risk factors assume a multiplicative relationship between factors and rely on subjective criteria to develop weightings for risk factors. In this paper we present an alternative approach to calculating the PAF based on sums of individual risk. It incorporates individual risk factor interrelationships and enables a range of assumptions about the way in which multiple risk factors will combine to affect dementia risk. Applying this method to global data demonstrates that the previous estimate of 40% is potentially too conservative an estimate of modifiable dementia risk and would necessitate subadditive interaction between risk factors. We calculate a plausible conservative estimate of 55.7% (95% confidence interval: 55.2, 56.1) based on additive risk factor interaction. [ABSTRACT FROM AUTHOR]

Recently, Mandal and Ghosh ["Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip," Phys. Fluids 35, 047121 (2023)] constructed perturbation solutions for viscous flow in porous channels with a slip condition and moving walls restricted to slow wall dilation–contraction rates. Herein, we show that this "slowness" assumption may be completely removed. In doing so, we develop a more widely applicable and more accurate perturbation scheme for all dilation–contraction rates. Our strategy involves generating new and exact solutions to the linear, inviscid problem with slip condition, and then we draw on this precise form to construct more accurate perturbation expansions for solutions to the nonlinear flow model than are currently available. [ABSTRACT FROM AUTHOR]

The purpose of this work is to explore alternative geometric pedagogical perspectives concerning justifications to 'fast' multiplication algorithms in a way that fosters opportunities for skill and understanding within younger, or less algebraically inclined, learners. Drawing on a visual strategy to justify these algorithms creates pedagogical alternatives to the presiding algebraic defences found in texts. Firstly, I formulate an overview of a particular 'fast' multiplication algorithm and situate it within historical and modern contexts. Next, I present a geometrical approach that offers a visual perspective of why this algorithm works and compare it with traditional algebraic approaches. Three examples are discussed throughout to give the ideas some concreteness. Finally, I furnish a personal reflection on the classroom experiences of sharing this geometric justification with younger learners. [ABSTRACT FROM AUTHOR]

The purpose of this work is to move towards a more complete understanding of the roles that shooting methods can play in the theory of discrete boundary value problems. I take the position that shooting methods can have an important function in exploring discrete boundary value problems due to: their mutually compatible characteristics; their accessibility; and the significance and flexibility of their conclusions. My position is supported through the establishment of new theory and via exemplification. [ABSTRACT FROM AUTHOR]

The purpose of this note is to sharpen Smirnov's recent work on existence and uniqueness of solutions to third-order ordinary differential equations that are subjected to two- and three-point boundary conditions. The advancement is achieved in the following ways. Firstly, we provide sharp and sharpened estimates for integrals regarding various Green's functions. Secondly, we apply these sharper estimates to problems in conjunction with Banach's fixed point theorem. Thirdly, we apply Rus's contraction mapping theorem in a metric space, where two metrics are employed. Our new results improve those of Smirnov by showing that a larger class of boundary value problems admit a unique solution. [ABSTRACT FROM AUTHOR]

Recently, Lima lamented on the 'time-consuming and tiresome' pedagogical nature of repeated integration by parts, throwing down the challenge of providing a pencil-and-paper solution to a related problem in a few seconds. Lima put forth a simple formula as an alternative. In this work, I offer a response to Lima's challenge and his formula. Drawing on the concepts of deconstruction and problematization, I offer alternative pedagogical perspectives regarding repeated integration by parts that involves tabular integration (aka Tic-Tac-Toe). I also explore the case of Lima's formula when an exponent in the integral is purely imaginary and compare this approach with repeated integration by parts. This paper forms alternative pedagogical constructs for students and educators to work with in the space of repeated integration by parts. [ABSTRACT FROM AUTHOR]

The aim of this article is to form new existence theory for global solutions to nonlinear fractional differential equations. Traditional approaches to existence, uniqueness and approximation of global solutions for initial value problems involving fractional differential equations have been unwieldy or intractable due to the limitations of previously used methods. This includes, for example, certain invariance conditions of the underlying local fixed point strategies. Herein we draw on an alternative tactics, applying the more modern ideas of continuation methods for contractive maps to fractional differential equations. In doing so, we shed new light on the situation, producing these new perspectives through a range of novel theorems that involve sufficient conditions under which global existence, uniqueness, approximation and location of solutions are ensured. [ABSTRACT FROM AUTHOR]

Recently, Gauthier introduced a method to construct solutions to the equations of motion associated with oscillating systems into the mathematics education research literature. In particular, Gauthier's approach involved certain manipulations of the differential equations; and drew on the theory of complex variables. Motivated by the work of Gauthier, we construct an alternative pedagogical approach for the learning and teaching of solution methods to these equations. The innovation lies in drawing on factorization techniques of differential equations and harmonizing them with Gauthier's approach of the theory of complex variables. When blended together to form a new approach, the significance lies in its accessibility, justifiability and transferrability to other problems. We pedagogically ground our approach in the educational development theory of Piaget, with the results informing the learning and teaching of solution methods to differential equations for lecturers, teachers and learners within universities, colleges, polytechnics and schools around the world. [ABSTRACT FROM AUTHOR]

Recently, Robin claimed to introduce clever innovations ('wrinkles') into the mathematics education literature concerning the solutions, and methods of solution, to differential equations. In particular, Robin formulated an iterative scheme in the form of a single integral representation. These ideas were applied to a range of examples involving differential equations. In this article, we respond to Robin's work by subjecting these claims, methods and applications to closer scrutiny. By outlining the historical development of Picard's iterative method for differential equations and drawing on relevant literature, we show that the iterative scheme of Robin has been known for some time. We introduce the need for a 'space for otherness' in mathematics education, by drawing on Foucault and posit alternative pedagogical approaches as heterotopias. We open a space for otherness and make it concrete by considering alternative perspectives to Robin's work. On a practical note, we see the importance of history and theory to be part of the pedagogical conversation when teaching and learning iterative methods; and provide a set of Maple code with which students and teachers can experiment, explore and learn. We also advocate more broadly for educators to open a space for otherness in their own pedagogical practice. [ABSTRACT FROM AUTHOR]

Recently, Wilmer III and Costa introduced a method into the mathematics education research literature which they employed to construct solutions to certain classes of ordinary differential equations. In this article, we build on their ideas in the following ways. We establish a link between their approach and the method of successive approximations. We show how applying the method of approximations naturally leads to the constructed approximation of Wilmer III and Costa. The new link is important because it enables us to respond to several challenges posed by Wilmer III and Costa. This includes addressing issues raised therein with convergence of their recursively constructed sequence of functions, and responding to their call regarding more mathematical rigour when relaxing the polynomial condition on the coefficients in the differential equation. Furthermore, the new link is pedagogically significant because it also opens up new pedagogical points of view. For example, the results in this paper provide potentially alternate, but overlapping, perspectives that are suitable for, and jointly inform, the learning and teaching of solution methods to differential equations. The value of this is supported by the presumption of Tisdell that teaching multiple ways to solve the same problem has academic and social value. [ABSTRACT FROM AUTHOR]

Recently, claims of a 'new and straightforward' method of solution to second-order linear difference equations have appeared in the mathematics education literature from Rivera-Figueroa and Rivera-Rebolledo. The claim of novelty is based on an assumption that 'since the equation is worked in its canonical form', the method within this context must be new. In addition, the assertion of straightforwardness is based on the position that 'the solution comes naturally' through this method, rather than artificially. In this article, we subject these claims and assumptions to closer scrutiny, examination and analysis. We note that the method has been published before, and we present the method in a more succinct form. We also discuss how the method can be extended to solve difference equations with non-constant coefficients, illustrating this via a discussion of an example. [ABSTRACT FROM AUTHOR]

This paper is based on the presumption that teaching multiple ways to solve the same problem has academic and social value. In particular, we argue that such a multifaceted approach to pedagogy moves towards an environment of more inclusive and personalized learning. From a mathematics education perspective, our discussion is framed around pedagogical approaches to triple integrals seen in a standard multivariable calculus curriculum. We present some critical perspectives regarding the dominant and long-standing approach to the teaching of triple integrals currently seen in hegemonic calculus textbooks; and we illustrate the need for more diverse pedagogical methods. Finally, we take a constructive position by introducing a new and alternate pedagogical approach to solve some of the classical problems involving triple integrals from the literature through a simple application of integration by parts. This pedagogical alternative for triple integrals is designed to question the dominant one-size-fits-all approach of rearranging the order of integration and the privileging of graphical methods; and to enable a shift towards a more inclusive, enhanced and personalized learning experience. [ABSTRACT FROM AUTHOR]

Despite the continuing interest in the transport of biological fluid within contracting or expanding vessels, our knowledge is yet to be fully developed, even in the two-dimensional case. For example, explicit solutions and close approximations to these models remain unknown, and the physical problem has been restricted to the "slow" expansion or contraction of the walls. Thus, the purpose of this short communication is to partially address such challenges and gaps by generating explicit solutions and improving approximations to the flow problem without the "slowness" restriction being imposed. We show that when the Reynolds number is zero (i.e., the inviscid case), the corresponding homogeneous differential equation under consideration may be completely solved. We then illustrate how this exact solution may be leveraged to form more precise approximations to the flow via perturbation techniques when the Reynolds number is small. Our perturbation approach is only in one parameter (the Reynolds number) instead of the usual two parameters (the Reynolds number and wall dilation rate), and thus we make no restriction regarding the "slowness" of wall expansion or contraction for our general perturbation scheme. Our act of "shining new light through old windows" improves and extends the results of Majdalani, Zhou and Dawson and, moreover, our method has significant potential to be applied by researchers to form more precise one-parameter perturbation approximations to flow problems in contrast to the limitations of the traditional two-parameter perturbation approaches that have dominated the literature. [ABSTRACT FROM AUTHOR]

This paper presents some critical perspectives regarding pedagogical approaches to the method of reversing the order of integration in double integrals from prevailing educational literature on multivariable calculus. First, we question the message found in popular textbooks that the traditional process of reversing the order of integration is necessary when solving well-known problems. Second, we illustrate that the method of integration by parts can be directly applied to many of the classic pedagogical problems in the literature concerning double integrals, without taking the well-worn steps associated with reversing the order of integration. Third, we examine the benefits and limitations of such a method. In our conclusion, we advocate for integration by parts to be a part of the pedagogical conversation in the learning and teaching of double integral methods; and call for more debate around its use in the learning and teaching of other areas of mathematics. Finally, we emphasize the need for critical approaches in the pedagogy of mathematics more broadly. [ABSTRACT FROM PUBLISHER]

For over 50 years, the learning of teaching ofa prioribounds on solutions to linear differential equations has involved a Euclidean approach to measuring the size of a solution. While the Euclidean approach toa prioribounds on solutions is somewhat manageable in the learning and teaching of the proofs involving second-order, linear problems with constant co-efficients, we believe it is not pedagogically optimal. Moreover, the Euclidean method becomes pedagogically unwieldy in the proofs involving higher-order cases. The purpose of this work is to propose a simpler pedagogical approach to establisha prioribounds on solutions by considering a different way of measuring the size of a solution to linear problems, which we refer to as the Uber size. The Uber form enables a simplification of pedagogy from the literature and the ideas are accessible to learners who have an understanding of the Fundamental Theorem of Calculus and the exponential function, both usually seen in a first course in calculus. We believe that this work will be of mathematical and pedagogical interest to those who are learning and teaching in the area of differential equations or in any of the numerous disciplines where linear differential equations are used. [ABSTRACT FROM AUTHOR]

The purpose of this work is to advance and simplify our understanding of some of the basic theory of linear dynamic equations and dynamic inequalities on time scales. Firstly, we revisit and simplify approaches to Gronwall's inequality on time scales. We provide new, simple and direct proofs that are accessible to those with only a basic understanding of calculus. Secondly, we apply the ideas to second and higher order linear dynamic equations on time scales. Part of the novelty herein involves a strategic choice of metric, notably the taxicab metric, to produce {\em a priori} bounds on solutions. This choice of metric significantly simplifies usual approaches and extends ideas from the literature. Thirdly, we examine mathematical applications of the aforementioned bounds. We form results concerning the non-multiplicity of solutions to linear problems; and error estimates on solutions to initial value problems when the initial conditions are imprecisely known. [ABSTRACT FROM AUTHOR]

Knowing an equation has a unique solution is important from both a modelling and theoretical point of view. For over 70 years, the approach to learning and teaching ‘well posedness’ of initial value problems (IVPs) for second- and higher-order ordinary differential equations has involved transforming the problem and its analysis to a first-order system of equations. We show that this excursion is unnecessary and present a direct approach regarding second- and higher-order problems that does not require an understanding of systems. [ABSTRACT FROM AUTHOR]

This article investigates nonlinear, second-order ordinary differential equations subject to various two-point boundary conditions. A condition is introduced that ensures a priori bounds on the derivatives of solutions to the problem. In particular, quadratic growth conditions on the right-hand side of the differential equation are not employed. The ideas are then applied to ensure the existence of at least one solution. The main tools involve differential inequalities and fixed-point methods. [ABSTRACT FROM AUTHOR]

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 admit solutions. For this, our choice of methods involves a monotone iterative technique and the method of successive approximations (a.k.a. Picard iterations) in the absence of Lipschitz conditions. Our existence results for the discrete BVP are of a constructive nature and are of independent interest in their own right. We then turn our attention to applying our existence results for the discrete BVP to the continuous BVP. We form new existence results for solutions to the continuous BVP with our methods involving linear interpolation of the data from the discrete BVP, combined with a priori bounds and the convergence Arzela-Ascoli theorem. Thus, our use of discrete BVPs to yield results for the continuous BVP may be considered as a discrete approach to continuous BVPs. [ABSTRACT FROM AUTHOR]

This article examines the qualitative properties of solutions to systems of boundary value problems involving fractional differential equations. Our primary interest is in forming new results that involve suficient conditions for the existence of solutions. To do this, we formulate some new ideas concerning a priori bounds on solutions, which are then applied to produce the novel existence results. The main techniques of the paper involve the introduction of novel fractional differential inequalities and the application of the fixed-point theorem of Schäfer. We conclude the work with several new results that link the number of solutions to our problem with a fractional initial value problem, akin to an abstract shooting method. [ABSTRACT FROM AUTHOR]

Abstract: This article examines two-point boundary value problems (BVPs) for second-order, singular ordinary differential equations where the right-hand-side of the differential equation may depend on the derivative of the solution. We introduce a method to obtain a priori bounds on all potential solutions, including their “derivatives”, to the singular BVP under consideration. The approach is based on the application of differential inequalities of singular type. The ideas are then applied to yield new existence results for solutions. [Copyright &y& Elsevier]

This work investigates a two-point boundary value problem (BVP) involving a first-order difference equation, known as the ‘discrete’ BVP. Some sufficient conditions are formulated under which the discrete BVP will possess a unique solution. The innovation herein involves a strategic choice of metric and utilization of Hölder's inequality. This approach enables the associated mappings to be contractive, which were previously non-contractive in traditional settings. This consequently enables an improved application of the fixed-point theorem of Stefan Banach by addressing a wider range of problems than those covered by the current literature. A YouTube video presentation by the author designed to complement this work is available at http://www.youtube.com/watch?v=luLuQ1KyXy8. [ABSTRACT FROM PUBLISHER]

Abstract: This article analyzes qualitative properties of solutions to two-point boundary value problems for singular ordinary differential equations. In particular, we form new approaches that ensure that all possible solutions satisfy certain a priori bounds. The methods involve differential inequalities. The ideas are then applied to generate new existence results for solutions to the boundary value problems under consideration. Many of the results are novel for both the singular and the nonsingular cases. The main concepts are further illustrated via applications to various equations arising in modeling, such as the Poisson–Boltzmann equation, and the Thomas–Fermi equation with an ionized atom. [Copyright &y& Elsevier]

We exhibit an alternative method for solving inhomogeneous second-order linear ordinary dynamic equations on time scales, based on reduction of order rather than variation of parameters. Our form extends recent (and long-standing) analysis on [image omitted] to a new form for difference equations, quantum equations and arbitrary dynamic equations on time scales. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple solutions of boundary value problems for second-order discrete equations [image omitted] , [image omitted] , [image omitted] , [image omitted] bounded on [image omitted] . The proofs are based on the upper and lower solutions, degree theory and sequential argument. [ABSTRACT FROM AUTHOR]

Abstract: We establish some new sufficient conditions under which periodic and non-periodic solutions exist for systems of nonlinear, first-order differential inclusions. Our new results are significant for two main reasons: they apply to differential inclusions that may have a right-hand side that grows super-linearly in the second variable and thus our ideas extend known linear-growth results; and our results also apply to systems of first-order differential inclusions, thus our work extends previous research on scalar-valued inclusions and may lead to a deeper understanding of higher-order differential inclusions. Our approach is based on novel differential inequalities and the well-known nonlinear alternative of Leray–Schauder. Some new results for ordinary differential equations with Carathéodory single-valued right-hand sides are also obtained. [Copyright &y& Elsevier]

Abstract: This article investigates both basic qualitative and basic quantitative properties of solutions to first- and higher-order dynamic equations on time scales and thus provides a foundation and framework for future advanced nonlinear studies in the field. Particular focus lies in the: existence; uniqueness; dependency; approximation; and explicit representation, of solutions to nonlinear initial value problems. The main tools used are from modern areas of nonlinear analysis, including: the fixed-point theorems of Banach and Schäfer; the method of successive approximations; a novel definition of measuring distance in metric spaces and normed spaces; and a “separation” of variables technique is introduced to the general time scale setting. The new results compliment and extend those of Stefan Hilger’s seminal paper of 1990. As an application of the new results we present and analyse a simple model from economics, known as the Keynesian–Cross model with “lagged” income, in the general time scale environment. Ideas suggesting further applications and possible new directions for the novel results are also presented. [Copyright &y& Elsevier]

In this paper we examine "terminal" value problems for dynamic equations on time scales -- that is, a dynamic equation whose solutions are asymptotic at infinity. We present a number of new theorems that guarantee the existence and uniqueness of solutions, as well as some comparison-type results. The methods we employ feature dynamic inequalities, weighted norms, and fixed-point theory. [ABSTRACT FROM AUTHOR]

Abstract: This article analyzes the solvability of second-order, nonlinear dynamic boundary value problems (BVPs) on time scales. New Bernstein–Nagumo conditions are developed that guarantee an a priori bound on the delta derivative of potential solutions to the BVPs under consideration. Topological methods are then employed to gain solvability. [Copyright &y& Elsevier]

Abstract: This paper investigates the solvability of discrete Dirichlet boundary value problems by the lower and upper solution method. Here, the second-order difference equation with a nonlinear right hand side is studied and can have a superlinear growth both in and in . Moreover, the growth conditions on are one-sided. We compute a priori bounds on solutions to the discrete problem and then obtain the existence of at least one solution. It is shown that solutions of the discrete problem will converge to solutions of ordinary differential equations. [Copyright &y& Elsevier]

Abstract: In this work we obtain some new results concerning the existence of solutions to an impulsive first-order, nonlinear ordinary differential equation with periodic boundary conditions. The ideas involve differential inequalities and Schaefer''s fixed-point theorem. [Copyright &y& Elsevier]

Abstract: We gain solvability of a system of nonlinear, second-order ordinary differential equations subject to a range of boundary conditions. The ideas involve differential inequalities and fixed point methods. In particular, maximum principles are not employed. [Copyright &y& Elsevier]

In this work we present some new results concerning the existence and uniqueness of solutions to an impulsive first-order, nonlinear ordinary differential equation with "non-periodic" boundary conditions. These boundary conditions include, as a special case, so-called "anti-periodic" boundary conditions. Our methods to prove the existence and uniqueness of solutions involve new differential inequalities, the classical fixed-point theorem of Schaefer, and the Nonlinear Alternative. Our new results apply to systems of impulsive differential equations where the right-hand side of the equation may grow linearly, or sub- or super-linearly in its second argument. [ABSTRACT FROM AUTHOR]

We consider the generalization of two classical periodic problems to the context of time scales. On the one hand, we generalize a celebrated result by Castro for the forced pendulum equation. On the other hand, we extend a well-known result by Nirenberg to a resonant system of equations on time scales. Furthermore, the results are new even for classical difference equations. [ABSTRACT FROM AUTHOR]

We are concerned with the existence and form of positive solutions to a nonlinear third-order three-point nonlocal boundary-value problem on general time scales. Using Green's functions, we prove the existence of at least one positive solution using the Guo-Krasnoselskii fixed point theorem. Due to the fact that the nonlinearity is allowed to change sign in our formulation, and the novelty of the boundary conditions, these results are new for discrete, continuous, quantum and arbitrary time scales. [ABSTRACT FROM AUTHOR]

Herein we consider the existence of solutions to second-order, two-point boundary value problems (BVPs) for systems of ordinary differential inclusions. Some new Bernstein–Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions. [ABSTRACT FROM AUTHOR]

This article analyzes a nonlinear system of first-order difference equations with periodic and non-periodic boundary conditions. Some sufficient conditions are presented under which: potential solutions to the equations will satisfy certain a priori bounds; and the equations will admit at least one solution. The methods involve new dynamic inequalities and use of Brouwer degree theory. The new results are compared with those featuring in the theory of solutions to boundary value problems for differential equations. [ABSTRACT FROM AUTHOR]

Abstract: This article investigates the existence of solutions to boundary value problems (BVPs) involving systems of first-order ordinary differential equations and two-point, periodic boundary conditions. The methods involve novel differential inequalities and fixed-point theory to yield new theorems guaranteeing the existence of at least one solution. [Copyright &y& Elsevier]

This article investigates the existence of solutions to first-order nonlinear boundary-value problems (BVPs) involving systems of ordinary differential equations and two-point boundary conditions. Some sufficient conditions are presented that will ensure solvability. The main tools employed are novel differential inequalities and fixed-point methods. [ABSTRACT FROM AUTHOR]

We give some necessary and sufficient conditions for the existence of C a 2 [0,1] and C 3 [0,1] positive solutions to the singular boundary value problem where: λ∈  (0,1) is given; and p   :  (0,1)→  [0,∞) can be singular at both ends t   =  0 and t   =  1. We also give a sufficient condition for the existence of C 1 [0,1] positive solutions to the above problem. The proofs are based upon the method of lower and upper solutions for singular fourth-order boundary value problems. [ABSTRACT FROM AUTHOR]

In this work, we investigate the boundedness and uniqueness of solutions to systems of dynamic equations on time scales. We define suitable Lyapunov-type functions and then formulate appropriate inequalities on these functions that guarantee all solutions to first-order initial value problems are uniformly bounded and/or unique. Several examples are given. [ABSTRACT FROM AUTHOR]

This work presents some results for the existence of solutions to boundary value problems on time scales. The ideas rely on the topological transversality of A. Granas. [Copyright &y& Elsevier]

The aim of this article is to establish sufficient conditions for the approximate controllability of fractional control systems with time delay in Hilbert spaces. By the technique of sequential approach, we prove that the fractional control systems with time delay are approximately controllable. Finally, an example is provided to illustrate our main results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we deal with the existence of at least three classical solutions for the following two point boundary value problemwhere , is a constant, is a positive parameter, . Our main tool is a recent three critical point theorem of Ricceri. Our emphasis here is constructing an appropriate separable and reflexive real Banach space. [Copyright &y& Elsevier]

Abstract: This paper focuses on the qualitative and quantitative properties of solutions to certain nonlinear dynamic equations on time scales. We present some new sufficient conditions under which these general equations admit a unique, positive solution. These positive (and hence non-oscillatory) solutions: extend across unbounded intervals; and tend to a finite limit as the independent variable becomes large and positive. Our methods include: Banach’s fixed-point theorem, including the method of Picard iterations; and weighted norms and metrics in the time scale setting. Due to the wide-ranging nature of dynamic equations on time scales our results are novel: for ordinary differential equations; for difference equations; for combinations of the two areas; and for general time scales — this is demonstrated via some examples. Furthermore, we state an open problem of interest. [Copyright &y& Elsevier]

An alternative pedagogical design is discussed that aims to guide engineering students to solve first-order ordinary differential equations (ODEs), and is based on students' learning weaknesses identified from previous teaching and learning activities. This approach supported student's self-enrichment through exploration of relevant resources in ODEs, and guided students towards the choice of their own effective ways for solving ODEs for different problems. This paper presents the practices on designing and delivering solution techniques for first-order linear ODEs using this approach for more than 400 undergraduate engineering students at a regional university in Australia during 2014–2017. The timeline involved initial experimentation in 2014 and 2015, followed by refinements to the pedagogy based on student's feedback. The refined pedagogy was then used for the advanced mathematics course in 2016 and 2017. Significant improvements were made in student's learning outcomes in effectively and accurately solving the first-order linear ODEs over this period. [ABSTRACT FROM AUTHOR]

We consider the following system of Volterra integral equations: and some of its particular cases that arise from physical problems. Criteria are offered for the existence of one and more constant-sign solutions of the system in . We say is of constant sign if for each for all , where is fixed. Examples are also included to illustrate the results obtained. [Copyright &y& Elsevier]