Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index

Dynamic Equations on Time Scales: An Introduction with Applications

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

Dedication We dedicate this book to our family members who complete us. In particular, M. Ben-chohra's dedication is to his wife, Kheira, and his children, Mohamed, Maroua, and Abdelillah; J. Henderson dedicates to his wife, Darlene, and his descendants, Kathy, Contents Preface xi 1. Preliminaries 1 1.1. Definitions and results for multivalued analysis 1 1.2. Fixed point theorems 4 1.3. Semigroups 7 1.4. Some additional lemmas and notions 9 2. Impulsive ordinary differential equations & inclusions 11 2.1. Introduction 11 2.2. Impulsive ordinary differential equations 12 2.3. Impulsive ordinary differential inclusions 24 2.4. Ordinary damped differential inclusions 49 2.5. Notes and remarks 62 3. Impulsive functional differential equations & inclusions 63 3.1. Introduction 63 3.2. Impulsive functional differential equations 63 3.3. Impulsive neutral differential equations 74 3.4. Impulsive functional differential inclusions 80 3.5. Impulsive neutral functional DIs 95 3.6. Impulsive semilinear functional DIs 107 3.7. Notes and remarks 118 4. Impulsive differential inclusions with nonlocal conditions 119 4.1. Introduction 119 4.2. Nonlocal impulsive semilinear differential inclusions 119 4.3. Existence results for impulsive functional semilinear differential inclusions with nonlocal conditions 136 4.4. Notes and remarks 145 5. Positive solutions for impulsive differential equations 147 5.1. Introduction 147 5.2. Positive solutions for impulsive functional differential equations 147 5.3. Positive solutions for impulsive boundary value problems 154 5.4. Double positive solutions for impulsive boundary value problems 159 5.5. Notes and remarks 165 viii Contents 6. Boundary value problems for impulsive differential inclusions 167 6.1. Introduction 167 6.2. First-order impulsive differential inclusions with periodic boundary conditions 167 6.3. Upper-and lower-solutions method for impulsive differential inclusions with nonlinear boundary conditions 184 6.4. Second-order boundary value problems 191 6.5. Notes and remarks 198 7. Nonresonance impulsive differential inclusions 199 7.1. Introduction 199 7.2. Nonresonance first-order impulsive functional differential inclusions with periodic boundary conditions 199 7.3. Nonresonance second-order impulsive functional differential inclusions with periodic boundary conditions 209 7.4. Nonresonance higher-order boundary value problems for impulsive functional differential inclusions 217 7.5. Notes and remarks 227 8. Impulsive differential equations & inclusions with variable times 229 8.1. Introduction 229 8.2. First-order impulsive differential equations with variable times 229 8.3. Higher-order impulsive differential equations with variable times 235 8.4. Boundary value problems for differential inclusions with variable times 241 8.5. Notes and remarks 252 9. Nondensely defined impulsive differential equations & inclusions 253 9.1. Introduction 253 9.2. Nondensely defined impulsive semilinear differential equations with nonlocal conditions 253 9.3. Nondensely defined …

Impulsive Differential Equations and Inclusions

高等学校計算数学学報 = Numerical mathematics

https://web.mst.edu/~bohner/papers/deotsas.pdf

Dynamic equations on time scales: a survey

We shall obtain the parameter region that
ensures the global asymptotical stability of the coexistence fixed
point of a Ricker-type competitive model. The parameter region can
be illustrated graphically and examples of such regions are
presented. Our result partially answers an open problem proposed by
Elaydi and Luis [3] and complements the very recent
work by Balreira, Elaydi and Luis [1].

https://www.aimsciences.org/article/exportPdf?id=1ae8595e-0673-4b7a-98bc-2dd9bae1bccb

Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model

Dengue fever has been prevalent for decades in human population and consequently, the human population is partially susceptible to the disease since partial immunity has been gained for those who recovered from dengue infection. We shall obtain an estimation formula for the reproduction number of dengue transmission when the human population is not completely susceptible. This is achieved by taking the number of infected cases periodically notified by surveillance organizations as the solution of a mathematical model for dengue transmission. As an application, the estimation formula is used to estimate the reproduction number of dengue transmission in Singapore in the years 2013 and 2014.

Estimation of reproduction number of dengue transmission in a partially susceptible population

In this chapter we shall offer two very general existence criteria for the initial value problem 

$$ \begin{gathered} y' = \;q(t)f(t,y),\;0 < t < T < \infty \hfill \\ y(0) = 0 \hfill \\ \end{gathered} $$

(1.1)

where the nonlinearity will be allowed to change sign, and can have singularities at y = 0 and/or t = 0.

First Order Initial Value Problems

This article presents a study on singularly perturbed 1D parabolic Dirichlet’s type differential equations with discontinuous source terms on an interior line. The time derivative is discretized using the Euler backward method, followed by the application of the streamline–diffusion finite element method (SDFEM) to solve locally one-dimensional stationary problems on a Shishkin mesh. Our proposed method is shown to achieve first-order convergence in time and second-order convergence in space. Our proposed method offers several advantages over existing techniques, including more accurate approximations of the solution on the boundary layer region, better efficiency, and robustness in dealing with discontinuous line source terms. The numerical examples presented in this paper demonstrate the effectiveness and efficiency of our method, which has practical applications in various fields, such as engineering and applied mathematics. Overall, our proposed method provides an effective and efficient solution to the challenging problem of solving singularly perturbed parabolic differential equations with discontinuous line source terms, making it a valuable tool for researchers and practitioners in various domains.

/pdf/streamline-diffusion-finite-element-method-for-singularly-kevtzz9x.pdf

Streamline Diffusion Finite Element Method for Singularly Perturbed 1D-Parabolic Convection Diffusion Differential Equations with Line Discontinuous Source

In this paper, we solve the fractional Bagley-Torvik equation by using discrete cubic spline and a fourth order approxi-mation based on weighted shifted Grunwald-Letnikov difference operator. By matrix analysis, the numerical scheme is proved to be uniquely solvable and third order accurate. An example is presented to verify the efficiency of the numerical scheme and to compare with other methods in the literature.In this paper, we solve the fractional Bagley-Torvik equation by using discrete cubic spline and a fourth order approxi-mation based on weighted shifted Grunwald-Letnikov difference operator. By matrix analysis, the numerical scheme is proved to be uniquely solvable and third order accurate. An example is presented to verify the efficiency of the numerical scheme and to compare with other methods in the literature.

Patricia J. Y. Wong

Papers

Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model

Estimation of reproduction number of dengue transmission in a partially susceptible population

First Order Initial Value Problems

Streamline Diffusion Finite Element Method for Singularly Perturbed 1D-Parabolic Convection Diffusion Differential Equations with Line Discontinuous Source

Numerical method for fractional Bagley-Torvik equation