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Full-Text Articles in Mathematics
Inequalities For Interval-Valued Riemann Diamond-Alpha Integrals, Martin Bohner, Linh Nguyen, Baruch Schneider, Tri Truong
Inequalities For Interval-Valued Riemann Diamond-Alpha Integrals, Martin Bohner, Linh Nguyen, Baruch Schneider, Tri Truong
Mathematics and Statistics Faculty Research & Creative Works
We propose the concept of Riemann diamond-alpha integrals for time scales interval-valued functions. We first give the definition and some properties of the interval Riemann diamond-alpha integral that are naturally investigated as an extension of interval Riemann nabla and delta integrals. With the help of the interval Riemann diamond-alpha integral, we present interval variants of Jensen inequalities for convex and concave interval-valued functions on an arbitrary time scale. Moreover, diamond alpha Hölder's and Minkowski's interval inequalities are proved. Also, several numerical examples are provided in order to illustrate our main results.
Delay Dynamic Equations On Isolated Time Scales And The Relevance Of One-Periodic Coefficients, Martin Bohner, Tom Cuchta, Sabrina Streipert
Delay Dynamic Equations On Isolated Time Scales And The Relevance Of One-Periodic Coefficients, Martin Bohner, Tom Cuchta, Sabrina Streipert
Mathematics and Statistics Faculty Research & Creative Works
We are motivated by the idea that certain properties of delay differential and difference equations with constant coefficients arise as a consequence of their one-periodic nature. We apply the recently introduced definition of periodicity for arbitrary isolated time scales to linear delay dynamic equations and a class of nonlinear delay dynamic equations. Utilizing a derived identity of higher order delta derivatives and delay terms, we rewrite the considered linear and nonlinear delayed dynamic equations with one-periodic coefficients as a linear autonomous dynamic system with constant matrix. As the simplification of a constant matrix is only obtained for one-periodic coefficients, dynamic …
Periodicity On Isolated Time Scales, Martin Bohner, Jaqueline Mesquita, Sabrina Streipert
Periodicity On Isolated Time Scales, Martin Bohner, Jaqueline Mesquita, Sabrina Streipert
Mathematics and Statistics Faculty Research & Creative Works
In this work, we formulate the definition of periodicity for functions defined on isolated time scales. The introduced definition is consistent with the known formulations in the discrete and quantum calculus settings. Using the definition of periodicity, we discuss the existence and uniqueness of periodic solutions to a family of linear dynamic equations on isolated time scales. Examples in quantum calculus and for mixed isolated time scales are presented.
Almost Oscillatory Three Dimensional Dynamic Systems, Elvan Akin, Zuzana Dosla, Bonita Lawrence
Almost Oscillatory Three Dimensional Dynamic Systems, Elvan Akin, Zuzana Dosla, Bonita Lawrence
Bonita Lawrence
In this article, we investigate oscillation and asymptotic properties for 3D systems of dynamic equations. We show the role of nonlinearities and we apply our results to the adjoint dynamic systems.
Sneak-Out Principle On Time Scales, Martin Bohner, Samir H. Saker
Sneak-Out Principle On Time Scales, Martin Bohner, Samir H. Saker
Mathematics and Statistics Faculty Research & Creative Works
In this paper, we show that the so-called "sneak-out principle" for discrete inequalities is valid also on a general time scale. In particular, we prove some new dynamic inequalities on time scales which as special cases contain discrete inequalities obtained by Bennett and Grosse-Erdmann. The main results also are used to formulate the corresponding continuous integral inequalities, and these are essentially new. The techniques employed in this paper are elementary and rely mainly on the time scales integration by parts rule, the time scales chain rule, the time scales Hölder inequality, and the time scales Minkowski inequality.
Higher Order Dynamic Equations On Measure Chains: Wronskians, Disconjugacy, And Interpolating Families Of Functions, Martin Bohner, Paul Eloe
Higher Order Dynamic Equations On Measure Chains: Wronskians, Disconjugacy, And Interpolating Families Of Functions, Martin Bohner, Paul Eloe
Paul W. Eloe
This paper introduces generalized zeros and hence disconjugacy of nth order linear dynamic equations, which cover simultaneously as special cases (among others) both differential equations and difference equations. We also define Markov, Fekete, and Descartes interpolating systems of functions. The main result of this paper states that disconjugacy is equivalent to the existence of any of the above interpolating systems of solutions and that it is also equivalent to a certain factorization representation of the operator. The results in this paper unify the corresponding theories of disconjugacy for nth order linear ordinary differential equations and for nth order linear difference …
Boundedness In Functional Dynamic Equations On Time Scales, Elvan Akin, Youssef N. Raffoul
Boundedness In Functional Dynamic Equations On Time Scales, Elvan Akin, Youssef N. Raffoul
Youssef N. Raffoul
Using nonnegative definite Lyapunov functionals, we prove general theorems for the boundedness of all solutions of a functional dynamic equation on time scales. We apply our obtained results to linear and nonlinear Volterra integro-dynamic equations on time scales by displaying suitable Lyapunov functionals.
Exponential Stability In Functional Dynamic Equations On Time Scales, Elvan Akin, Youssef Raffoul, Christopher Tisdell
Exponential Stability In Functional Dynamic Equations On Time Scales, Elvan Akin, Youssef Raffoul, Christopher Tisdell
Youssef N. Raffoul
We are interested in the exponential stability of the zero solution of a functional dynamic equation on a time scale, a nonempty closed subset of real numbers. The approach is based on suitable Lyapunov functionals and certain inequalities. We apply our results to obtain exponential stability in Volterra integrodynamic equations on time scales.
Oscillatory Behavior Of Solutions Of Third-Order Delay And Advanced Dynamic Equations, Murat Adivar, Elvan Akin, Raegan Higgins
Oscillatory Behavior Of Solutions Of Third-Order Delay And Advanced Dynamic Equations, Murat Adivar, Elvan Akin, Raegan Higgins
Mathematics and Statistics Faculty Research & Creative Works
In this paper, we consider oscillation criteria for certain third-order delay and advanced dynamic equations on unbounded time scales. A time scale T is a nonempty closed subset of the real numbers. Examples will be given to illustrate some of the results.
Almost Oscillatory Three Dimensional Dynamic Systems, Elvan Akin, Zuzana Dosla, Bonita Lawrence
Almost Oscillatory Three Dimensional Dynamic Systems, Elvan Akin, Zuzana Dosla, Bonita Lawrence
Mathematics and Statistics Faculty Research & Creative Works
In this article, we investigate oscillation and asymptotic properties for 3D systems of dynamic equations. We show the role of nonlinearities and we apply our results to the adjoint dynamic systems.