Measure functional differential equations and functional dynamic equations on time scales

We study measure functional differential equations and clarify their relation to generalized ordinary differential equations. We show that functional dynamic equations on time scales represent a special case of measure functional differential equations. For both types of equations, we obtain results on the existence and uniqueness of solutions, continuous dependence, and periodic averaging.     

Maximum principles for second order dynamic equations on time scales

This paper establishes some new maximum principles for second order dynamic equations on time scales, including: a strong maximum principle; a generalized maximum principle; and a boundary point lemma. The new results include, as special cases, well-known ideas for ordinary differential equations and difference equations.     

Sign properties of Green's functions for disconjugate dynamic equations on time scales

This paper continues the development of disconjugacy of higher order dynamic equations on time scales. Two-point conjugate type boundary value problems for general disconjugate dynamic equations on time scales are studied and the sign properties of associated Green's functions are established. As expected, the results unify known results from the theories of ordinary differential equations and finite difference equations.     

Prolongation of solutions of measure differential equations and dynamic equations on time scales

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations.     

Non-periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses

We present a non-periodic averaging principle for measure functional differential equations and, using the correspondence between solutions of measure functional differential equations and solutions of functional dynamic equations on time scales (see Federson et al., 2012 [8]), we obtain a non-periodic averaging result for functional dynamic equations on time scales. Moreover, using the relation between measure functional differential equations and impulsive measure functional differential equations, we get a non-periodic averaging theorem for these equations. Also, it is a known fact that we can relate impulsive measure functional differential equations and impulsive functional dynamic equations on time scales (see Federson et al., 2013 [9]). Therefore, applying this correspondence to our averaging principle, we obtain a non-periodic averaging theorem for impulsive functional dynamic equations on time scales.     

Periodicity of scalar dynamic equations and applications to population models

Easily verifiable sufficient criteria are established for the existence of periodic solutions of a class of nonautonomous scalar dynamic equations on time scales, which incorporate as special cases many single species models governed by ordinary differential and difference equations when the time scale is the set of all real and all integer numbers, respectively.     

Ulam-type stability for differential equations driven by measures

Based on a notion of Stieltjes derivative of a function with respect to another function, we provide Ulam–Hyers type stability results for nonlinear differential equations driven by measures on compact or on unbounded intervals, in the lack of Lipschitz continuity assumptions. In particular, one can deduce stability results for generalized differential equations, dynamic equations on time scales or impulsive differential equations (including the case of an infinite number of impulses that accumulate in the considered interval, thus allowing the study of Zeno hybrid systems).     

Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients

Unifying ordinary differential and difference equations, we consider linear dynamic equations on measure chains or time scales, which possess an exponential dichotomy uniformly in a parameter, and show that this dichotomy is robust, if the mentioned parameter changes slowly in time. Here, the equations can be infinite dimensional and are not assumed to be invertible.     

Applications of maximum principles to dynamic equations on time scales

We apply the strong maximum principle to obtain a priori bounds and uniqueness of solutions for some initial value and boundary value problems as well as to establish oscillation results for second-order dynamic equations on time scales. Our comparison, uniqueness, and oscillation results are new and are extensions of results for ordinary differential equations to the times scale setting.     

Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales

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.     

Impulsive nonlocal differential equations through differential equations on time scales

We propose a non-standard approach to impulsive differential equations in Banach spaces by embedding this type of problems into differential (dynamic) problems on time scales. We give an existence result for dynamic equations and, as a consequence, we obtain an existence result for impulsive differential equations.     

Dichotomies for generalized ordinary differential equations and applications

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation.     

时间尺度上具阻尼项的二阶半线性时滞动力方程的振动准则

利用广义Riccati变换和不等式技巧,研究了一类具阻尼项的二阶半线性时滞动力方程解的振动性质,在一定条件下,建立了四个新的振动准则,其结果不仅推广和改进了已知的一些结果,而且在时间尺度上统一了具阻尼项的二阶半线性时滞微分方程和差分方程解的振动性质.     

Oscillation criteria for certain third-order variable delay functional dynamic equations on time scales

The oscillation of certain third-order nonlinear variable delay functional dynamic equations with nonlinear neutral on time scales is discussed in this article. By using the generalized Riccati transformation and a lot of inequality techniques, some new oscillation criteria for the equations are established. Many of the results are new for the corresponding third-order difference equations and differential equations are as special cases. Some examples are given to illustrate the importance of our results.     

Oscillation of even order nonlinear delay dynamic equations on time scales

One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor’s Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.     

Periodic solutions of functional dynamic equations with infinite delay

In this paper, sufficient criteria are established for the existence of periodic solutions of some functional dynamic equations with infinite delays on time scales, which generalize and incorporate as special cases many known results for differential equations and for difference equations when the time scale is the set of the real numbers or the integers, respectively. The approach is mainly based on the Krasnosel’ski? fixed point theorem, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in studying dynamic equations on time scales. This study shows that one can unify such existence studies in the sense of dynamic equations on general time scales.     

Phase spaces and periodic solutions of set functional dynamic equations with infinite delay

Very recently, a new theory known as set dynamic equations on time scales has been built. In this paper, a phase space is built for set functional dynamic equations with infinite delay on time scales and sufficient criteria are established for the existence of periodic solutions of such equations, which generalize and incorporate as special cases some known results for set differential equations and for set difference equations when the time scale is the real number set or the integer set, respectively, moreover, for differential inclusions and difference inclusions if the variable under consideration is a single valued mapping. Our results show that one can unify the study of some continuous or discrete problems in the sense of (set) dynamic equations on general time scales.     

A unified analysis of linear quaternion dynamic equations on time scales

Over the last years, considerable attention has been paid to the role of the quaternion differential equations (QDEs) which extend the ordinary differential equations. The theory of QDEs was recently well established and has wide applications in physics and life science. This paper establishes a systematic frame work for the theory of linear quaternion dynamic equations on time scales (QDETS), which can be applied to wave phenomena modeling, fluid dynamics and filter design. The algebraic structure of the solutions to the QDETS is actually a left- or right- module, not a linear vector space. On the non-commutativity of the quaternion algebra, many concepts and properties of the classical dynamic equations on time scales (DETS) can not be applied. They should be redefined accordingly. Using $q$-determinant, a novel definition of Wronskian is introduced under the framework of quaternions which is different from the standard one in DETS. Liouville formula for QDETS is also analyzed. Upon these, the solutions to the linear QDETS are established. The Putzer''s algorithms to evaluate the fundamental solution matrix for homogeneous QDETS are presented. Furthermore, the variation of constants formula to solve the nonhomogeneous QDETs is given. Some concrete examples are provided to illustrate the feasibility of the proposed algorithms.     

The differential equations on time scales through impulsive differential equations

In this paper we investigate differential equations on certain time scales with transition conditions (DETC) on the basis of reduction to the impulsive differential equations (IDE). DETC are in some sense more general than dynamic equations on time scales [M. Bohner, A. Peterson, Dynamic equations on time scales, in: An Introduction With Applications, Birkhäuser Boston, Inc., Boston, MA, 2001, p. x+358; V. Laksmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamical Systems on Measure Chains, in: Math. and its Appl., vol. 370, Kluwer Academic, Dordrecht, 1996]. The basic properties of linear systems, the existence and stability of periodic solutions, and almost periodic solutions are considered. Appropriate examples are given to illustrate the theory.     

Nonlinear oscillation of second-order dynamic equations on time scales

Interval oscillation criteria are established for a second-order nonlinear dynamic equation on time scales by utilizing a generalized Riccati technique and the Young inequality. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.