Unbounded Time Scales Research Articles

Hyers–Ulam–Rassias Stability of Nonlinear Implicit Higher-Order Volterra Integrodifferential Equations from above on Unbounded Time Scales

In this paper, we present sufficient conditions for Hyers-Ulam-Rassias stability of nonlinear implicit higher-order Volterra-type integrodifferential equations from above on unbounded time scales. These new sufficient conditions result by reducing Volterra-type integrodifferential equations to Volterra-type integral equations, using the Banach fixed point theorem, and by applying an appropriate Bielecki type norm, the Lipschitz type functions, where Lipschitz coefficient is replaced by unbounded rd-continuous function.

Nonlinear fourth-order dynamic equations on unbounded time scales

"In this paper, we investigate nonlinear fourth-order dynamic equations on unbounded time scales. The existence and uniqueness of the solutions for these problems are obtained."

Nonlinear Volterra Integrodifferential Equations from above on Unbounded Time Scales

The paper is devoted to studying the existence, uniqueness and certain growth rates of solutions with certain implicit Volterra-type integrodifferential equations on unbounded from above time scales. We consider the case where the integrand is estimated by the Lipschitz type function with respect to the unknown variable. Lipschitz coefficient is an unbounded rd-function and the Banach fixed-point theorem at a functional space endowed with a suitable Bielecki-type norm.

Bilinear state systems on an unbounded time scale

We demonstrate the existence and uniqueness of solutions to a bilinear state system with locally essentially bounded coefficients on an unbounded time scale. We obtain a Volterra series representation for these solutions which is norm convergent and uniformly convergent on compact subsets of the time scale. We show the associated state transition matrix has a similarly convergent Peano-Baker series representation and identify a necessary and sufficient condition for its invertibility. Finally, we offer numerical applications for dynamic bilinear systems – a frequency modulated signal model and a two-compartment cancer chemotherapy model.

On Expansion in Eigenfunction for Dirac Systems on the Unbounded Time Scales

In this work, we consider one dimensional Dirac operators on unbounded time scales. We construct a spectral function of such an operator. Using this function, we establish a Parseval equality and an expansion formula in eigenfunctions for the Dirac operator on unbounded time scales.

Investigation of the spectrum of singular Sturm–Liouville operators on unbounded time scales

In this article, we consider the self-adjoint singular operators associated with the Sturm–Liouville expression $$\begin{aligned} Ly:=-\left[ p\left( t\right) y^{\Delta }\left( t\right) \right] ^{\nabla }+q\left( t\right) y\left( t\right) ,\ t\in (-\infty ,\infty )_{\mathbb {T}}. \end{aligned}$$on time scale $$\mathbb {T}$$. Some conditions are given for this operator to have a discrete spectrum. Further, we investigate the continuous spectrum of this operator. We also prove that the regular Sturm–Liouville operator on time scale is semi-bounded from below which is not studied in literature yet.

Some properties of the resolvent of Sturm-Liouville operators on unbounded time scales

Some properties of the resolvent of Sturm-Liouville operators on unbounded time scales

Limit behaviors of nonoscillatory solutions of three-dimensional time scale systems

In this article, we investigate the oscillatory behavior of a three-dimensional system of dynamic equations on an unbounded time scale. A time scale $\T$ is a nonempty closed subset of real numbers. An example is given to illustrate some of the results.

On extensions of singular fourth order dynamic operators on time scales

In this work, we consider a singular fourth order dynamic operator on time scales. We construct a space of boundary values. Later, we give a description of all maximal dissipative, self-adjoint and other extensions of singular fourth order differential operators on unbounded time scales.

Changing-periodic time scales and decomposition theorems of time scales with applications to functions with local almost periodicity and automorphy

In this work, we address the periodic coverage phenomenon on arbitrary unbounded time scales and initiate a new idea, namely, we introduce the concept of changing-periodic time scales. We discuss some properties of this new concept and illustrate several examples. Specially, we establish a basic decomposition theorem of time scales which provides bridges between periodic time scales and an arbitrary time scale with a bounded graininess function μ. Based on this result, we introduce local-almost periodic and local-almost automorphic functions on changing-periodic time scales and study some related properties. The concept of changing-periodic time scales introduced in this paper will help in understanding and removing the serious deficiency which arises in the study of classical functions on time scales.

Oscillatory behavior of solutions of third-order delay and advanced dynamic equations

Abstract In this paper, we consider oscillation criteria for certain third-order delay and advanced dynamic equations on unbounded time scales. A time scale "Equation missing" is a nonempty closed subset of the real numbers. Examples will be given to illustrate some of the results. MSC:34N05, 39A10, 39A21.

Krein-von Neumann and Friedrichs extensions for second order operators on time scales

We consider an operator defined by the second order Sturm-Liouville equation on an unbounded time scale. For such an operator we give characterisations of the domains of its Krein-von Neumann and Friedrichs extensions by using the recessive solution. This generalises and unifies similar results obtained for operators connected with the second order Sturm-Liouville differential and difference equations.

Transversality conditions for infinite horizon variational problems on time scales

We consider problems of the calculus of variations on unbounded time scales. We prove the validity of the Euler–Lagrange equation on time scales for infinite horizon problems, and a new transversality condition.

Existence of positive solutions of a Sturm-Liouville BVP on an unbounded time scale

A fixed point theorem of Guo-Krasnoselskii type is used to establish existence results for the nonlinear Sturm-Liouville dynamic equation with the boundary conditions on an unbounded time scale. Later on the positivity and the boundedness of the solutions are obtained by imposing some conditions on f.

Realizations of Nonlinear Control Systems on Time Scales

Nonlinear partially defined systems on an arbitrary unbounded time scale are studied. They include continuous-time and discrete-time systems. The main problem is to find necessary and sufficient conditions for an abstract input/output map to have a realization as a nonlinear system of a specific class on the time scale. The obtained results extend criteria of realizability of continuous-time polynomial systems. A simple construction of a realization is provided.

On the Asymptotic Integration of Nonlinear Dynamic Equations

The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class of second-order nonlinear dynamic equations on unbounded time scales. Four different results are obtained by using the Banach fixed point theorem, the Boyd and Wong fixed point theorem, the Leray-Schauder nonlinear alternative, and the Schauder fixed point theorem. For each theorem, an illustrative example is presented. The results provide unification and some extensions in the time scale setup of the theory of asymptotic integration of nonlinear equations both in the continuous and discrete cases.

Global asymptotic behavior for delay dynamic equations

We give conditions under which the trivial solution of a first-order nonlinear variable-delay dynamic equation is asymptotically stable, for arbitrary time scales that are unbounded above. In an example, we apply our techniques to a logistic dynamic equation on isolated, unbounded time scales.