Existence of solutions to second-order BVPs on time scales

We consider a boundary value problem (BVP) for systems of second-order dynamic equations on time scales. Using methods involving dynamic inequalities, we formulate conditions under which all solutions to a certain family of systems of dynamic equations satisfy certain a priori bounds. These results are then applied to guarantee the existence of solutions to BVPs for systems of dynamic equations on time scales.     

Variational approach for a p-Laplacian boundary value problem on time scales

In this paper, we study a second-order p-Laplacian dynamic equation on a periodic time scale subject to certain boundary value conditions. The variational structure of the above-mentioned boundary value problem is presented. Some new results on the existence of at least one or three distinct periodic solutions are established by means of the saddle point theorem, the least action principle as well as the three critical point theorem.     

Variational approach for Sturm-Liouville boundary value problems on time scales

In this paper, we consider a class of second order Sturm-Liouville boundary value problems with positive parameter λ on time scales. By using variational method and critical point theory, we obtain that the boundary value problem has solutions for λ being in some different intervals. Recent results in the literature are generalized and improved.     

Solvability for a third-order three-point BVP on time scales

We are concerned with the existence of a nontrivial solution to a nonlinear third-order three-point boundary-value problem on general time scales. Using the corresponding Green function, without a non-negative or monotone-type assumption on the nonlinearity, we obtain sufficient conditions for the existence of at least one nontrivial solution, using the Leray–Schauder nonlinear alternative theorem. This paper extends recent results in differential equations to difference equations, quantum equations, and general dynamic equations on time scales.     

Existence of positive solutions for nonlinear three-point problems on time scales

In this paper, by using fixed point theorems in cones, we study the existence of at least one, two and three positive solutions of a nonlinear second-order three-point boundary value problem for dynamic equations on time scales. As an application, we also give some examples to demonstrate our results.     

A classification of time scales and analysis of the general delays on time scales with applications

In this paper, we present five classes/categories of time scales. Then, on each class, we introduce and analyze delays that not only lead to new types of delay systems on time scales but also reveal the limitations of the known results in the literature. To show the importance and significance of our analysis, several examples are illustrated. We conclude our paper with some interesting open problems. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Noether's theorem on time scales

We show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective Euler-Lagrange extremals.     

Weighted pseudo‐almost periodic functions on time scales with applications to cellular neural networks with discrete delays

In this paper, we first propose a concept of weighted pseudo‐almost periodic functions on time scales and study some basic properties of weighted pseudo‐almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo‐almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo‐almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new. Copyright © 2016 John Wiley & Sons, Ltd.     

Titchmarsh-Sims-Weyl theory for complex Hamiltonian systems on Sturmian time scales

We study non-self-adjoint Hamiltonian systems on Sturmian time scales, defining Weyl-Sims sets, which replace the classical Weyl circles, and a matrix-valued M-function on suitable cone-shaped domains in the complex plane. Furthermore, we characterize realizations of the corresponding dynamic operator and its adjoint, and construct their resolvents. Even-order scalar equations and the Orr-Sommerfeld equation on time scales are given as examples illustrating the theory, which are new even for difference equations. These results unify previous discrete and continuous theories to dynamic equations on Sturmian time scales.     

Triple solutions to boundary value problems on time scales

Criteria are developed for the existence of three nonnegative solutions to nonlinear differential equations on time scales.     

Lower bounds for the eigenvalues of first-order nonlinear Hamiltonian systems on time scales

We study first-order nonlinear planar Hamiltonian boundary value problems on time scales. Estimates on lower bounds for the eigenvalues of the problems are established by way of the Lyapunov inequality method. Our results are interpreted to nonlinear differential and difference planar Hamiltonian boundary value problems. As a special case, an estimate on lower bounds for eigenvalues of half-linear dynamic equations is obtained which generalizes and improves the existing ones to nonlinear Hamiltonian systems. Based on the main results, we establish existence and uniqueness of solutions of a related linear boundary value problem.     

Existence results for second-order impulsive boundary value problems on time scales

In this paper, using a fixed point theorem due to Krasnoselskii and Zabreiko and the Leggett–Williams fixed point theorem respectively, we investigate the existence of one solution and three nonnegative solutions to second-order impulsive equations on time scales.     

Nonlinear two-point boundary value problems on time scales

This work gives a criterion for the existence of positive solutions for nonlinear second order ordinary differential equations with two-point boundary conditions on time scales. Moreover, for some source terms which are in the sense of sublinear or superlinear, we also formulate corollaries and examples for applications and we improve previous results related with the first eigenvalue discussed by Professor Li.     

Existence of periodic solutions for a predator-prey system with sparse effect and functional response on time scales

In this paper, we systematically investigates the existence of periodic solutions of a predator-prey system with sparse effect and Beddington-DeAngelis or Holling III functional response on time scales. By using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of periodic solutions for the systems. Moreover, when the time scale T is chosen as R or Z, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.     

Parameter-dependent integral on time scales

In this paper, we focus on the parameter-dependent integral on time scales. By combining multiple integration theory on time scales and usual parameter-dependent integration theory in classical mathematical analysis, both normal parameter-dependent integral on time scales and improper parameter-dependent integral on time scales are investigated efficiently. Many criteria for uniform convergence and important analytical properties are developed, respectively. Finally, a lot of examples are also presented to substantiate the theoretical results.     

Existence of solutions to boundary value problems for dynamic systems on time scales

Here, we investigate systems of boundary value problems for dynamic equations on time scales. Using a generalized relationship between the boundary conditions and a certain subset of the solution space, the existence of solutions is established through topological arguments. The main tools used are Leray-Schauder and Brouwer degree theory.     

On existence of optimal control governed by a class of the first-order linear dynamic systems on time scales

In this paper, by using the approximation of classical solution, we introduce the definition of solution and prove the existence and uniqueness of solutions of the first-order linear dynamic systems on time scales. Existence of Lagrange optimal control problem governed by the first-order linear dynamic systems on time scales is also presented. For illustration, some examples of optimal control problems on time scales are also discussed.     

Necessary and sufficient conditions for local Pareto optimality on time scales

We study a multiobjective variational problem on time scales. For this problem, necessary and sufficient conditions for weak local Pareto optimality are given. We also prove a necessary optimality condition for the isoperimetric problem with multiple constraints on time scales.     

Limit circle invariance for two differential systems on time scales

In this paper we consider two linear differential systems on a time scale. Both systems depend linearly on a complex spectral parameter λ. We prove that if all solutions of these two systems are square integrable with respect to a given weight matrix for one value λ₀, then this property is preserved for all complex values λ. This result extends and improves the corresponding continuous time statement, which was derived by Walker (1975) for two non‐hermitian linear Hamiltonian systems, to appropriate differential systems on arbitrary time scales. The result is new even in the purely discrete case, or in the scalar time scale case, as well as when both time scale systems coincide. The latter case also generalizes a limit circle invariance criterion for symplectic systems on time scales, which was recently derived by the authors.