On some dynamic inequalities of Steffensen type on time scales

In the present article, we investigate some new inequalities of Steffensen type on an arbitrary time scale using the diamond‐α dynamic integrals, which are defined as a linear combination of the delta and nabla integrals. The obtained inequalities extend some known dynamic inequalities on time scales and unify and extend some continuous inequalities and their discrete analogues.     

Basics of diamond- partial dynamic calculus on time scales

In this paper we introduce partial diamond-α dynamic derivatives for two-variable functions and the double integral calculus via the diamond-α dynamic integral on time scales. Also we establish a two-dimensional weighted Hardy–Knopp type integral inequality on time scales.     

Ramsey Theory and Bandwidth of Graphs

 The bandwidth of a graph is the minimum, over vertex labelings with distinct integers, of the maximum difference between labels on adjacent vertices. Kuang and McDiarmid proved that almost all n-vertex graphs have bandwidth . Thus the sum of the bandwidths of a graph and its complement is almost always at least ; we prove that it is always at most 2n−4 log ₂ n+o(log n). The proofs involve improving the bounds on the Ramsey and Turán numbers of the “halfgraph”. Received: September 2, 1998?Final version received: November 29, 1999     

Halanay type inequalities on time scales with applications

This paper aims to introduce Halanay type inequalities on time scales. By means of these inequalities we derive new global stability conditions for nonlinear dynamic equations on time scales. Giving several examples we show that besides generalization and extension to q-difference case, our results also provide improvements for the existing theory regarding differential and difference inequalities, which are the most important particular cases of dynamic inequalities on time scales.     

Interactions of fast and slow waves in hyperbolic systems with two time scales

We consider certain symmetric hyperbolic systems of nonlinear partial differential equations whose solutions vary on two time scales. The large part of the spatial operator is assumed to have constant coefficients, but a nonlinear term multiplying the time derivatives is allowed. We show that if the initial data are not prepared correctly for the suppression of the fast scale motion, but contain errors of amplitude O(?), then the perturbation in the solution will also be of amplitude O(?). Further, if the large part of the spatial operator is nonsingular, we show that the error introduced in the slow scale motion will be of amplitude O(?²), even though fast scale waves of amplitude O(?) will be present in the solution.     

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.     

Universality and extremal aging for dynamics of spin glasses on subexponential time scales

We consider random hopping time (RHT) dynamics of the Sherrington‐Kirkpatrick (SK) model and p‐spin models of spin glasses. For any of these models and for any inverse temperature β > 0 we prove that, on time scales that are subexponential in the dimension, the properly scaled clock process (time‐change process) of the dynamics converges to an extremal process. Moreover, on these time scales, the system exhibits aging‐like behavior, which we call extremal aging. In other words, the dynamics of these models ages as the random energy model (REM) does. Hence, by extension, this confirms Bouchaud's REM‐like trap model as a universal aging mechanism for a wide range of systems that, for the first time, includes the SK model. © 2011 Wiley Periodicals, Inc.     

Refined discrete regular variation and its applications

We introduce a new class of the so‐called regularly varying sequences with respect to τ and state its properties. This class, on one hand, generalizes regularly varying sequences. On the other hand, it refines them and makes it possible to do a more sophisticated analysis in applications. We show a close connection with regular variation on time scales; thanks to this relation, we can use the existing theory on time scales to develop discrete regular variation with respect to τ. We reveal also a connection with generalized regularly varying functions. As an application, we study asymptotic behavior of solutions to linear difference equations; we obtain generalization and extension of known results. The theory also yields, in some way, a new view on the tests for convergence and divergence of series; we establish the statement that generalizes Raabe test and Bertrand test.     

Combined dynamic Grüss inequalities on time scales

We prove a more general version of the Grüss inequality by using the recent theory of combined dynamic derivatives on time scales and the more general notions of diamond-α derivative and integral. For the particular case where α = 1, one obtains the delta-integral Grüss inequality on time scales in (see M. Bohner and T. Matthews [5]); for α = 0 a nabla-integral Grüss inequality is derived. If we further restrict ourselves by fixing the time scale to the real (or integer) numbers, then the standard continuous (discrete) inequalities are obtained.     

Duality principle of time scales and inequalities

Here we adopt, develop further and use the principle of duality in time scales Caputo, [Time scales: from nabla calculus to delta calculus and vice versa via duality, arxiv: 0910.0085v1 [math.OC] 1 Oct. 2009]. Using this principle and based on a variety of important delta inequalities we produce the corresponding nabla ones. We give several applications.     

A model for an age-structured population with two time scales

In the modelisation of the dynamics of a sole population, an interesting issue is the influence of daily vertical migrations of the larvae on the whole dynamical process. As a first step towards getting some insight on that issue, we propose a model that describes the dynamics of an age-structured population living in an environment divided into N different spatial patches. We distinguish two time scales: at the fast time scale, we have migration dynamics and at the slow time scale, the demographic dynamics. The demographic process is described using the classical McKendrick model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process. Assuming that the migration process is conservative with respect to the total population and some additional technical assumptions, we proved in a previous work that the semigroup associated to our problem has the property of positive asynchronous exponential growth and that the characteristic elements of that asymptotic behaviour can be approximated by those of a scalar classical McKendrick model. In the present work, we develop the study of the nature of the convergence of the solutions of our problem to the solutions of the associated scalar one when the ratio between the time scales is ε (0 < ε ⪡ 1). The main result decomposes the action of the semigroup associated to our problem into three parts:

• 1.(1) the semigroup associated to a demographic scalar problem times the vector of the equilibrium distribution of the migration process;
• 2.(2) the semigroup associated to the transitory process which leads to the first part; and
• 3.(3) an operator, bounded in norm, of order ε.

     

A critical oscillation constant as a variable of time scales for half-linear dynamic equations

We present criteria of Hille-Nehari type for the half-linear dynamic equation (r(t)Φ(y ^Δ))^Δ+p(t)Φ(y ^σ ) = 0 on time scales. As a particular important case we get that there is a a (sharp) critical constant which may be different from what is known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. As applications we state criteria for strong (non)oscillation, examine generalized Euler type equations, and establish criteria of Kneser type. Examples from q-calculus, a Hardy type inequality with weights, and further possibilities for study are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete case.     

Asymptotic analysis of the M/G/1 queue with a time‐dependent arrival rate

We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(εt) where ε is a small parameter. In the asymptotic limit ε → 0, we construct approximations to the probability p _n(t)that η customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale Τ= εt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching. This revised version was published online in June 2006 with corrections to the Cover Date.     

Convergence in almost periodic cooperative systems with a first integral

This paper is to investigate the asymptotic dynamics in almost periodic cooperative systems with a first integral. By appealing to the theory of skew-product semiflows we establish the asymptotic almost periodicity of bounded solutions to such systems, which extends the existing convergence results for time independent and periodic cooperative systems with a first integral and proves a conjecture of B. Tang, Y. Kuang and H. Smith in SIAM J. Math. Anal., 24 (1993), 1331-1339.

     

On qualitative analysis of delay systems and xΔ = f(t, x, xσ) on time scales

Here we solve two problems presented in paper [9] (C C Tisdell and A Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal. 68 (2008) 3504–3524). We study existence and uniqueness of solutions for delay systems and first-order dynamic equations of the form x ^Δ = f (t,x,x ^σ ) on time scales by using the Banach’s fixed-point theorem. Some examples are presented to illustrate the efficiency of the proposed results.     

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.     

Several existence theorems of nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales

In this paper, several existence theorems of positive solutions are established for nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales, as an application, an example to demonstrate our results is given. The conditions we used in the paper are different from those in [H.R. Sun, W.T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004) 508–524; H.R. Sun, W.T. Li, Positive solutions for nonlinear m-point boundary value problems on time scales, Acta Math. Sinica 49 (2006) 369–380 (in Chinese); Y. Wang, C. Hou, Existence of multiple positive solutions for one-dimensional p-Laplacian, J. Math. Anal. Appl. 315 (2006) 144–153; Y. Wang, W. Ge, Positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian, Nonlinear Appl. 66 (6) (2007) 1246–1256].     

On the existence of positive solutions of the p‐Laplacian dynamic equations on time scales

In this paper, we investigate the existence of positive solutions for a nonlinear m‐point boundary value problem for the p‐Laplacian dynamic equations on time scales, by applying a Krasnosel'skii's fixed point theorem. As an application, an example is included to demonstrate the main results. Copyright © 2017 John Wiley & Sons, Ltd.     

Homogenization of eigenvalue problem for Laplace–Beltrami operator on Riemannian manifold with complicated ‘bubble‐like’ microstructure

We study the asymptotic behavior of the eigenvalues and the eigenfunctions of the Laplace–Beltrami operator on a Riemannian manifold M^ε depending on a small parameter ε>0 and whose structure becomes complicated as ε→0. Under a few assumptions on scales of M^ε we obtain the homogenized eigenvalue problem. In addition we study the behavior of the heat equation on M^ε and investigate the large time behavior of the homogenized equation. Copyright © 2009 John Wiley & Sons, Ltd.