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.     

Discrete-space partial dynamic equations on time scales and applications to stochastic processes

We consider a general class of discrete-space linear partial dynamic equations. The basic properties of solutions are provided (existence and uniqueness, sign preservation, maximum principle). Above all, we derive the following main results: first, we prove that the solutions depend continuously on the choice of the time scale. Second, we show that, under certain conditions, the solutions describe probability distributions of nonhomogeneous Markov processes, and that their time integrals remain the same for all underlying regular time scales.     

On strongly α-preinvex functions

In this paper, by means of a series of counterexamples, we study in a systematic way the relationships among (pseudo, quasi) α-preinvexity, (strict, strong, pseudo, quasi) α-invexity and (strict, strong, pseudo, quasi) αη-monotonicity. Results obtained in this paper can be viewed as a refinement and improvement of the results of Noor and Noor [M.A. Noor, K.I. Noor, Some characterizations of strongly preinvex functions, J. Math. Anal. Appl. 316 (2006) 697–706].     

Some nonlinear dynamic inequalities on time scales

The aim of this paper is to investigate some nonlinear dynamic inequalities on time scales, which provide explicit bounds on unknown functions. The inequalities given here unify and extend some inequalities in (B G Pachpatte, On some new inequalities related to a certain inequality arising in the theory of differential equation, J. Math. Anal. Appl. 251 (2000) 736–751).     

h-Stability for linear dynamic equations on time scales

We study the h-stability for linear dynamic equations on time scales and their perturbations by using the Bihari type inequality on time scales and the unified time scale quadratic Lyapunov functions.     

A generalized mixed vector variational-like inequality problem

In this paper, we introduce relaxed η-α-P-monotone mapping, and by utilizing KKM technique and Nadler’s Lemma we establish some existence results for the generalized mixed vector variational-like inequality problem. Further, we give the concepts of η-complete semicontinuity and η-strong semicontinuity and prove the solvability for generalized mixed vector variational-like inequality problem without monotonicity assumption by applying the Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.     

Delay dynamic equations on time scales

This article is concerned with delay dynamic equations on time scales. Linear and nonlinear delay dynamic equations are discussed. By using a different theorem to that used S. Hilger, Analysis on measure chains. A unified approach to continous and discrete calculus. Results Math. 18 (1990), pp. 18–56], some criteria for the existence, uniqueness and continuous dependence of the solution for the nonlinear delay dynamic equations are established.     

Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function fC(Y,Z) with YY^′ and ZZ^′, find , or at least , such that ; sometimes Z=Z^′ is demanded. In this spirit the authors prove several quite general theorems in the context Y^′=(X_I)_κ=∏_iIX_i in the κ-box topology (that is, with basic open sets of the form ∏_iIU_i with U_i open in X_i and with U_i≠X_i for <κ-many iI). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this. Theorem Let ωκα (that means: κ<α, and [β<α and λ<κ]β^λ<α) with α regular, be a set of non-empty spaces with each d(X_i)<α, π[Y]=X_J for each non-empty JI such that |J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets of Z×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for every fC(Y,Z) there is J[I]^<α such that f(x)=f(y) whenever x_J=y_J, and (c) every such f extends to .     

New comparison principle and stability criteria for impulsive hybrid systems on time scales

In this paper, we develop a new comparison principle on time scales, and give some new stability criteria for a class of impulsive hybrid systems.     

Mean value in invexity analysis

In this paper, a generalization of the mean value theorem is considered in the case of functions defined on an invex set with respect to η (which is not necessarily connected).     

Oscillation criteria for second-order dynamic equations on time scales

Erbe’s and Hassan’s contributions regarding oscillation criteria are interesting in the development of oscillation theory of dynamic equations on time scales. The objective of this paper is to amend these results.     

G-pre-invex functions in mathematical programming

In the present paper, we introduce the concept of G-pre-invex functions with respect to η defined on an invex set with respect to η. These function unify the concepts of nondifferentiable convexity, pre-invexity and r-pre-invexity. Furthermore, relationships of G-pre-invex functions to various introduced earlier pre-invexity concepts are also discussed. Some (geometric) properties of this class of functions are also derived. Finally, optimality results are established for optimization problems under appropriate G-pre-invexity conditions.     

A countable Fréchet–Urysohn space of uncountable character

We shall construct a countable Fréchet–Urysohn α₄ not α₃ space X such that all finite powers of X are Fréchet–Urysohn.     

Almost automorphic solutions of dynamic equations on time scales

In the present work, we introduce the concept of almost automorphic functions on time scales and present the first results about their basic properties. Then, we study the nonautonomous dynamic equations on time scales given by x^Δ(t)=A(t)x(t)+f(t)$x^{\mathrm{\Delta }}(t)=A(t)x(t)+f(t)$ and x^Δ(t)=A(t)x(t)+g(t,x(t))$x^{\mathrm{\Delta }}(t)=A(t)x(t)+g(t,x(t))$, t∈T$t\in \mathbb{T}$ where T$\mathbb{T}$ is a special case of time scales that we define in this article. We prove a result ensuring the existence of an almost automorphic solution for both equations, assuming that the associated homogeneous equation of this system admits an exponential dichotomy. Also, assuming that the function g satisfies the global Lipschitz type condition, we prove the existence and uniqueness of an almost automorphic solution of the nonlinear dynamic equation on time scales. Further, we present some applications of our results for some new almost automorphic time scales. Finally, we present some interesting models in which our main results can be applied.     

Oscillation criteria for half-linear dynamic equations on time scales

This paper is concerned with oscillation of the second-order half-linear dynamic equation

(r(t)(xΔγ⁾Δ⁾+p(t)xγ(t)=0,     

On a Theorem of T. Gneiting on α-Symmetric Multivariate Characteristic Functions

T. Gneiting (1998, J. Multivariate Analysis64, 131–147) proved a relation between the primitives of the classes Φ_d(2) and Φ_d(1) of 2- and 1-symmetric characteristic functions on ^d, respectively. We will give a straightforward proof of his relation, answering a question of his. To do this we use the calculus of generalized hypergeometric functions.     

The asymptotic behavior of quadratic forms in -mixing random variables

Let {Z_i,i≥1} be a linear process defined by with {d_j,j≥0} being a regular varying sequence of real numbers and {ξ_t,−∞<t<∞} being a sequence of -mixing random variables. The present paper studies the asymptotic behavior of the quadratic form under some mild assumptions on d_j and ξ_t. Meanwhile, the similar results of α-mixing random variables are presented.     

The α-limit sets of a unimodal map without homtervals

Let f:I→I be a unimodal map on I without homtervals. We characterize the α-limit set of each point in I by considering the consecutive renormalization process of f.     

Notes on the diamond-α dynamic derivative on time scales

Various dynamic derivative formulae have been proposed in the development of a time scales calculus, with the goal of unifying continuous and discrete analysis. Recent discussions of combined dynamic derivatives, in particular the α_? derivative defined as a linear combination of the Δ and the ∇ derivatives, have promised improved approximation formulae for computational applications. This paper presents an equivalent definition of the α_? functions without reference to the existing Δ and ∇ derivatives, examines the status of the α_? as a dynamic derivative and its properties relative to the Δ and ∇ derivatives, and compares data obtained using the various dynamic derivatives as approximation formulae in computational experiments. A α_? integral case is investigated.