Dolores R. Vivero

Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral; application to the calculus of Δ-antiderivatives

This paper is devoted to the study of the Lebesgue @D-integral on time scales. We give a formula in which such an integral is obtained as a sum of adequate real Lebesgue integrals. By using the relationship between these kinds of integrals and the Riemann ones, we rewrite the given expression to obtain the @D-antiderivatives of functions defined on time scales. The results obtained are illustrated with some examples.

Criterions for absolute continuity on time scales

In this paper, we establish the concept of absolutely continuous function on and we prove a characterization of such functions, which generalizes the one given for the real case in the classical Banach-Zarecki Theorem. Moreover, we prove that this kind of functions satisfy the Fundamental Theorem of Calculus. A criterion for absolute continuity of the inverse function of any strictly monotone and absolutely continuous function on is also obtained. †Research partially supported by D. G. I. and F.E.D.E.R. project BFM2001-3884-C02-01, and by Xunta of Galicia and F.E.D.E.R. project PGIDIT020XIC20703PN, Spain.

BASIC PROPERTIES OF SOBOLEV'S SPACES ON TIME SCALES

We study the theory of Sobolevs spaces of functions defined on a closed subinterval of an arbitrary time scale endowed with the Lebesgue Δ-measure; analogous properties to that valid for Sobolevs spaces of functions defined on an arbitrary open interval of the real numbers are derived.

Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations

We prove existence and uniqueness results in the presence of coupled lower and upper solutions for the general n th problem in time scales with linear dependence on the i th Δ-derivatives for i = 1,2,…,n, together with antiperiodic boundary value conditions. Here the nonlinear right-hand side of the equation is defined by a function f(t,x) which is rd-continuous in t and continuous in x uniformly in t. To do that, we obtain the expression of the Greens function of a related linear operator in the space of the antiperiodic functions.

Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations

This paper is devoted to using perturbation and variational techniques to derive some sufficient conditions for the existence of multiple positive solutions in the sense of distributions to a singular second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes those problems related to the negative exponent Emden-Fowler equation.

Wirtinger's inequalities on time scales

This paper is devotedto the study of Wirtinger-type inequalities for the Lebesgue�-integral on an arbitrary time scale T. We prove a general inequality for a class of absolutely continuous func- tions on closed subintervals of an adequate subset of T. By using this expression and by assuming that T is bounded, we deduce that a general inequality is valid for every absolutely continuous function on T such that its �-derivative belongs to L2 ((a,b) ∩ T) and at most it vanishes on the boundary of T.

Existence of extremal solutions by approximation to a first-order initial dynamic equation with Carathéodory's conditions and discontinuous non-linearities*

This paper is devoted to the study of the existence of extremal solutions to a first-order initial value problem on an interval of an arbitrary time scale. We prove the existence of extremal solutions for problems satisfying Carathéodorys conditions. Moreover, they are approximated uniformly by a sequence of lower and upper solutions to this problem, respectively. We also can warrant the existence and approximation of extremal solutions for the problem by relaxing their continuity properties.