Ahmet Yantir

EXISTENCE OF SOLUTIONS OF THE DYNAMIC CAUCHY PROBLEM IN BANACH SPACES

Abstract In this paper we obtain the existence of solutions and Carathéodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales xΔ(t)=f(t,x(t)),x(0)=x0,t∈Ia,

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

\matrix{{x^\Delta (t) = f(t,x(t)),} \hfill & {} \hfill \cr {x(0) = x_0 ,} \hfill & {t \in I_a ,} \hfill }

On continuous dependence of solutions of dynamic equations

where f is continuous or f satisfies Carathéodory conditions and some conditions expressed in terms of measures of noncompactness. The Mönch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.

Measure on time scales with mathematica

Stefan Hilger introduced the calculus on time scales in order to unify continuous and discrete analysis in 1988. The study of dynamic equations is an active area of research since time scales unifies both discrete and continuous processes, besides many others. In this paper we give many examples on derivative and integration on time scales calculus with Mathematica. We conclude with solving the first order linear dynamic equation NΔ(t) = N(t), and show that the solution is a generalized exponential function with Mathematica.

Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

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.

Weak solutions for the dynamic Cauchy problem in Banach spaces

Mathematical modeling of time dependent systems are always interesting for applied mathematicians. First continuous and then discrete mathematical modeling are built during the mathematical development from ancient to the modern times. By the discovery of the time scales, the problem of irregular controlling of time dependent systems is solved in 1990s. In this paper, we explain the derivative of functions on time scales and the solutions of some basic calculus problems by using Mathematica.

Positive Solutions of a Second Order m-point BVP on Time Scales

The main goal of the paper is to present a new approach to the problem of continuous dependence of solutions of differential or dynamic problems on their domains. This is of particular interests when we use dynamic (difference, in particular) equations as discretization of a given one. We cover a standard construction based of difference approximations for the continuous one, but we are not restricted only to this case. For a given differential equation we take a sequence of time scales and we study the convergence of time scales to the domain of the considered problem. We choose a kind of convergence of such approximated solutions to the exact solution. This is a step for creating numerical analysis on time scales and we propose to replace in such a situation the difference equations by dynamic ones. In the proposed approach we are not restricted to the case of classical numerical algorithms. Moreover, this allows us to find an exact solution for considered problems as a limit of a sequence of solutions for appropriate time scales instead of solving it analytically or calculating approximated solutions for the original problems.

Nonlinear Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

In this paper we study the Lebesgue Δ-measure on time scales. We refer to [3, 4] for the main notions and facts from the general measure and Lebesgue Δ integral theory. The objective of this paper is to show how the main concepts of Mathematica can be applied to fundamentals of Lebesgue Δ- and Lebesgue