and Applied Analysis 3 We note that 1.1 in its general form involves some different types of differential and difference equations depending on the choice of the time scale T . For example: 1 for T R, we have σ t t, μ t 0, and xΔ t x′ t , and 1.1 becomes the Cauchy integrodifferential equation: x′ t f ( t, x t , ∫ t 0 k t, s, x s ds ) , t ∈ R,

We prove an existence theorem for the equation x = f(t, xt), x(Θ) = φ(Θ), where xt(Θ) = x(t + Θ), for −r ≤ Θ < 0, t ∈ Ia, Ia = [0, a], a ∈ R+ in a Banach space, using the Henstock-KurzweilPettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose tha...

T he concept of fuzzy integral was initiated by Dubois and Prade [11] and then investigated by Kaleva [21], Goetschel and Voxman [20], Nanda [23] and others. In [33], the Henstock integral of fuzzy-valued functions is defined, while the fuzzy Riemann integral and its numerical integration was investigated byWu in [34]. In [7], the authors introduced some quadrature rules for the integral of fuz...

In this paper we introduce the interval-valued APHenstock-Stieltjes integral and investigate some properties of the these integrals.

The main purpose of this paper is to establish different types of convergence theorems for fuzzy Henstock integrable functions, introduced by  Wu and Gong cite{wu:hiff}. In fact, we have proved fuzzy uniform convergence theorem, convergence theorem for fuzzy uniform Henstock integrable functions and fuzzy monotone convergence theorem. Finally, a necessary and sufficient condition under which th...