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Linear Stieltjes integral equations in Banach spaces

Štefan Schwabik (1999)

Mathematica Bohemica

Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in [5]. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [3]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. Here basic results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) are presented on the basis of...

Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral

Márcia Federson, Ricardo Bianconi (2001)

Archivum Mathematicum

In 1990, Hönig proved that the linear Volterra integral equation x t - ( K ) ∫ a , t α t , s x s d s = f t , t ∈ a , b , where the functions are Banach space-valued and f is a Kurzweil integrable function defined on a compact interval a , b of the real line ℝ , admits one and only one solution in the space of the Kurzweil integrable functions with resolvent given by the Neumann series. In the present paper, we extend Hönig’s result to the linear Volterra-Stieltjes integral equation x t - ( K ) ∫ a , t α t , s x s d g s = f t , t ∈ a , b , in a real-valued context.

Mathematical analysis for the peridynamic nonlocal continuum theory

Qiang Du, Kun Zhou (2011)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We develop a functional analytical framework for a linear peridynamic model of a spring network system in any space dimension. Various properties of the peridynamic operators are examined for general micromodulus functions. These properties are utilized to establish the well-posedness of both the stationary peridynamic model and the Cauchy problem of the time dependent peridynamic model. The connections to the classical elastic models are also provided.

Mathematical analysis for the peridynamic nonlocal continuum theory*

Qiang Du, Kun Zhou (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We develop a functional analytical framework for a linear peridynamic model of a spring network system in any space dimension. Various properties of the peridynamic operators are examined for general micromodulus functions. These properties are utilized to establish the well-posedness of both the stationary peridynamic model and the Cauchy problem of the time dependent peridynamic model. The connections to the classical elastic models are also provided.

On generalized d'Alembert functional equation.

Mohamed Akkouchi, Allal Bakali, Belaid Bouikhalene, El Houcien El Qorachi (2006)

Extracta Mathematicae

Let G be a locally compact group. Let σ be a continuous involution of G and let μ be a complex bounded measure. In this paper we study the generalized d'Alembert functional equationD(μ)    ∫G f(xty)dμ(t) + ∫G f(xtσ(y))dμ(t) = 2f(x)f(y) x, y ∈ G;where f: G → C to be determined is a measurable and essentially bounded function.

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