Large time behavior of the solution to the nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. Furthermore, the rate of convergence is given. Initial-boundary value problem with mixed boundary conditions is considered.

n this paper we investigate systems of linear integral equations in the space ${𝔾}_L^n$ of $n$-vector valued functions which are regulated on the closed interval $[0,1]$ (i.e. such that can have only discontinuities of the first kind in $[0,1]$) and left-continuous in the corresponding open interval $(0,1).$ In particular, we are interested in systems of the form x(t) - A(t)x(0) - 01B(t,s)[d x(s)] = f(t), where $f\in {𝔾}_L^n$, the columns of the $n\times n$-matrix valued function $A$ belong to ${𝔾}_L^n$, the entries of $B(t,.)$ have a bounded variation on $[0,1]$ for any...

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...

This paper is a continuation of [9]. In [9] results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) were presented. The Kurzweil type Stieltjes integration in the setting of [6] for Banach space valued functions was used. Here we consider operator valued solutions of the homogeneous problem (t) = I +dt [A(s)](s) as well as the variation-of-constants formula for the former equation.

In 1990, Hönig proved that the linear Volterra integral equation $$x\left(t\right)-\left(K\right){\int }_{\left[a,t\right]}\alpha \left(t,s\right)x\left(s\right)ds=f\left(t\right),t\in \left[a,b\right],$$ where the functions are Banach space-valued and $f$ is a Kurzweil integrable function defined on a compact interval $\left[a,b\right]$ 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 $$x\left(t\right)-\left(K\right){\int }_{\left[a,t\right]}\alpha \left(t,s\right)x\left(s\right)dg\left(s\right)=f\left(t\right),t\in \left[a,b\right],$$ in a real-valued context.

Error estimates in L∞(0,T;L2(Ω)), L∞(0,T;L2(Ω)2), L∞(0,T;L∞(Ω)), L∞(0,T;L∞(Ω)2), Ω in ${ℝ}^2$, are derived for a mixed finite element method for the initial-boundary value problem for integro-differential equation $$u_t=\mathrm{div}\{a▽u+{\int }_0^t{b}_1▽u\mathrm{d}\tau +{\int }_0^t𝐜u\mathrm{d}\tau \}+f$$ based on the Raviart-Thomas space Vh x Wh ⊂ H(div;Ω) x L2(Ω). Optimal order estimates are obtained for the approximation of u,ut in L∞(0,T;L2(Ω)) and the associated velocity p in L∞(0,T;L2(Ω)2), divp in L∞(0,T;L2(Ω)). Quasi-optimal order estimates are obtained for the approximation...

The existence and attractivity of a local center manifold for fully nonlinear parabolic equation with infinite delay is proved with help of a solutions semigroup constructed on the space of initial conditions. The result is applied to the stability problem for a parabolic integrodifferential equation.