Advanced Search

We slightly modify the definition of the Kurzweil integral and prove that it still gives the same integral.

We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, ${\int }_R\mathrm{d}\alpha \left(t\right)f\left(t\right)$, where $R$ is a compact interval of ${ℝ}^n$, $\alpha$ and $f$ are functions with values on $L(Z,W)$ and $Z$ respectively, and $Z$ and $W$ are monotone ordered normed spaces. Analogous results can be obtained for the Kurzweil vector integral, ${\int }_R\alpha \left(t\right)\mathrm{d}f\left(t\right)$, as well as to unbounded intervals $R$.

Results on integration by parts and integration by substitution for the variational integral of Henstock are well-known. When real-valued functions are considered, such results also hold for the Generalized Riemann Integral defined by Kurzweil since, in this case, the integrals of Kurzweil and Henstock coincide. However, in a Banach-space valued context, the Kurzweil integral properly contains that of Henstock. In the present paper, we consider abstract vector integrals of Kurzweil and prove Substitution...

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.

We consider a large class of impulsive retarded functional differential equations (IRFDEs) and prove a result concerning uniqueness of solutions of impulsive FDEs. Also, we present a new result on continuous dependence of solutions on parameters for this class of equations. More precisely, we consider a sequence of initial value problems for impulsive RFDEs in the above setting, with convergent right-hand sides, convergent impulse operators and uniformly convergent initial data. We assume that the...

We present a variation-of-constants formula for functional differential equations of the form $$\dot{y}=ℒ\left(t\right)y_t+f(y_t,t),y_{t_0}=\varphi ,$$ where $ℒ$ is a bounded linear operator and $\varphi$ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application $t\mapsto f(y_t,t)$ is Kurzweil integrable with $t$ in an interval of $ℝ$, for each regulated function $y$. This means that $t\mapsto f(y_t,t)$ may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain...