Differential and integral equations in the space of regulated functions. (English) Zbl 1081.34504

The report (104 pages) under consideration is a self-contained exposition of generalised linear differential and integral equations whose solutions possess discontinuities of the first kind. The study covers, as a special case, linear problems with impulses. For generalised differential equations, the problem on continuous dependence of solutions of the initial value problem (IVP) on a parameter \(k\in\mathbb{N}\cup\{0\}\) is considered. Further, the problem on unique solvability of generalised integral equations is considered. The report is divided into five chapters and ends with an exhaustive bibliography which ranges between the paper of A. J. Ward (1936) and the paper of M. Tvrdy (2000). The chapterwise division is as follows. Chapter 1 includes preliminaries, Vitali and strongly bounded variation, the Kurzweil integral, and linear operators and their useful properties. Chapter 2 considers Perron-Stieltjes integrals, and the \(\sigma\)-Riemann-Stieltjes integral defined with respect to impulsive behaviour. Basic results like the integration by parts formula, the convergence theorem, substitution theorems and the unsymmetric Fubini theorem are established for studying Stieltjes integral equations. Chapter 3 treats the IVP for the linear homogeneous generalized differential equation \(x(t)-x(0)-\int_0^td[A(s)]x(s)=0\), \(t\in[0,1]\), \(x(0)=\overline x\), where \(A\in BV^{n\times n}\) and \(\overline x\in \mathbb{R}^n\) are given. Here, the problem of continuous dependence of solutions on a parameter \(k\in\mathbb{N}\cup\{0\}\) is discussed. The next chapter continues to study the above equation with a nonhomogeneous term \(f(t)-f(0)\) on the right-hand side and with a boundary condition of the form \[ Mx(0)+\int^t_0 K(s)d[(s)]=r. \] The corresponding controllability problems are dealt with. These concepts are further extended to include interface conditions. The last chapter treats the linear integral equations (operator equations) with impulsive behaviour of the form \((Lx)(t)=A(t)x(t)+\int_0^1B(t,s)d [x(s)]=f(t)\), \(t\in[0, 1)\). Here, the author proves basic existence and uniqueness results for the given equation and obtains an explicit form of its adjoint equation. The entire treatment of the report is based on modern analysis and leads the reader to the frontiers of the latest developments in the specialised topic presented.


34A36 Discontinuous ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
45A05 Linear integral equations
45B05 Fredholm integral equations