We present a direct proof of a known result that the Hardy operator Hf(x) = 1/x ∫0x f(t) dt in the space L2 = L2(0, ∞) can be written as H = I - U, where U is a shift operator (Uen = en+1, n ∈ Z) for some orthonormal basis {en}. The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y' - 1/x y = g and point out some generalizations to the case with weighted Lw2(a, b) spaces.

We study left n-invertible operators introduced in two recent papers. We show how to construct a left n-inverse as a sum of a left inverse and a nilpotent operator. We provide refinements for results on products and tensor products of left n-invertible operators by Duggal and Müller (2013). Our study leads to improvements and different and often more direct proofs of results of Duggal and Müller (2013) and Sid Ahmed (2012). We make a conjecture about tensor products of left n-invertible operators...

In this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory....

We consider a quasistatic frictional contact problem between a viscoelastic body with long memory and a deformable foundation. The contact is modelled with normal compliance in such a way that the penetration is limited and restricted to unilateral constraint. The adhesion between contact surfaces is taken into account and the evolution of the bonding field is described by a first order differential equation. We derive a variational formulation and prove the existence and uniqueness result of the...

In this paper, we use a modification of Krasnoselskii’s fixed point theorem introduced by Burton (see [Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem. Nonlinear Stud., 9 (2002), 181–190.] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equation with variable delay $$x^{\text{'}}\left(t\right)=-a\left(t\right)h\left(x\left(t\right)\right)+\frac{d}{dt}Q\left(t,x\left(t-\tau \left(t\right)\right)\right)+G\left(t,x\left(t\right),x\left(t-\tau \left(t\right)\right)\right).$$ The stability of the zero solution of this eqution provided that $h\left(0\right)=Q\left(t,0\right)=G\left(t,0,0\right)=0$. The Caratheodory condition is used for the functions $Q$ and $G$.

The class of Sturm-Liouville systems is defined. It appears to be a subclass of Riesz-spectral systems, since it is shown that the negative of a Sturm-Liouville operator is a Riesz-spectral operator on L^2(a,b) and the infinitesimal generator of a C_0-semigroup of bounded linear operators.

Let $A$ be an open subset of ${ℝ}^n$, $W_m(A)$ the linear space of $m$-vector valued functions defined on $A$, $G\equiv \{\gamma \}$ a group of orthogonal matrices mapping $A$ onto itself and $T\equiv \{T_{\gamma }\}$ a linear representation of order $m$ of $G$. A suitable group $𝒯(G,T)$ of linear operators of $W_m(A)$ is introduced which leads to a general definition of $T$-invariant linear operator with respect to $G$. When $G$ is a finite group, projection operators are explicitly obtained which define a "maximal" decomposition of the function space into a direct sum of subspaces...

We study general continuity properties for an increasing family of Banach spaces $S_w^p$ of classes for pseudo-differential symbols, where $S_w^{\infty }=S_w$ was introduced by J. Sjöstrand in 1993. We prove that the operators in $\mathrm{Op}\left(S_w^p\right)$ are Schatten-von Neumann operators of order $p$ on $L^2$. We prove also that $\mathrm{Op}\left(S_w^p\right)\mathrm{Op}\left(S_w^r\right)\subset \mathrm{Op}\left(S_w^r\right)$ and $S_w^p·S_w^q\subset S_w^r$, provided $1/p+1/q=1/r$. If instead $1/p+1/q=1+1/r$, then $S_w^{p}w*S_w^q\subset S_w^r$. By modifying the definition of the $S_w^p$-spaces, one also obtains symbol classes related to the $S(m,g)$ spaces.