Operational calculus and differential equations with infinitely smooth coefficients.

*(English)*Zbl 0701.44006The field \(M\) of Mikusiński operators, provided with addition and convolution (denoted by \(+\) and \(*\)) has been used in solving different problems. The inability to define a suitable product of a function and an operator has had a limiting effect, specially if we apply Mikusiński operators to differential equations. Many authors tried to introduce a product of an element belonging to \(M\) and some special functions, as polynomials and exponential functions, but with restricted effects. The author of this paper introduces the product of a smooth function and an element of \(M^ k\), subring of \(M\).

Let \(C\) be the ring of continuous functions on \([0,\infty)\) and \(\ell =\{1\}\in C\). For \(k=0,1,...\), let \(M^ k=\{x\in M\), \(\ell^ k*x\in C\}\). \(M^ k\) is a Fréchet space with the family of seminorms: \(p_{k,m}(x)=\{\sup | (\ell^ k*x)(t)|\), \(0\leq t\leq m\}\), \(m\in N\). \(M_ F\) is the countable union space \(M_ F=\cup^{\infty}_{k=0}M^ k\) with the corresponding convergent class. If \(x\in M_ F\) it belongs to a \(M^ k\). For \(f\in C^{\infty}\) and \(x\in M^ k\) the product is defined by \[ f\cdot x=\sum^{k}_{j=0}(- 1)^ j\left( \begin{matrix} k\\ j\end{matrix} \right)f^{(j)}(\ell^ k*x)/\ell^{k-j}\quad (g/\ell^ 0=g). \] The author proves properties of the defined product and applies it to differential equations proving that the corresponding equation in M to the differential equation \(y''+fy'+gy=0\) with \(y(0)=\alpha\), \(y'(0)=\beta\), where \(\alpha\),\(\beta\in {\mathbb R}\) and \(f,g\in C^{\infty}\) has only the classical solution.

Let \(C\) be the ring of continuous functions on \([0,\infty)\) and \(\ell =\{1\}\in C\). For \(k=0,1,...\), let \(M^ k=\{x\in M\), \(\ell^ k*x\in C\}\). \(M^ k\) is a Fréchet space with the family of seminorms: \(p_{k,m}(x)=\{\sup | (\ell^ k*x)(t)|\), \(0\leq t\leq m\}\), \(m\in N\). \(M_ F\) is the countable union space \(M_ F=\cup^{\infty}_{k=0}M^ k\) with the corresponding convergent class. If \(x\in M_ F\) it belongs to a \(M^ k\). For \(f\in C^{\infty}\) and \(x\in M^ k\) the product is defined by \[ f\cdot x=\sum^{k}_{j=0}(- 1)^ j\left( \begin{matrix} k\\ j\end{matrix} \right)f^{(j)}(\ell^ k*x)/\ell^{k-j}\quad (g/\ell^ 0=g). \] The author proves properties of the defined product and applies it to differential equations proving that the corresponding equation in M to the differential equation \(y''+fy'+gy=0\) with \(y(0)=\alpha\), \(y'(0)=\beta\), where \(\alpha\),\(\beta\in {\mathbb R}\) and \(f,g\in C^{\infty}\) has only the classical solution.

Reviewer: B.Stanković

##### MSC:

44A40 | Calculus of Mikusiński and other operational calculi |

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |