NONLOCAL BY TIME PROBLEM FOR SOME DIFFERENTIAL-OPERATOR EQUATION IN SPACES OF S AND S TYPES

Abstract

In the theory of fractional integro-differentiation the operator $A := \displaystyle \Big(I-\frac{\partial^2}{\partial x^2}\Big)$ is often used. This operator called the Bessel operator of fractional differentiation of the order of $ 1/2 $. This paper investigates the properties of the operator $B := \displaystyle \Big(I-\frac{\partial^2}{\partial x^2}+\frac{\partial^4}{\partial x^4}\Big)$, which can be understood as a certain analogue of the operator $A$. It is established that $B$ is a self-adjoint operator in Hilbert space $L_2(\mathbb{R})$, the narrowing of which to a certain space of $S$ type (such spaces are introduced in \cite{lit_bodn_2}) matches the pseudodifferential operator $F_{\sigma \to x}^{-1}[a(\sigma) F_{x\to \sigma}]$ constructed by the function-symbol $a(\sigma) = (1+\sigma^2+\sigma^4)^{1/4}$, $\sigma \in \mathbb{R}$ (here $F$, $F^{-1}$ are the Fourier transforms). This approach allows us to apply effectively the Fourier transform method in the study of the correct solvability of a nonlocal by time problem for the evolution equation with the specified operator. The correct solvability for the specified equation is established in the case when the initial function, by means of which the nonlocal condition is given, is an element of the space of the generalized function of the Gevrey ultradistribution type. The properties of the fundamental solution of the problem was studied, the representation of the solution in the form of a convolution of the fundamental solution of the initial function is given.