Solvability Conditions for the Nonlocal Boundary-Value Problem for a Differential-Operator Equation with Weak Nonlineari
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SOLVABILITY CONDITIONS FOR THE NONLOCAL BOUNDARY-VALUE PROBLEM FOR A DIFFERENTIAL-OPERATOR EQUATION WITH WEAK NONLINEARITY IN THE REFINED SOBOLEV SCALE OF SPACES OF FUNCTIONS OF MANY REAL VARIABLES V. S. Il’kiv,1 N. I. Strap,2 and I. I. Volyanska3, 4
UDC 517.946+511.37
We study the solvability of the nonlocal boundary-value problem for a differential equation with weak nonlinearity. By using the Nash–Mozer iterative scheme, we establish the solvability conditions for the posed problem in the Hilbert H¨ormander spaces of functions of several real variables, which form a refined Sobolev scale.
1. Introduction In recent years, the interest of the researchers in the investigation of various problems for partial differential equations in various classes of function spaces (both in the Sobolev spaces [1, 2] and in the H¨ormander spaces [3, 4]) has been considerably increased. Thus, the theory of nonlocal boundary-value problems for partial differential equations [5] was extended, e.g., from the class of Sobolev spaces to the class of Hilbert H¨ormander spaces, which form a refined Sobolev scale of spaces [6, 7]. In the present paper, we study the nonlocal boundary-value problem for a differential-operator equation with nonlinear right-hand side in a refined Sobolev scale of functions of many real variables. In these scales of spaces, a numerical parameter specifies the main smoothness, while a functional parameter specifies the auxiliary smoothness. The solvability of the problem is proved by using the Nash–Mozer iterative scheme [8, 9]. The most important feature of this scheme is the possibility of estimation of norms in the corresponding spaces of inverse linearized operators appearing in each iteration. The problem of construction of these estimates is connected with the problem of small denominators, which is solved with the help of the metric approach in the set of parameters of the problem. In [10], we studied the conditions of solvability of this nonlocal problem in the Sobolev spaces of functions of many real variables. 2. Main Notation and Statement of the Problem Let X be a separable Hilbert space and let Aˆi : X ! X, i = 1, . . . , p, be linear operators with common spectral representation, i.e., there exists a complete orthonormal system of elements xk 2 X, k 2 N, such that the equalities Aˆi xk = ↵ik xk , i = 1, . . . , p, k 2 N, are true for some complex numbers ↵ik . Further, we use the notation ↵k = (↵1k , . . . , ↵pk ), k↵k k2 = |↵1k |2 + . . . + |↵pk |2 , and the assumption that k↵k k > Ck β0 , C > 0, β0 2 R. 1
“Lvivs’ka Politekhnika” National University, Lviv, Ukraine; e-mail: [email protected]. “Lvivs’ka Politekhnika” National University, Lviv, Ukraine; e-mail: [email protected]. 3 “Lvivs’ka Politekhnika” National University, Lviv, Ukraine; e-mail: [email protected]. 4 Corresponding author. 2
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 452–466, April, 2020. Original article submitted December 15, 2019. 0041-5995/20/7204–0515
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