Nonlocal Problems with Partially Integral Conditions for Fourth-Order Sobolev-Type Differential Equations

The article is devoted to the study of the solvability of nonlocal boundary value problems with conditions integral with respect to the distinguished variable \(t\) for the differential equations

* \(\left(\frac{\partial^{2}}{\partial t^{2}}+a(t)\right)\Delta u+b(t)u=f(x,t)\)

with the Laplace operator \(\Delta\) with respect to the spatial variables \(x_{1},\ldots,x_{n}\) . Recently in the literature, equations \((*)\) have been called Sobolev-type equations. The article aims to prove existence and uniqueness theorems for regular solutions to the problems under study, i.e., for solutions having all weak derivatives occurring in \((*)\) .