On the nonlocal boundary value problem for the elliptic differential equations with integral type Samarskii–Ionkin conditions

The present paper is devoted to the study of the abstract nonlocal boundary value problem with integral type Samarskii–Ionkin conditions for the differential equation of elliptic type
$$\hspace{-6em} -u''(t)+Au(t)=f(t) (0\leq t\leq T), u\left( 0\right) =\varphi, u'\left( 0\right) =u'\left( T\right) +\int\limits_{0}^{T}\alpha \left( s\right) u(s)ds+\psi. $$
in an arbitrary Banach space $E$ with the positive operator $A$. The well-posedness of this problem in various Banach spaces is established. In applications, theorems on the well-posedness of several nonlocal boundary value problems for elliptic equations with integral type Samarskii–Ionkin conditions are proved.