<p>In this paper, we study one abstract nonlocal boundary-value problem with Samarskii–Ionkin conditions of integral type for the differential equation of elliptic type <Equation ID="Equ22"> <EquationSource Format="TEX">\(\begin{aligned} -u^{\prime \prime }(t)+Au(t)=f(t)\quad (0\le t\le T),~u\left( 0\right) =\varphi ,~u^{{\prime }}\left( 0\right) =u^{{\prime }}\left( T\right) +\int \limits _{0}^{T}\alpha \left( s\right) u(s)ds+\psi \quad \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <msup> <mi>u</mi> <mo>″</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>A</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="3.33333pt" /> <mi>u</mi> <mfenced close=")" open="("> <mn>0</mn> </mfenced> <mo>=</mo> <mi>φ</mi> <mo>,</mo> <mspace width="3.33333pt" /> <msup> <mi>u</mi> <mo>′</mo> </msup> <mfenced close=")" open="("> <mn>0</mn> </mfenced> <mo>=</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mfenced close=")" open="("> <mi>T</mi> </mfenced> <mo>+</mo> <munderover> <mo movablelimits="false">∫</mo> <mrow> <mn>0</mn> </mrow> <mi>T</mi> </munderover> <mi>α</mi> <mfenced close=")" open="("> <mi>s</mi> </mfenced> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <mi>s</mi> <mo>+</mo> <mi>ψ</mi> <mspace width="1em" /> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in an arbitrary Banach space <i>E</i>,&#xa0; where the operator <i>A</i> is positive. We establish well-posedness of this problem in various Banach spaces. We prove theorems on the well-posedness of several nonlocal boundary-value problems for elliptic equations with Samarskii–Ionkin conditions of integral type as an application.</p>