<p>A nonlocal boundary value problem for the elliptic differential equation <Equation ID="Equ25"> <EquationSource Format="TEX">\(\begin{aligned} -v^{\prime \prime }(t)+Av(t)=f(t)\quad (0\le t\le T),v(0)=v(T)+\varphi ,\int \limits _{0}^{T}v(s)ds=\psi \quad \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <msup> <mi>v</mi> <mo>″</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>A</mi> <mi>v</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> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>φ</mi> <mo>,</mo> <munderover> <mo movablelimits="false">∫</mo> <mrow> <mn>0</mn> </mrow> <mi>T</mi> </munderover> <mi>v</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> with the positive operator <i>A</i> is considered. The second order of approximation two-step difference scheme is presented. The well-posedness of this difference scheme in Hölder spaces is established. In applications, the coercive stability estimates in Hölder norms for the solutions of three type elliptic difference nonlocal problems are obtained.</p>