We study the initial value problem <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column Subscript upper C Baseline normal upper Delta Superscript alpha Baseline u left-parenthesis n right-parenthesis 2nd Column a m p semicolon equals upper A u left-parenthesis n plus 1 right-parenthesis comma n element-of double-struck upper N 0 semicolon 2nd Row 1st Column u left-parenthesis 0 right-parenthesis 2nd Column a m p semicolon equals u 0 element-of upper X comma EndLayout EndLayout"> <mml:semantics> <mml:mtable side="left" displaystyle="false"> <mml:mlabeledtr> <mml:mtd> <mml:mrow> <mml:mtext>(</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mtext>)</mml:mtext> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="right left left" rowspacing="4pt" columnspacing="1em"> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi/> <mml:mi>C</mml:mi> </mml:msub> <mml:msup> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>;</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence="true" stretchy="true" symmetric="true"/> </mml:mrow> </mml:mtd> </mml:mlabeledtr> </mml:mtable> <mml:annotation encoding="application/x-tex">\begin{equation*} \tag {$*$} \left \{\begin {array}{rll} _C\Delta ^{\alpha } u(n) &amp;= Au(n+1), \quad n \in \mathbb {N}_0; \\ u(0) &amp;= u_0 \in X, \end{array}\right . \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a closed linear operator with domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">D(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined on a Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We introduce a method based on the Poisson distribution to show existence and qualitative properties of solutions for the problem <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>∗</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(*)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, using operator-theoretical conditions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show how several properties for fractional differences, including their own definition, are connected with the continuous case by means of sampling using the Poisson distribution. We prove necessary conditions for stability of solutions, that are only based on the spectral properties of the operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the case of Hilbert spaces.