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• W3100759767 abstract "Abstract We consider in a Banach space E the inverse problem <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:msubsup> <m:mi>𝐃</m:mi> <m:mi>t</m:mi> <m:mi>α</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi mathvariant=script>ℱ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>f</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace=12.5pt>,</m:mo> <m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=false>[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy=false>]</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mn>0</m:mn> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msup> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>T</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mi>u</m:mi> <m:mi>T</m:mi> </m:msup> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mn> 0</m:mn> <m:mo><</m:mo> <m:mi>α</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> (mathbf{D}_{t}^{alpha}u)(t)=Au(t)+mathcal{F}(t)f,quad tin[0,T],u(0)=u^{0}% ,u(T)=u^{T},,0<alpha<1 with operator A , which generates the analytic and compact α-times resolvent family <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mi>α</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>A</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=false>}</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> {{S_{alpha}(t,A)}_{tgeq 0}} , the function <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi mathvariant=script>ℱ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo rspace=4.2pt stretchy=false>(</m:mo> <m:mo rspace=4.2pt>⋅</m:mo> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy=false>]</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {mathcal{F}(,cdot,)in C^{1}[0,T]} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msup> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msup> <m:mo>,</m:mo> <m:msup> <m:mi>u</m:mi> <m:mi>T</m:mi> </m:msup> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>A</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {u^{0},u^{T}in D(A)} are given and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mi>E</m:mi> </m:mrow> </m:math> {fin E} is an unknown element. Under natural conditions we have proved the Fredholm solvability of this problem. In the special case for a self-adjoint operator A , the existence and uniqueness theorems for the solution of the inverse problem are proved. The semidiscrete approximation theorem for this inverse problem is obtained." @default.
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• W3100759767 title "Inverse problem with final overdetermination for time-fractional differential equation in a Banach space" @default.
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