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- W2034392743 abstract "Using Hausdorff measure of noncompactness and a fixed-point argument we prove the existence of mild solutions for the semilinear integrodifferential equation subject to nonlocal initial conditions<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mi>B</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy=false>[</mml:mo><mml:mrow><mml:mn>0,1</mml:mn></mml:mrow><mml:mo stretchy=false>]</mml:mo></mml:mrow></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:math>, where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mi>A</mml:mi><mml:mo>:</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi><mml:mo>→</mml:mo><mml:mi>X</mml:mi></mml:math>, and for every<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=false>[</mml:mo><mml:mn>0,1</mml:mn><mml:mo stretchy=false>]</mml:mo></mml:math>the maps<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mi>B</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>:</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo stretchy=false>)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi><mml:mo>→</mml:mo><mml:mi>X</mml:mi></mml:math>are linear closed operators defined in a Banach space<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:math>. We assume further that<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mi>D</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo stretchy=false>)</mml:mo></mml:math>for every<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M9><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=false>[</mml:mo><mml:mn>0,1</mml:mn><mml:mo stretchy=false>]</mml:mo></mml:math>, and the functions<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M10><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mo stretchy=false>[</mml:mo><mml:mn>0,1</mml:mn><mml:mo stretchy=false>]</mml:mo><mml:mo>×</mml:mo><mml:mi>X</mml:mi><mml:mo>→</mml:mo><mml:mi>X</mml:mi></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M11><mml:mi>g</mml:mi><mml:mo>:</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mo stretchy=false>[</mml:mo><mml:mn>0,1</mml:mn><mml:mo stretchy=false>]</mml:mo><mml:mo>;</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>→</mml:mo><mml:mi>X</mml:mi></mml:math>are<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M12><mml:mrow><mml:mi>X</mml:mi></mml:mrow></mml:math>-valued functions which satisfy appropriate conditions." @default.
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- W2034392743 title "Existence of Mild Solutions for a Semilinear Integrodifferential Equation with Nonlocal Initial Conditions" @default.
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