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- W3206645271 abstract "Abstract In the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019), the authors have used the Krasnoselskii fixed point theorem for showing the existence of mild solutions of an abstract class of conformable fractional differential equations of the form: $frac{d^{alpha }}{dt^{alpha }}[frac{d^{alpha }x(t)}{dt^{alpha }}]=Ax(t)+f(t,x(t))$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mfrac> <mml:msup> <mml:mi>d</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mrow> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>[</mml:mo> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>d</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mi>x</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>]</mml:mo> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:math> , $tin [0,tau ]$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>]</mml:mo> </mml:math> subject to the nonlocal conditions $x(0)=x_{0}+g(x)$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>x</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>g</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math> and $frac{d^{alpha }x(0)}{dt^{alpha }}=x_{1}+h(x)$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>d</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mi>x</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math> , where $frac{d^{alpha }(cdot)}{dt^{alpha }}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>d</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mo>(</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> </mml:math> is the conformable fractional derivative of order $alpha in, ]0,1]$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mspace /> <mml:mo>]</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:math> and A is the infinitesimal generator of a cosine family $({C(t),S(t)})_{tin mathbb{R}}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>{</mml:mo> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>}</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:msub> </mml:math> on a Banach space X . The elements $x_{0}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and $x_{1}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> are two fixed vectors in X , and f , g , h are given functions. The present paper is a continuation of the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) in order to use the Darbo–Sadovskii fixed point theorem for proving the same existence result given in (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) [Theorem 3.1] without assuming the compactness of the family $(S(t))_{t>0}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> and any Lipschitz conditions on the functions g and h ." @default.
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- W3206645271 title "On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative" @default.
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