Existence and uniqueness of solutions of the fractional integro-differential equations in vector-valued function space

Bahloul Rachid

Archivum Mathematicum (2019)

  • Volume: 055, Issue: 2, page 97-108
  • ISSN: 0044-8753

Abstract

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The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations d d t [ x ( t ) - L ( x t ) ] = A [ x ( t ) - L ( x t ) ] + G ( x t ) + 1 Γ ( α ) ∫ - ∞ t ( t - s ) α - 1 ( ∫ - ∞ s a ( s - ξ ) x ( ξ ) d ξ ) d s + f ( t ) , ( α > 0 ) with the periodic condition x ( 0 ) = x ( 2 π ) , where a ∈ L 1 ( ℝ + ) . Our approach is based on the R-boundedness of linear operators L p -multipliers and UMD-spaces.

How to cite

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Rachid, Bahloul. "Existence and uniqueness of solutions of the fractional integro-differential equations in vector-valued function space." Archivum Mathematicum 055.2 (2019): 97-108. <http://eudml.org/doc/294389>.

@article{Rachid2019,
abstract = {The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations $\frac\{d\}\{dt\}[x(t) - L(x_\{t\})]= A[x(t)- L(x_\{t\})]+G(x_\{t\})+ \frac\{1\}\{\Gamma (\alpha )\} \int _\{- \infty \}^\{t\} (t-s)^\{\alpha - 1\} ( \int _\{- \infty \}^\{s\}a(s-\xi )x(\xi ) d \xi )ds+f(t)$, ($\alpha > 0$) with the periodic condition $x(0) = x(2\pi )$, where $a \in L^\{1\}(\mathbb \{R\}_\{+\})$ . Our approach is based on the R-boundedness of linear operators $L^\{p\}$-multipliers and UMD-spaces.},
author = {Rachid, Bahloul},
journal = {Archivum Mathematicum},
keywords = {periodic solution; $L^\{p\}$-multipliers; UMD-spaces},
language = {eng},
number = {2},
pages = {97-108},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Existence and uniqueness of solutions of the fractional integro-differential equations in vector-valued function space},
url = {http://eudml.org/doc/294389},
volume = {055},
year = {2019},
}

TY - JOUR
AU - Rachid, Bahloul
TI - Existence and uniqueness of solutions of the fractional integro-differential equations in vector-valued function space
JO - Archivum Mathematicum
PY - 2019
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 055
IS - 2
SP - 97
EP - 108
AB - The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations $\frac{d}{dt}[x(t) - L(x_{t})]= A[x(t)- L(x_{t})]+G(x_{t})+ \frac{1}{\Gamma (\alpha )} \int _{- \infty }^{t} (t-s)^{\alpha - 1} ( \int _{- \infty }^{s}a(s-\xi )x(\xi ) d \xi )ds+f(t)$, ($\alpha > 0$) with the periodic condition $x(0) = x(2\pi )$, where $a \in L^{1}(\mathbb {R}_{+})$ . Our approach is based on the R-boundedness of linear operators $L^{p}$-multipliers and UMD-spaces.
LA - eng
KW - periodic solution; $L^{p}$-multipliers; UMD-spaces
UR - http://eudml.org/doc/294389
ER -

References

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