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Well-posedness of fractional degenerate differential equations in Banach spaces

  • Shangquan Bu EMAIL logo and Gang Cai
Published/Copyright: May 11, 2019
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 22 Issue 2

Abstract

We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces Bp,qs (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 < β < α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and Bp,qs-well-posedness of above equation.

MSC 2010: Primary 34K13; Secondary 34K30; 47D06; 47A10
Key Words and Phrases: well-posedness; fractional degenerate differential equation; Fourier multiplier; Lebesgue-Bochner spaces; Besov spaces

Acknowledgements

This work was supported by the NSF of China (Grant No. 11571194, 11731010, 11771063), the Natural Science Foundation of Chongqing (Grant No. cstc2017jcyjAX0006), Science and Technology Project of Chongqing Education Committee (Grant No. KJ1703041), the University Young Core Teacher Foundation of Chongqing (Grant No. 020603011714), the Talent Project of Chongqing Normal University (Grant No. 02030307-00024). G. Cai is the corresponding author.

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Received: 2017-10-18
Revised: 2018-12-27
Published Online: 2019-05-11
Published in Print: 2019-04-24

© 2019 Diogenes Co., Sofia

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  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–2–2019)
  4. Discussion Paper
  5. The flaw in the conformable calculus: It is conformable because it is not fractional
  6. The failure of certain fractional calculus operators in two physical models
  7. Research Paper
  8. Inverse problems for a class of degenerate evolution equations with Riemann – Liouville derivative
  9. On Riesz derivative
  10. A note on fractional powers of strongly positive operators and their applications
  11. Semi-fractional diffusion equations
  12. Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations
  13. Well-posedness of fractional degenerate differential equations in Banach spaces
  14. Structure factors for generalized grey Browinian motion
  15. Linear stationary fractional differential equations
  16. Perfect control for right-invertible Grünwald-Letnikov plants – an innovative approach to practical implementation
  17. On fractional differential inclusions with Nonlocal boundary conditions
  18. On solutions of linear fractional differential equations and systems thereof
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  20. A CAD-based algorithm for solving stable parameter region of fractional-order systems with structured perturbations
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https://doi.org/10.1515/fca-2019-0023
Keywords for this article
well-posedness; fractional degenerate differential equation; Fourier multiplier; Lebesgue-Bochner spaces; Besov spaces

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–2–2019)
  4. Discussion Paper
  5. The flaw in the conformable calculus: It is conformable because it is not fractional
  6. The failure of certain fractional calculus operators in two physical models
  7. Research Paper
  8. Inverse problems for a class of degenerate evolution equations with Riemann – Liouville derivative
  9. On Riesz derivative
  10. A note on fractional powers of strongly positive operators and their applications
  11. Semi-fractional diffusion equations
  12. Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations
  13. Well-posedness of fractional degenerate differential equations in Banach spaces
  14. Structure factors for generalized grey Browinian motion
  15. Linear stationary fractional differential equations
  16. Perfect control for right-invertible Grünwald-Letnikov plants – an innovative approach to practical implementation
  17. On fractional differential inclusions with Nonlocal boundary conditions
  18. On solutions of linear fractional differential equations and systems thereof
  19. Existence of mild solution of a class of nonlocal fractional order differential equation with not instantaneous impulses
  20. A CAD-based algorithm for solving stable parameter region of fractional-order systems with structured perturbations
  21. On representation and interpretation of Fractional calculus and fractional order systems
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