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On the existence of solutions of fractional differential equations in Banach spaces

Dušan Oberta

TL;DR

This work develops a local existence theory for fractional differential equations in Banach spaces by combining measures of non-compactness with Kamke functions of order $\alpha$. It establishes a local solvability result under a Kamke-type bound on the non-compactness measure and a singleton property, and it also provides a Lipschitz-based uniqueness theorem. The results are then specialized to Banach sequence spaces, yielding local solutions for countable systems arising from semi-discretisations of fractional PDEs with a $p$-Laplacian, including an explicit treatment in the space $c_0$. The paper further demonstrates Kamke-function constructions and discusses avenues for extending the framework to broader settings and higher-order fractional dynamics.

Abstract

Utilising the notion of measures of non-compactness and Kamke function of order $α$, we address the question of solvability of fractional differential equations in Banach spaces. In particular, we provide sufficient conditions ensuring the existence of a local solution. Our main existence theorem is then applied on countable systems of fractional differential equations arising from semi-discretisation of fractional PDEs with $p$-Laplacian.

On the existence of solutions of fractional differential equations in Banach spaces

TL;DR

This work develops a local existence theory for fractional differential equations in Banach spaces by combining measures of non-compactness with Kamke functions of order . It establishes a local solvability result under a Kamke-type bound on the non-compactness measure and a singleton property, and it also provides a Lipschitz-based uniqueness theorem. The results are then specialized to Banach sequence spaces, yielding local solutions for countable systems arising from semi-discretisations of fractional PDEs with a -Laplacian, including an explicit treatment in the space . The paper further demonstrates Kamke-function constructions and discusses avenues for extending the framework to broader settings and higher-order fractional dynamics.

Abstract

Utilising the notion of measures of non-compactness and Kamke function of order , we address the question of solvability of fractional differential equations in Banach spaces. In particular, we provide sufficient conditions ensuring the existence of a local solution. Our main existence theorem is then applied on countable systems of fractional differential equations arising from semi-discretisation of fractional PDEs with -Laplacian.

Paper Structure

This paper contains 15 sections, 21 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Auxiliary Results
  3. Abstract functions
  4. Integrals of abstract functions
  5. Fractional calculus for abstract functions
  6. Initial value problem for fractional differential equations
  7. Basic results from functional analysis
  8. Measures of non-compactness
  9. Kamke function of order $\alpha$
  10. Main Results
  11. Applications
  12. Kamke function of order $\alpha$
  13. Existence of solutions in Banach sequence spaces
  14. Semi-discretisation of fractional PDEs
  15. Concluding remarks

Key Result

Theorem 1

Let $f\in L^1\left(\left[a,b\right],E\right)$. Then it holds that $\norm{f\left(\cdot\right)}_E\in L^1\left(\left[a,b\right],\mathbb{R}\right)$. Moreover

Theorems & Definitions (55)

  • Theorem 1: Mikusinski, Theorem $3.1$ in Chapter III
  • Theorem 2: Mikusinski, Theorem $2.2$ in Chapter XIII
  • Theorem 3: Mikusinski, Theorem $2.3$ in Chapter XIII
  • Definition 1: Riemann-Liouville fractional abstract integral operator of order $\alpha$
  • Remark 1
  • Remark 2
  • Theorem 4: Anastassiou, Theorem $2.3$
  • Theorem 5: Anastassiou, Remark $2.4$ and Theorem $2.5$
  • Theorem 6: Anastassiou, Theorem $2.6$
  • Remark 3
  • ...and 45 more