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Existence and uniqueness of mild solutions for a class of psi-Caputo time-fractional systems of order from one to two

Hamza Ben Brahim, Fatima-Zahrae El Alaoui, Asmae Tajani, Delfim F. M. Torres

TL;DR

We address existence/uniqueness of mild solutions for a time-fractional evolution system driven by a psi-Caputo derivative of order $1<\alpha\leq 2$ in Banach spaces, i.e. $^{{\mathcal{CD}}}^{\alpha,\psi}_{0^{+}}\Theta(t)=A\Theta(t)+f(t,\Theta(t))$ with $\Theta(0)=\Theta_0$, $\Theta'(0)=\Theta_1$. The authors construct a mild-solution formula using cosine/sine operator families and Mainardi's Wright-type function via the generalized Laplace transform in the variable $\psi$, enabling an explicit integral representation that extends the classical $(0,1)$ case to $(1,2]$. They establish existence and uniqueness through two fixed-point frameworks under different hypotheses (Lipschitz vs growth/compactness), derive a Mittag-Leffler-based expansion for a second-order elliptic operator, and illustrate applicability with a concrete diffusion example. Together, the results extend mild-solution theory to higher-order psi-Caputo dynamics, providing explicit kernels and representations applicable to memory-influenced diffusion and wave-like processes.

Abstract

We prove the existence and uniqueness of mild solutions for a specific class of time-fractional $ψ$-Caputo evolution systems with a derivative order ranging from 1 to 2 in Banach spaces. By using the properties of cosine and sine family operators, along with the generalized Laplace transform, we derive a more concise expression for the mild solution. This expression is formulated as an integral, incorporating Mainardi's Wright-type function. Furthermore, we provide various valuable properties associated with the operators present in the mild solution. Additionally, employing the fixed-point technique and Grönwall's inequality, we establish the existence and uniqueness of the mild solution. To illustrate our results, we conclude with an example of a time-fractional equation, presenting the expression for its corresponding mild solution.

Existence and uniqueness of mild solutions for a class of psi-Caputo time-fractional systems of order from one to two

TL;DR

We address existence/uniqueness of mild solutions for a time-fractional evolution system driven by a psi-Caputo derivative of order in Banach spaces, i.e. with , . The authors construct a mild-solution formula using cosine/sine operator families and Mainardi's Wright-type function via the generalized Laplace transform in the variable , enabling an explicit integral representation that extends the classical case to . They establish existence and uniqueness through two fixed-point frameworks under different hypotheses (Lipschitz vs growth/compactness), derive a Mittag-Leffler-based expansion for a second-order elliptic operator, and illustrate applicability with a concrete diffusion example. Together, the results extend mild-solution theory to higher-order psi-Caputo dynamics, providing explicit kernels and representations applicable to memory-influenced diffusion and wave-like processes.

Abstract

We prove the existence and uniqueness of mild solutions for a specific class of time-fractional -Caputo evolution systems with a derivative order ranging from 1 to 2 in Banach spaces. By using the properties of cosine and sine family operators, along with the generalized Laplace transform, we derive a more concise expression for the mild solution. This expression is formulated as an integral, incorporating Mainardi's Wright-type function. Furthermore, we provide various valuable properties associated with the operators present in the mild solution. Additionally, employing the fixed-point technique and Grönwall's inequality, we establish the existence and uniqueness of the mild solution. To illustrate our results, we conclude with an example of a time-fractional equation, presenting the expression for its corresponding mild solution.
Paper Structure (7 sections, 17 theorems, 121 equations)

This paper contains 7 sections, 17 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mild solution, resolvent operator and properties
  4. Existence and uniqueness of mild solution
  5. Particular case
  6. Example
  7. Conclusion

Key Result

proposition thmcounterproposition

Let us consider ${(S(t))}_{t \in \mathbb{R}}$ the sine family associated with the strongly continuous cosine family ${(C(t))}_{t \in \mathbb{R}}$. The following assertions hold:

Theorems & Definitions (33)

  • definition thmcounterdefinition: See 2
  • proposition thmcounterproposition: See 2
  • proposition thmcounterproposition: See 2
  • lemma thmcounterlemma: See 2
  • definition thmcounterdefinition: See Almeidaa
  • definition thmcounterdefinition: See Almeidaa
  • proposition thmcounterproposition: See Almeidaa
  • definition thmcounterdefinition: See T.Abdeljawadd
  • proposition thmcounterproposition: See T.Abdeljawadd
  • definition thmcounterdefinition: See T.Abdeljawadd
  • ...and 23 more