Fractional Evolution Equation with Nonlocal Multi-Point Condition: Application to Fractional Ginzburg–Landau Equation
Abstract
:1. Introduction
- : The state function in Banach space equipped with the norm ;
- : A Hilfer–Katugampola fractional derivative of order and types and ;
- : A Katugampola fractional integral of order with ;
- A: An infinitesimal generator of semigroup in a Banach space ;
- : The continuous function in a Banach space ;
- are real constants;
- are positive real constants such that .
2. Preliminaries
2.1. Fractional Calculus
- The infinite series is called the Neumann series, and it is also an element in .
- The operator has a bounded inverse operator on .
- .
- (Eigenvalue) There is a non-trivial solution to the equation .
- (Bounded resolvent) The operator has a bounded inverse on .
- u and v are nonnegative;
- g is nonnegative and nondecreasing.
2.2. -Laplace Transform
2.3. Semigroup
- where (the semigroup property);
- , where I is the identity operator in .
3. Construct of Mild Solution and Some Properties
- 1.
- For any fixed , the operators and are linear and satisfy
- 2.
- The operators and are strongly continuous for all ;
- 3.
- If is a compact set, then the operators and are also compact operators for every .
- 1.
- The linearity of the semi group operator Q implies the linearity of and . Assume that H1 is satisfied. If letting in the Definition 9. Direct calculating gives:From the Definition 1, Lemma 1 and inequality (8), we have
- 2.
- Let for all and , whereHence, for any and , we obtainHere, we use the well-know formula for all and . Under the assumption H1 and based on Lebesgue dominated convergence Theorem 1, we see that approaches zero as . Now, since and , then which implies thatConsequently, we have
- 3.
- Define . It is clear that is a closed and bounded subset in .Now, we need to prove the relative compactness of the two sets and in , where
4. Construct of Mild Solution with Multi-Point Conditions
- C1:
- There exist some real numbers such that
- C2:
- The operator is compact for each and the homogeneous linear nonlocal problem
5. Existence and Uniqueness of Mild Solution
- (I)
- The function ;
- (II)
- There exists a positive constant such that
6. An Application: The Real Ginzburg–Landau Equation
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Salem, A.; Al-Maalwi, R. Fractional Evolution Equation with Nonlocal Multi-Point Condition: Application to Fractional Ginzburg–Landau Equation. Axioms 2025, 14, 205. https://doi.org/10.3390/axioms14030205
Salem A, Al-Maalwi R. Fractional Evolution Equation with Nonlocal Multi-Point Condition: Application to Fractional Ginzburg–Landau Equation. Axioms. 2025; 14(3):205. https://doi.org/10.3390/axioms14030205
Chicago/Turabian StyleSalem, Ahmed, and Rania Al-Maalwi. 2025. "Fractional Evolution Equation with Nonlocal Multi-Point Condition: Application to Fractional Ginzburg–Landau Equation" Axioms 14, no. 3: 205. https://doi.org/10.3390/axioms14030205
APA StyleSalem, A., & Al-Maalwi, R. (2025). Fractional Evolution Equation with Nonlocal Multi-Point Condition: Application to Fractional Ginzburg–Landau Equation. Axioms, 14(3), 205. https://doi.org/10.3390/axioms14030205