[Paper Review] The New Existence and Uniqueness Results for Complex Nonlinear Fractional Differential Equation

Müfit Şan, Kamal N. Soltanov|arXiv (Cornell University)|Dec 15, 2015

Nonlinear Differential Equations Analysis14 references3 citations

TL;DR

This paper establishes new existence and uniqueness theorems for complex nonlinear fractional differential equations (FDEs) in the unit disc using a complex generalization of the Riemann-Liouville fractional derivative. By applying Schwarz Lemma and analytic continuation, it proves that initial value problems (IVPs) with Caputo or Riemann-Liouville derivatives admit at least one local solution continuous on $[0,R]$ and real analytic on $(0,R)$, with $R \leq 1$, under mild analyticity and growth conditions on the nonlinear term.

ABSTRACT

In this article, we obtain existence and uniqueness results to some problems involving complex nonlinear fractional differential equations (FDEs) in the closed unit disc of C. By help of these results, we prove that some IVPs for some fractional differential equations with Caputo or Riemann-Liouville derivative admit at least one local (or unique) solution continuous on a closed interval $[0,R]$ and real analytic on $(0,R),$ where $0

Motivation & Objective

• To establish existence and uniqueness results for complex nonlinear fractional differential equations (FDEs) in the closed unit disc $\mathbb{U}$.
• To extend classical existence theorems for real fractional IVPs to the complex domain using analytic function theory.
• To prove that solutions are real analytic on $(0,R)$ and continuous on $[0,R]$ for $R \leq 1$, under suitable growth and analyticity conditions.
• To demonstrate that the complex approach yields stronger and more general results than prior real-variable methods.
• To connect complex FDE solutions to real fractional IVPs via real part extraction, ensuring analyticity and uniqueness of real solutions.

Proposed method

• Utilizes a complex generalization of the Riemann-Liouville fractional derivative defined on $\mathbb{U}$, with $0 < a < 1$.
• Imposes conditions (I) and (II) on the nonlinear term $f(z,t)$: analyticity of $z^a f(z,t)$ on $\mathbb{U} \times \mathbb{C}$ and initial condition $z^a f(z,b)|_{z=0} = b / \Gamma(1-a)$.
• Applies Schwarz Lemma to prove uniqueness under a Lipschitz-type condition: $|f(z,\eta) - f(z,\nu)| < \kappa |z|^{-a} |\eta - \nu|$ with $\kappa < 1 / \Gamma(2-a)$.
• Uses the analytic continuation of real FDE right-hand sides to define complex FDEs, ensuring that the real part of the complex solution satisfies the original real IVP.
• Establishes existence via fixed-point arguments in a Banach space of analytic functions on $\overline{\mathbb{U}}_R$, under a growth bound $|z^a f(z,t)| \leq c|t-b|^{n_0} + |h(z)|$.
• Leverages the relation between Caputo and Riemann-Liouville derivatives via $^C D^a u(x) = D^a (u(x) - u(0))$ to extend results to Caputo-type problems.

Experimental results

Research questions

• RQ1 Under what conditions does a complex nonlinear FDE with a Riemann-Liouville-type fractional derivative admit a solution analytic on $\mathbb{U}$ and continuous on $\overline{\mathbb{U}}$?
• RQ2 How can the complex variable approach via analytic continuation and Schwarz Lemma yield stronger existence and uniqueness results than prior real-variable methods?
• RQ3 What conditions ensure that the real part of a complex FDE solution satisfies a corresponding real fractional IVP with Riemann-Liouville or Caputo derivative?
• RQ4 Does the univalence of $z^a f(z,t)$ imply univalence of the solution to the complex FDE?
• RQ5 Can the existence of real analytic solutions to real fractional IVPs be guaranteed under broader conditions than previously known?

Key findings

• The problem $D_z^a u(z) = f(z,u(z))$ with $u(0) = b$ admits at least one solution $u \in \mathcal{B}_R$ analytic on $\mathbb{U}_R$ and continuous on $\overline{\mathbb{U}}_R$ for some $R \in (0,1]$, provided $f$ satisfies conditions (I) and (II).
• Under a Lipschitz-type condition with $\kappa < 1 / \Gamma(2-a)$, the solution to the same problem is unique in $\mathcal{B}_R$.
• For the problem $D_z^a (u(z) - u(0)) = f(z,u(z))$, existence and uniqueness are established under similar conditions, with $z^a f(z,b)|_{z=0} = 0$.
• The real part of the complex solution $u(z)$ satisfies the real fractional IVP $\mathcal{D}^a u(x) = f(x, u(x))$ with Riemann-Liouville or Caputo derivative, provided $f(x,y) = \Re(f(z,t))$.
• When $z^a f(z,t)$ is linear in $z$ and $t$, the real solution to the IVP is unique and real analytic on $(0,1)$, continuous on $[0,1]$.
• The use of Schwarz Lemma allows for a broader class of $f$ in the uniqueness result than in earlier studies, improving on prior results (Remark 3.9(i)).