Existence and uniqueness theorems for nonlinear fractional differential equations. (English) Zbl 0964.34004

Summary: The authors study the following Cauchy-type problem for the nonlinear differential equation of fractional order \(\alpha\in \mathbb{C}\), \(\text{Re}(\alpha)> 0\), \[ (D^\alpha_{a+}y)(x)= f[x, y(x)],\quad n-1< \text{Re}(\alpha)\leq n,\quad n= -[-\text{Re}(\alpha)], \]
\[ (D^{\alpha- k}_{a+} y)(a+)= b_k,\quad b_k\in \mathbb{C},\quad k= 1,2,\dots, n, \] containing the Riemann-Liouville fractional derivative \(D^\alpha_{a+}y\), on a finite interval \([a,b]\) of the real axis \(\mathbb{R}= (-\infty, \infty)\) in the space of summable functions \(L(a,b)\). An equivalence of this problem and a nonlinear Volterra integral equation are established. The existence and uniqueness of the solution \(y(x)\) to the above Cauchy-type problem are proved by using the method of successive approximations. Corresponding assertions for the ordinary differential equations are presented. Examples are given.


34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
26A33 Fractional derivatives and integrals
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations