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.

MSC:

 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