Picard’s Iterative Method for Singular Fractional Differential Equations

Fractional differential equations have gained considerable importance due to their applications in various fields, such as physics, mechanics, chemistry, engineering, etc. It has been found that the differential equations involving fractional derivatives are more realistic and practical to describe many phenomena in nature[1–3]. We also refer the reader to some other works [4–7] on the fractional differential problems. For more details about fractional calculus and fractional differential equations, we refer to the books by Podlubrry [8], Kilbas et al. [9], Lakshmikantham and Vatsala [10] and Agarwal et al. [11]. Recently, there are some works about the existence of solutions for singular fractional differential equations, see [12–16]. The iterative technique is a powerful tool for proving the existence of solutions for differential equations, see, for example, [17–23] and references therein. In [18], Picard’s iterative method is employed to obtain the existence of solutions to fractional differential equations with both Riemann-Liouville and Caputo fractional derivatives. Recently, Yang and Liu discussed the singular fractional differential equation (1) in [24]. The existence of local solutions was obtained by u sing Picard’s iterative method. However, there is a spurious computation in the proof of Lemma 2.4 in [24], which is essential for the main result. Motivated by the above comment, in this paper, we discuss the singular fractional differential equation (1). A modified Picard iterative method is employed to investigate the existence and uniqueness of global solutions to (1). Compared with the earlier results, we obtain the global existence results without any restrictions on the Lipschitz constant. The rest of this paper is organized as follows. In Section 2, we give some definitions and facts about fractional derivative and integral. The main results, the existence and uniqueness of global solutions to (1), is obtained in Section 3. In Section 4, an example is presented to illustrate the main results.