Basic theory of initial value problems of conformable fractional differential equations

• Published in: Advances in difference equations Vol. 2018; no. 1; pp. 1 - 14
• Main Authors: Zhong, Wenyong, Wang, Lanfang
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 12.09.2018; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Boundary value problems; Conformable fractional derivatives; Difference and Functional Equations; Differential equations; Fixed point theorems; Fixed points (mathematics); Fractional calculus; Fractional differential equations; Functional Analysis; Initial value problems; Mathematical analysis; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Fixed point theorems; Initial value problems; Fractional differential equations; Conformable fractional derivatives
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-018-1778-5

In this paper, we discuss the basic theory of the conformable fractional differential equation T α a x ( t ) = f ( t , x ( t ) ) , t ∈ [ a , ∞ ) , subject to the local initial condition x ( a ) = x a or the nonlocal initial condition x ( a ) + g ( x ) = x a , where 0 < α < 1 , T α a x ( t ) denotes the conformable fractional derivative of a function x ( t ) of order α , f : [ a , ∞ ) × R ↦ R is continuous and g is a given functional defined on an appropriate space of functions. The theory of global existence, extension, boundedness, and stability of solutions is considered; by virtue of the theory of the conformable fractional calculus and by the use of fixed point theorems, some criteria are established. Several concrete examples are given to illustrate the possible application of our analytical results.