Basic theory of initial value problems of conformable fractional differential equations

Abstract

In this paper, we discuss the basic theory of the conformable fractional differential equation Tαax(t)=f(t,x(t)),t∈[a,∞), subject to the local initial condition x(a) = xa or the nonlocal initial condition x(a) + g(x) = xa, where 0 < α< 1 , Tαax(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.