On a fractional higher order initial value problem

Existence of Positive Solutions for a Higher-Order Fractional Differential Equation with Multi-Term Lower-Order Derivatives

“…Lemma 8. If δ − α n−1 < 1, the Green's function K(t, s) defined in (19) has the following properties:…”

“…In view of Corollaries 2 and 3, we know that K, K i have properties almost identical to those of K, K i . In combination with (19) and Corollary 1, we can unify the expressions of K, K i and K, K i , and we will uniformly write them as K, K i (i = 1, 2) below.…”

“…Fractional differential equations with various initial or boundary value conditions have been widely discussed, meanwhile, a variety of techniques have been applied to obtain the existence of solutions, uniqueness, multiplicity, etc. In all studies of higher-order differential equations that depend on lower-order derivatives of either integer or fractional order, there is a limitation, i.e., the difference between the highest derivative and adjacent lower-order derivative is greater than or equal to 1 (see [13][14][15][16][17][18][19][20][21][22][23][24]).…”

“…As for Riemann-Liouville-type fractional differential equations, the current result is that the difference between the highest derivative and the adjacent lower derivative is greater than or equal to 1, as can be seen in [17][18][19][20][21][22][23][24]. For example, in [17], applying Schauder's fixed point theorem and upper and lower solutions method, Zhang established the existence of positive solutions to a singular higher-order fractional differential equation involving fractional derivatives:…”

“…By means of the Guo-Krasnoselskii fixed point theorem, under sublinearity conditions, Ref. [19] investigated the existence of at least one positive solution of the following initial value problem with the higher-order Riemann-Liouville-type fractional differential equation:…”