Existence and uniqueness of positive solutions for first-order nonlinear Liouville–Caputo fractional differential equations

Abstract

We study the existence and uniqueness of positive solutions of the first-order nonlinear Liouville-Caputo fractional differential equation C D α (x (t) − g(t, x(t))) = f (t, x (t)) , 0 < t ≤ 1, x (0) = x 0 > g (0, x 0) > 0,

Upper and Lower Solution method for Positive solution of generalized Caputo fractional differential equations

“…Proof. Consider the operator Π dened by (10). Then we shall prove that this operator is a contraction in [ψ(t) − ψ(0)] θ Γ(θ + 1)…”

“…On the other hand, there has been much more focus paid in developing the theory of existence and uniqueness of positive solutions for nonlinear FDEs have been investigated by using Leray-Schauder, coincidence degree theory, xed point index theory, xed point theorems in cones and so on, we refer the readers to [10,11,13,12,15,23,6,22,20,30]. For instance, N. Li and C. Wang in [22] studied the existence and uniqueness of positive solution for nonlinear FDE D θ 0+ φ(t) = f (t, φ(t)), 0 < t < 1, φ(0) = 0,…”

A decreasing operator method for a fractional initial value problem on infinite interval

“…It is known that there have a large of fractional derivatives, in which Caputo fractional derivative is important, and the theory of Caputo fractional differential equations also greatly attracted the attention for their wide applications; see, e.g. [1,2,7,10,15,17,19,21,23] and the related references therein.…”

Positive solutions for fractional boundary value problems under a generalized fractional operator

“…On the other hand, there has been much more contemplation paid in developing the theory of existence and uniqueness of positive solutions for nonlinear FDEs, through standard fixed‐point techniques; for details see the literature 17–29 …”