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.…”

On Nonlinear Fractional Relaxation Differential Equations

“…Ardjouni and Djoudi [1] investigated the positivity of solutions of a nonlinear Caputo fractional differential equation of the type:…”