Positive solutions of conformable fractional differential equations with integral boundary conditions

Abstract

In this paper, we discuss the existence of positive solutions of the conformable fractional differential equation T α x(t) + f (t, x(t)) = 0, t ∈ [0, 1], subject to the boundary conditions x(0) = 0 and x(1) = λ 1 0 x(t) dt, where the order α belongs to (1, 2], T α x(t) denotes the conformable fractional derivative of a function x(t) of order α, and f : [0, 1] × [0, ∞) → [0, ∞) is continuous. By use of the fixed point theorem in a cone, some criteria for the existence of at least one positive solution are established. The obtained conditions are generally weaker than those derived by using the classical norm-type expansion and compression theorem. A concrete example is given to illustrate the possible application of the obtained results.

Separation and stability of solutions to nonlinear systems involving Caputo–Fabrizio derivatives

“…Moreover, the definition of the conformable fractional derivative was also suggested in [22] such that the chain rule holds for the new derivative, and some results for this type of derivative have been already achieved [23][24][25][26].…”

A new class of mixed fractional differential equations with integral boundary conditions

“…Integral boundary conditions of fractional problems are approached by various researchers by applying different fixed point theorems, iterative technique, and upper and lower solutions method. We refer the reader to a series of articles [22,23,37,29,28,35,6,33,3,10,9,32,20,34,31,5]. Another aspect of stability analysis has been taken up by a number of mathematicians called "Ulam-Hyers stability" this concept was introduced in the mid of 19 th century and recently it is a one of the most important subjects in the mathematical analysis area.…”

Existence of positive solutions for a class of conformable fractional differential equations with integral boundary conditions and a parameter

“…Very recently, in 2018, W. Zhong and L. Wang [17] discussed the existence of positive solutions of the conformable fractional differential equation…”

“…For the case of η = 1, problem (1.1) and (1.2) reduces to the problem studied by Zhong and Wang in [17]. Our approach is similar to that used in [17], i.e., fixed point theorem in a cone, lower and upper bounds for the Greens function are employed as the main tool of analysis.…”

“…For the case of η = 1, problem (1.1) and (1.2) reduces to the problem studied by Zhong and Wang in [17]. Our approach is similar to that used in [17], i.e., fixed point theorem in a cone, lower and upper bounds for the Greens function are employed as the main tool of analysis. It should noticed that our results seem more natural than those in [17], and in this case, the results in [17] are special cases of those in this paper.…”