Positive solution of a fractional differential equation with integral boundary conditions

Abstract

In this paper, we prove the existence and uniqueness of a positive solution for a boundary value problem of nonlinear fractional differential equations involving a Caputo fractional operator with integral boundary conditions. The technique used to prove our results depends on the upper and lower solution, the Schauder fixed point theorem and the Banach contraction principle. The result of existence obtained through constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Illustrative examples are provided. MSC 2010: 34A08, 34B15, 34B18, 47H10 Keywords: fractional derivative and integral, positive solutions of nonlinear boundary value problems, upper and lower solutions, fixed point theorem

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

“…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,…”

Positive solutions for first-order nonlinear Caputo-Hadamard fractional differential equations

“…In [2], Abdo, Wahash and Panchat discussed the existence and uniqueness of the positive solution of the following nonlinear fractional differential equation with integral boundary conditions…”

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

“…Fractional differential equations with and without delay arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1][2][3][4][5][6][7][8][9][10][11][12]14] and the references therein.…”

“…By employing the method of the upper and lower solutions and Schauder and Banach fixed point theorems, the authors obtained positivity results. In Abdo et al [2] discussed the existence and uniqueness of the positive solution of the following nonlinear fractional differential equation with integral boundary conditions…”

“…In this paper, we are interested in the analysis of qualitative theory of the problems of the positive solutions to fractional differential equations. Inspired and motivated by the works mentioned above and the papers [1][2][3][4][5][6][7][8][9][10][11][12]14] and the references therein, we concentrate on the positivity of solutions for the first-order nonlinear Liouville-Caputo…”