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

Abstract

In this research paper, the nonlinear fractional relaxation equation involving the generalized Caputo derivative is reduced to an equivalent integral equation via the generalized Laplace transform. Moreover, the upper and lower solutions method combined with some xed point theorems, and the properties of the Mittag-Leer function are applied to investigate the existence and uniqueness of positive solutions for the problem at hand. At the end, to illustrate our results, we give an example.

On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator

“…As we know, the existence theory is meat-and-potatoes in every field of science, as it is very applicative to comprehend whether there is a solution to a given differential equation beforehand; otherwise, all the attempts to find a numerical or analytic solution will become valueless. The analysis of fractional differential equations has been carried out by various authors (see, for example, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]).…”

Numerical analysis of coupled systems of ODEs and applications to enzymatic competitive inhibition by product

“…Also, the model may be applied to describe the reaction of β-galactosidase [24]. The system under consideration may be further studied in the form of fractional dierential equations, such as Caputo fractional dierential equations [25,26].…”

On a time fractional diffusion with nonlocal in time conditions

“…For example, when the initial temperature or the final temperature for heat equation is not given immediately, but there is information regarding the temperature over a given period of time that can be described by a nonlocal initial condition. PDEs with nonlocal conditions were considered in many works, for example, see [25] for reaction-diffusion equations and [26][27][28][29][30][31][32][33][34][35] for some other PDEs. As we said before, there are not any results for considering our model (1.1) with the nonlocal final condition and the integral condition…”