Abstract
In this paper, we study the positive solutions of a higher-order singular fractional differential system with coupled integral boundary conditions. The conditions for the existence of at least one positive solution are established together with the estimates of the lower and upper bounds of the solution at any instant of time. Our results are based on the method of upper and lower solutions and the Schauder fixed point theorem. In the end, an example is worked out to illustrate our main results.
Highlights
The fractional order differential equations has been gaining much attention due to their various applications in science and engineering such as fluid dynamics, heat conduction, control theory, electroanalytical chemistry, economics, fractal theory, fractional biological neurons, etc
It is proved that the fractional order differential equation is a better tool for the description of hereditary properties of various materials and processes than the corresponding integer order differential equation
Westerlund [ ] utilized the fractional differential equation to depict the transmission of electromagnetic waves; the one dimensional model is με
Summary
The fractional order differential equations has been gaining much attention due to their various applications in science and engineering such as fluid dynamics, heat conduction, control theory, electroanalytical chemistry, economics, fractal theory, fractional biological neurons, etc. Was discussed in [ ], in which < α, β ≤ , Dα, Dβ are standard Riemann-Liouville fractional derivatives, a, b : ( , ) → [ , +∞) are continuous, κ, μ : [ , ] → [ , +∞) are nonnegative and integrable functions, and f , g : [ , ] × [ , +∞) → [ , +∞) are continuous. Very few authors studied the existence of positive solutions for the singular phenomena in coupled integral condition for fractional differential system, and this work improves and further develops results of previous work in this field to a certain degree.
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