Abstract
This manuscript provides some results concerning the sign of solutions for linear fractional integral equations with constant coefficients. This information is later used to prove the existence of solutions to some nonlinear problems, together with underestimates and overestimates. These results are obtained after applying suitable modifications in the classical process of monotone iterative techniques. Finally, we provide an example where we prove the existence of solutions, and we compute some estimates.
Highlights
The theory of equations of arbitrary order has been proposed as an adequate framework to deal with the heterogeneity and memory effects present in the physical phenomena [1,2]
For nonlinear problems, one interesting approach is the development of iterative techniques based on the use of upper and lower solutions [5]
We introduce some basic concepts involving fractional calculus, together with some fundamental and useful results
Summary
The theory of equations of arbitrary order has been proposed as an adequate framework to deal with the heterogeneity and memory effects present in the physical phenomena [1,2]. Linear problems for fractional equations can be addressed by passing to integer order equations [3,4]. For nonlinear problems, one interesting approach is the development of iterative techniques based on the use of upper and lower solutions [5]. The main purpose of this manuscript is to provide some results concerning estimations of solutions to nonlinear fractional integral problems. We introduce some basic concepts involving fractional calculus, together with some fundamental and useful results. We describe several theorems providing conditions ensuring that certain linear fractional integral equations with constant coefficients have nonnegative solutions. We use the previous results to adapt the classical idea of the monotone iterative technique [5,6] for this case. We give an example of application in a specific nonlinear equation
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