Abstract
The main object of the present paper is to state and prove a general nonnegativity principle in the framework of multiterm fractional differential equations, which we use to investigate some iterative monotone sequences of lower and upper solutions to a certain fractional eigenvalue problem. The obtained results can be easily extended to fractional differential equations of distributed orders since the latter are the natural extension of multiterm fractional differential equations.
Highlights
Fractional calculus is present in different fields of science and technology
Regarding fractional differential equations of distributed orders, they were first used by Caputo in 1967 to study elastic media as well as to model dielectric induction and diffusion [5, 6]
Our main concern in this paper is first the extension of the nonnegativity principle stated in Lemma 3.3 [2] to multiterm fractional differentials, and its application to the investigation of some iterative monotone sequences of lower and upper solutions to the following multiterm fractional eigenvalue problem:
Summary
Fractional calculus is present in different fields of science and technology. We may encounter it in medicine, ecology, seismology, physics, electronics, mechanics, viscoelasticity, etc. Our main concern in this paper is first the extension of the nonnegativity principle stated in Lemma 3.3 [2] to multiterm fractional differentials, and its application to the investigation of some iterative monotone sequences of lower and upper solutions to the following multiterm fractional eigenvalue problem: Lu (t) = −λq (t, u) , t ∈ J = (a, b) ,. We apply the obtained nonnegativity principle in the comparison between any two lower and upper solutions to a certain multiterm fractional boundary value problem. The following theorem gives the uniqueness of the solution whenever it exists as well as the order character of any lower and upper solutions. We conclude as before that the condition λ ≥ −Λ is necessary for the existence of an eigenfunction of problem (4)
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