Abstract
The purpose of this paper is to establish some results on the existence and nonexistence of positive solutions for a type of nonlinear fractional-order two-point boundary value problems. The main tool is a fixed point theorem of the cone expansion and compression of functional type due to Avery et al. Some examples are presented to illustrate the availability of the main results.
Highlights
This paper investigates the fractional boundary value problem (FBVP for short): ⎧⎨cDα + u(t) + f (t, u(t)) =, t ∈ (, ), ⎩u( ) = u ( ) = u ( ) = · · · = u(n– )( ) =, u ( ) =, ( . )where cDα + is Caputo’s fractional derivative
Fractional differential equations can be extensively applied to various disciplines such as physics, mechanics, chemistry, engineering, and many other branches of science
There have been some papers dealing with the existence and multiplicity of solutions of nonlinear fractional differential equations with various boundary conditions
Summary
This paper investigates the fractional boundary value problem (FBVP for short):. ⎨cDα + u(t) + f (t, u(t)) = , t ∈ ( , ), ⎩u( ) = u ( ) = u ( ) = · · · = u(n– )( ) = , u ( ) = ,. Fixed point theorems have been applied to various boundary value problems to show the existence and multiplicity of positive solutions in the last two decades. Motivated greatly by the above-mentioned work, by constructing a special cone and using the fixed point theorem of cone expansion and compression of functional type due to Avery et al, in this paper, we obtain some sufficient conditions for the existence of positive solutions for FBVP To the author’s best knowledge, no paper in the existing literature can be found using this fixed point theorem to prove the existence of a positive solution to the boundary value problem of nonlinear fractional-order differential equations. The approach used in proving the existence results in this paper is the following fixed point theorem of cone expansion and compression of functional type due to Avery et al [ ]. FBVP ( . ) has at least one positive and nondecreasing solution u∗ satisfying r ≤ min u∗(t) and max u∗(t) ≤ R
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