Abstract
We establish the existence, uniqueness, and global behavior of a positive solution for the following superlinear fractional boundary value problem: D α u(x )= u(x)ϕ(x, u(x)), x ∈ (0, 1), limx→0+ D α-1 u(x )=- a, u(1) = b ,w here 1 0a ndϕ(x, t) is a nonnegative continuous function in (0, 1) × (0, ∞ )t hat is required to satisfy some appropriate conditions related to a certain class of functions Kα. Our approach is based on estimates of the Green's function and on perturbation arguments. MSC: 34A08; 34B18; 34B27
Highlights
Fractional differential equations are gaining much importance and attention since they can be applied in various fields of science and engineering
Motivated by the above mentioned work, we study in this paper the existence, uniqueness, and global behavior of a positive continuous solution for the following superlinear fractional boundary value problem: Dαu(x) = u(x)φ(x, u(x)), x ∈ (, ), limx→ + Dα– u(x) = –a, u( ) = b, ( . )
In order to simplify our statements, we introduce some convenient notations
Summary
Fractional differential equations are gaining much importance and attention since they can be applied in various fields of science and engineering. For sufficiently small positive constant λ, the following problem: Dαu(x) = λp(x)u(x)f (u(x)), x ∈ ( , ), limx→ + Dα– u(x) = –a, u( ) = b, has a unique positive solution u in C –α([ , ]) satisfying c ω(x) ≤ u(x) ≤ ω(x), where c is a constant in ( , ). ) and the dominated convergence theorem that the function (x, t) → x –αGn(x, t) is continuous on [ , ] × [ , ]
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