Abstract
This paper concerns the investigation of an eigenvalue problem for a nonlinear fractional differential equation. Using the properties of the Green function, Banach contraction principle, Leray-Schauder nonlinear alternative and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation are considered. Some sufficient conditions for the existence of at least one positive solution is established. Some examples are presented to illustrate the main results.
Highlights
Using the Banach fixed point theorem, the nonlinear alternative of Leray Schauder type and Guokrasnosel’skii fixed point theorem on cone, we investigate the eigenvalue interval for the existence and uniqueness of positive solutions
Let f ∈ C([a, b] × R), u ∈ E is a solution of the fractional boundary value problem (P) if and only if T u(t) = u(t), for any t ∈ [0, 1]
According to Theorem 6 T has a fixed point in P ∩ (Ω2/Ω1), that means that the problem (P2) has at least one positive solution in P ∩ (Ω2/Ω1)
Summary
We recall some necessary definitions. Definition 2.1. [12] If g ∈ L1 ([a, b]) and α > 0, the Riemann -Liouville fractional integral is defined by t. [12] If g ∈ L1 ([a, b]) and α > 0, the Riemann -Liouville fractional integral is defined by t. If g ∈ ACn ([a, b]) the Caputo fractional derivative of order α of g is defined by t cDaα+ g(t). [12] For α > 0, g(t) ∈ C ([a, b]), the homogenous fractional differential equation cDaα+g(t) = 0 has a solution g(t) = c1 + c2t + c3t2 + ... The following lemmas gives some properties of Riemann -Liouville fractional integrals and Caputo fractional derivative. The following lemma is fundamental in the proof of the existence Theorems. Let F be a Banach space and Ω be a bounded open subset of F , 0 ∈ Ω and let T : Ω −→ F be a completely continuous operator.
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