Abstract
In this paper, by using the properties of the Green function, u_{0}-positive operator and Gelfand’s formula, some properties of the first eigenvalue corresponding to the relevant operator are obtained. Based on these properties, the fixed point index of the nonlinear operator is calculated explicitly and some sufficient conditions for the existence and uniqueness results of positive solution are established.
Highlights
1 Introduction The boundary value problems (BVPs) for fractional differential equations arise from the studies of models of aerodynamics, fluid flows, electrodynamics of complex medium, electrical networks, rheology, polymer rheology, economics, biology chemical physics, control theory, signal and image processing
Inspired by the above work, we study the existence and uniqueness of the positive solutions for the following two-term fractional differential equations BVP:
Proof It is not difficult to prove that T : P → P is completely continuous and the existence of positive solution tα–2x for BVP (1.2) is equivalent to that of fixed point x of T in P
Summary
The boundary value problems (BVPs) for fractional differential equations arise from the studies of models of aerodynamics, fluid flows, electrodynamics of complex medium, electrical networks, rheology, polymer rheology, economics, biology chemical physics, control theory, signal and image processing. Inspired by the above work, we study the existence and uniqueness of the positive solutions for the following two-term fractional differential equations BVP:. Lemma 2.10 ([31]) Let L be a completely continuous u0-bounded operator, λ1 > 0 is the first eigenvalue of L, y0 is a positive eigenfunction which belongs to P \ {θ }. Proof It is not difficult to prove that T : P → P is completely continuous and the existence of positive solution tα–2x for BVP (1.2) is equivalent to that of fixed point x of T in P. BVP (1.2) has at least one positive solution Proof It follows from (4.1) that there exist r > 0 and ε1 > 0, such that f (t, y) ≥ (λ1 + ε1)y, 0 ≤ y ≤ r, t ∈ [0, 1]. F (t, y) ≤ (λ1 – ε2)y + M0, for any y ≥ 0, t ∈ [0, 1]
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