Eigenvalue Criteria for Existence of Positive Solutions to Fractional Boundary Value Problem

The existence andmultiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) Dα0+yðxÞ + f ðx, yðxÞÞ = 0, 0 < x < 1, yð0Þ = y′ð1Þ = y′′ð0Þ = 0 is established, where 2 < α ≤ 3, D0+ is the Caputo fractional derivative, and f : 1⁄20, 1 × 1⁄20,∞Þ⟶ 1⁄20,∞Þ is a continuous function. The conclusion relies on the fixed-point index theory and the Leray-Schauder degree theory. The growth conditions of the nonlinearity with respect to the first eigenvalue of the related linear operator is given to guarantee the existence and multiplicity.


Introduction
In this paper, we concentrate on the existence and multiplicity of positive solutions for the following problem: where 2 < α ≤ 3, C D α 0+ is the Caputo fractional derivative, and f : ½0, 1 × ½0,+∞Þ ⟶ ½0,+∞Þ is a continuous function.
In the past twenty years, the fractional differential equation has aroused great consideration  not only in its application in mathematics but also in other applications in science and engineering, for example, fluid mechanics, viscoelastic mechanics, electroanalytical chemistry, and biological engineering. Bai and Qiu [22,23] have investigated the existence and multiplicity of positive solutions of (1) and (2) by using the nonlinear alternative of the Leray-Schauder type and Krasnoselskii's fixed-point theorem in a cone, but they did not consider its eigenvalue criteria.
The rest of the paper is organized as follows. In Section 2, we recall some concepts relative to fractional calculus and give some lemmas with respect to the corresponding Green function. In Section 3, with the use of the fixed-point theory, some existence and multiplicity results of positive solutions are obtained. At last, two examples are given.

Background Materials
For the convenience of the reader, we give some definitions and lemmas.
Lemma 2 (see [15]). Given h ∈ C½0, 1, the unique solution of is given by where Lemma 3 (see [23]). The Green function Gðx, tÞ defined by (6) satisfies the following properties: Lemma 4 (see [24]). Let K be a cone in a Banach space X, and Ω be a bounded open set in K. Suppose that T : Ω ⟶ K is a completely continuous operator. If there exists y 0 ∈ K \ fθg such that then the fixed-point index iðT, Ω, KÞ = 0.
Lemma 5 (see [24]). Let K be a cone in a Banach space X. Suppose that T : K ⟶ K is a completely continuous operator. If there exists a bounded open set Ω such that each solution of satisfies y ∈ Ω, then the fixed-point index iðT, Ω, KÞ = 1.

Existence and Multiplicity
Let C½0, 1 be endowed with the maximum norm kuk = max 0≤x≤1 juðxÞj and the ordering u ≤ v if uðxÞ ≤ vðxÞ for all x ∈ ½0, 1. Define Given f ∈ Cð½0, 1 × ½0,∞Þ, ½0,∞ÞÞ. Let T, A : K ⟶ C ½0, 1 be the operators defined by and It is well known that T, A : K ⟶ K are all completely continuous [23]. Denote where M, N are positive constants. (11), then the spectral radius rðAÞ > 0 and A has a positive eigenfunction φ 1 corresponding to its first eigenvalue λ 1 = ðrðAÞÞ −1 .
Theorem 8. Suppose the following conditions hold: where λ 1 is the first eigenvalue of the operator A defined by (11). Then, BVP (1) and (2) have at least one positive solution.
Proof. By condition ðI 0 Þ, there exists r 1 > 0 small enough such that
For every φ ∈ K r 1 , for x ∈ ½0, 1 Suppose without loss of generality that T has no fixed point on ∂K r 1 (otherwise, the proof is completed). We claim that In fact, if there exist φ 1 ∈ ∂K r 1 and μ 0 > 0 such that Let It is easy to see that +∞>μ * ≥ μ 0 > 0 and φ 1 ≥ μ * φ * . Taking into account that A is a linear positive operator, we have Therefore, by (17), which contradicts the definition of μ * . Hence (18) holds and we have from Lemma 3 that On the other hand, by ðS ∞ Þ, there exist 0 < σ < 1 and r 2 > r 1 such that f x, y ð Þ≤ σλ 1 y, for all y ≥ r 2 , x ∈ 0, 1 ½ : ð24Þ It is clear that B < +∞. Let In the following, we firstly prove that the set W is bounded.
Choose r 3 > max fr 2 , kðI − A 1 Þ −1 Bkg. Then by Lemma 4, we have By (23) and (29), one has Then, T has at least one fixed point on K r 3 \ K r 1 . This means that problem (1) and (2) have at least one positive solution. The proof is complete. Theorem 9. Suppose the following conditions are met: where λ 1 is the first eigenvalue of the operator A defined by (11). Then BVP (1) and (2) have at least one positive solution.The proof is similar to Theorem 8.

Theorem 10.
Suppose there exist two numbers b > a > 0 such that the following conditions are met: Then, BVP (1) and (2) have at least one positive solution.

Journal of Function Spaces
Proof. If C 1 and C 2 hold, similar to Lemma 3 [6], we have Consequently, the additivity of the fixed-point index implies Consequently, A has a fixed point yðxÞ in Theorem 11. The problem in (1) and (2) has at least two positive solutions if conditions ðI 0 Þ, ðI ∞ Þ, and C 1 hold, where λ 1 is the first eigenvalue of the operator A defined by (11).
Proof. Because ðI 0 Þ and ðI ∞ Þ hold, there exist 0 < r < a < R such that On the other hand, C 1 implies iðA, K a , KÞ = 1. So we have therefore, A has two fixed points, y 1 ∈ K a \ K r , y 2 ∈ K R \ K a .
Theorem 12. The problem in (1) and (2) has at least two positive solutions if conditions ðS 0 Þ, ðS ∞ Þ, and C 2 hold, where λ 1 is the first eigenvalue of the operator A defined by (10).
Proof. Because ðS 0 Þ and ðS ∞ Þ hold, there exist 0 < r < b < R such that On the other hand C 2 implies iðA, K b , KÞ = 0. So we have Therefore, A has two fixed points,

Example
To illustrate the main points, we give two examples.
Example 13. Let Consider the BVP It is not difficult to see that Then liminf y→0 + f x, y ð Þ y = +∞ > λ 1 , where λ 1 is the first eigenvalue of the operator A defined by (11). By Theorem 11, BVP (40) and (41) have at least one positive solution.

Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.