Positive solutions for a class of fractional differential equations with infinite-point boundary conditions on infinite intervals

In this paper, the existence of the multiple positive solutions for a class of higher-order fractional differential equations on infinite intervals with infinite-point boundary value conditions is mainly studied. First, we construct the Green function and analyze its properties, and then by using the Leggett–Williams fixed point theorem, some new results on the existence of positive solutions for the boundary value problem are obtained. Finally, we illustrate the application of our conclusion by an example.


Introduction
Boundary value problems of fractional differential equations have always been of great interest to researchers and are of great importance in the fields of physics, biology, chemistry, control theory, fluid mechanics, aerodynamics, complex medium electrodynamics, and other areas of engineering and science [1][2][3][4][5][6].The Guo-Krasnoselskii fixed point theorem, Avery-Peterson fixed point theorem, Leggett-Williams fixed point theorem, etc., are important research tools in solving fractional differential equations of boundary value conditions [7][8][9].In recent years, the study of finite multipoint boundary value problems for fractional differential equations on finite intervals has yielded more significant results [10][11][12][13][14][15][16][17].However, the existence of multiple positive solutions for fractional differential equations with infinite multipoint boundary conditions on infinite intervals is relatively rare.
In [10], the authors investigated the fractional differential equation boundary value problems at resonance: © The Author(s) 2023.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
In [11], the authors investigated the following m-point p-Laplacian fractional boundary value problem involving Riemann-Liouville fractional integral boundary conditions on the half-line: where D γ 0 + and D α 0 + are the standard Riemann-Liouville fractional derivatives and p + 1 q = 1.Motivated by the above papers, in this work, we consider the following fractional differential equations with infinite-point boundary conditions on an infinite interval: where For boundary value problem (1), we will first construct the Green function and then use some properties of the Green function to obtain at least three positive solutions to boundary value problem (1) by using the Leggett-Williams fixed point theorem.
The research in this paper is different from the existing studies.In [11], the boundary condition contained finite integral terms, and the authors obtained existence of one positive solution by using the Leray-Schauder nonlinear alternative theorem.In this paper, the boundary condition of boundary value problem (1) contains infinite integral terms and infinite points, and the order of the fractional derivative is higher.The method which we use is the Leggett-Williams fixed point theorem, and we get the existence of three positive solutions.The new results of this paper can be considered as a contribution to this field.
The organization of this paper is as follows.In Sect.2, we show some necessary definitions and lemmas from fractional calculus theory.In Sect.3, we prove the existence of multiple positive solutions of boundary value problem (1).In Sect.4, we will give an example to illustrate the applicability of our conclusions.Now we list some conditions for convenience: is not constant to 0 on any subinterval of [0, +∞), and +∞ 0 a(s) ds < +∞.

Preliminaries
In this section, some definitions and lemmas are introduced.
where n is the smallest integer greater than or equal to α.
Lemma 2. 6 The function G(t, s) in Lemma 2.4 satisfies the following properties: Proof By calculation we can give According to the definition of G(t, s) and ∂G(t,s) ∂t , it is clear that (i) and (ii) hold.Next we will prove that (iii) and (iv) hold.
In summary, min In conclusion, min 1 and O (t) are equicontinuous on any finite subinterval of [0, +∞) and equiconvergent at +∞, that is, for any > 0 there Lemma 2.8 ([12]) Let P be a cone in a real Banach space.Assume that there exists a concave nonnegative continuous functional θ on P, with θ (u) ≤ u , ∀u ∈ P c .Letting a, b, c > 0 be constants, we define

Main results
Define a cone P ⊂ E ∞ by P = {u ∈ E ∞ : u(t) ≥ 0, u (t) ≥ 0}.We introduce an operator T : P → E ∞ as follows: By Lemma 2.4, we can know that the fixed point of T is the solution of the boundary value problem (1) and vice versa.Now we make the following assumption: +∞ 0 a(s)p(s) ds < +∞, where p(s) is defined as (4).
Lemma 3.1 Suppose that (H 1 )-(H 4 ) hold.Then T : P → P is a completely continuous operator.
Proof First of all, we will show that T : P → P.
In view of the properties of G(t, s) and ∂G(t,s) ∂t and the nonnegativity of f , it is easy to know that Tu(t) ≥ 0, Tu (t) ≥ 0, ∀t ∈ [0, +∞).
Secondly, we will prove that T : It can be known from Lebesgue's dominated convergence theorem and the continuity of In the same way, we have It is known from Lebesgue's dominated convergence theorem and the continuity of f that Hence, T : P → P is continuous.Now let ⊂ P be bounded.Then there exists a positive constant κ such that u ≤ κ, ∀u ∈ .Next we will prove that T( ) is bounded.
For any u ∈ , by (H 4 ), we get +∞ 0 p(s)a(s)f s, u(s), u (s) ds It can be known from Lemma 2.6 that lim t→+∞ Tu(t) Therefore, { Tu(t) 1+t α-1 : u ∈ } and { (Tu(t)) 1+t α-1 : u ∈ } are equiconvergent at t → +∞.By Lemma 2.7, we can know that T is relatively compact.So T is completely continuous.Now in the following part of the paper, we take k = (α -2) 1 α-1 .We will use the Leggett-Williams fixed point theorem to prove that there are at least three positive solutions to boundary value problem (1).For convenience, we denote We denote a nonnegative concave functional on P by θ (u) = min min and suppose that the function f satisfies the following conditions: Then boundary value problem (1) has at least three positive solutions u 1 , u 2 , Proof We will show that all conditions of Lemma 2.8 are satisfied for T defined by (7).For all u ∈ P c , we have u ≤ c, that is, 0 ≤ u(t) 1+t α-1 ≤ c, 0 ≤ u (t) 1+t α-1 ≤ c, ∀t ∈ [0, +∞).By using assumption (C 1 ), we can get f (t, u(t), u (t)) = f (t, (1 For all u ∈ P c , we have Tu < c.Thus, T : P c → P c .By Lemma 3.1, we can know that T is completely continuous.Using the above argument, it follows from assumption (C 3 ) that if u ∈ P a , then Tu < a.Hence, condition (ii) of Lemma 2.8 holds.Next we will show that condition (i) of Lemma 2.8 holds.