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Introduction

In this paper, we will devote to considering the following infinite-point singular p-Laplacian fractional differential equation:

with infinite-point boundary condition

Where Laplacian operator φp is defined as and 0 1 and

has singularity at is the standard Riemann-Liouville derivative, is the estandard Caputo derivative.

Fractional-order system may have additional attractive feature over the integer-ordersystem. For example, the analytical solutions of the systems

are respectively. Obviously, the integer-ordersystem is unstable for a , the fractional dynamic system is stable as 0 < a$\le 1$.Moreover, fractional-order systems have been shown to be more accurate and realisticthan integer-order models, and it also provides an excellent tool to describe the hereditaryproperties of material and processes, particularly, in viscoelasticity, electrochemistry,porous media, and so on. As a result, there has been a significant development in thestudy of fractional differential equations in recent years, readers can refer to [2, 4–10,15–17, 21–24]. Jong [12] studied the following p-Laplacian fractional differential equations:

with m-point boundary condition

Where is a fixed integer, permits singularities with respect to both thetime and space variables. According to introducing height functions, the author obtainedthe existence and multiplicity of positive solution theorems, and Zhang and Zhai obtainedthe existence and uniqueness of positive solution for this equation in [18]. In [19], Zhangand Liu investigated the following infinite-point fractional differential equation:

Laplacian operator$\textcolor{red}{}\phi p$ is defined as

The authors obtained the existence and uniqueness of solutions byusing the fixed point theorem for mixed monotone operators. Jong [11] obtained theexistence and uniqueness of positive solutions by the Banach contraction mapping principlefor equation (3), (4). In [20], the author considered following fractional differential equation:

with infinite-point boundary condition

where is a Caratheodory function, and are two monotic sequence with lim is the standard Riemann–Liouville derivative. Theauthors established the existence of at least one solution for this equation by Mawhin’s continuation theorem. Motivated by the excellent results above, in this paper, the existence of multiplepositive solutions are obtained for a singular infinite-point p-Laplacian boundary valueproblems. Compared with [19], the equation in this paper is p-Laplacian fractional differentialequation, and the method which we used in this paper is Avery–Peterson fixed pointtheorem. Compared with [12], fractional derivative is involved in the nonlinear terms for BVP (1), (2), and multiple positive solutions are obtained for the BVP (1), (2).

Preliminaries and lemmas

In this section, we introduce definitions and preliminary results, which are used through out this paper. First, we let[0; 1] be the Banach space with the maximum norm

In this section, we introduce definitions and preliminary results, which are used throughout this paper. First, we let [0; 1] be the Banach space with the maximum norm then we list a condition below to be used later in the paper.

(HO) and for all there exists an funcion , such that

Now, we state some lemmas, which are basic and used in this paper.

(See [13, 16].) Assume that , then

Lemma 2. Assume that , then

where n is the least integer greater than or equal to

Lemma 3. See [3, Thm. 1.2.7].) Let, then H is a relatively compact set i fand only if

• H´ is equicontinuous, and H´(t) is a relatively compact set for any

  There exists to such that is a relatively compact set on E.

i) H is equicontinuous, and H´(t) is a relatively compact set for any;

(ii) There exists such that is a relatively compact set on E.

Lemma 4. (See [1, 14].) Let P be a cone of be nonnegative continuous concave funcional on P be nonnegative continuous concave funcional on P, and be a nonnegative continuos funcional on P such that for some positive numbers L Let A: is completely continuous and there exist positive nembers e, c, d, with e< c such that following conditions are satisfied:

Then A has at least three fixed points _{x1, x2, x3} such that and

Lemma 5. Given y then the solution of the BVP

with boundary condition (2) can be expressed by

where

in which

Proof: By Means of Lemma I we reduce(5) to an equivalent equation

for From

we have Consequently, we get.

hence, we have

On the other hand, , combining with (10),we get

Therefore, G(t; s) is as (7). By simple calculation we have

Lemma6. let then the BVP (1), (2) has a unique solution.

where G(t; s) is as (6), and

in which

Proof. Let Consider the boundary value problem

By means of the Lemma 2 we reduce (15) to an equivalent integral equation

for From we have Consequently,we get

for . From we have . Con sequently, we get

On the other hand, and combining with (16), we get

where is as (13), P(s) is as 14. Hence,

Therefore, H(t,s) is as 12

Lemma7.Take with thenwhere

Proof. By direct calculation we get and so is nondecreasing with respect to s. For we get and obviously,

Hence, for we have furthermore,for we get and obviously, for . we get on the other hand, for we have

and obviously holds for For

and for we have

Therefore, the proof of Lemma 7 is completed.

Lemma 8. Let then the Green functions defined by (12) satisfies:

• is continuous and for all

• for all in which

Proof. The proof is similar to Lemma 3 of [20], we omit it here.

Now we define a cone K on C1[0; 1] and an operator A : Kas follows:

where and

Problems (1), (2) has a positive solution if and only if u is a fixed point of A in K.

3 Main results

Lemma 9. The operator is continuous.

Proof. First, by the integrability of f and (H0) we get

so we have that A is well defined on K. Moreover, it follows from the uniform continuityof G(t; s) on [0; 1] [0; 1] and (H0)

Thus, we have that Furthermore, by the uniform continuity of, we get

Let Since are uniformly continous, there exists M>0 such that

On the other hand, since , there exists A > 0 such that.and thenFurthermore, by (H0) we have

so, we have

hence, for any there exists such, for any by (17) and Lebesgue controls convergence theorem, we have

By for the above there exists N such that for all n>N, we get Hence, for any for any hence, for any by (18) we obtain

Thus, for n> bye (19) we have

and

and hence, we get That is, namely, A is continuous in the space E .

Lemma 10. is completely continuous.

Proof. From Lemma 7 we have y

thus,

consequently,

On the other hand, for all by Lemma 7 we have

Hence

Thus,

Now we will prove that AV is relatively compact for bounded Since V isbounded, there exists such that for any ,we have

Similarly, for we derive

which shows that AV is bounded. Next, we will verify that is equicontinuous. Let we get

Furthermore,

Thus, we obtain

From above and the uniform continuity of, and together with Lemma 3, wecan derive that AV is relatively compact in C1[0; 1], and so we get that is completely continuous

Let be nonnegative continuous convex functionals on bea nonnegative continuous concave functional on P, and be a nonnegative continuous functional on P. Then for nonnegative numbers e, c, d, h, we define the following convex sets:.

We will apply the following fixed point theorem of Avery and Peterson to solve problem (1), (2).

Let the convex functions on P, and define a concave function where are the same as in Lemma 7.

Theorem 1:Assume that there exist positive numbers e,c,d,h with and such that

Then problem (1), (2) has at least three fixed points u1,u2 , u3 satisfying

y

Proof. Let. By condition H1 we get

Consequently, we obtain This together with Lemmas 9 and 10, means that: completely continuous.

Take By simple calculation we have and so

which shows that condition (S1) is satisfied.

Take Since we obtain

which implies that condition (S2) holds.

Next, we will verify that condition (S3) holds. For we have Let and by (H3) we get

and

Consequently, we have Thus, condition (S3) holds.By Lemma 4 we get that (1), (2) has at least three positive solutions u1, u2, u3 satisfying (20)–(22).

4 An example

Consider the following infinite-point p-Laplacian fractional differential equations:

where

Cearly

and

so is integrable, condition (H0) holds.

We take by simple calculation we have

Hence, we have

and as Let Then for we for

so condition (H1) of Therem 1 hold. For by MATLAB software we have

therefore, condition (H2) of Theorem 1 hold. By the same method with proofing (H1) we get (H3) hold, so all the conditions of Theorem 1 hold. Hence, the BVP (23) has at least three positive solutions u1, u2, u3 satisfying

Acknowledgments

This research was supported by the National Natural Science Foundation of China (12101086, 11871302),Changzhou Science and technology planning project (CJ20210133), and Project of Shandong Province Higher Educational Science and Technology Program (J18KA217)

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