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The Eigenvalue Problem for Caputo Type Fractional Differential Equation with Riemann-Stieltjes Integral Boundary Conditions.

1. Introduction

The experience of the last few years has fully borne out the fact that the integer order calculus is not as widely used as fractional order calculus in some fields such as chemistry, control theory, and signal processing. On the remarkable survey of Agarwal, Benchohra, and Hamani [1] it is pointed out that fractional differential equations constitute a fundamental tool in the modeling of some phenomena (see also [2-4]). The use of fractional order is more accurate for the description of phenomena, so the study of fractional differential equations becomes the mainstream with the help of techniques of nonlinear analysis. We refer the reader to [536] for recent results. For example, in [9], the author studied the following fractional differential equation:

[sup.c][D.sup.[alpha]] x(y) + f(y, x(y)) = 0, 0 < y < 1, (1)

with boundary conditions

x'(0) = x" (0) = 0,

x(1) = [mu] [[integral].sup.1.sub.0] x(s)ds, (2)

where [alpha] [member of] (2,3], [mu] [member of] [0,1), and [sup.c][D.sup.[alpha]] is the Caputo derivative. They solved the above problem by means of classical fixed point theorems.

In [5], the boundary value problem for the following nonlinear fractional differential equation was discussed:

[mathematical expression not reproducible], (3)

with boundary conditions

x (0) = x' (0) = x" (0) = x" (1) = 0, (4)

where [mathematical expression not reproducible] is the Riemann-Liouville differentiation, a [member of] (3,4]. By using a fixed point theorem, a new result of the existence of three positive solutions is obtained.

In [15], the authors investigated the following class of BVP:

[mathematical expression not reproducible], (5)

with boundary conditions

x (0) = x" (0) = 0,

x' (1) = [alpha]x" (1), (6)

where q [member of] (2, 3), [sigma] [member of] (1,2), f: [0,1] x R x R [right arrow] R is a given function, and [mathematical expression not reproducible] denotes the Caputo differentiation. The author investigated this problem by using Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green's function, and Guo-Krasnoselskii fixed point theorem on cone. Similar problems can be referred to in [25].

In this paper, we investigate the eigenvalue problem for Caputo fractional boundary value problem with Riemann-Stieltjes integral boundary conditions

[mathematical expression not reproducible], (7)

where [theta] [member of] (2, 3), f: [0, 1] x [0, [infinity]) [right arrow] (0, [infinity]) is continuous, [mu] > 0 is a parameter, CD9+ is the Caputo fractional derivative, and A is a bounded variation function with positive measures with

B = [[integral].sup.1.sub.0] y dA(y) < 1. (8)

Our proof is based on the properties of the Green's function and the Guo-Krasnosel'skii fixed point theorem on cone.

2. Preliminaries

In order to solve problem (7), we provide the properties related to problem (7).

Definition 1 (see [3]). The Caputo's fractional derivative of order 0 >0 for a function x [member of] [C.sup.n] [0, +[infinity]) is defined as

[mathematical expression not reproducible], (9)

where n is the smallest integer greater than or equal to [theta].

Lemma 2 (see [3]). Let [theta] > 0. If we assume x [member of] C(0, 1) [intersection] L(0, 1), then the fractional differential equation

[mathematical expression not reproducible] (10)

has the general solution x(y) = [C.sub.0] + [C.sub.1]y + ... + [C.sub.n-1][y.sup.n-1], [C.sub.k] [member of] R, k = 0, 1 ..., n - 1, where n is the smallest integer greater than or equal to [theta].

Lemma 3 (see [3]). Given that x [member of] C(0, 1) [intersection] L(0, 1) with a fractional derivative of order [theta] that belongs to C(0, 1) [intersection] L(0, 1). Then

[mathematical expression not reproducible], (11)

where n is the smallest integer greater than or equal to d.

Firstly, we consider the following linear Caputo fractional differential equation:

[mathematical expression not reproducible]. (12)

Lemma 4. Let [theta] e (2,3] and assume that [sigma] [member of] C[0, 1]. Then p is the solution of the above boundary value problem (12), if and only if p satisfies the following integral equation:

p(y) = [[integral].sup.1.sub.0] G(y, s) [sigma] (s)ds, (13)

where

[mathematical expression not reproducible], (14)

and

B = [[integral].sup.1.sub.0] ydA (y) < 1. (15)

Proof. Applying the fractional integral of order [theta] to both sides of (12) for j [member of] [0,1],we get the following formula:

[mathematical expression not reproducible]. (16)

According to p(0) = p"(0) = 0 and Lemma 3, we obtain

P(y) = cy - 1/[GAMMA]([theta]) [[integral].sup.y.sub.0] [(y - s).sup.[theta]-1] [sigma](s)ds, (17)

where c e R. Since p(1) = [[integral].sup.1.sub.0] p(y)dA(y), we deduce that

[mathematical expression not reproducible]. (18)

Therefore,

[mathematical expression not reproducible]. (19)

Substituting the above equality into (17), one has

[mathematical expression not reproducible] (20)

and

[mathematical expression not reproducible]. (21)

The proof is completed.

Lemma 5. The Green's function G(y, s) has the following properties:

(i) [GAMMA]([theta])G(y, s) [less than or equal to] (1/(1 - B))[(1 - s).sup.[theta]-1], for y, s [member of] [0, 1j;

(ii) [GAMMA]([theta])G(y, s) [greater than or equal to] N[(1 - s).sup.[theta]-1],for y [member of] [1/4,3/4] and s [member of] [0,1], where

[mathematical expression not reproducible]. (22)

Proof. (i) Obviously, the inequality [GAMMA]([theta])G(y, s) [less than or equal to] (1/(1 B)) [(1 - s).sup.[theta]-1] holds from the representation of G(y, s).

(ii) In view of B = [[integral].sup.1.sub.0] ydA(y) < 1 and [theta] [member of] (2,3), we have 1 - [[integral].sup.1.sub.0] [y.sup.[theta]-1] dA(y) > 0.

For s [less than or equal to] y, we have

[mathematical expression not reproducible]. (23)

For s [less than or equal to] y, we have

[mathematical expression not reproducible]. (24)

Thus, the above two inequalities yield the inequality in (ii). The proof is completed.

Let X = C[0,1], [parallel]p[parallel] = [maxy[member of][0,1]] [absolute value of p(y)]; then (X, [parallel] * [parallel]) is a Banach space. We define the cone P [subset] X by

[mathematical expression not reproducible]. (25)

Let [A.sub.[mu]] : X [right arrow] X be the operator defined as

([A.sub.[mu]] p) (y) = [mu] [[integral].sup.1.sub.0] G(y, s) f(s, p(s)) ds. (26)

Thus, the fixed point of the above integral equation is equivalent to the solution of the BVP (7).

Lemma 6. [A.sub.[mu]](P) [subset] P and [A.sub.[mu]] : P [right arrow] P is a completely continuous operator, where [A.sub.[mu]] is defined in (26).

Proof. By Lemma 5, for [for all]p [member of] P, we have

[mathematical expression not reproducible]. (27)

Hence we have [A.sub.[mu]](P) [subset] P. Let [OMEGA] [subset] P be bounded. Then there exists a constant M > 0 such that [parallel] p [parallel] [less than or equal to] M for [for all]p [member of] [OMEGA]. Let [L.sub.1] = [max.sub.y[member of][0,1],p[member of][0,M]]([mu]f(y,P) + 1). Then

[mathematical expression not reproducible]. (28)

Thus, [A.sub.[mu]]([OMEGA]) is bounded. Put p [member of] [OMEGA] and [y.sub.1], [y.sub.2] [member of] [0,1]. We deduce that

[mathematical expression not reproducible]. (29)

Since y, y[(y - 1)[theta]], [y.sup.[theta]+1], [y.sup.[theta]+1] are uniformly continuous on [0,1], [A.sub.[mu]]([OMEGA]) is equicontinuous, by using Arzela-Ascoli's theorem, we can prove that [A.sub.[mu]] : P [right arrow] P is completely continuous. The proof is completed.

The following Guo-Krasnoselskii's fixed point theorem is used to prove the existence of positive solution of (7).

Theorem 7 (see [37]). Let P be a cone of real Banach space X and let [[OMEGA].sub.1] and [[OMEGA].sub.2] be two bounded open sets in X such that 0 [member of] [[OMEGA].sub.1] [member of] [[bar.[OMEGA]].sub.1] [member of] [[OMEGA].sub.2]. Let operator A : P [intersection] ([[bar.[OMEGA]].sub.2] \ [[OMEGA].sub.1]) [right arrow] P be completely continuous operator. If one of the following two conditions holds:

(1) [mathematical expression not reproducible],

(2) [mathematical expression not reproducible],

then A has at least one fixed point in P [intersection] ([[bar.[OMEGA]].sub.1] \ [[OMEGA].sub.1]).

3. Existence of Positive Solutions

In this section, we investigate the existence of positive solutions for integral boundary value problems of fractional differential equation (7).

For convenience, we denote them by

[mathematical expression not reproducible]. (30)

Theorem 8. Suppose that [Ch.sub.0] < [Eh.sup.*.sub.[infinity]] holds; then for [mu] [member of] (1/[h.sup.*.sub.[infinity]]E, 1 /[Ch.sub.0]), problem (7) has a positive solution. Here we impose [h.sup.-1.sub.0] = +[infinity] if [h.sub.0] = 0 and [[[h.sup.*.sub.[infinity]].sup.-1] = 0 if [h.sup.*.sub.[infinity]] = +[infinity].

Proof. Let [] [member of] (1/[h.sup.*.sub.[infinity]]E, 1/[Ch.sub.0]) and [epsilon] > 0 satisfy

[mathematical expression not reproducible]. (31)

According to the definition of [h.sub.0], we know that there exists a constant [E.sub.1] > 0 such that

f (y, u) < ([h.sub.0] + [epsilon])u, for u [member of] [0, [E.sub.1]], y [member of] [0,1]. (32)

Put [[OMEGA].sub.1] = {p [member of] P : [parallel] p [parallel] < [E.sub.1]}. Let p [member of] P [intersection] [partial derivative] [[OMEGA].sub.1]. We have [parallel] p [parallel] = [E.sub.1] and

[mathematical expression not reproducible]. (33)

Therefore, [parallel] [A.sub.[mu]]p [parallel] [less than or equal to] [parallel] p [parallel] for p [member of] P [intersection] [partial derivative] [[OMEGA].sub.1].

By the definition of [h.sup.*.sub.[infinity]], we know that there exists [E.sub.2] > 0 such that

f(y,u) [greater than or equal to] ([h.sup.*.sub.[infinity]] - [epsilon]) u, for u [greater than or equal to] [E.sub.2] and y [member of] [1/4], [3/4]. (34)

Let [E.sub.3] = max{2[E.sub.1], [E.sub.2]/N(1 - B)}, [[OMEGA].sub.2] = {p [member of] P : [parallel] p [parallel] < [E.sub.3]}. Then for p [member of] P [intersection] [partial derivative] [[OMEGA].sub.1], by (25) we have

[mathematical expression not reproducible], (35)

and thus

[mathematical expression not reproducible]. (36)

Therefore, [parallel] [A.sub.[mu]]p [parallel] [greater than or equal to] [parallel] p [parallel] for p [member of] P [intersection] [partial derivative] [[OMEGA].sub.2].

By Theorem 7, if [mu] [member of] (1/[h.sup.*.sub.[infinity]]E, 1/[Ch.sub.0]), we assert that [A.sub.[mu]] has a fixed point in P [intersection] ([[bar.[OMEGA]].sub.2] \ [[OMEGA].sub.1]) and therefore problem (7) has at least one positive solution. The proof is completed.

Theorem 9. Assume that [h.sub.[infinity]]C < E[h.sup.*.sub.0] holds. Then for [mu] [member of] (1/[Eh.sup.*.sub.0], 1/[h.sub.[infinity]]C), the problem (7) has at least a positive solution. Here we impose [[[h.sup.*.sub.0]].sup.-1] = 0 if [h.sup.*.sub.0] = +[infinity] and [[[h.sub.[infinity]]].sup.- 1] = +[infinity] if [h.sub.[infinity]] = 0.

Proof. Let [mu] [member of] (1/[Eh.sup.*.sub.0], 1/[h.sub.[infinity]]C) and [epsilon] > 0 such that

1/([Eh.sup.*.sub.0] - [epsilon])E [less than or equal to] [mu] [less than or equal to] 1/C([h.sub.[infinity]] + [epsilon]). (37)

According to the definition of [h.sup.*.sub.0], there exists a constant [E.sub.4] > 0 such that

f(y, u) [greater than or equal to] ([h.sup.*.sub.0] - [epsilon])u,

for u [member of] (0, [E.sub.4]] and y [member of] [1/4, 3/4]. (38)

Put [[OMEGA].sub.3] = {p [member of] P : [parallel] p [parallel] < [E.sub.4]}. Let p [member of] P [intersection] [partial derivative] [[OMEGA].sub.3]; we have

[mathematical expression not reproducible]. (39)

Therefore, [parallel] [A.sub.[mu]]p [parallel] [greater than or equal to] [parallel] p [parallel] for p [member of] P [intersection] [partial derivative] [[OMEGA].sub.3].

It follows from the definition of [h.sub.[infinity]] that there exists a constant [E.sub.5] > 0 such that

f(y, u) [less than or equal to] ([h.sub.[infinity]] + [epsilon]/2)u, for u [greater than or equal to] [E.sub.5] and y [member of] [0,1]. (40)

This together with the continuity of f implies that

f(y, u) [less than or equal to] ([h.sub.[infinity]] + [epsilon]/2) u + M

for u [member of] R and y [member of] [0, 1] (41)

holds for some M > 0.

Let [E.sub.6] = max{2[E.sub.4], [E.sub.5], 2M/[epsilon]}, [[OMEGA].sub.4] = {p [member of] P : [parallel] p [parallel] < [E.sub.6]}. For [for all] p [member of] P [intersection] [partial derivative] [[OMEGA].sub.4], we conclude that

[mathematical expression not reproducible]. (42)

Therefore, [parallel] [A.sub.[mu]]p [parallel] [less than or equal to] [parallel] p [parallel] for p [member of] P [intersection] [partial derivative] [[OMEGA].sub.2].

By Theorem 7, if [mu] [member of] (1/[Eh.sup.*.sub.0], 1/[h.sub.[infinity]]C), we conclude that [A.sub.[mu]] has a fixed point in P [intersection] [[bar.[OMEGA]].sub.4] [[OMEGA].sub.3], and so problem (7) has one positive solution. The proof is completed. ?

4. Nonexistence of Positive Solutions

In this section, we present some sufficient conditions for nonexistence of positive solution to integral boundary value problems of fractional differential equation (7).

Theorem 10. If [h.sub.0] < +[infinity] and [h.sub.[infinity]] < +[infinity], then there exists a [[mu].sup.*] > 0 such that problem (7) has no positive solution for [mu] [member of] (0, [[mu].sup.*]).

Proof. Since [h.sub.0] < +[infinity] and [h.sub.[infinity]] < +[infinity] < +[infinity], we have f(y, u) [less than or equal to] [n.sub.1]u for u [member of] [0, [r.sub.1]], and f(y, u) [less than or equal to] [n.sub.2]u for u [member of] [[r.sub.2], +[infinity]), where [n.sub.1], [r.sub.2], [r.sub.1], [r.sub.2] are positive numbers with [r.sub.1] < [r.sub.2]. Let [mathematical expression not reproducible]; then we have f(y, u) [less than or equal to] nu for u [member of] [0, +[infinity]). Suppose [p.sub.0](y) is a positive solution of problem (7); then we are going to prove that this leads to a contradiction for 0 < [mu] < [[mu].sup.*] := 1/Cn. Since ([A.sub.[mu]][p.sub.0])(y) = [p.sub.0](y), for y [member of] [0,1], then

[mathematical expression not reproducible], (43)

which is a contradiction. Therefore this completes the proof.

Theorem 11. If [h.sup.*.sub.0] > 0, [h.sup.*.sub.[infinity]] > 0, f(t, u) > 0 for t [member of] [1/4, 3/4] and u > 0, then there exists a [[mu].sup.*] > 0 such that problem (7) has no positive solution for all [mu] > [[mu].sup.*].

Proof. Since [h.sup.*.sub.0] > +[infinity], [h.sup.*.sub.[infinity]], > +[infinity], we have f(y, u) [greater than or equal to] [m.sub.1]u for u [member of] [0, [r.sub.1]], and f(y, u) > [m.sub.2]u for u [member of] [[r.sub.2], +[infinity]), where [m.sub.1], [m.sub.2], [r.sub.1], [r.sub.2] are positive numbers and [r.sub.1] < [r.sub.2]. Let [mathematical expression not reproducible]; then we have f(y, u) [greater than or equal to] mu for u [member of] [0, +[infinity]). Suppose [p.sub.1](y) is a positive solution of problem (7); then we are going to prove that this leads to a contradiction for [mu] > [[mu].sup.*] := 1/EmN(1 - B). Since [A.sub.[mu]][p.sub.1] = [p.sub.1], then

[mathematical expression not reproducible], (44)

which is a contradiction. Therefore this completes the proof

5. Example

Example 1. We consider the following fractional equation:

[mathematical expression not reproducible], (45)

where [theta] = 5/2, f(y, p) = [yp.sup.3] - p + [e.sup.p] - 1, A(y) = (1/2)y, B = 1/4 < 1. We obtain C = 8/15[GAMMA](5/2), N = 3/64, E = 27(9[square root of 5] - 1)/1310720[GAMMA](5/2).

It is easy to see that, for all p > 0,

[mathematical expression not reproducible] (46)

and

[mathematical expression not reproducible]. (47)

Then [h.sub.0] = 0, [h.sup.0.sub.[infinity]] = [infinity]; from Theorem 8, problem (45) has a positive solution.

Example 2. We consider the following fractional equation:

[mathematical expression not reproducible], (48)

where [theta] = 5/2, f(y, p) = [[cube root of p] + y ln(p + 3), A(y) = (1/2)y, and B = 1/4 < 1. We obtain C = 8/15[GAMMA](5/2), N = 3/64, and E = 27(9 [square root of 3] - 1)/1310720[GAMMA](5/2).

One can easily see that, for all p > 0,

[mathematical expression not reproducible] (49)

and

[mathematical expression not reproducible]. (50)

Then [h.sub.[infinity]] = 0, [h.sup.*.sub.0] = [infinity]; from Theorem 9, problem (48) has a positive solution.

Example 3. We consider the following fractional equation:

[mathematical expression not reproducible], (51)

where [theta] = 5/2, f(y, p) = yp, A(y) = (1/2)y, and B = 1/4 < 1. We obtain C = 8/15[GAMMA](5/2), N = 3/64, and E = 27(9 [square root of 3] - 1)/1310720[GAMMA] (5/2).

By direct calculation, we obtain that

[mathematical expression not reproducible] (52)

and

[mathematical expression not reproducible]. (53)

Take n = 1 and [[mu].sup.*] = 45[square root of [pi]]/32 > 0. By Theorem 10, problem (51) has no positive solution for 0 < [mu] < [[mu].sup.*].

Example 4. We consider the following fractional equation:

[mathematical expression not reproducible], (540

where [theta] = 5/2, f(y, p) = yp, A(y) = (1/2)y, and B = 1/4 < 1. We obtain C = 8/15[GAMMA] (5/2), N = 3/64, and E = 27(9 [square root of 3 - 1)/1310720[GAMMA](5/2).

Obviously, we can infer that

[mathematical expression not reproducible] (55)

and

[mathematical expression not reproducible]. (56)

Take m = 1/4 and [[mu]* = 3.4 [square root of [pi]]/81(9 [square root of 3] - 1) x [10.sup.8]. By Theorem 11, problem (54) has no positive solution for [mu] > [[mu].sup.*].

https://doi.org/10.1155/2018/2176809

Data Availability

The dataset supporting the conclusions of this article is included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research is supported by the National Natural Science Foundation (NNSF) of China (11371221, 11571207), Shandong Natural Science Foundation (Z[R.SUB.2]018MA011), and the Tai'shan Scholar Engineering Construction Fund of Shandong Province of China.

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Wenjie Ma (1) and Yujun Cui (iD) (2)

(1) Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China

(2) State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Yujun Cui; cyj720201@163.com

Received 13 June 2018; Accepted 30 July 2018; Published 12 August 2018

Academic Editor: Liguang Wang
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Title Annotation:Research Article
Author:Ma, Wenjie; Cui, Yujun
Publication:Journal of Function Spaces
Date:Jan 1, 2018
Words:4423
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