Positive Solutions for Singular Semipositone Fractional Differential

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Hindawi Journal of Function Spaces Volume 2017, Article ID 5892616, 7 pages https://doi.org/10.1155/2017/5892616

Research Article Positive Solutions for Singular Semipositone Fractional Differential Equation Subject to Multipoint Boundary Conditions Rui Pu,1 Xingqiu Zhang,1 Yujun Cui,2 Peilong Li,1 and Weiwei Wang1 1

School of Medical Information Engineering, Jining Medical College, Rizhao, Shandong 276826, China Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China

2

Correspondence should be addressed to Xingqiu Zhang; [email protected] Received 5 April 2017; Accepted 1 June 2017; Published 2 July 2017 Academic Editor: Xinguang Zhang Copyright ยฉ 2017 Rui Pu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Existence result together with multiplicity result of positive solutions of higher-order fractional multipoint boundary value problems is given by considering the integrations of height functions on some special bounded sets. The nonlinearity may change its sign and may possess singularities on the time and the space variables at the same time.

1. Introduction Very recently, Henderson and Luca [1] obtained the existence result of positive solution for the following singular differential equation with fractional derivative: ๐›ผ ๐‘ข (๐‘ก) + ๐‘“ (๐‘ก, ๐‘ข (๐‘ก)) = 0, ๐ท0+

0 < ๐‘ก < 1,

(1)

subject to multipoint BCs ๐‘ข (0) = ๐‘ข๓ธ€  (0) = โ‹… โ‹… โ‹… = ๐‘ข(๐‘›โˆ’2) (0) = 0, ๐‘

๐‘š

๐‘ž

๐ท0+ ๐‘ข (1) = โˆ‘๐‘Ž๐‘– ๐ท0+ ๐‘ข (๐œ‰๐‘– ) .

For example, people have realized that electrical conductance of cell membranes of biological organism took on fractionalorder style [2]. As a consequence, fractional-order ordinary differential equations can give a more accurate description on spread process of some infectious diseases such as HIV, handfoot-mouth disease [3], malaria, tuberculosis, and measles. Arafa et al. [4] formulated a model with orders ๐›ผ1 , ๐›ผ2 , ๐›ผ3 > 0 to describe the infection of CD4+ T cells. In 2015, Wang et al. [5] concentrated on the positive solutions for a class of modified HIV-1 population dynamics model presented in [6]

(2)

๐‘–=1

๐›ผ is the traditional Riemann-Liouville derivative Here ๐ท0+ with order ๐›ผ, ๐‘› โˆ’ 1 < ๐›ผ โ‰ค ๐‘› (๐‘› โ‰ฅ 3), ๐‘Ž๐‘– โ‰ฅ 0, ๐‘– = 1, 2, . . . , ๐‘š (๐‘š โˆˆ N+ ), 0 < ๐œ‰1 < ๐œ‰2 < โ‹… โ‹… โ‹… < ๐œ‰๐‘š < 1, ๐‘, ๐‘ž โˆˆ R, 1 โ‰ค ๐‘ โ‰ค ๐‘› โˆ’ 2, and 0 โ‰ค ๐‘ž โ‰ค ๐‘ with ฮ” = ฮ“(๐›ผ)/ฮ“(๐›ผ โˆ’ ๐›ผโˆ’๐‘žโˆ’1 > 0. The nonlinear term ๐‘) โˆ’ (ฮ“(๐›ผ)/ฮ“(๐›ผ โˆ’ ๐‘ž)) โˆ‘๐‘š ๐‘–=1 ๐‘Ž๐‘– ๐œ‰๐‘– ๐‘“ permits sign-changing and singularities at ๐‘ก = 0, 1. Fractional derivative gives a more precise exhibition on long-memory behavior owing to its nonlocal characteristic. Nowadays, researchers have reached a consensus that fractional calculus is one of the effective tools to describe phenomena in almost every field of science and technology.

๐›ฝ

๐ท๐‘ก๐›ผ ๐‘ข (๐‘ก) + ๐œ†๐‘“ (๐‘ก, ๐‘ข (๐‘ก) , ๐ท๐‘ก ๐‘ข (๐‘ก) , V (๐‘ก)) = 0, ๐›พ

๐ท๐‘ก V (๐‘ก) + ๐œ†๐‘” (๐‘ก, ๐‘ข (๐‘ก)) = 0, ๐›ฝ

0 < ๐‘ก < 1,

๐›ฝ+1

๐ท๐‘ก ๐‘ข (0) = ๐ท๐‘ก ๐‘ข (0) = 0, 1

๐›ฝ

๐›ฝ

๐ท๐‘ก ๐‘ข (1) = โˆซ ๐ท๐‘ก ๐‘ข (๐‘ ) d๐ด (๐‘ ) , 0

๓ธ€ 

V (0) = V (0) = 0, 1

V (1) = โˆซ V (๐‘ ) d๐ต (๐‘ ) , 0

(3)

2

Journal of Function Spaces

where 2 < ๐›ผ, ๐›พ โ‰ค 3, 0 < ๐›ฝ < 1, ๐œ† > 0 is a parameter, ๐›ฝ ๐›พ ๐›ผ โˆ’ ๐›ฝ > 2, and ๐ท๐‘ก๐›ผ , ๐ท๐‘ก , and ๐ท๐‘ก are traditional R-L derivatives. 1 ๐›ฝ 1 โˆซ0 ๐ท๐‘ก ๐‘ข(๐‘ )d๐ด(๐‘ ) and โˆซ0 V(๐‘ )d๐ต(๐‘ ) represent R-S integrals and ๐ด, ๐ต are bounded variations, and the nonlinearities permit sign-changing and singularities at ๐‘ก = 0, 1. Other relative papers on nonlocal boundary value problems can be found in [7โ€“20]. Motivated by the above papers, we devote ourselves to the existence result as well as multiplicity result of positive solutions for BVP (1)-(2). This article admits some new features. First, the nonlinear term ๐‘“ may take negative infinity and change its sign. Compared with [1], ๐‘“ permits singularities on the time and the space variables at the same time. Second, the method exploited in this paper is different from that in [1] in essence. More precisely, height functions with their integrations on some special bounded set are utilized to get the existence result for positive solutions of BVP (1)-(2). Thirdly, a result of multiple positive solutions is also given in this paper. Conditions employed in this paper are easy to be verified.

2. Preliminaries and Several Lemmas Two traditional Banach spaces ๐ธ = ๐ถ[0, 1] and ๐ฟ1 (0, 1) are involved in this article, where ๐ธ = ๐ถ[0, 1] and ๐ฟ1 (0, 1) represent the spaces of the continuous functions and Lebesgue integrable functions equipped with the norms 1 โ€–๐‘ขโ€– = max0โ‰ค๐‘กโ‰ค1 |๐‘ข(๐‘ก)| and โ€–๐‘ขโ€–1 = โˆซ0 |๐‘ข(๐‘ก)|d๐‘ก, respectively. 1

๐บ2 (๐‘ก, ๐‘ ) =

1 ฮ“ (๐›ผ โˆ’ ๐‘ž)

๐›ผโˆ’๐‘žโˆ’1 (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 โˆ’ (๐‘ก โˆ’ ๐‘ )๐›ผโˆ’๐‘žโˆ’1 , {๐‘ก โ‹…{ ๐‘ก๐›ผโˆ’๐‘žโˆ’1 (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 , {

๐‘š

๐‘

Lemma 2 (see [1]). Assume that ๐‘Ž๐‘– > 0 (๐‘– = 1, 2, . . . , ๐‘š) and ฮ” > 0. Then the Green function ๐บ of (4) given by (6) is continuous on [0, 1] ร— [0, 1] and meets (a) ๐บ(๐‘ก, ๐‘ ) โ‰ค ๐ฝ(๐‘ ), โˆ€๐‘ก, ๐‘  โˆˆ [0, 1], and here ๐ฝ(๐‘ ) = โ„Ž1 (๐‘ ) + ๐›ผโˆ’๐‘โˆ’1 (1 โˆ’ (1 โˆ’ (1/ฮ”) โˆ‘๐‘š ๐‘–=1 ๐‘Ž๐‘– ๐บ2 (๐œ‰๐‘– , ๐‘ ), โ„Ž1 (๐‘ ) = (1 โˆ’ ๐‘ ) ๐‘ ๐‘ ) )/ฮ“(๐›ผ), and ๐‘  โˆˆ [0, 1]; (b) ๐บ(๐‘ก, ๐‘ ) โ‰ฅ ๐‘ก๐›ผโˆ’1 ๐ฝ(๐‘ ), โˆ€๐‘ก, ๐‘  โˆˆ [0, 1]; (c) ๐บ(๐‘ก, ๐‘ ) โ‰ค ๐œŽ๐‘ก๐›ผโˆ’1 โˆ€๐‘ก, ๐‘  โˆˆ [0, 1], and here ๐œŽ = 1/ฮ“(๐›ผ) + ๐›ผโˆ’๐‘žโˆ’1 /(ฮ”ฮ“(๐›ผ โˆ’ ๐‘ž)). โˆ‘๐‘š ๐‘–=1 ๐‘Ž๐‘– ๐œ‰๐‘– Lemma 3. Let ๐‘ข โˆˆ ๐ถ[0, 1] satisfy (4), where ๐‘ฆ โˆˆ ๐ฟ1 (0, 1), ๐‘ฆ(๐‘ก) โ‰ฅ 0, and 0 โ‰ค ๐‘ก โ‰ค 1. Then, ๐‘ข(๐‘ก) โ‰ฅ ๐‘ก๐›ผโˆ’1 โ€–๐‘ขโ€–, and 0 โ‰ค ๐‘ก โ‰ค 1. Proof. It can be easily seen from Lemma 5 in [1]; we omit the details. Lemma 4. Suppose that ๐‘ค(๐‘ก) โˆˆ ๐ถ[0, 1] be the solution of ๐›ผ ๐‘ข (๐‘ก) + ๐‘˜ (๐‘ก) = 0, ๐ท0+

0 < ๐‘ก < 1,

๐‘ข (0) = ๐‘ข๓ธ€  (0) = โ‹… โ‹… โ‹… = ๐‘ข(๐‘›โˆ’2) (0) = 0, ๐‘š

๐‘

(0) = 0,

(4)

๐‘ž

๐‘ž

where ๐‘˜ โˆˆ ๐ฟ1 (0, 1) and ๐‘˜(๐‘ก) > 0. Then, ๐‘ค(๐‘ก) โ‰ค ๐œŽโ€–๐‘˜โ€–1 ๐‘ก๐›ผโˆ’1 and 0 โ‰ค ๐‘ก โ‰ค 1. Proof. For any 0 โ‰ค ๐‘ก โ‰ค 1, we get by Lemma 2

๐‘–=1

1

1

๐‘ค (๐‘ก) = โˆซ ๐บ (๐‘ก, ๐‘ ) ๐‘˜ (๐‘ ) d๐‘  โ‰ค ๐œŽ๐‘ก๐›ผโˆ’1 โˆซ ๐‘˜ (๐‘ ) d๐‘ 

satisfies

0

1

๐‘ข (๐‘ก) = โˆซ ๐บ (๐‘ก, ๐‘ ) ๐‘ฆ (๐‘ ) d๐‘ , 0

0

๐›ผโˆ’1

๐‘ก โˆˆ [0, 1] ,

(5)

and here ๐บ (๐‘ก, ๐‘ ) = ๐บ1 (๐‘ก, ๐‘ ) +

(9)

๐‘–=1

(๐‘›โˆ’2)

๐ท0+ ๐‘ข (1) = โˆ‘๐‘Ž๐‘– ๐ท0+ ๐‘ข (๐œ‰๐‘– )

๐บ1 (๐‘ก, ๐‘ ) =

0 โ‰ค ๐‘ก โ‰ค ๐‘  โ‰ค 1,

๐ท0+ ๐‘ข (1) = โˆ‘๐‘Ž๐‘– ๐ท0+ ๐‘ข (๐œ‰๐‘– ) ,

๐›ผ ๐ท0+ ๐‘ข (๐‘ก) + ๐‘ฆ (๐‘ก) = 0, 0 < ๐‘ก < 1,

๐‘ข (0) = ๐‘ข (0) = โ‹… โ‹… โ‹… = ๐‘ข

(8)

โˆ€(๐‘ก, ๐‘ ) โˆˆ [0, 1] ร— [0, 1].

Lemma 1 (see [1]). If ฮ” =ฬธ 0. Given ๐‘ฆ โˆˆ ๐ถ(0, 1) โˆฉ ๐ฟ (0, 1), the solution of the following differential equation:

๓ธ€ 

0 โ‰ค ๐‘  โ‰ค ๐‘ก โ‰ค 1,

๐‘ก๐›ผโˆ’1 ๐‘š โˆ‘๐‘Ž ๐บ (๐œ‰ , ๐‘ ) , ฮ” ๐‘–=1 ๐‘– 2 ๐‘–

(6)

1 ฮ“ (๐›ผ)

๐›ผโˆ’๐‘โˆ’1 ๐›ผโˆ’1 โˆ’ (๐‘ก โˆ’ ๐‘ )๐›ผโˆ’1 , 0 โ‰ค ๐‘  โ‰ค ๐‘ก โ‰ค 1, {๐‘ก (1 โˆ’ ๐‘ ) โ‹…{ ๐‘ก๐›ผโˆ’1 (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 , 0 โ‰ค ๐‘ก โ‰ค ๐‘  โ‰ค 1, {

= ๐œŽ โ€–๐‘˜โ€–1 ๐‘ก

(10)

.

Lemma 5 (see [21]). Let ฮฉ1 and ฮฉ2 be two bounded open sets in Banach space ๐ธ such that ๐œƒ โˆˆ ฮฉ1 and ฮฉ1 โŠ‚ ฮฉ2 and ๐ด : ๐‘ƒ โˆฉ (ฮฉ2 \ ฮฉ1 ) โ†’ ๐‘ƒ a completely continuous operator, where ๐œƒ denotes the zero element of ๐ธ and ๐‘ƒ a cone of ๐ธ. Suppose that one of the two conditions holds: (i) โ€–๐ด๐‘ขโ€– โ‰ค โ€–๐‘ขโ€–, โˆ€๐‘ข โˆˆ ๐‘ƒโˆฉ๐œ•ฮฉ1 ; โ€–๐ด๐‘ขโ€– โ‰ฅ โ€–๐‘ขโ€–, โˆ€๐‘ข โˆˆ ๐‘ƒโˆฉ๐œ•ฮฉ2 .

(7)

(ii) โ€–๐ด๐‘ขโ€– โ‰ฅ โ€–๐‘ขโ€–, โˆ€๐‘ข โˆˆ ๐‘ƒโˆฉ๐œ•ฮฉ1 ; โ€–๐ด๐‘ขโ€– โ‰ค โ€–๐‘ขโ€–, โˆ€๐‘ข โˆˆ ๐‘ƒโˆฉ๐œ•ฮฉ2 . Then ๐ด has a fixed point in ๐‘ƒ โˆฉ (ฮฉ2 \ ฮฉ1 ).

Journal of Function Spaces

3 The operator ๐‘‡๐‘› is given by

3. Main Results

(๐‘‡๐‘› ๐‘ข) (๐‘ก)

Let ๐พ = {๐‘ข : ๐‘ข (๐‘ก) โ‰ฅ ๐‘ก๐›ผโˆ’1 โ€–๐‘ขโ€–} .

(11)

1

= โˆซ ๐บ (๐‘ก, ๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘ , 0

Obviously, ๐พ is a cone in ๐ธ and (๐ธ, ๐พ) is a partial ordering Banach space. (A0 ) ๐‘“ โˆˆ ๐ถ((0, 1) ร— (0, +โˆž), (โˆ’โˆž, +โˆž)), and there exists a function ๐‘˜ โˆˆ ๐ฟ1 (0, 1) and ๐‘˜(๐‘ก) > 0, such that ๐‘“(๐‘ก, ๐‘ข) โ‰ฅ โˆ’๐‘˜(๐‘ก), โˆ€๐‘ก โˆˆ (0, 1), ๐‘ข > 0. (A1 ) For any positive numbers ๐‘Ÿ1 < ๐‘Ÿ2 , there exists a nonnegative function ๐›พ๐‘Ÿ1 ,๐‘Ÿ2 โˆˆ ๐ฟ1 (0, 1) such that ๓ต„จ๓ต„จ ๓ต„จ ๐›ผโˆ’1 โ‰ค ๐‘ข โ‰ค ๐‘Ÿ2 ๓ต„จ๓ต„จ๐‘“ (๐‘ก, ๐‘ข)๓ต„จ๓ต„จ๓ต„จ โ‰ค ๐›พ๐‘Ÿ1 ,๐‘Ÿ2 (๐‘ก) , 0 < ๐‘ก < 1, ๐‘Ÿ1 ๐‘ก

(12)

0 โ‰ค ๐‘ก โ‰ค 1. Next, we will give the proof from three steps. (I) For Any ๐œŽโ€–๐‘˜โ€–1 < ๐‘Ÿ1 < ๐‘Ÿ2 , ๐‘‡๐‘› : ๐พโˆฉ(ฮฉ๐‘Ÿ2 \ฮฉ๐‘Ÿ1 ) โ†’ ๐พ, ๐‘› > 1/๐‘Ÿ1 , Is Completely Continuous. For any 0 < ๐‘ก < 1, considering that ๐‘ข(๐‘ก) โˆ’ ๐‘ค(๐‘ก) โ‰ฅ (๐‘Ÿ1 โˆ’ ๐œŽโ€–๐‘˜โ€–1 )๐‘ก๐›ผโˆ’1 > 0, one has 1 (๐‘Ÿ1 โˆ’ ๐œŽ โ€–๐‘˜โ€–1 ) ๐‘ก๐›ผโˆ’1 โ‰ค max {๐‘ข (๐‘ก) โˆ’ ๐‘ค (๐‘ก) , } โ‰ค ๐‘Ÿ2 , ๐‘› 1 ๐‘›> , ๐‘Ÿ1

with 1

โˆซ (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 ๐›พ๐‘Ÿ1 ,๐‘Ÿ2 (๐‘ ) d๐‘  < +โˆž. 0

(13)

โˆซ ๐ฝ (๐‘ ) ๐œ‘ (๐‘ , ๐‘Ÿฬƒ1 ) d๐‘  < ๐‘Ÿฬƒ1 ,

(14)

0

1

โˆซ ๐ฝ (๐‘ ) ๐œ“ (๐‘ , ๐‘Ÿฬƒ2 ) d๐‘  > ๐‘Ÿฬƒ2 ,

(15)

0

where ๐œ“(๐‘ก, ๐‘Ÿฬƒ2 ) = min{๐‘“(๐‘ก, ๐‘ข) : (ฬƒ๐‘Ÿ2 โˆ’ ๐œŽโ€–๐‘˜โ€–1 )๐‘ก๐›ผโˆ’1 โ‰ค ๐‘ข โ‰ค ๐‘Ÿฬƒ2 } + ๐‘˜(๐‘ก). Theorem 6. Suppose that conditions (A0 ), (A1 ), (A2 ), and (A3 ) hold. Then BVP (1)-(2) admits at least one positive solution. Proof. First of all, we concentrate on the following modified approximating BVP (MABVP for short) to overcome difficulties caused by singularities: ๐›ผ ๐‘ข (๐‘ก) + ๐‘“ (๐‘ก, ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ก)) + ๐‘˜ (๐‘ก) = 0, ๐ท0+

1

โ‰ค โˆซ ๐ฝ (๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘  0

1

โ‰คโˆซ [ 0

+

(1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 (1 โˆ’ (1 โˆ’ ๐‘ )๐‘ ) ฮ“ (๐›ผ)

๐‘š 1 ๐›ผโˆ’๐‘žโˆ’1 โˆ‘๐‘Ž๐‘– ๐œ‰๐‘– (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 ] ฮ”ฮ“ (๐›ผ โˆ’ ๐‘ž) ๐‘–=1

(21)

โ‹… [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘  1

โ‰คโˆซ [ 0

+

(1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 (1 โˆ’ (1 โˆ’ ๐‘ )๐‘ ) ฮ“ (๐›ผ)

๐‘š 1 ๐›ผโˆ’๐‘žโˆ’1 โˆ‘๐‘Ž๐‘– ๐œ‰๐‘– (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 ] ฮ”ฮ“ (๐›ผ โˆ’ ๐‘ž) ๐‘–=1

โ‹… (๐›พ๐‘Ÿ1 โˆ’๐œŽโ€–๐‘˜โ€–1 ,๐‘Ÿ2 (๐‘ ) + ๐‘˜ (๐‘ )) d๐‘  < +โˆž. 0 < ๐‘ก < 1,

๐‘š

1

(๐‘‡๐‘› ๐‘ข) (๐‘ก) = โˆซ ๐บ (๐‘ก, ๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘  0

where ๐œ‘(๐‘ก, ๐‘Ÿฬƒ1 ) = max{๐‘“(๐‘ก, ๐‘ข) : (ฬƒ๐‘Ÿ1 โˆ’ ๐œŽโ€–๐‘˜โ€–1 )๐‘ก๐›ผโˆ’1 โ‰ค ๐‘ข โ‰ค ๐‘Ÿฬƒ1 } + ๐‘˜(๐‘ก). (A3 ) There exists a ๐‘Ÿฬƒ2 > ๐‘Ÿฬƒ1 such that

๐›ฝ

(20)

Thus, one gets from (A0 ), (A1 ), and Lemma 2 that

1

๐‘ข (0) = ๐‘ข๓ธ€  (0) = โ‹… โ‹… โ‹… = ๐‘ข(๐‘›โˆ’2) (0) = 0,

(19)

which means (๐‘Ÿ1 โˆ’ ๐œŽ โ€–๐‘˜โ€–1 ) ๐‘ก๐›ผโˆ’1 โ‰ค ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ก) โ‰ค ๐‘Ÿ2 , 0 < ๐‘ก < 1.

(A2 ) There exists a ๐‘Ÿฬƒ1 > ๐œŽโ€–๐‘˜โ€–1 such that

(18)

(16)

So, ๐‘‡๐‘› : ๐พ โˆฉ (ฮฉ๐‘Ÿ2 \ ฮฉ๐‘Ÿ1 ) โ†’ ๐ธ is well defined. At the same time, for given ๐‘ข โˆˆ ๐พ โˆฉ (ฮฉ๐‘Ÿ2 \ ฮฉ๐‘Ÿ1 ), ๐‘ก โˆˆ [0, 1], one gets from (21) and Lemma 2 that 1

๐‘ž

(๐‘‡๐‘› ๐‘ข) (๐‘ก) = โˆซ ๐บ (๐‘ก, ๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘ 

๐ท0+ (1) = โˆ‘๐‘Ž๐‘– ๐ท0+ ๐‘ข (๐œ‰๐‘– ) ,

0

๐‘–=1

1

(22)

โ‰ค โˆซ ๐ฝ (๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘ .

and, here, 1 { {๐‘ข, ๐‘ข โ‰ฅ ๐‘› , ๐œ’๐‘› (๐‘ข) = { 1 { , ๐‘ข < 1. {๐‘› ๐‘›

0

Hence, (17)

1

๓ต„ฉ๓ต„ฉ ๓ต„ฉ๓ต„ฉ ๓ต„ฉ๓ต„ฉ๐‘‡๐‘› ๐‘ข๓ต„ฉ๓ต„ฉ โ‰ค โˆซ ๐ฝ (๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘ . 0

(23)

4

Journal of Function Spaces

Given ๐‘ข โˆˆ ๐พ, it follows from Lemma 2 that

Equicontinuity of ๐‘‡๐‘› (๐ท) can be derived from the absolute continuity of Lebesgue integral and (25). Thus, the fact that ๐‘‡๐‘› : ๐พ โˆฉ (ฮฉ๐‘Ÿ2 \ ฮฉ๐‘Ÿ1 ) โ†’ ๐พ, ๐‘› > 1/๐‘Ÿ1 , is completely continuous has been proved.

1

(๐‘‡๐‘› ๐‘ข) (๐‘ก) = โˆซ ๐บ (๐‘ก, ๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘  0

1

โ‰ฅ ๐‘ก๐›ผโˆ’1 โˆซ ๐ฝ (๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘ 

(24)

0

๓ต„ฉ ๓ต„ฉ โ‰ฅ ๐‘ก๐›ผโˆ’1 ๓ต„ฉ๓ต„ฉ๓ต„ฉ๐‘‡๐‘› ๐‘ข๓ต„ฉ๓ต„ฉ๓ต„ฉ ,

which means that ๐‘‡๐‘› : ๐พ โˆฉ (ฮฉ๐‘Ÿ2 \ ฮฉ๐‘Ÿ1 ) โ†’ ๐พ. Given bounded set ๐ท โŠ‚ ๐พ โˆฉ (ฮฉ๐‘Ÿ2 \ ฮฉ๐‘Ÿ1 ), we can see from (21) that ๐‘‡๐‘š (๐ท) is uniformly bounded. By Ascoli-Arzela theorem, in order to show the complete continuity, we need only to prove that ๐‘‡๐‘› (๐ท) is equicontinuous. It follows from (6), (7), (8), and (A1 ) that ๓ต„จ๓ต„จ 1 ๓ต„จ๓ต„จ๓ต„จ(๐‘‡ ๐‘ข)๓ธ€  (๐‘ก)๓ต„จ๓ต„จ๓ต„จ = ๓ต„จ๓ต„จ๓ต„จโˆซ ๐œ• ๐บ (๐‘ก, ๐‘ ) [๐‘“ (๐‘ , ๐œ’ (๐‘ข โˆ’ ๐‘ค) (๐‘ )) ๓ต„จ๓ต„จ ๐‘› ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ ๐‘› ๓ต„จ๓ต„จ 0 ๐œ•๐‘ก ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ 1 ๐œ• ๓ต„จ ๓ต„จ + ๐‘˜ (๐‘ )] d๐‘ ๓ต„จ๓ต„จ๓ต„จ๓ต„จ โ‰ค ๓ต„จ๓ต„จ๓ต„จ๓ต„จโˆซ ๐บ (๐‘ก, ๐‘ ) ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ 0 ๐œ•๐‘ก 1 ๓ต„จ๓ต„จ ๓ต„จ โ‹… [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘ ๓ต„จ๓ต„จ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ 1 ๓ต„จ๓ต„จ (๐›ผ โˆ’ 1) ๐‘ก๐›ผโˆ’2 ๐‘š + ๓ต„จ๓ต„จ๓ต„จโˆซ โˆ‘๐‘Ž๐‘– ๐บ2 (๐œ‰๐‘– , ๐‘ ) ๓ต„จ๓ต„จ 0 ฮ” ๐‘–=1 ๓ต„จ ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ โ‹… [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘ ๓ต„จ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ ๓ต„จ ๓ต„จ๓ต„จ ๐‘ก 1 ๓ต„จ๓ต„จ ๓ต„จ๓ต„จโˆซ [(๐›ผ โˆ’ 1) ๐‘ก๐›ผโˆ’2 (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 โˆ’ (๐›ผ โˆ’ 1) โ‰ค ฮ“ (๐›ผ) ๓ต„จ๓ต„จ๓ต„จ 0 โ‹… (๐‘ก โˆ’ ๐‘ )

๐›ผโˆ’2

] [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘ 

๐‘ก

๓ต„จ๓ต„จ ๓ต„จ + ๐‘˜ (๐‘ )] d๐‘ ๓ต„จ๓ต„จ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ 1 ๓ต„จ๓ต„จ (๐›ผ โˆ’ 1) ๐‘š ๐›ผโˆ’๐‘žโˆ’1 + ๓ต„จ๓ต„จ๓ต„จโˆซ โˆ‘๐‘Ž ๐œ‰ (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 ๓ต„จ๓ต„จ 0 ฮ”ฮ“ (๐›ผ โˆ’ ๐‘ž) ๐‘–=1 ๐‘– ๐‘– ๓ต„จ ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ โ‹… [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘ ๓ต„จ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ ๓ต„จ โ‰ค

1

1 [โˆซ [(๐›ผ โˆ’ 1) (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 + (๐›ผ โˆ’ 1) ฮ“ (๐›ผ) 0

โ‹… (1 โˆ’ ๐‘ )๐›ผโˆ’2 ] (๐›พ๐‘Ÿ1 โˆ’๐œŽโ€–๐‘˜โ€–1 ,๐‘Ÿ2 (๐‘ ) + ๐‘˜ (๐‘ )) d๐‘ ] ๓ต„จ๓ต„จ 1 ๓ต„จ๓ต„จ (๐›ผ โˆ’ 1) ๐‘š ๐›ผโˆ’๐‘žโˆ’1 + ๓ต„จ๓ต„จ๓ต„จโˆซ โˆ‘๐‘Ž ๐œ‰ (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 ๓ต„จ๓ต„จ 0 ฮ”ฮ“ (๐›ผ โˆ’ ๐‘ž) ๐‘–=1 ๐‘– ๐‘– ๓ต„จ ๓ต„จ๓ต„จ ๓ต„จ๓ต„จ โ‹… (๐›พ๐‘Ÿ1 โˆ’๐œŽโ€–๐‘˜โ€–1 ,๐‘Ÿ2 (๐‘ ) + ๐‘˜ (๐‘ )) d๐‘ ๓ต„จ๓ต„จ๓ต„จ < +โˆž, ๐‘ก โˆˆ [0, 1] . ๓ต„จ๓ต„จ ๓ต„จ

(ฬƒ๐‘Ÿ1 โˆ’ ๐œŽ โ€–๐‘˜โ€–1 ) ๐‘ก๐›ผโˆ’1 โ‰ค ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ก) โ‰ค ๐‘Ÿฬƒ1 , 0 < ๐‘ก < 1.

(26)

From the definition of ๐œ‘(๐‘ก, ๐‘Ÿฬƒ1 ), we have ๐‘“ (๐‘ก, ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ก)) + ๐‘˜ (๐‘ก) โ‰ค ๐œ‘ (๐‘ก, ๐‘Ÿฬƒ1 ) .

(27)

By (A2 ) and Lemma 2, one gets ๓ต„ฉ๓ต„ฉ ๓ต„ฉ๓ต„ฉ ๓ต„ฉ๓ต„ฉ๐‘‡๐‘› ๐‘ข๓ต„ฉ๓ต„ฉ 1

= max โˆซ ๐บ (๐‘ก, ๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘  0โ‰ค๐‘กโ‰ค1 0

(28)

1

โ‰ค โˆซ ๐ฝ (๐‘ ) ๐œ‘ (๐‘ , ๐‘Ÿฬƒ1 ) d๐‘  < ๐‘Ÿฬƒ1 ; 0

that is, ๓ต„ฉ๓ต„ฉ ๓ต„ฉ๓ต„ฉ ๓ต„ฉ๓ต„ฉ๐‘‡๐‘› ๐‘ข๓ต„ฉ๓ต„ฉ โ‰ค โ€–๐‘ขโ€– ,

โˆ€๐‘ข โˆˆ ๐พ โˆฉ ๐œ•ฮฉ (ฬƒ๐‘Ÿ1 ) .

(29)

Given ๐‘ข โˆˆ ๐พ โˆฉ ๐œ•ฮฉ(ฬƒ๐‘Ÿ2 ), one has ๐‘ข(๐‘ก) โ‰ฅ ๐‘Ÿฬƒ2 ๐‘ก๐›ผโˆ’1 , 0 โ‰ค ๐‘ก โ‰ค 1. Thus, for ๐‘› > 1/ฬƒ๐‘Ÿ2 , we get ๐‘Ÿฬƒ2 โ‰ฅ ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ก) โ‰ฅ (ฬƒ๐‘Ÿ2 โˆ’ ๐œŽ โ€–๐‘˜โ€–1 ) ๐‘ก๐›ผโˆ’1 .

(30)

According to the definition of ๐œ“(๐‘ก, ๐‘Ÿฬƒ2 ), one gets

1

+ โˆซ (๐›ผ โˆ’ 1) ๐‘ก๐›ผโˆ’2 (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ ))

(II) For Sufficiently Large ๐‘›, MABVP (16) Admits at Least One Positive Solution. Given ๐‘ข โˆˆ ๐พโˆฉ๐œ•ฮฉ(ฬƒ๐‘Ÿ1 ), one has ๐‘ข(๐‘ก) โ‰ฅ ๐‘Ÿฬƒ1 ๐‘ก๐›ผโˆ’1 , 0 โ‰ค ๐‘ก โ‰ค 1. Thus, for ๐‘› > 1/ฬƒ๐‘Ÿ1 , similar to (20), we get

๐‘“ (๐‘ก, ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ก)) + ๐‘˜ (๐‘ก) โ‰ฅ ๐œ“ (๐‘ก, ๐‘Ÿฬƒ2 ) .

(25)

(31)

By Lemma 2, (A3 ), and (31), we have ๓ต„ฉ๓ต„ฉ ๓ต„ฉ๓ต„ฉ ๓ต„ฉ๓ต„ฉ๐‘‡๐‘› ๐‘ข๓ต„ฉ๓ต„ฉ 1

= max โˆซ ๐บ (๐‘ก, ๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘  0โ‰ค๐‘กโ‰ค1 0

1

โ‰ฅ max ๐‘ก๐›ผโˆ’1 โˆซ ๐ฝ (๐‘ ) [๐‘“ (๐‘ , ๐œ’๐‘› (๐‘ข โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )] d๐‘  0โ‰ค๐‘กโ‰ค1

(32)

0

1

โ‰ฅ โˆซ ๐ฝ (๐‘ ) ๐œ“ (๐‘ , ๐‘Ÿฬƒ2 ) d๐‘  โ‰ฅ ๐‘Ÿฬƒ2 ; 0

that is, ๓ต„ฉ๓ต„ฉ ๓ต„ฉ๓ต„ฉ ๓ต„ฉ๓ต„ฉ๐‘‡๐‘› ๐‘ข๓ต„ฉ๓ต„ฉ โ‰ฅ โ€–๐‘ขโ€– ,

โˆ€๐‘ข โˆˆ ๐พ โˆฉ ๐œ•ฮฉ (ฬƒ๐‘Ÿ2 ) .

(33)

Take ๐‘š0 = max{1/ฬƒ๐‘Ÿ1 , 1/ฬƒ๐‘Ÿ2 }. Let ๐‘ = {๐‘š0 , ๐‘š0 + 1, . . .}. Then, for ๐‘› โˆˆ ๐‘, both (29) and (33) hold. This together with Lemma 5 shows that ๐‘‡๐‘› (๐‘› โˆˆ ๐‘) has at least one fixed point ๐‘ขฬƒ๐‘› โˆˆ ๐พ โˆฉ ฮฉ๐‘Ÿฬƒ2 \ ฮฉ๐‘Ÿฬƒ1 .

Journal of Function Spaces

5

(III) BVP (1)-(2) Admits at Least One Positive Solution. By Lemma 4, one has ๓ต„ฉ ๓ต„ฉ ๐‘ขฬƒ๐‘› (๐‘ก) โ‰ฅ ๓ต„ฉ๓ต„ฉ๓ต„ฉ๐‘ขฬƒ๐‘› ๓ต„ฉ๓ต„ฉ๓ต„ฉ ๐‘ก๐›ผโˆ’1 โ‰ฅ ๐‘Ÿฬƒ1 ๐‘ก๐›ผโˆ’1 > ๐œŽ โ€–๐‘˜โ€–1 ๐‘ก๐›ผโˆ’1 โ‰ฅ ๐‘ค (๐‘ก) , (34) ๐‘š > ๐‘, ๐‘ก โˆˆ [0, 1] , 1

๐‘ข๐‘› โˆ’ ๐‘ค) (๐‘ )) + ๐‘˜ (๐‘ )) d๐‘ , ๐‘ขฬƒ๐‘› (๐‘ก) = โˆซ ๐บ (๐‘ก, ๐‘ ) (๐‘“ (๐‘ , ๐œ’๐‘› (ฬƒ 0

(35)

0 < ๐‘ก < 1. By (A1 ), we know {ฬƒ ๐‘ข๐‘› : ๐‘› > ๐‘} are bounded and equicontinuous sets [0, 1]. It follows from Arzela-Ascoli theorem that there exist a subsequence ๐‘0 of ๐‘ and a function ๐‘ขฬƒ โˆˆ ๐ถ[0, 1] such that ๐‘ขฬƒ๐‘› converges to ๐‘ขฬƒ uniformly on [0, 1] as ๐‘› โ†’ โˆž through ๐‘0 . Taking limit as ๐‘› โ†’ โˆž on both sides of (35) together with the fact ๐‘ขฬƒ๐‘› (๐‘ก) โ‰ฅ ๐‘ก๐›ผโˆ’1 โ€–ฬƒ ๐‘ข๐‘› โ€– โ‰ฅ ๐‘Ÿฬƒ1 ๐‘ก๐›ผโˆ’1 , we get 1

๐‘ขฬƒ (๐‘ก) = โˆซ ๐บ (๐‘ก, ๐‘ ) (๐‘“ (๐‘ , ๐‘ขฬƒ (๐‘ ) โˆ’ ๐‘ค (๐‘ )) + ๐‘˜ (๐‘ )) d๐‘ , 0

(36)

0 < ๐‘ก < 1. Let ๐‘ข(๐‘ก) = ๐‘ขฬƒ(๐‘ก) โˆ’ ๐‘ค(๐‘ก), and then from (36) one has that ๐‘ข(๐‘ก) is a positive solution for BVP (1)-(2). A multiplicity result follows if we replace (A2 ) and (A3 ) by the following: (A๓ธ€ 2 ) There exists ๐‘Ÿฬƒ๐‘– > ๐œŽโ€–๐‘˜โ€–1 (๐‘– = 1, 2, . . . , ๐‘š) such that 1

โˆซ ๐ฝ (๐‘ ) ๐œ‘ (๐‘ , ๐‘Ÿฬƒ๐‘– ) d๐‘  < ๐‘Ÿฬƒ๐‘– ,

(37)

0

where ๐œ‘(๐‘ก, ๐‘Ÿฬƒ๐‘– ) = max{๐‘“(๐‘ก, ๐‘ข) : (ฬƒ๐‘Ÿ๐‘– โˆ’๐œŽโ€–๐‘˜โ€–1 )๐‘ก๐›ผโˆ’1 โ‰ค ๐‘ข โ‰ค ๐‘Ÿฬƒ๐‘– }+๐‘˜(๐‘ก). ฬƒ ๐‘– (๐‘– = 1, 2, . . . , ๐‘š) with (A๓ธ€ 3 ) There exists ๐‘… ฬƒ 1 < ๐‘Ÿฬƒ2 < ๐‘… ฬƒ 2 < โ‹… โ‹… โ‹… < ๐‘Ÿฬƒ๐‘š < ๐‘… ฬƒ๐‘š 0 < ๐‘Ÿฬƒ1 < ๐‘…

(38)

such that 1

ฬƒ ๐‘– ) d๐‘  > ๐‘… ฬƒ๐‘– , โˆซ ๐ฝ (๐‘ ) ๐œ“ (๐‘ , ๐‘…

(39)

0

ฬƒ ๐‘– โˆ’ ๐œŽโ€–๐‘˜โ€–1 )๐‘ก๐›ผโˆ’1 โ‰ค ๐‘ข โ‰ค ๐‘… ฬƒ๐‘– } + ฬƒ ๐‘– ) = min{๐‘“(๐‘ก, ๐‘ข) : (๐‘… where ๐œ“(๐‘ก, ๐‘… ๐‘˜(๐‘ก).

It is clear that (40)-(41) has the form of (1)-(2), where ๐›ผ = 8/3, ๐‘› = 3, ๐‘ = 1/2, ๐‘ž = 1/3, ๐‘š = 3, ๐œ‰1 = 1/2, ๐œ‰2 = 1/4, ๐œ‰3 = 4/5, ๐‘Ž1 = 1/2, ๐‘Ž2 = 2/3, and ๐‘Ž3 = 1/4. Obviously, (A0 ) holds for ๐‘˜(๐‘ก) = 1/12โˆš3 ๐‘ก. After direct calculation, we have ๐›ผ โˆ’ ๐‘ โˆ’ 1 = 7/6, ๐›ผ โˆ’ ๐‘ž โˆ’ 1 = 4/3, ฮ“(๐›ผ) = 1.5046, ฮ“(๐›ผ โˆ’ ๐‘) = 1.0823, ฮ“(๐›ผ โˆ’ ๐›ผโˆ’๐‘žโˆ’1 = 0.5290, ฮ” = 0.7217, ๐œŽ = 1.2803, ๐‘ž) = 1.1906, โˆ‘3๐‘–=1 ๐‘Ž๐‘– ๐œ‰๐‘– โ€–๐‘˜โ€–1 = 0.1250, and ๐œŽโ€–๐‘˜โ€–1 = 0.1600. It is clear that (A1 ) is valid for ๐›พ๐‘Ÿ1 ,๐‘Ÿ2 = (1/16โˆš3 ๐‘ก2 (1 โˆ’ ๐‘ก))[๐‘Ÿ2 6 + (๐‘Ÿ1 ๐‘ก5/3 )โˆ’1/8 ] + 1/12โˆš3 ๐‘ก. Next, we check (A2 ) and (A3 ). Take ๐‘Ÿฬƒ1 = 1 > 0.1600 = ๐œŽโ€–๐‘˜โ€–1 , and then by Lemma 2 we have 1

1

โˆซ ๐ฝ (๐‘ ) ๐œ‘ (๐‘ , 1) d๐‘  โ‰ค โˆซ [ 0

0

+

๐‘š

1 ๐›ผโˆ’๐‘žโˆ’1 โˆ‘๐‘Ž ๐œ‰ (1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 ] ฮ”ฮ“ (๐›ผ โˆ’ ๐‘ž) ๐‘–=1 ๐‘– ๐‘– 1

โ‹…

16โˆš๐‘ 2 3

max {(๐‘ข6 +

(1 โˆ’ ๐‘ )

1 ) : 0.8400๐‘ 5/3 โ‰ค ๐‘ข โˆš๐‘ข 8

1

(1 โˆ’ ๐‘ )7/6 (1 โˆ’ (1 โˆ’ ๐‘ )1/2 )

0

ฮ“ (๐›ผ)

โ‰ค 1} d๐‘  = โˆซ [ +

(1 โˆ’ ๐‘ )๐›ผโˆ’๐‘โˆ’1 (1 โˆ’ (1 โˆ’ ๐‘ )๐‘ ) ฮ“ (๐›ผ)

โˆ‘3๐‘–=1 ๐‘Ž๐‘– ๐œ‰๐‘–4/3 (1 โˆ’ ๐‘ )7/6 ] ฮ”ฮ“ (๐›ผ โˆ’ ๐‘ž)

1 1 max {(๐‘ข6 + 8 ) : 0.8400๐‘ 5/3 โ‰ค ๐‘ข โˆš๐‘ข 16โˆš3 ๐‘ 2 (1 โˆ’ ๐‘ ) (42) 1 1 7/6 โ‰ค 1} d๐‘  < โˆซ [0.6646 (1 โˆ’ ๐‘ ) โˆ’ 0.6646 (1 16 0

โ‹…

โˆ’ ๐‘ )5/3 + 0.6156 ร— (1 โˆ’ ๐‘ )7/6 ] ๐‘ โˆ’2/3 (1 โˆ’ ๐‘ )โˆ’1/3 (1 +

1 โˆš0.8400๐‘ 5/3 8

) d๐‘ 
20. (43)

[10]

Thus, (A3 ) is valid. According to Theorem 6, BVP (40)-(41) admits at least one positive solution.

Conflicts of Interest The authors declare that they have no conflicts of interest.

[11]

[12]

Authorsโ€™ Contributions [13]

The authors contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgments The project is supported financially by the Innovative Training Program for College Students (Mathematical Modelling on Spread, Prevention and Control of HFMD 201610443081, cx2015072), a Project of Shandong Province Higher Educational Science and Technology Program (J15LI16), the Natural Science Foundation of Shandong Province of China (ZR2015AL002), the Foundation for Jining Medical College Natural Science (JY2015BS07), and the National Natural Science Foundation of China (11571197, 11571296, 11371221, and 11071141).

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