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: ๐ผ ๐ข (๐ก) + ๐ (๐ก, ๐ข (๐ก)) = 0, ๐ท0+
0 < ๐ก < 1,
(1)
subject to multipoint BCs ๐ข (0) = ๐ข๓ธ (0) = โ
โ
โ
= ๐ข(๐โ2) (0) = 0, ๐
๐
๐
๐ท0+ ๐ข (1) = โ๐๐ ๐ท0+ ๐ข (๐๐ ) .
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 ๐ผ1 , ๐ผ2 , ๐ผ3 > 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
๐ผ is the traditional Riemann-Liouville derivative Here ๐ท0+ with order ๐ผ, ๐ โ 1 < ๐ผ โค ๐ (๐ โฅ 3), ๐๐ โฅ 0, ๐ = 1, 2, . . . , ๐ (๐ โ N+ ), 0 < ๐1 < ๐2 < โ
โ
โ
< ๐๐ < 1, ๐, ๐ โ R, 1 โค ๐ โค ๐ โ 2, and 0 โค ๐ โค ๐ with ฮ = ฮ(๐ผ)/ฮ(๐ผ โ ๐ผโ๐โ1 > 0. The nonlinear term ๐) โ (ฮ(๐ผ)/ฮ(๐ผ โ ๐)) โ๐ ๐=1 ๐๐ ๐๐ ๐ permits sign-changing and singularities at ๐ก = 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.
๐ฝ
๐ท๐ก๐ผ ๐ข (๐ก) + ๐๐ (๐ก, ๐ข (๐ก) , ๐ท๐ก ๐ข (๐ก) , V (๐ก)) = 0, ๐พ
๐ท๐ก V (๐ก) + ๐๐ (๐ก, ๐ข (๐ก)) = 0, ๐ฝ
0 < ๐ก < 1,
๐ฝ+1
๐ท๐ก ๐ข (0) = ๐ท๐ก ๐ข (0) = 0, 1
๐ฝ
๐ฝ
๐ท๐ก ๐ข (1) = โซ ๐ท๐ก ๐ข (๐ ) d๐ด (๐ ) , 0
๓ธ
V (0) = V (0) = 0, 1
V (1) = โซ V (๐ ) d๐ต (๐ ) , 0
(3)
2
Journal of Function Spaces
where 2 < ๐ผ, ๐พ โค 3, 0 < ๐ฝ < 1, ๐ > 0 is a parameter, ๐ฝ ๐พ ๐ผ โ ๐ฝ > 2, and ๐ท๐ก๐ผ , ๐ท๐ก , and ๐ท๐ก are traditional R-L derivatives. 1 ๐ฝ 1 โซ0 ๐ท๐ก ๐ข(๐ )d๐ด(๐ ) and โซ0 V(๐ )d๐ต(๐ ) represent R-S integrals and ๐ด, ๐ต are bounded variations, and the nonlinearities permit sign-changing and singularities at ๐ก = 0, 1. Other relative papers on nonlocal boundary value problems can be found in [7โ20]. 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 ๐ธ = ๐ถ[0, 1] and ๐ฟ1 (0, 1) are involved in this article, where ๐ธ = ๐ถ[0, 1] and ๐ฟ1 (0, 1) represent the spaces of the continuous functions and Lebesgue integrable functions equipped with the norms 1 โ๐ขโ = max0โค๐กโค1 |๐ข(๐ก)| and โ๐ขโ1 = โซ0 |๐ข(๐ก)|d๐ก, respectively. 1
๐บ2 (๐ก, ๐ ) =
1 ฮ (๐ผ โ ๐)
๐ผโ๐โ1 (1 โ ๐ )๐ผโ๐โ1 โ (๐ก โ ๐ )๐ผโ๐โ1 , {๐ก โ
{ ๐ก๐ผโ๐โ1 (1 โ ๐ )๐ผโ๐โ1 , {
๐
๐
Lemma 2 (see [1]). Assume that ๐๐ > 0 (๐ = 1, 2, . . . , ๐) and ฮ > 0. Then the Green function ๐บ of (4) given by (6) is continuous on [0, 1] ร [0, 1] and meets (a) ๐บ(๐ก, ๐ ) โค ๐ฝ(๐ ), โ๐ก, ๐ โ [0, 1], and here ๐ฝ(๐ ) = โ1 (๐ ) + ๐ผโ๐โ1 (1 โ (1 โ (1/ฮ) โ๐ ๐=1 ๐๐ ๐บ2 (๐๐ , ๐ ), โ1 (๐ ) = (1 โ ๐ ) ๐ ๐ ) )/ฮ(๐ผ), and ๐ โ [0, 1]; (b) ๐บ(๐ก, ๐ ) โฅ ๐ก๐ผโ1 ๐ฝ(๐ ), โ๐ก, ๐ โ [0, 1]; (c) ๐บ(๐ก, ๐ ) โค ๐๐ก๐ผโ1 โ๐ก, ๐ โ [0, 1], and here ๐ = 1/ฮ(๐ผ) + ๐ผโ๐โ1 /(ฮฮ(๐ผ โ ๐)). โ๐ ๐=1 ๐๐ ๐๐ Lemma 3. Let ๐ข โ ๐ถ[0, 1] satisfy (4), where ๐ฆ โ ๐ฟ1 (0, 1), ๐ฆ(๐ก) โฅ 0, and 0 โค ๐ก โค 1. Then, ๐ข(๐ก) โฅ ๐ก๐ผโ1 โ๐ขโ, and 0 โค ๐ก โค 1. Proof. It can be easily seen from Lemma 5 in [1]; we omit the details. Lemma 4. Suppose that ๐ค(๐ก) โ ๐ถ[0, 1] be the solution of ๐ผ ๐ข (๐ก) + ๐ (๐ก) = 0, ๐ท0+
0 < ๐ก < 1,
๐ข (0) = ๐ข๓ธ (0) = โ
โ
โ
= ๐ข(๐โ2) (0) = 0, ๐
๐
(0) = 0,
(4)
๐
๐
where ๐ โ ๐ฟ1 (0, 1) and ๐(๐ก) > 0. Then, ๐ค(๐ก) โค ๐โ๐โ1 ๐ก๐ผโ1 and 0 โค ๐ก โค 1. Proof. For any 0 โค ๐ก โค 1, we get by Lemma 2
๐=1
1
1
๐ค (๐ก) = โซ ๐บ (๐ก, ๐ ) ๐ (๐ ) d๐ โค ๐๐ก๐ผโ1 โซ ๐ (๐ ) d๐
satisfies
0
1
๐ข (๐ก) = โซ ๐บ (๐ก, ๐ ) ๐ฆ (๐ ) d๐ , 0
0
๐ผโ1
๐ก โ [0, 1] ,
(5)
and here ๐บ (๐ก, ๐ ) = ๐บ1 (๐ก, ๐ ) +
(9)
๐=1
(๐โ2)
๐ท0+ ๐ข (1) = โ๐๐ ๐ท0+ ๐ข (๐๐ )
๐บ1 (๐ก, ๐ ) =
0 โค ๐ก โค ๐ โค 1,
๐ท0+ ๐ข (1) = โ๐๐ ๐ท0+ ๐ข (๐๐ ) ,
๐ผ ๐ท0+ ๐ข (๐ก) + ๐ฆ (๐ก) = 0, 0 < ๐ก < 1,
๐ข (0) = ๐ข (0) = โ
โ
โ
= ๐ข
(8)
โ(๐ก, ๐ ) โ [0, 1] ร [0, 1].
Lemma 1 (see [1]). If ฮ =ฬธ 0. Given ๐ฆ โ ๐ถ(0, 1) โฉ ๐ฟ (0, 1), the solution of the following differential equation:
๓ธ
0 โค ๐ โค ๐ก โค 1,
๐ก๐ผโ1 ๐ โ๐ ๐บ (๐ , ๐ ) , ฮ ๐=1 ๐ 2 ๐
(6)
1 ฮ (๐ผ)
๐ผโ๐โ1 ๐ผโ1 โ (๐ก โ ๐ )๐ผโ1 , 0 โค ๐ โค ๐ก โค 1, {๐ก (1 โ ๐ ) โ
{ ๐ก๐ผโ1 (1 โ ๐ )๐ผโ๐โ1 , 0 โค ๐ก โค ๐ โค 1, {
= ๐ โ๐โ1 ๐ก
(10)
.
Lemma 5 (see [21]). Let ฮฉ1 and ฮฉ2 be two bounded open sets in Banach space ๐ธ such that ๐ โ ฮฉ1 and ฮฉ1 โ ฮฉ2 and ๐ด : ๐ โฉ (ฮฉ2 \ ฮฉ1 ) โ ๐ a completely continuous operator, where ๐ denotes the zero element of ๐ธ and ๐ a cone of ๐ธ. Suppose that one of the two conditions holds: (i) โ๐ด๐ขโ โค โ๐ขโ, โ๐ข โ ๐โฉ๐ฮฉ1 ; โ๐ด๐ขโ โฅ โ๐ขโ, โ๐ข โ ๐โฉ๐ฮฉ2 .
(7)
(ii) โ๐ด๐ขโ โฅ โ๐ขโ, โ๐ข โ ๐โฉ๐ฮฉ1 ; โ๐ด๐ขโ โค โ๐ขโ, โ๐ข โ ๐โฉ๐ฮฉ2 . Then ๐ด has a fixed point in ๐ โฉ (ฮฉ2 \ ฮฉ1 ).
Journal of Function Spaces
3 The operator ๐๐ is given by
3. Main Results
(๐๐ ๐ข) (๐ก)
Let ๐พ = {๐ข : ๐ข (๐ก) โฅ ๐ก๐ผโ1 โ๐ขโ} .
(11)
1
= โซ ๐บ (๐ก, ๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ , 0
Obviously, ๐พ is a cone in ๐ธ and (๐ธ, ๐พ) is a partial ordering Banach space. (A0 ) ๐ โ ๐ถ((0, 1) ร (0, +โ), (โโ, +โ)), and there exists a function ๐ โ ๐ฟ1 (0, 1) and ๐(๐ก) > 0, such that ๐(๐ก, ๐ข) โฅ โ๐(๐ก), โ๐ก โ (0, 1), ๐ข > 0. (A1 ) For any positive numbers ๐1 < ๐2 , there exists a nonnegative function ๐พ๐1 ,๐2 โ ๐ฟ1 (0, 1) such that ๓ตจ๓ตจ ๓ตจ ๐ผโ1 โค ๐ข โค ๐2 ๓ตจ๓ตจ๐ (๐ก, ๐ข)๓ตจ๓ตจ๓ตจ โค ๐พ๐1 ,๐2 (๐ก) , 0 < ๐ก < 1, ๐1 ๐ก
(12)
0 โค ๐ก โค 1. Next, we will give the proof from three steps. (I) For Any ๐โ๐โ1 < ๐1 < ๐2 , ๐๐ : ๐พโฉ(ฮฉ๐2 \ฮฉ๐1 ) โ ๐พ, ๐ > 1/๐1 , Is Completely Continuous. For any 0 < ๐ก < 1, considering that ๐ข(๐ก) โ ๐ค(๐ก) โฅ (๐1 โ ๐โ๐โ1 )๐ก๐ผโ1 > 0, one has 1 (๐1 โ ๐ โ๐โ1 ) ๐ก๐ผโ1 โค max {๐ข (๐ก) โ ๐ค (๐ก) , } โค ๐2 , ๐ 1 ๐> , ๐1
with 1
โซ (1 โ ๐ )๐ผโ๐โ1 ๐พ๐1 ,๐2 (๐ ) d๐ < +โ. 0
(13)
โซ ๐ฝ (๐ ) ๐ (๐ , ๐ฬ1 ) d๐ < ๐ฬ1 ,
(14)
0
1
โซ ๐ฝ (๐ ) ๐ (๐ , ๐ฬ2 ) d๐ > ๐ฬ2 ,
(15)
0
where ๐(๐ก, ๐ฬ2 ) = min{๐(๐ก, ๐ข) : (ฬ๐2 โ ๐โ๐โ1 )๐ก๐ผโ1 โค ๐ข โค ๐ฬ2 } + ๐(๐ก). 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: ๐ผ ๐ข (๐ก) + ๐ (๐ก, ๐๐ (๐ข โ ๐ค) (๐ก)) + ๐ (๐ก) = 0, ๐ท0+
1
โค โซ ๐ฝ (๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ 0
1
โคโซ [ 0
+
(1 โ ๐ )๐ผโ๐โ1 (1 โ (1 โ ๐ )๐ ) ฮ (๐ผ)
๐ 1 ๐ผโ๐โ1 โ๐๐ ๐๐ (1 โ ๐ )๐ผโ๐โ1 ] ฮฮ (๐ผ โ ๐) ๐=1
(21)
โ
[๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ 1
โคโซ [ 0
+
(1 โ ๐ )๐ผโ๐โ1 (1 โ (1 โ ๐ )๐ ) ฮ (๐ผ)
๐ 1 ๐ผโ๐โ1 โ๐๐ ๐๐ (1 โ ๐ )๐ผโ๐โ1 ] ฮฮ (๐ผ โ ๐) ๐=1
โ
(๐พ๐1 โ๐โ๐โ1 ,๐2 (๐ ) + ๐ (๐ )) d๐ < +โ. 0 < ๐ก < 1,
๐
1
(๐๐ ๐ข) (๐ก) = โซ ๐บ (๐ก, ๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ 0
where ๐(๐ก, ๐ฬ1 ) = max{๐(๐ก, ๐ข) : (ฬ๐1 โ ๐โ๐โ1 )๐ก๐ผโ1 โค ๐ข โค ๐ฬ1 } + ๐(๐ก). (A3 ) There exists a ๐ฬ2 > ๐ฬ1 such that
๐ฝ
(20)
Thus, one gets from (A0 ), (A1 ), and Lemma 2 that
1
๐ข (0) = ๐ข๓ธ (0) = โ
โ
โ
= ๐ข(๐โ2) (0) = 0,
(19)
which means (๐1 โ ๐ โ๐โ1 ) ๐ก๐ผโ1 โค ๐๐ (๐ข โ ๐ค) (๐ก) โค ๐2 , 0 < ๐ก < 1.
(A2 ) There exists a ๐ฬ1 > ๐โ๐โ1 such that
(18)
(16)
So, ๐๐ : ๐พ โฉ (ฮฉ๐2 \ ฮฉ๐1 ) โ ๐ธ is well defined. At the same time, for given ๐ข โ ๐พ โฉ (ฮฉ๐2 \ ฮฉ๐1 ), ๐ก โ [0, 1], one gets from (21) and Lemma 2 that 1
๐
(๐๐ ๐ข) (๐ก) = โซ ๐บ (๐ก, ๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐
๐ท0+ (1) = โ๐๐ ๐ท0+ ๐ข (๐๐ ) ,
0
๐=1
1
(22)
โค โซ ๐ฝ (๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ .
and, here, 1 { {๐ข, ๐ข โฅ ๐ , ๐๐ (๐ข) = { 1 { , ๐ข < 1. {๐ ๐
0
Hence, (17)
1
๓ตฉ๓ตฉ ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ๐๐ ๐ข๓ตฉ๓ตฉ โค โซ ๐ฝ (๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ . 0
(23)
4
Journal of Function Spaces
Given ๐ข โ ๐พ, it follows from Lemma 2 that
Equicontinuity of ๐๐ (๐ท) can be derived from the absolute continuity of Lebesgue integral and (25). Thus, the fact that ๐๐ : ๐พ โฉ (ฮฉ๐2 \ ฮฉ๐1 ) โ ๐พ, ๐ > 1/๐1 , is completely continuous has been proved.
1
(๐๐ ๐ข) (๐ก) = โซ ๐บ (๐ก, ๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ 0
1
โฅ ๐ก๐ผโ1 โซ ๐ฝ (๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐
(24)
0
๓ตฉ ๓ตฉ โฅ ๐ก๐ผโ1 ๓ตฉ๓ตฉ๓ตฉ๐๐ ๐ข๓ตฉ๓ตฉ๓ตฉ ,
which means that ๐๐ : ๐พ โฉ (ฮฉ๐2 \ ฮฉ๐1 ) โ ๐พ. Given bounded set ๐ท โ ๐พ โฉ (ฮฉ๐2 \ ฮฉ๐1 ), we can see from (21) that ๐๐ (๐ท) is uniformly bounded. By Ascoli-Arzela theorem, in order to show the complete continuity, we need only to prove that ๐๐ (๐ท) is equicontinuous. It follows from (6), (7), (8), and (A1 ) that ๓ตจ๓ตจ 1 ๓ตจ๓ตจ๓ตจ(๐ ๐ข)๓ธ (๐ก)๓ตจ๓ตจ๓ตจ = ๓ตจ๓ตจ๓ตจโซ ๐ ๐บ (๐ก, ๐ ) [๐ (๐ , ๐ (๐ข โ ๐ค) (๐ )) ๓ตจ๓ตจ ๐ ๓ตจ๓ตจ ๓ตจ๓ตจ ๐ ๓ตจ๓ตจ 0 ๐๐ก ๓ตจ๓ตจ ๓ตจ๓ตจ 1 ๐ ๓ตจ ๓ตจ + ๐ (๐ )] d๐ ๓ตจ๓ตจ๓ตจ๓ตจ โค ๓ตจ๓ตจ๓ตจ๓ตจโซ ๐บ (๐ก, ๐ ) ๓ตจ๓ตจ ๓ตจ๓ตจ 0 ๐๐ก 1 ๓ตจ๓ตจ ๓ตจ โ
[๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจ 1 ๓ตจ๓ตจ (๐ผ โ 1) ๐ก๐ผโ2 ๐ + ๓ตจ๓ตจ๓ตจโซ โ๐๐ ๐บ2 (๐๐ , ๐ ) ๓ตจ๓ตจ 0 ฮ ๐=1 ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจ โ
[๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ ๓ตจ๓ตจ ๐ก 1 ๓ตจ๓ตจ ๓ตจ๓ตจโซ [(๐ผ โ 1) ๐ก๐ผโ2 (1 โ ๐ )๐ผโ๐โ1 โ (๐ผ โ 1) โค ฮ (๐ผ) ๓ตจ๓ตจ๓ตจ 0 โ
(๐ก โ ๐ )
๐ผโ2
] [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐
๐ก
๓ตจ๓ตจ ๓ตจ + ๐ (๐ )] d๐ ๓ตจ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจ 1 ๓ตจ๓ตจ (๐ผ โ 1) ๐ ๐ผโ๐โ1 + ๓ตจ๓ตจ๓ตจโซ โ๐ ๐ (1 โ ๐ )๐ผโ๐โ1 ๓ตจ๓ตจ 0 ฮฮ (๐ผ โ ๐) ๐=1 ๐ ๐ ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจ โ
[๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ ๓ตจ๓ตจ๓ตจ ๓ตจ๓ตจ ๓ตจ โค
1
1 [โซ [(๐ผ โ 1) (1 โ ๐ )๐ผโ๐โ1 + (๐ผ โ 1) ฮ (๐ผ) 0
โ
(1 โ ๐ )๐ผโ2 ] (๐พ๐1 โ๐โ๐โ1 ,๐2 (๐ ) + ๐ (๐ )) d๐ ] ๓ตจ๓ตจ 1 ๓ตจ๓ตจ (๐ผ โ 1) ๐ ๐ผโ๐โ1 + ๓ตจ๓ตจ๓ตจโซ โ๐ ๐ (1 โ ๐ )๐ผโ๐โ1 ๓ตจ๓ตจ 0 ฮฮ (๐ผ โ ๐) ๐=1 ๐ ๐ ๓ตจ ๓ตจ๓ตจ ๓ตจ๓ตจ โ
(๐พ๐1 โ๐โ๐โ1 ,๐2 (๐ ) + ๐ (๐ )) d๐ ๓ตจ๓ตจ๓ตจ < +โ, ๐ก โ [0, 1] . ๓ตจ๓ตจ ๓ตจ
(ฬ๐1 โ ๐ โ๐โ1 ) ๐ก๐ผโ1 โค ๐๐ (๐ข โ ๐ค) (๐ก) โค ๐ฬ1 , 0 < ๐ก < 1.
(26)
From the definition of ๐(๐ก, ๐ฬ1 ), we have ๐ (๐ก, ๐๐ (๐ข โ ๐ค) (๐ก)) + ๐ (๐ก) โค ๐ (๐ก, ๐ฬ1 ) .
(27)
By (A2 ) and Lemma 2, one gets ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ๐๐ ๐ข๓ตฉ๓ตฉ 1
= max โซ ๐บ (๐ก, ๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ 0โค๐กโค1 0
(28)
1
โค โซ ๐ฝ (๐ ) ๐ (๐ , ๐ฬ1 ) d๐ < ๐ฬ1 ; 0
that is, ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ๐๐ ๐ข๓ตฉ๓ตฉ โค โ๐ขโ ,
โ๐ข โ ๐พ โฉ ๐ฮฉ (ฬ๐1 ) .
(29)
Given ๐ข โ ๐พ โฉ ๐ฮฉ(ฬ๐2 ), one has ๐ข(๐ก) โฅ ๐ฬ2 ๐ก๐ผโ1 , 0 โค ๐ก โค 1. Thus, for ๐ > 1/ฬ๐2 , we get ๐ฬ2 โฅ ๐๐ (๐ข โ ๐ค) (๐ก) โฅ (ฬ๐2 โ ๐ โ๐โ1 ) ๐ก๐ผโ1 .
(30)
According to the definition of ๐(๐ก, ๐ฬ2 ), one gets
1
+ โซ (๐ผ โ 1) ๐ก๐ผโ2 (1 โ ๐ )๐ผโ๐โ1 [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ ))
(II) For Sufficiently Large ๐, MABVP (16) Admits at Least One Positive Solution. Given ๐ข โ ๐พโฉ๐ฮฉ(ฬ๐1 ), one has ๐ข(๐ก) โฅ ๐ฬ1 ๐ก๐ผโ1 , 0 โค ๐ก โค 1. Thus, for ๐ > 1/ฬ๐1 , similar to (20), we get
๐ (๐ก, ๐๐ (๐ข โ ๐ค) (๐ก)) + ๐ (๐ก) โฅ ๐ (๐ก, ๐ฬ2 ) .
(25)
(31)
By Lemma 2, (A3 ), and (31), we have ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ๐๐ ๐ข๓ตฉ๓ตฉ 1
= max โซ ๐บ (๐ก, ๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ 0โค๐กโค1 0
1
โฅ max ๐ก๐ผโ1 โซ ๐ฝ (๐ ) [๐ (๐ , ๐๐ (๐ข โ ๐ค) (๐ )) + ๐ (๐ )] d๐ 0โค๐กโค1
(32)
0
1
โฅ โซ ๐ฝ (๐ ) ๐ (๐ , ๐ฬ2 ) d๐ โฅ ๐ฬ2 ; 0
that is, ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ ๓ตฉ๓ตฉ๐๐ ๐ข๓ตฉ๓ตฉ โฅ โ๐ขโ ,
โ๐ข โ ๐พ โฉ ๐ฮฉ (ฬ๐2 ) .
(33)
Take ๐0 = max{1/ฬ๐1 , 1/ฬ๐2 }. Let ๐ = {๐0 , ๐0 + 1, . . .}. Then, for ๐ โ ๐, both (29) and (33) hold. This together with Lemma 5 shows that ๐๐ (๐ โ ๐) has at least one fixed point ๐ขฬ๐ โ ๐พ โฉ ฮฉ๐ฬ2 \ ฮฉ๐ฬ1 .
Journal of Function Spaces
5
(III) BVP (1)-(2) Admits at Least One Positive Solution. By Lemma 4, one has ๓ตฉ ๓ตฉ ๐ขฬ๐ (๐ก) โฅ ๓ตฉ๓ตฉ๓ตฉ๐ขฬ๐ ๓ตฉ๓ตฉ๓ตฉ ๐ก๐ผโ1 โฅ ๐ฬ1 ๐ก๐ผโ1 > ๐ โ๐โ1 ๐ก๐ผโ1 โฅ ๐ค (๐ก) , (34) ๐ > ๐, ๐ก โ [0, 1] , 1
๐ข๐ โ ๐ค) (๐ )) + ๐ (๐ )) d๐ , ๐ขฬ๐ (๐ก) = โซ ๐บ (๐ก, ๐ ) (๐ (๐ , ๐๐ (ฬ 0
(35)
0 < ๐ก < 1. By (A1 ), we know {ฬ ๐ข๐ : ๐ > ๐} are bounded and equicontinuous sets [0, 1]. It follows from Arzela-Ascoli theorem that there exist a subsequence ๐0 of ๐ and a function ๐ขฬ โ ๐ถ[0, 1] such that ๐ขฬ๐ converges to ๐ขฬ uniformly on [0, 1] as ๐ โ โ through ๐0 . Taking limit as ๐ โ โ on both sides of (35) together with the fact ๐ขฬ๐ (๐ก) โฅ ๐ก๐ผโ1 โฬ ๐ข๐ โ โฅ ๐ฬ1 ๐ก๐ผโ1 , we get 1
๐ขฬ (๐ก) = โซ ๐บ (๐ก, ๐ ) (๐ (๐ , ๐ขฬ (๐ ) โ ๐ค (๐ )) + ๐ (๐ )) d๐ , 0
(36)
0 < ๐ก < 1. Let ๐ข(๐ก) = ๐ขฬ(๐ก) โ ๐ค(๐ก), and then from (36) one has that ๐ข(๐ก) 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 ๐ฬ๐ > ๐โ๐โ1 (๐ = 1, 2, . . . , ๐) such that 1
โซ ๐ฝ (๐ ) ๐ (๐ , ๐ฬ๐ ) d๐ < ๐ฬ๐ ,
(37)
0
where ๐(๐ก, ๐ฬ๐ ) = max{๐(๐ก, ๐ข) : (ฬ๐๐ โ๐โ๐โ1 )๐ก๐ผโ1 โค ๐ข โค ๐ฬ๐ }+๐(๐ก). ฬ ๐ (๐ = 1, 2, . . . , ๐) with (A๓ธ 3 ) There exists ๐
ฬ 1 < ๐ฬ2 < ๐
ฬ 2 < โ
โ
โ
< ๐ฬ๐ < ๐
ฬ๐ 0 < ๐ฬ1 < ๐
(38)
such that 1
ฬ ๐ ) d๐ > ๐
ฬ๐ , โซ ๐ฝ (๐ ) ๐ (๐ , ๐
(39)
0
ฬ ๐ โ ๐โ๐โ1 )๐ก๐ผโ1 โค ๐ข โค ๐
ฬ๐ } + ฬ ๐ ) = min{๐(๐ก, ๐ข) : (๐
where ๐(๐ก, ๐
๐(๐ก).
It is clear that (40)-(41) has the form of (1)-(2), where ๐ผ = 8/3, ๐ = 3, ๐ = 1/2, ๐ = 1/3, ๐ = 3, ๐1 = 1/2, ๐2 = 1/4, ๐3 = 4/5, ๐1 = 1/2, ๐2 = 2/3, and ๐3 = 1/4. Obviously, (A0 ) holds for ๐(๐ก) = 1/12โ3 ๐ก. After direct calculation, we have ๐ผ โ ๐ โ 1 = 7/6, ๐ผ โ ๐ โ 1 = 4/3, ฮ(๐ผ) = 1.5046, ฮ(๐ผ โ ๐) = 1.0823, ฮ(๐ผ โ ๐ผโ๐โ1 = 0.5290, ฮ = 0.7217, ๐ = 1.2803, ๐) = 1.1906, โ3๐=1 ๐๐ ๐๐ โ๐โ1 = 0.1250, and ๐โ๐โ1 = 0.1600. It is clear that (A1 ) is valid for ๐พ๐1 ,๐2 = (1/16โ3 ๐ก2 (1 โ ๐ก))[๐2 6 + (๐1 ๐ก5/3 )โ1/8 ] + 1/12โ3 ๐ก. Next, we check (A2 ) and (A3 ). Take ๐ฬ1 = 1 > 0.1600 = ๐โ๐โ1 , and then by Lemma 2 we have 1
1
โซ ๐ฝ (๐ ) ๐ (๐ , 1) d๐ โค โซ [ 0
0
+
๐
1 ๐ผโ๐โ1 โ๐ ๐ (1 โ ๐ )๐ผโ๐โ1 ] ฮฮ (๐ผ โ ๐) ๐=1 ๐ ๐ 1
โ
16โ๐ 2 3
max {(๐ข6 +
(1 โ ๐ )
1 ) : 0.8400๐ 5/3 โค ๐ข โ๐ข 8
1
(1 โ ๐ )7/6 (1 โ (1 โ ๐ )1/2 )
0
ฮ (๐ผ)
โค 1} d๐ = โซ [ +
(1 โ ๐ )๐ผโ๐โ1 (1 โ (1 โ ๐ )๐ ) ฮ (๐ผ)
โ3๐=1 ๐๐ ๐๐4/3 (1 โ ๐ )7/6 ] ฮฮ (๐ผ โ ๐)
1 1 max {(๐ข6 + 8 ) : 0.8400๐ 5/3 โค ๐ข โ๐ข 16โ3 ๐ 2 (1 โ ๐ ) (42) 1 1 7/6 โค 1} d๐ < โซ [0.6646 (1 โ ๐ ) โ 0.6646 (1 16 0
โ
โ ๐ )5/3 + 0.6156 ร (1 โ ๐ )7/6 ] ๐ โ2/3 (1 โ ๐ )โ1/3 (1 +
1 โ0.8400๐ 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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