Abstract

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:subject to multipoint BCs

Here is the traditional Riemann-Liouville derivative with order ,   , , , , , , and with . The nonlinear term permits sign-changing and singularities at .

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. For example, people have realized that electrical conductance of cell membranes of biological organism took on fractional-order 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, hand-foot-mouth disease [3], malaria, tuberculosis, and measles. Arafa et al. [4] formulated a model with orders to describe the infection of CD 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] where , , , is a parameter, , and , , and are traditional R-L derivatives. and represent R-S integrals and , are bounded variations, and the nonlinearities permit sign-changing and singularities at Other relative papers on nonlocal boundary value problems can be found in [7–20].

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 may take negative infinity and change its sign. Compared with [1], 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 and are involved in this article, where and represent the spaces of the continuous functions and Lebesgue integrable functions equipped with the norms and , respectively.

Lemma 1 (see [1]). If Given , the solution of the following differential equation:satisfies and here

Lemma 2 (see [1]). Assume that and . Then the Green function of (4) given by (6) is continuous on and meets, , and here , , and ;, ;  , and here

Lemma 3. Let satisfy (4), where , , and Then, , and

Proof. It can be easily seen from Lemma in [1]; we omit the details.

Lemma 4. Suppose that be the solution ofwhere and Then, and

Proof. For any , we get by Lemma 2

Lemma 5 (see [21]). Let and be two bounded open sets in Banach space such that and and a completely continuous operator, where denotes the zero element of and a cone of . Suppose that one of the two conditions holds:, ; , , ; , Then has a fixed point in .

3. Main Results

LetObviously, is a cone in and is a partial ordering Banach space.

  , and there exists a function and , such that , , .

For any positive numbers , there exists a nonnegative function such that with

There exists a such that where .

There exists a such that where .

Theorem 6. Suppose that conditions , , , and 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:and, here,The operator is given by Next, we will give the proof from three steps.
(I) For Any , , , Is Completely Continuous. For any , considering that , one has which meansThus, one gets from , , and Lemma 2 thatSo, is well defined.
At the same time, for given , , one gets from (21) and Lemma 2 thatHence,Given , it follows from Lemma 2 thatwhich means that .
Given bounded set , we can see from (21) that is uniformly bounded. By Ascoli-Arzela theorem, in order to show the complete continuity, we need only to prove that is equicontinuous. It follows from (6), (7), (8), and thatEquicontinuity of can be derived from the absolute continuity of Lebesgue integral and (25). Thus, the fact that , , is completely continuous has been proved.
(II) For Sufficiently Large , MABVP (16) Admits at Least One Positive Solution. Given , one has , Thus, for , similar to (20), we getFrom the definition of , we have By and Lemma 2, one gets that is, Given , one has , Thus, for , we get According to the definition of , one getsBy Lemma 2, , and (31), we havethat is,Take . Let Then, for , both (29) and (33) hold. This together with Lemma 5 shows that has at least one fixed point .
(III) BVP (1)-(2) Admits at Least One Positive Solution. By Lemma 4, one hasBy , we know are bounded and equicontinuous sets . It follows from Arzela-Ascoli theorem that there exist a subsequence of and a function such that converges to uniformly on as through . Taking limit as on both sides of (35) together with the fact , we getLet , and then from (36) one has that is a positive solution for BVP (1)-(2).
A multiplicity result follows if we replace and by the following:
There exists such that where .
There exists with such thatwhere .