Multiple solutions for singular semipositone boundary value problems of fourth-order differential systems with parameters

<jats:p>The aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters: <jats:disp-formula><jats:alternatives><jats:tex-math>$$ \textstyle\begin{cases} u^{(4)}(t)+\beta _{1}u''(t)-\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\quad 0< t< 1; \\ v^{(4)}(t)+\beta _{2}v''(t)-\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                </mml:math></jats:alternatives></jats:disp-formula> where <jats:inline-formula><jats:alternatives><jats:tex-math>$f_{1},f_{2}\in C[(0,1)\times \mathbb{R}^{+}_{0}\times \mathbb{R}, \mathbb{R}]$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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                </mml:math></jats:alternatives></jats:inline-formula>. By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results.</jats:p>


Introduction
In the recent decades, the topic about the existence of solutions of nonlinear boundary value problems (BVPs for short) has received considerable popularity due to its wide applications in biology, hydrodynamics, physics, chemistry, control theory, and so forth. Some progress has also been made in the study of solutions for various types of equations or systems including differential equation [13,21,25,27], integro-differential equation [2,19,27], evolution equations [1,7], fractional systems [3, 15, 17, 22-24, 30, 31], impulsive systems [14,18,28], and delay systems [14]. In consequence, many meaningful results have been obtained in these fields. For more details, please see Lakshmikantham et al. [8], Podlubny [16], and the references therein.
As we all know, fourth-order boundary value problems have important practical applications in physics and engineering, and, for instance, they are usually used to describe the deformation of an elastic beam supported at the end points [4,9,20]. Wang et al. [20] investigated the boundary value problems of a class of fourth-order differential systems with parameters as follows: The existence results of positive solutions were proved by using the fixed point theory under two novel cones being constructed. Unfortunately, the result obtained in [20] is only the existence of at least one nontrivial positive solution when the nonlinear terms have no singularity. It should be stressed also in [20] that the solutions of BVPs (1.1) are all positive and the nonlinear terms must be nonnegative, which is limited to a certain extent in some cases. Besides, we know that there is always some connection between the nonlinear terms in practical applications, but the description of this connection is rarely mentioned and studied in the present literature. To our best knowledge, there is no paper considering SBVPs (1.1) when f 1 (t, u, v) and f 2 (t, u, v) are singular at t = 0, t = 1, and u = 0, and also no result is available about the existence of multiple solutions for such boundary value problems.
Motivated by all the above analyses, in this paper we discuss the existence and multiplicity of solutions to SBVPs (1.1) when the parameters β i , α i ∈ R (i = 1, 2) satisfy condition (1.2). In addition, , v) and f 2 (t, u, v) may be singular at t = 0, t = 1 and u = 0, and f 1 , f 2 are semipositone rather than positive with some connection imposed between them. Our approaches are based on the approximation method and the well-known fixed point index theory.
Obviously, what we consider is more different from [20] and [32]. The main features of the present work are as follows. Firstly, f 1 (t, u, v) and f 2 (t, u, v) may be singular at both t = 0, t = 1 and u = 0, and under some suitable assumptions, the multiple nontrivial solutions for SBVPs (1.1) are established. Secondly, f 1 may be negative for some values of t, u, and v; f 2 is also allowed to change sign. Moreover, f 2 is controlled by f 1 . Thirdly, in the obtained solution (u, v), the component u is positive, but the component v is allowed to have different signs, even may be negative.
The rest of the present work is organized as follows. Section 2 contains some preliminaries. In Sect. 3, some transformations are introduced to convert SBVPs (1.1) into the corresponding approximate boundary value problems. The main results will be given and proved in Sect. 4. Finally, in Sect. 5, an example is given to demonstrate the main result.

Preliminaries
In view of condition (1.2), as in [9], denote and let G i,j (t, s) (i, j = 1, 2) be the Green function of the linear boundary value problem Then, for h i ∈ C[0, 1], the solution to the following linear boundary value problem can be expressed as

Lemma 2.1
The function G i,j (t, s) (i = 1, 2) has the following properties: Proof (1)-(3) can be seen from [9]. In addition, by careful calculation and Lemma 2.1 in [9], it is not difficult to prove that N j := sup 0<t,s<1 The main tool used here is the following fixed point index theory.

Lemma 2.2 ([6]
) Let E 1 be a Banach space and P be a cone in E 1 . Denote P r = {u ∈ P : u < r} and ∂P r = {u ∈ P : u = r} (∀r > 0). Let T : P → P be a complete continuous mapping, then the following conclusions are valid.

Conversion of boundary value problem (1.1)
In order to overcome the difficulties arising from singularity and semipositone, we convert boundary value problem (1.1) into another form (see (3.4)). For simplicity and convenience, set Then C 1 and M i,j (i, j = 1, 2) are positive numbers. Now let us list the following assumptions which will be satisfied throughout the paper.
(H1) There exist functions p ∈ L 1 [J, R + ] such that , and there exists N 3 > 0 such that In this paper, the basic space is E : Moreover, let Define a function w : [0, 1] → R + by Applying (3.1) and Lemma 2.1, one can easily obtain that This together with (2.1) guarantees that w(t) is the positive solution of the following boundary value problem: Now we are in a position to convert SBVPs (1.1) into an approximate boundary value problem. For this matter, it will be carried out in two steps.
Firstly, in order to overcome the difficulties arising from semipositone, consider the following singular nonlinear differential boundary value problem: Then we can obtain the following conclusion.
Secondly, in order to overcome the singularity associated with SBVPs (1.1), consider the following approximate boundary value problem: In the following, we shall mainly discuss the existence results for BVPs (3.7) by using the fixed point index theory. For this matter, first we define the following mappings: (3.8) Obviously, it is easy to see that the existence of nontrivial solutions for BVPs (3.7) is equivalent to the existence of the nontrivial fixed point of T j . Therefore, we just need to find the nontrivial fixed point of T j in the following work.
For the sake of obtaining the nontrivial fixed point of operator T j , set 1 (t, t) and N = N 1 N 2 N 3 . N 1 , N 2 , and N 3 are defined in Lemma 2.1 and (H2), respectively.
Evidently, P is a nonempty, convex, and closed subset of E. Furthermore, one can prove that P is a cone of Banach space E. For simplicity, denote Then, by the definition of cone P and the norm (u, v) , one can see that ∂P r := (u, v) ∈ P : (u, v) = r = (u, v) ∈ P : u = r N , Clearly, for each r > 0, P r is a relatively open and bounded set of P.

Main results
In this section, we present the main results of this paper. To do this, first we need to investigate the properties of mapping T j (j ∈ N).

Lemma 4.1
Assume that (H1) and (H2) hold. Then, for any j ∈ N, T j : P → P is completely continuous and T j (P) ⊂ P.
Proof For (u, v) ∈ P, by virtue of Lemma 2.1, one can easily get that Moreover, (H2) together with Lemma 2.1 implies that Therefore, T j (u, v) ∈ P, namely T j (P) ⊂ P. In addition, notice that f 1 , f 2 , and G i,j are continuous, one can deduce that T j is completely continuous for each j ∈ N by using normal methods such as Ascoli-Arzela theorem, etc.
For convenience of expression, for each R 1 > r 1 > Nr p , take where r p is defined in Next, let us list the following assumptions which will be used in what follows.
(H1 ) For each R 1 > r 1 > Nr p , there exists r 1 ,R 1 ∈ L 1 (J) such that (H3) There exist R > r > Nr p and function r such that (2) ( R,R ) < R N , ( r ) > r N . Now we are in a position to give the following two lemmas to calculate the fixed point index of T j (j ∈ N) in P r . (i) For any j ∈ N, i(T j , P r , P) = 0; (ii) For any j ∈ N, i(T j , P R , P) = 1.
Proof (i) For the sake of obtaining the desired result, we firstly prove that In fact, if it is not true, then there exist μ 0 ≥ 1 and (u 0 , v 0 ) ∈ ∂P r such that (u 0 , v 0 ) = μ 0 T j (u 0 , v 0 ). By (3.1), (3.2), and the definition of cone P, one can obtain that That is, Moreover, by the definition of function [·] * j , we have which means Hence, applying (u 0 , v 0 ) = μ 0 T j (u 0 , v 0 ) and (H3), we obtain immediately that (4.5) Taking the maximum for both sides of (4.5) in [0, 1], we get This is in contradiction with (u 0 , v 0 ) ∈ ∂P r . Besides, it is clear that inf (u,v)∈∂P r T j (u, v) > 0 by (4.5), and then (4.3) holds.
(ii) Next, we claim that Suppose on the contrary that there exist μ λ ∈ (0, 1] and ( Using a similar process of the proof as (i), we immediately get that which indicates In addition, (u λ , v λ ) = μ λ T j (u λ , v λ ) together with (4.2), (4.7), and (H3) deduces that This is in contradiction with (u λ , v λ ) ∈ ∂P R . Therefore, (4.6) holds. To sum up, the proof is complete. Then there exists a constant R * > R such that i(T j , P R * , P) = 0 for each j ∈ N.
Proof First, choose a positive number ϒ satisfying Then, by (H4), it is easy to see that there exists > R N such that Let R * be a positive number satisfying R * > 2N min t∈[α,β] σ (t) . Then In fact, if it is not true, then there exist μ 0 ≥ 1 and (u 0 , v 0 ) ∈ ∂P R * such that (u 0 , v 0 ) = μ 0 T j (u 0 , v 0 ). Therefore, for any t ∈ [α, β], by (3.1), (3.2), and (4.10), one can easily get that σ (t) > > 0. (4.12) Hence, from (4.9) and (4.12), we have (4.13) Consequently, by (4.8) and (4.13), we immediately obtain that This is in contradiction with (u 0 , v 0 ) ∈ ∂P R * . Moreover, in view of (4.13) we know that inf (u,v)∈∂P R * T j (u, v) > 0. So, by Lemma 2.2, the conclusion of this lemma follows. Now, we are in a position to prove the main theorem of the present paper. Proof This proof will be carried out in four steps.

Claim 1 System (3.7) has at least two nontrivial solutions.
In fact, applying Lemmas 4.2-4.3 and the additivity of the fixed point index, one can get for any j ∈ N that Namely, system (3.7) has at least two nontrivial solutions satisfying (4.14) Notice that the boundedness is obvious. To prove the equicontinuity, let us prove that {u j } j∈N are equicontinuous on [0, 1] first. Since applying (4.2) and (4.15), we get that, for any 0 < t 1 < t 2 < 1 and j ∈ N,   Since (u j n , v j n ) satisfies the integral equations  Therefore, (u 0 , v 0 ) is a nontrivial solution of BVPs (3.4). Similarly, we also get that (U 0 , V 0 ) is a nontrivial solution of BVPs (3.4). In addition, it is obvious that u 0 (t)w(t) ≥ 0 and U 0 (t)w(t) ≥ 0. Then, from Lemma 3.1, we know that (u 0w(t), v 0 ) and (U 0 (t)w(t), V 0 (t)) are the nontrivial solution of SBVPs (1.1).