Existence of symmetric positive solutions for a singular system with coupled integral boundary conditions

In this paper, we study a class of nonlinear singular system with coupled integral boundary condition. Based on the Guo–Krasnosel’skii fixed point theorem, some new results on the existence of symmetric positive solutions for the coupled singular system are obtained. The impact of the two different parameters on the existence of symmetric positive solutions is also investigated. Finally, an example is then given to demonstrate the applicability of our results.

We notice that a type of symmetric problem has received much attention, for instance, [3,7,9,[11][12][13][14]16] and the references therein. At the same time, a class of boundary value problems with integral BCs appeared in heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. For earlier contributions on problems with Lebesgue integral BCs, we refer the reader to [3,13,14,16] and the more general nonlocal Riemann-Stieltjes integral BCs, we refer the reader to [2,5,6,10] and references therein, such integral BCs are a general type of nonlocal boundary conditions and cover multi-point and integral BCs as special cases. Infante, Minhós, Pietramala [5] gave a general method for dealing with these problems in the important case when p = 2. Ma [14] studied the existence of a symmetric positive solution for the following singular fourth-order nonlocal boundary value problem is continuous, symmetric on (0, 1) and may be singular at t = 0 and t = 1, f : 1] for all x ∈ [0, +∞). The existence of at least one symmetric positive solution was obtained by the application of the fixed point index in cones.
In [16], Zhang, Feng, Ge studied fourth-order boundary value problem with integral BCs: . By using of fixed point theorem in cones, the existence and multiplicity of symmetric positive solutions were obtained, and the nonexistence of a positive solution was also studied.
Inspired and motivated by the above mentioned work and wide applications of coupled BCs in various fields of sciences and engineering, we study the existence of symmetric positive solutions to a singular system (1.1). Of necessity 1)), so the problem can be handled by considering the simpler problem (2.1) on [0, 1], then using symmetry u Our work presented in this paper has the following new features. First of all, we discuss the system (1.1) subject to coupled BCs with p-Laplacian operators, Riemann-Stieltjes integral BCs are a general type of nonlocal boundary conditions and cover multi-point and integral BCs as special cases, these are different from [3,7,[11][12][13][14]16]. The second new feature is that the system (1.1) possesses singularity, that is, the nonlinear terms may be singular at t = −1, 1. Thirdly, we involve the parameter λ i (i = 1, 2) in the model and obtain the sufficient conditions for the existence of symmetric positive solutions of system (1.1) within certain interval of λ i (i = 1, 2). To the best knowledge of the authors, there is no earlier literature studying the existence of symmetric positive solutions for boundary value system with coupled integral BCs.
The rest of the paper is organized as follows. In Section 2, we present a positive cone, a fixed point theorem which will be used to prove existence of symmetric positive solutions, Green's function for the modified system and some related lemmas. In Section 3, we present main results of the paper and in Section 4 an example is given to illustrate the application of our main results.

Preliminaries and lemmas
We recall that the function ω is said to be concave on [a, b] if , τ ∈ (0, 1), and the function ω is said to be symmetric on ) is a symmetric positive solution of the singular system (1.1), obviously, u (0) = 0, v (0) = 0, u (0) = 0 and v (0) = 0 are necessary. So the problem (1.1) can be handled by considering the following simpler problem 1], then using symmetry u(−t) = u(t) to extend the solution to [−1, 1]. In view of the above, we will concentrate our study on the system (2.1).
The basic space used in this paper is E = C[0, 1] × C[0, 1]. Obviously, the space E is a Banach space if it is endowed with the norm as follows: In the rest of the paper, we make the following assumptions:

Lemma 2.2.
Assume that (H 1 ) holds. Then for any x, y ∈ L 1 (0, 1) ∩ C(0, 1), the system of BVPs consisting of the equations and the BCs has a unique integral representation where Proof. Let where c 1 , c 2 , c 3 and c 4 are constants to be determined. Clearly, u(t) and v(t) satisfy (2.4). In the following, we determine c i (1 ≤ i ≤ 4) so that u(t) and v(t) satisfy (2.5). Substituting (2.10) and (2.11) into (2.5), we obtain c 3 = c 4 = 0 and Thus, the system (2.12)-(2.13) has a unique solution for c i (1 ≤ i ≤ 2). By the Cramer's rule and simple calculations, it follows that Then from (2.10) and (2.11), it is obvious that (2.6) and (2.7) hold.
Similar to the proof of Lemma 2.2, we have , v(t)), t ∈ (0, 1), has a unique integral representation Proof. It follows from ( Proof. First, we will show that (2.14) is true. By (2.3), the first equalities of (2.8) and (2.9), we obtain On the other hand, by (2.3), the first equalities of (2.8) and (2.9), we also have Next we show that (2.15) holds. In fact, using (2.3), the second equalities of (2.8) and (2.9), we get On the other hand, by (2.3), the second equalities of (2.8) and (2.9), we also have Similar to the proof of (2.14) and ( Employing Lemmas 2.2 and 2.3, the system (2.1) can be expressed as , v(τ))dτ ds.
Clearly, (u, v) ∈ K is a fixed point of T if and only if (u, v) is a solution of system (2.1). Lemma 2.7. Assume that (H 1 )-(H 3 ) hold. Then T : K → K is well defined. Furthermore, T : K → K is a completely continuous operator.
Next, we prove that T : K → K is completely continuous. For any natural number n, we set , v(τ))dτ ds, i = 1, 2, i + j = 3. Similar to [14], by the approximating theorem of completely continuous operators, we can prove T : K → K is a completely continuous operator.

Lemma 2.8 ([4]
). Let X be a real Banach space, P be a cone in X. Assume that Ω 1 and Ω 2 are two bounded open sets of X with θ ∈ Ω 1 and Ω 1 ⊂ Ω 2 . Let T : P ∩ (Ω 2 \Ω 1 ) → P be a completely continuous operator such that either Then T has at least one fixed point in P ∩ (Ω 2 \ Ω 1 ).
Similar to the proof of Theorem 3.3, we have the following result.
can get