1 Introduction

In this paper, we discuss the existence of positive solutions for the following second-order coupled differential system with coupled integral boundary value conditions and nonlinearities depending on the first derivatives:

$$\begin{aligned} \left\{ \begin{array}{l}{-u^{\prime \prime }(t)=f_{1}\left( t, u(t), v(t), u^{\prime }(t), v^{\prime }(t)\right) , t \in [0,1]}, \\ {-v^{\prime \prime }(t)=f_{2}\left( t, u(t), v(t), u^{\prime }(t), v^{\prime }(t)\right) , t \in [0,1]}, \\ {u(0)=\alpha [v], \quad u^{\prime }(1)=\beta [u]}, \\ {v(0)=\beta [u], \quad v^{\prime }(1)=\alpha [v]},\end{array}\right. \end{aligned}$$
(1.1)

where \(\alpha \) and \(\beta \) denote linear functionals given by

$$\begin{aligned} \alpha [u]=\int _{0}^{1}u(t){\text {d}}A(t),\ \ \beta [u]=\int _{0}^{1}u(t){\text {d}}B(t) \end{aligned}$$

involving Stieltjes integrals with suitable functions A, B of bounded variation.

The existence of positive solutions for second-order differential systems with either coupled boundary value conditions or coupled nonlinear terms have been studied by many researchers. Cui and Zou in [6] studied the existence and uniqueness of positive solutions for second-order differential systems with coupled integral boundary value conditions using prior estimation method and maximum principle. In [12], Henderson and Luca investigated the existence of positive solutions of coupled systems for second-order ordinary differential equations with multi-point boundary conditions using Guo–Krasnosel’skii fixed point theorem. In these works mentioned above, the nonlinear terms do not depend on derivatives. Also see [5].

In other papers, the systems are coupled in nonlinear terms of differential equations and the nonlinear terms depend on derivatives, but are not coupled in boundary conditions. For example, Minhós and Sousa in [21] studied the existence of positive solutions for coupled systems of three-order boundary value problems using Guo–Krasnosel’skii fixed point theorem.

And finally, there are some papers, and the systems are coupled both in nonlinear terms of differential equations and in boundary conditions, but the nonlinearities do not depend on derivatives. In [7], Cui and Zou proved the existence of solutions for second-order differential systems with coupled integral boundary value conditions using monotone iterative technique combined with upper and lower solutions. Goodrich in [10] studied a coupled system of second-order boundary value problems with asymptotically superlinear and nonlocal boundary conditions using Guo–Krasnosel’skii fixed point theorem. For this situation, we can also cite [1,2,3,4, 9, 18, 20].

In all of the above works with Stieltjes integral boundary conditions or with multi-point boundary conditions, the vast majority of Stieltjes integral boundary value problem is limited to the positive measure, the coefficients of multi-point boundary conditions are limited to be positive, and only a small number of them allowed sign-changing measure and sign-changing coefficients. It is noted that the Stieltjes integral boundary conditions include two special cases: multi-point and integral boundary conditions.

Motivated by those previous works, we investigate the existence of positive solutions for the system (1.1) that is coupled in both the nonlinearities of the differential equations and the boundary conditions with Stieltjes integrals of sign-changing measure, and whose nonlinearities depend on the derivative terms. We convert (1.1) into a coupled system of integral equations. By the theory of fixed point index on a special cone in \(C^1[0,1]\times C^1[0,1]\), the existence of the positive solutions of the system (1.1) is obtained through posing some inequality conditions and the spectral radius conditions on the nonlinearities. The conditions and method in this paper are different from those in references. We refer to some references in which the more general cases of integral equations are discussed; see [13,14,15,16,17, 22, 23, 25]. Especially, it should be noted that Infante and Minhós in [15] obtained the existence and nonexistence of nontrivial solutions of coupled systems of integral equations by constructing a cone of sign-changing functions and using the fixed point index theory. Most recent works due to Infante [13] (where the cone of non-negative, non-decreasing functions is used to study, via fixed point index, second-order ODEs with nonlocal boundary conditions with derivative dependence) and [14] (where the Krein–Rutman theorem, combined with fixed point index, is used to study systems with nonlinearities involving derivative dependence and coupled functional boundary conditions) also deserve attention. The results of this paper complement those in [13, 14]. In addition, we refer to [24, 26].

The structure of this paper is as follows. In Sect. 2, we give some useful lemmas for our main results. In Sect. 3, we discuss the existence results of positive solutions. In Sect. 4, two examples are given that are second-order coupled systems with mixed boundary conditions including multi-point one of sign-changing coefficients and integral one of sign-changing kernel.

2 Preliminaries

Let \(C^1[0,1]\) denote the Banach space of all continuously differentiable functions on [0, 1] with the norm

$$\begin{aligned} \Vert {u}\Vert _{C^{1}}=\max \{\Vert u\Vert _{C}, \Vert u'\Vert _{C}\}=\max \Bigg \{\max _{0\le t\le 1}|u(t)|, \max _{0\le t\le 1}|u'(t)|\Bigg \}. \end{aligned}$$

For \((u,v)\in C^1[0,1]\times C^1[0,1]\), define

$$\begin{aligned} \Vert (u, v)\Vert _{C^{1}}=\max \left\{ \Vert u\Vert _{C^{1}}, \Vert v\Vert _{C^{1}}\right\} ,\ \Vert (u, v)\Vert _{C}=\max \left\{ \Vert u\Vert _{C},\Vert v\Vert _{C}\right\} . \end{aligned}$$

We make the assumptions:

\((C_{1})\) \(f_i\): [0,1]\(\times {\mathbb {R}}^4_{+}\rightarrow {\mathbb {R}}_{+}\) is continuous \((i=1,2)\); here, \({\mathbb {R}}_{+}=[0,\infty )\);

\((C_2)\) A and B are of bounded variation and for \(s\in [0,1]\)

$$\begin{aligned} {\mathcal {K}}_{A}(s):=\int _{0}^{1}k(t,s){\text {d}}A(t)\ge 0,\ {\mathcal {K}}_{B}(s):=\int _{0}^{1}k(t,s){\text {d}}B(t)\ge 0, \end{aligned}$$

where

$$\begin{aligned} k(t,s)=\left\{ \begin{array}{l}s,\quad 0\le s\le t\le 1,\\ t,\quad 0\le t\le s\le 1;\end{array}\right. \end{aligned}$$
(2.1)

\((C_3)\) \(0\le \alpha [t]<1,\ \alpha [1]\ge 0,\ 0\le \beta [t]<1,\ \beta [1]\ge 0\), and

$$\begin{aligned} D:=(1-\alpha [t])(1-\beta [t])-\alpha [1] \beta [1]>0. \end{aligned}$$

It is easy to verify the following lemmas.

Lemma 2.1

For \(y_1,y_2\in C[0,1]\), the second-order system

$$\begin{aligned} \left\{ \begin{array}{l}{-u^{\prime \prime }(t)=y_{1}(t), t \in [0,1]}, \\ {-v^{\prime \prime }(t)=y_{2}(t), t \in [0,1]}, \\ {u(0)=\alpha [v], \quad u^{\prime }(1)=\beta [u]}, \\ {v(0)=\beta [u], \quad v^{\prime }(1)=\alpha [v]}, \end{array}\right. \end{aligned}$$
(2.2)

has a unique solution

$$\begin{aligned} \left\{ \begin{array}{l}u(t)=\int _{0}^{1} G_{1}(t, s) y_{1}(s) {\text {d}}s+\int _{0}^{1} H_{1}(t, s) y_{2}(s) {\text {d}}s, \\ v(t)=\int _{0}^{1} H_{2}(t, s) y_{1}(s) {\text {d}}s+\int _{0}^{1} G_{2}(t, s) y_{2}(s) {\text {d}}s,\end{array}\right. \end{aligned}$$
(2.3)

where

$$\begin{aligned} \left\{ \begin{array}{l}G_{1}(t, s)=\frac{\alpha [1]+t(1-\alpha [t])}{D} {\mathcal {K}}_{B}(s)+k(t, s),\\ H_{1}(t, s)=\frac{(1-\beta [t])+t \beta [1]}{D}{\mathcal {K}}_{A}(s), \\ G_{2}(t, s)=\frac{\beta [1]+t(1-\beta [t])}{D}{\mathcal {K}}_{A}(s)+k(t, s),\\ {H_{2}(t, s)=\frac{(1-\alpha [t])+t \alpha [1]}{D} {\mathcal {K}}_{B}(s)}.\end{array}\right. \end{aligned}$$
(2.4)

Lemma 2.2

If \((C_2)\) and \((C_3)\) hold, then the following inequalities are satisfied for \(t, s\in [0,1]\):

$$\begin{aligned} t \Phi _{1}(s)\le & {} G_{1}(t, s) \le \Phi _{1}(s),\ t \Phi _{2}(s) \le G_{2}(t, s) \le \Phi _{2}(s),\\ t \Phi _{3}(s)\le & {} H_{1}(t, s) \le \Phi _{3}(s),\ t \Phi _{4}(s) \le H_{2}(t, s) \le \Phi _{4}(s), \end{aligned}$$

where

$$\begin{aligned} \Phi _{1}(s)= & {} \frac{\alpha [1]+(1-\alpha [t])}{D} {\mathcal {K}}_{B}(s)+s,\ \Phi _{2}(s)=\frac{\beta [1]+(1-\beta [t])}{D} {\mathcal {K}}_{A}(s)+s,\\ \Phi _{3}(s)= & {} \frac{(1-\beta [t])+\beta [1]}{D} {\mathcal {K}}_{A}(s),\ \Phi _{4}(s)=\frac{(1-\alpha [t])+\alpha [1]}{D} {\mathcal {K}}_{B}(s) \end{aligned}$$

are non-negative functions.

Define two cones in \(C^{1}[0,1]\times C^{1}[0,1]\)

$$\begin{aligned} P= & {} P_{1} \times P_{2}=\left\{ \left( w_{1}, w_{2}\right) \in C^{1}[0,1] \times C^{1}[0,1] : w_{i} \in P_{i}, i=1,2\right\} , \end{aligned}$$
(2.5)
$$\begin{aligned} K= & {} K_{1} \times K_{2}=\left\{ \left( w_{1}, w_{2}\right) \in C^{1}[0,1] \times C^{1}[0,1] : w_{i} \in K_{i}, i=1,2\right\} ,\nonumber \\ \end{aligned}$$
(2.6)

where \(P_{i}=\left\{ w \in C^{1}[0,1] : w(t) \ge 0, w^{\prime }(t) \ge 0, t \in [0,1]\right\} (i=1,2)\) and

$$\begin{aligned} K_{i}=\left\{ w \in P_i : w(t) \ge t\Vert w\Vert _C, t \in [0,1],\ \alpha _{i}[w] \ge 0\right\} \end{aligned}$$

\((i=1,2,\ \alpha _{1}=\alpha , \alpha _{2}=\beta )\), and an operator as follows:

$$\begin{aligned} S(u, v)=\left( S_{1}(u, v), S_{2}(u, v)\right) , \end{aligned}$$
(2.7)

where

$$\begin{aligned} (S_{1}(u, v))(t)= & {} \int _{0}^{1} G_{1}(t, s) f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&+\int _{0}^{1} H_{1}(t, s) f_{2}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) d s,\\ (S_{2}(u, v))(t)= & {} \int _{0}^{1} G_{2}(t, s) f_{2}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&+\int _{0}^{1} H_{2}(t, s) f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) d s. \end{aligned}$$

By Lemma 2.1 (u, v) is a positive solution of (1.1) if and only if \((u,v)\in K\) is a fixed point of S. Define a linear operator

$$\begin{aligned} L(u, v)=\left( L_{1}(u, v), L_{2}(u, v)\right) , \end{aligned}$$
(2.8)

where

$$\begin{aligned} (L_{1}(u, v))(t)= & {} \int _{0}^{1} G_{1}(t, s)\left[ c_{1} u(s)+c_{2} v(s)+c_{3} u^{\prime }(s)+c_{4} v^{\prime }(s)\right] {\text {d}}s\\&+\int _{0}^{1} H_{1}(t, s)\left[ d_{1} u(s)+d_{2} v(s)+d_{3} u^{\prime }(s)+d_{4} v^{\prime }(s)\right] {\text {d}}s,\\ (L_{2}(u, v))(t)= & {} \int _{0}^{1} H_{2}(t, s)\left[ c_{1} u(s)+c_{2} v(s)+c_{3} u^{\prime }(s)+c_{4} v^{\prime }(s)\right] {\text {d}}s\\&+\int _{0}^{1} G_{2}(t, s)\left[ d_{1} u(s)+d_{2} v(s)+d_{3} u^{\prime }(s)+d_{4} v^{\prime }(s)\right] {\text {d}}s, \end{aligned}$$

where \(a_i,b_i,c_i,d_i\) are non-negative constants \((i=1,2,3,4)\). We write \((u_1,v_1)\preceq (u_2,v_2)\), or \((u_2,v_2)\succeq (u_1,v_1)\), if and only if \((u_2,v_2)-(u_1,v_1)\in P\), to denote the cone ordering induced by P.

Lemma 2.3

If \((C_1)\)-\((C_3)\) hold, then \(S:P\rightarrow K\) and \(L: C^{1}[0,1]\times C^{1}[0,1]\rightarrow C^{1}[0,1]\times C^{1}[0,1]\) are completely continuous operators with \(L(P)\subset K\).

Proof

It is clear that \(S:P\rightarrow P\). By Lemma 2.2 for \((u,v)\in P\) and \(t\in [0,1]\)

$$\begin{aligned} \Vert S_{1}(u, v)\Vert _C= & {} \max _{t\in [0,1]}(S_{1}(u, v))(t)\\\le & {} \int _{0}^{1} \Phi _{1}(s) f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&+\int _{0}^{1} \Phi _3(s) f_{2}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s, \end{aligned}$$

and hence

$$\begin{aligned} (S_{1}(u, v))(t)&\ge t \int _{0}^{1} \Phi _{1}(s) f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&\quad +t\int _{0}^{1} \Phi _3(s) f_{2}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&\ge t\Vert S_{1}(u, v)\Vert _C; \end{aligned}$$

moreover

$$\begin{aligned} \alpha [S_{1}(u, v)]= & {} \int _0^1(S_{1}(u, v))(t){\text {d}}A(t)\\= & {} \int _{0}^{1}\left( \int _{0}^{1}\frac{\alpha [1]+t(1-\alpha [t])}{D} {\mathcal {K}}_{B}(s) f_{1}\left( s, u, v, u', v'\right) d s\right) {\text {d}}A(t)\\&+\int _{0}^{1}\left( \int _{0}^{1}k(t, s) f_{1}\left( s, u, v, u', v'\right) {\text {d}}s\right) {\text {d}}A(t)\\&+\int _{0}^{1}\left( \int _{0}^{1}\frac{(1-\beta [t])+t \beta [1]}{D}{\mathcal {K}}_{A}(s)f_{2}\left( s, u, v, u', v'\right) d s\right) {\text {d}}A(t)\\= & {} \frac{(\alpha [1])^2+(1-\alpha [t])\alpha [t]}{D}\int _{0}^{1} {\mathcal {K}}_{B}(s) f_{1}\left( s, u, v, u', v'\right) {\text {d}}s\\&+\int _{0}^{1}{\mathcal {K}}_{A}(s)f_{1}\left( s, u, v, u', v'\right) {\text {d}}s\\&+\frac{(1-\beta [t])\alpha [1]+\beta [1]\alpha [t]}{D}\int _{0}^{1}{\mathcal {K}}_{A}(s)f_{2}\left( s, u, v, u', v'\right) {\text {d}}s\ge 0, \end{aligned}$$

so \(S_{1}(u, v)\in K_1\). Similarly, we also have \(S_2(u, v)\in K_2\), and thus, \(S:P\rightarrow K\). It is easy to see from \((C_1)\) that S is continuous.

Let F be a bounded set in P, and then, there exists \(M>0\), such that \(\Vert (u,v)\Vert _{C^1}\le M\) for all \((u,v)\in F\). By \((C_{1})\) and Lemma 2.2, we have that \(\forall (u,v)\in F\) and \(t\in [0,1]\)

$$\begin{aligned} (S_{1}(u, v))(t)\le & {} \Bigg (\max _{[0,1]\times [0,M]^4}f_1(s,x,y,z,w)\Bigg )\int _0^1\Phi _1(s){\text {d}}s\\&+\Bigg (\max _{[0,1]\times [0,M]^4}f_2(s,x,y,z,w)\Bigg )\int _0^1\Phi _3(s){\text {d}}s,\\ (S_{1}(u, v))'(t)\le & {} \Bigg (\max _{[0,1]\times [0,M]^4}f_1(s,x,y,z,w)\Bigg )\int _0^1\left( \frac{1-\alpha [t]}{D} {\mathcal {K}}_{B}(s)+1\right) {\text {d}}s\\&+\Bigg (\max _{[0,1]\times [0,M]^4}f_2(s,x,y,z,w)\Bigg )\int _0^1\frac{\beta [1]}{D}{\mathcal {K}}_{A}(s){\text {d}}s, \end{aligned}$$

then \(S_1(F)\) is uniformly bounded in \(C^{1}[0,1]\). Moreover, \(\forall (u,v)\in F\) and \(t_1,t_2\in [0,1]\) with \(t_1<t_2\)

$$\begin{aligned}&|(S_{1}(u, v))(t_1)-(S_{1}(u, v))(t_2)|\\&\quad \le \int _{0}^{1}|G_1(t_1,s)-G_1(t_2,s)|f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&\qquad +\int _{0}^{1}|H_1(t_1,s)-H_1(t_2,s)|f_2\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&\quad \le \Bigg (\max _{[0,1]\times [0,M]^4}f_1(s,x,y,z,w)\Bigg )\int _{0}^{1}|G_1(t_1,s)-G_1(t_2,s)|{\text {d}}s\\&\qquad +\Bigg (\max _{[0,1]\times [0,M]^4}f_2(s,x,y,z,w)\Bigg )\int _{0}^{1}|H_1(t_1,s)-H_1(t_2,s)|{\text {d}}s,\\&\quad |(S_{1}(u, v))'(t_1)-(S_{1}(u, v))'(t_2)|\\&\quad \le \int _0^1\big |G'_1(t_1,s)-G'_1(t_2,s)\big |f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&\qquad +\int _0^1\big |H'_1(t_1,s)-H'_1(t_2,s)\big |f_2\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&\quad = \int _{t_1}^{t_2}f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&\quad \le \Bigg (\max _{[0,1]\times [0,M]^4}f_1(s,x,y,z,w)\Bigg )|t_2-t_1|; \end{aligned}$$

thus, \(S_1(F)\) and \(S'_1(F):=\{w': w'(t)=(S_1(u,v))'(t), (u,v)\in F\}\) are equicontinuous. It follows from Arzelà-Ascoli theorem that \(\overline{S_1(F)}\) is a compact set in \(C^1[0,1]\).

In the same way, \(\overline{S_2(F)}\) is a compact set in \(C^1[0,1]\). Therefore, \(\overline{S(F)}=\overline{S_1(F)}\times \overline{S_2(F)}\) is compact set in \(C^1[0,1]\times C^1[0,1]\), and hence, \(S: P\rightarrow K\) is completely continuous. A similar argument works for the operator L. \(\square \)

To prove the main theorems, we need the following properties of fixed point index, see [8, 11].

Lemma 2.4

Let \(\Omega \) be a bounded open subset of X with \(0\in \Omega \) and K be a cone in X. If \(A:K\cap {\overline{\Omega }}\rightarrow K\) is a completely continuous operator and \(\mu Au\ne u\) for \(u\in K\cap \partial \Omega \) and \(\mu \in [0,1],\) then the fixed point index \(i(A,K\cap \Omega ,K)=1.\)

Lemma 2.5

Let \(\Omega \) be a bounded open subset of X and K be a cone in X. If \(A:K\cap {\overline{\Omega }}\rightarrow K\) is a completely continuous operator and there exists \(v_{0}\in {K{\setminus }\{0\}}\), such that \(u-Au\ne \nu v_{0}\) for \(u\in K\cap \partial \Omega \) and \(\nu \ge 0,\) then the fixed point index \(i(A,K\cap \Omega ,K)=0.\)

3 Main Results

Recall that a cone P in Banach space X is said to be reproducing if \(X=P-P\).

Lemma 3.1

(Krein–Rutman). Let P be a reproducing cone in Banach space X and \(L:X\rightarrow X\) be a completely continuous linear operator with \(L(P)\subset P.\) If the spectral radius \(r(L)>0\), then there exists \(\varphi \in {P{\setminus }\{0\}}\), such that \(L\varphi =r(L)\varphi ,\) where 0 denotes the zero element in X.

Lemma 3.2

([19]). Let P be a cone in Banach space X and \(L:X\rightarrow X\) be a completely continuous linear operator with \(L(P)\subset P.\) If there exist \(v_{0}\in P{\setminus }\{0\}\) and \(\lambda _0>0\), such that \(Lv_0\ge \lambda _0v_0\) in the sense of partial ordering induced by P, then there exist \(u_0\in P{\setminus }\{0\}\) and \(\lambda _1\ge \lambda _0\), such that \(Lu_0=\lambda _1u_0\).

Lemma 3.3

The cone P, which is defined by (2.5), is solid in \(C^1[0,1]\times C^{1}[0,1]\), i.e., the interior point set \(\mathring{P}\ne \emptyset \). Actually, \(\mathring{P}=\mathring{P}_1\times \mathring{P}_2\), where

$$\begin{aligned} \mathring{P}_i=\left\{ w \in C^{1}[0,1] : w(t)>0, w'(t)>0, t\in [0,1]\right\} (i=1,2). \end{aligned}$$

Proof

Let \(\left( w_{1}^{(0)}, w_{2}^{(0)}\right) \in \mathring{P}_1\times \mathring{P}_2\), and then, \(w_i^{(0)}\in \mathring{P}_i (i=1,2)\) and

$$\begin{aligned} r=\min \left\{ \min _{0\le t\le 1}w_{1}^{(0)}(t),\ \min _{0\le t\le 1}w_2^{(0)}(t),\ \min _{0\le t\le 1}\big (w_1^{(0)}(t)\big )',\ \min _{0\le t\le 1}\big (w_2^{(0)}(t)\big )'\right\} \end{aligned}$$

with \(r>0\). If \(\big (w_1, w_2\big )\in X\) and \(\big \Vert \big (w_1, w_2\big )-\big (w_1^{(0)}, w_2^{(0)}\big )\big \Vert _{C^1}<r\), we have

$$\begin{aligned}&\max _{0\le t\le 1}\big |w_1(t)-w_{1}^{(0)}(t)\big |<r,\ \max _{0\le t\le 1}\big |w_2(t)-w_2^{(0)}(t)\big |<r,\\&\max _{0\le t\le 1}\big |w_1'(t)-\big (w_1^{(0)}(t)\big )'\big |<r,\ \max _{0\le t\le 1}\big |w_2'(t)-\big (w_2^{(0)}(t)\big )'\big |<r; \end{aligned}$$

therefore, for \(t\in [0,1]\)

$$\begin{aligned}&w_1(t)>w_{1}^{(0)}(t)-r\ge 0,\ w_2(t)>w_2^{(0)}(t)-r\ge 0,\\&w_1'(t)>\big (w_1^{(0)}(t)\big )'-r\ge 0,\ w_2'(t)>\big (w_2^{(0)}(t)\big )'-r\ge 0. \end{aligned}$$

Therefore, \(\big (w_1, w_2\big )\in P\) which means that \(\left( w_{1}^{(0)}, w_{2}^{(0)}\right) \in \mathring{P}\).

Conversely, Let \(\left( w_{1}^{(0)}, w_{2}^{(0)}\right) \in \mathring{P}\), and then, there exists \(r>0\), such that \(\big (w_1, w_2\big )\in P\) when \(\big (w_1, w_2\big )\in X\) and \(\Vert \big (w_1, w_2\big )-\big (w_1^{(0)}, w_2^{(0)}\big )\Vert _{C^1}\le r\). For \(t\in [0,1]\), take

$$\begin{aligned} v_1(t)=w_{1}^{(0)}(t)-\frac{1}{2}rt-\frac{1}{2}r\ \text {and}\ v_2(t)=w_2^{(0)}(t)-\frac{1}{2}rt-\frac{1}{2}r; \end{aligned}$$

thus, \(\big (v_1, v_2\big )\in X\) and \(\big \Vert \big (v_1, v_2\big )-\big (w_1^{(0)}, w_2^{(0)}\big )\big \Vert _{C^1}=r\). Hence, \(\big (v_1, v_2\big )\in P\) and for \(t\in [0,1]\)

$$\begin{aligned} w_{1}^{(0)}(t)\ge & {} \frac{1}{2}rt+\frac{1}{2}r>0,\ w_2^{(0)}(t)\ge \frac{1}{2}rt+\frac{1}{2}r>0,\\ \big (w_1^{(0)}(t)\big )'\ge & {} \frac{1}{2}r>0,\ \big (w_2^{(0)}(t)\big )'\ge \frac{1}{2}r>0. \end{aligned}$$

Consequently, \(\left( w_{1}^{(0)}, w_{2}^{(0)}\right) \in \mathring{P}_1\times \mathring{P}_2\), i.e., \(\mathring{P}=\mathring{P}_1\times \mathring{P}_2\).\(\square \)

Let \(X=C^1[0,1]\times C^{1}[0,1]\) and for \(r>0\) denote

$$\begin{aligned} \Omega _{r}=\{(u,v)\in X: \Vert (u,v)\Vert _{C^{1}}<r\}. \end{aligned}$$

Theorem 3.4

Under the hypotheses \((C_1)\)-\((C_3)\), suppose that

\((F_{1})\) there exist constants \(a_{1},\ b_{1}>0,\ \delta \ge 0\), such that

$$\begin{aligned} \left\{ \begin{array}{l}{f_{1}\left( t, x_{1}, x_{2}, x_{3}, x_{4}\right) \le a_{1}\left( x_{1}+x_{2}+x_{3}+x_{4}+\delta \right) }, \\ {f_{2}\left( t, x_{1}, x_{2}, x_{3}, x_{4}\right) \le b_{1}\left( x_{1}+x_{2}+x_{3}+x_{4}+\delta \right) }\end{array}\right. \end{aligned}$$
(3.1)

for all \((t,x_{1},x_{2},x_{3},x_{4})\in [0,1]\times {\mathbb {R}}^4_{+}\), where

$$\begin{aligned}&a_{1} \int _{0}^{1}\left[ \Phi _{1}(s)+\Phi _{4}(s)\right] {\text {d}}s+b_{1} \int _{0}^{1}\left[ \Phi _{2}(s)+\Phi _{3}(s)\right] d s+\chi _{1}+\chi _{2}<1, \nonumber \\&\quad \chi _{1}=a_{1} \int _{0}^{1} \frac{\alpha [1]}{D} {\mathcal {K}}_{B}(s) {\text {d}}s+b_{1}\left[ \int _{0}^{1}\frac{1-\beta [t]}{D} {\mathcal {K}}_{A}(s) {\text {d}}s+1\right] ,\nonumber \\&\quad \chi _{2}=a_{1}\left[ \int _{0}^1\frac{1-\alpha [t]}{D} {\mathcal {K}}_{B}(s) {\text {d}}s+1\right] +b_{1} \int _{0}^{1} \frac{\beta [1]}{D} {\mathcal {K}}_{A}(s) {\text {d}}s; \end{aligned}$$
(3.2)

\((F_{2})\) there exist constants \(c_1, d_1>0, c_i, d_i\ge 0 (i=2,3,4)\) and \(r>0\), such that

$$\begin{aligned} \left\{ \begin{array}{l}{f_{1}\left( t, x_{1}, x_{2}, x_{3}, x_{4}\right) \ge c_{1} x_{1}+c_{2} x_{2}+c_{3} x_{3}+c_{4} x_{4}}, \\ {f_{2}\left( t, x_{1}, x_{2}, x_{3}, x_{4}\right) \ge d_{1} x_{1}+d_{2} x_{2}+d_{3} x_{3}+d_{4} x_{4}}\end{array}\right. \end{aligned}$$
(3.3)

for all \((t,x_{1},x_{2},x_{3},x_{4})\in [0,1]\times [0,r]^4\), moreover the spectral radius \(r(L)\ge 1\), where L is defined by (2.8).

Then BVP (1.1) has at least one positive solution.

Proof

Let \(W=\{(u,v)\in K: (u,v)=\mu S(u,v),\ \mu \in [0,1]\}\) where K and S are, respectively, defined in (2.6) and (2.7).

We first assert that W is a bounded set. In fact, if \((u,v)\in W,\) then \((u,v)=\mu S(u,v)\) for some \(\mu \in [0,1].\) From Lemma 2.2, (3.1), and (3.2), we have that

$$\begin{aligned} \Vert u\Vert _C&=\mu \max _{0\le t\le 1}\left( (S_{1}(u, v))(t)\right) \\&= \mu \max _{0\le t\le 1}\left( \int _{0}^{1} G_{1}(t, s) f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\right. \\&\quad \left. +\int _{0}^{1} H_{1}(t, s) f_{2}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\right) \\&\le \int _{0}^{1}\Phi _{1}(s) f_{1}(s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)){\text {d}}s\\&\quad +\int _{0}^{1}\Phi _{3}(s) f_{2}(s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)) {\text {d}}s\\&\le \int _{0}^{1} \Phi _{1}(s) a_{1}\left[ u(s)+v(s)+u^{\prime }(s)+v^{\prime }(s)+\delta \right] {\text {d}}s\\&\quad +\int _{0}^{1} \Phi _{3}(s) b_{1}\left[ u(s)+v(s)+u^{\prime }(s)+v^{\prime }(s)+\delta \right] {\text {d}}s\\&\le a_{1}\left( \Vert u\Vert _{C}+\Vert v\Vert _{C}+\left\| u^{\prime }\right\| _{C}+\left\| v^{\prime }\right\| _{C}+\delta \right) \int _{0}^{1} \Phi _{1}(s) {\text {d}}s\\&\quad +b_{1}\left( \Vert u\Vert _{C}+\Vert v\Vert _{C}+\left\| u^{\prime }\right\| _{C}+\left\| v^{\prime }\right\| _{C}+\delta \right) \int _{0}^{1} \Phi _{3}(s) {\text {d}}s\\&= \left( a_{1}\int _{0}^{1} \Phi _{1}(s) {\text {d}}s+b_{1}\int _{0}^{1} \Phi _{3}(s) {\text {d}}s\right) \\&\quad \times \left( \Vert u\Vert _{C}+\Vert v\Vert _{C}+\left\| u^{\prime }\right\| _{C}+\left\| v^{\prime }\right\| _{C}+\delta \right) ,\\ \Vert u'\Vert _C&=\mu \max _{0\le t\le 1}\left( (S_{1}(u, v))'(t)\right) \\&= \mu \left( \int _{0}^{1} \frac{1-\alpha [t]}{D} {\mathcal {K}}_{B}(s) f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) d s\right. \\&\quad +\int _{t}^{1} f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) {\text {d}}s\\&\quad +\left. \int _{0}^{1} \frac{\beta [1]}{D} {\mathcal {K}}_{A}(s) f_{2}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) d s\right) \\&\le \int _{0}^{1} \frac{1-\alpha [t]}{D} {\mathcal {K}}_{B}(s)a_{1}\left[ u(s)+v(s)+u^{\prime }(s)+v^{\prime }(s)+\delta \right] {\text {d}}s\\&\quad +\int _{t}^{1} a_{1}\left[ u(s)+v(s)+u^{\prime }(s)+v^{\prime }(s)+\delta \right] {\text {d}}s\\&\quad +\int _{0}^{1} \frac{\beta [1]}{D} {\mathcal {K}}_{A}(s) b_{1}\left[ u(s)+v(s)+u^{\prime }(s)+v^{\prime }(s)+\delta \right] d s\\&\le a_{1}\left( \Vert u\Vert _{C}+\Vert v\Vert _{C}+\left\| u^{\prime }\right\| _{C}+\left\| v^{\prime }\right\| _{C}+\delta \right) \left[ \int _{0}^{1} \frac{1-\alpha [t]}{D} {\mathcal {K}}_{B}(s) {\text {d}}s+1\right] \\&\quad +b_{1}\left( \Vert u\Vert _{C}+\Vert v\Vert _{C}+\left\| u^{\prime }\right\| _{C}+\left\| v^{\prime }\right\| _{C}+\delta \right) \int _{0}^{1}\frac{\beta [1]}{D} {\mathcal {K}}_{A}(s) {\text {d}}s\\&\le \chi _2\left( \Vert u\Vert _{C}+\Vert v\Vert _{C}+\left\| u^{\prime }\right\| _{C}+\left\| v^{\prime }\right\| _{C}+\delta \right) ; \end{aligned}$$

we have again

$$\begin{aligned} \Vert v\Vert _C&\le \left( a_{1}\int _{0}^{1} \Phi _{4}(s) d s+b_{1}\int _{0}^{1} \Phi _{2}(s) {\text {d}}s\right) \\&\quad \times \left( \Vert u\Vert _{C}+\Vert v\Vert _{C}+\left\| u^{\prime }\right\| _{C}+\left\| v^{\prime }\right\| _{C}+\delta \right) ,\\ \Vert v'\Vert _C&\le \chi _1\left( \Vert u\Vert _{C}+\Vert v\Vert _{C}+\left\| u^{\prime }\right\| _{C}+\left\| v^{\prime }\right\| _{C}+\delta \right) . \end{aligned}$$

By solving the inequalities system above, it follows that:

$$\begin{aligned} \Vert u\Vert _C\le T_1,\ \Vert u'\Vert _C\le T_2,\ \Vert v\Vert _C\le T_3,\ \Vert v'\Vert _C\le T_4, \end{aligned}$$

where

$$\begin{aligned} T_1= & {} \frac{\left( a_1\int _{0}^{1}\Phi _1(s){\text {d}}s+b_1\int _{0}^{1}\Phi _3(s){\text {d}}s\right) \delta }{1-\chi _{1}-\chi _{2}-a_{1} \int _{0}^{1}\left[ \Phi _{1}(s)+\Phi _{4}(s)\right] {\text {d}}s-b_{1} \int _{0}^{1}\left[ \Phi _{2}(s)+\Phi _{3}(s)\right] {\text {d}}s},\\ T_2= & {} \frac{\chi _{2}\delta }{1-\chi _{1}-\chi _{2}-a_{1} \int _{0}^{1}\left[ \Phi _{1}(s)+\Phi _{4}(s)\right] {\text {d}}s-b_{1} \int _{0}^{1}\left[ \Phi _{2}(s)+\Phi _{3}(s)\right] {\text {d}}s},\\ T_3= & {} \frac{\left( a_1\int _{0}^{1}\Phi _4(s){\text {d}}s+b_1\int _{0}^{1}\Phi _2(s){\text {d}}s\right) \delta }{1-\chi _{1}-\chi _{2}-a_{1} \int _{0}^{1}\left[ \Phi _{1}(s)+\Phi _{4}(s)\right] {\text {d}}s-b_{1} \int _{0}^{1}\left[ \Phi _{2}(s)+\Phi _{3}(s)\right] {\text {d}}s},\\ T_4= & {} \frac{\chi _{1}\delta }{1-\chi _{1}-\chi _{2}-a_{1} \int _{0}^{1}\left[ \Phi _{1}(s)+\Phi _{4}(s)\right] {\text {d}}s-b_{1} \int _{0}^{1}\left[ \Phi _{2}(s)+\Phi _{3}(s)\right] {\text {d}}s}, \end{aligned}$$

and thus, \(\Vert (u,v)\Vert _{C^1}\le \max \{T_1,T_2,T_3,T_4\}\), i.e., W is bounded.

Now, select \(R>\max \{r,\sup W\}\), then \(\mu S(u,v)\ne (u,v)\) for \((u,v)\in K\cap \partial \Omega _{R}\) and \(\mu \in [0,1],\) and \(i(S,K\cap \Omega _{R}, K)=1\) follows from Lemma 2.4.

It follows from Lemma 3.3 that P is a solid cone, and then, P is reproducing (cf. [8, 11, 19]). Since \(L: P\rightarrow K\subset P\) and \(r(L)\ge 1\), we have from Lemma 3.1 that there exists \(\varphi _0=(\varphi _{01},\varphi _{02})\in P{\setminus }\{(0,0)\}\), such that \(L(\varphi _{01},\varphi _{02})=r(L)(\varphi _{01},\varphi _{02}).\) Furthermore, \((\varphi _{01},\varphi _{02})=(r(L))^{-1}L(\varphi _{01},\varphi _{02})\in K\).

We may suppose that S has no fixed points in \(K\cap \partial \Omega _{r}\) and will show that

$$\begin{aligned} (u,v)-S(u,v)\ne \tau (\varphi _{01},\varphi _{02}) \end{aligned}$$

for \((u,v)\in K\cap \partial \Omega _{r}\) and \(\tau \ge 0\).

Otherwise, there exist \((u_0,v_0)\in K\cap \partial \Omega _{r}\) and \(\tau _{0}\ge 0\), such that

$$\begin{aligned} (u_0,v_0)-S(u_0,v_0)=\tau _{0}(\varphi _{01},\varphi _{02}), \end{aligned}$$

and it is clear that \(\tau _{0}>0.\) Since \((u_0,v_0)\in K\cap \partial \Omega _{r},\) we have

$$\begin{aligned} 0\le u_0(t), u_0'(t), v_0(t), v_0'(t)\le r, \forall t\in [0,1]. \end{aligned}$$

It follows from (2.4), (2.1), (2.8), and (3.3) that:

$$\begin{aligned} (S_1(u_0,v_0))(t)\ge (L_1(u_0,v_0))(t) \end{aligned}$$

and

$$\begin{aligned}&(S_{1}(u_0,v_0))'(t)\\&\quad = \int _{0}^{1} \frac{1-\alpha [t]}{D} {\mathcal {K}}_{B}(s) f_{1}\left( s, u_0(s), v_0(s), u_0^{\prime }(s), v_0^{\prime }(s)\right) {\text {d}}s\\&\qquad +\int _{t}^{1} f_{1}\left( s, u_0(s), v_0(s), u_0^{\prime }(s), v_0^{\prime }(s)\right) {\text {d}}s\\&\qquad +\int _{0}^{1} \frac{\beta [1]}{D} {\mathcal {K}}_{A}(s) f_{2}\left( s, u_0(s), v_0(s), u_0^{\prime }(s), v_0^{\prime }(s)\right) {\text {d}}s\\&\quad \ge \int _{0}^{1} \frac{1-\alpha [t]}{D} {\mathcal {K}}_{B}(s)\left[ c_1u_0(s)+c_2v_0(s)+c_3u_0^{\prime }(s)+c_4v_0^{\prime }(s)\right] {\text {d}}s\\&\qquad +\int _{t}^{1} \left[ c_1u_0(s)+c_2v_0(s)+c_3u_0^{\prime }(s)+c_4v_0^{\prime }(s)\right] {\text {d}}s\\&\qquad +\int _{0}^{1} \frac{\beta [1]}{D} {\mathcal {K}}_{A}(s) \left[ d_1u_0(s)+d_2v_0(s)+d_3u_0^{\prime }(s)+d_4v_0^{\prime }(s)\right] {\text {d}}s\\&\quad = (L_1(u_0,v_0))'(t),\ \forall t\in [0,1]; \end{aligned}$$

similarly

$$\begin{aligned} \left( S_2\left( u_0,v_0\right) \right) (t)\ge \left( L_2\left( u_0,v_0\right) \right) (t),\ \left( S_2\left( u_0,v_0\right) \right) '(t)\ge \left( L_2\left( u_0,v_0\right) \right) '(t) \end{aligned}$$

for \(t\in [0,1]\). These imply that

$$\begin{aligned} \left( u_0,v_0\right)&=\tau _{0}\left( \varphi _{01},\varphi _{02}\right) +S\left( u_0,v_0\right) \nonumber \\&\succeq \tau _{0}\left( \varphi _{01},\varphi _{02}\right) + L(u_0,v_0)\succeq \tau _{0}\left( \varphi _{01},\varphi _{02}\right) . \end{aligned}$$
(3.4)

Set \(\tau ^{*}=\sup \{\tau >0: (u_0,v_0)\succeq \tau (\varphi _{01},\varphi _{02})\},\) and then, \(\tau _0\le \tau ^{*}<+\infty \) and \((u_0,v_0)\succeq \tau ^*(\varphi _{01},\varphi _{02})\). Thus, it follows from (3.4) that:

$$\begin{aligned} \left( u_0,v_0\right) \succeq&\tau _{0}(\varphi _{01},\varphi _{02})+ L\left( u_0,v_0\right) \\ \succeq&\tau _{0}(\varphi _{01},\varphi _{02})+\tau ^{*}L\left( \varphi _{01},\varphi _{02}\right) =\tau _{0}\left( \varphi _{01},\varphi _{02}\right) +\tau ^{*}r(L)\left( \varphi _{01},\varphi _{02}\right) . \end{aligned}$$

Since \(r(L)\ge 1,\) \((u_0,v_0)\succeq (\tau _{0}+\tau ^{*})(\varphi _{01},\varphi _{02}),\) which is a contradiction to the definition of \(\tau ^*\). Therefore, \((u,v)-S(u,v)\ne \tau (\varphi _{01},\varphi _{02})\) for \((u,v)\in K\cap \partial \Omega _{r}\) and \(\tau \ge 0.\)

From Lemma 2.5, it follows that \(i(S,K\cap \Omega _{r}, K)=0.\)

Making use of the properties of fixed point index, we have that

$$\begin{aligned} i\left( S,K\cap \left( \Omega _{R}{\setminus }{\overline{\Omega }}_{r}\right) ,K\right) =i\left( S,K\cap \Omega _{R}, K\right) -i\left( S,K\cap \Omega _{r}, K\right) =1, \end{aligned}$$

and hence, S has at least one fixed point in K. Therefore, BVP (1.1) has at least one positive solution. \(\square \)

Theorem 3.5

Under the hypotheses \((C_1)\)–\((C_3)\), suppose that

\((F_{3})\) there exist positive constants \(a_{i},\ b_{i} (i=1,2,3)\) satisfying

$$\begin{aligned} \min \left\{ \begin{array}{l}\min \{a_1,a_2\}\int _0^1s\Phi _1(s){\text {d}}s+\min \{b_1,b_2\}\int _0^1s\Phi _3(s){\text {d}}s,\\ \min \{a_1,a_2\}\int _0^1s\Phi _4(s){\text {d}}s+\min \{b_1,b_2\}\int _0^1s\Phi _2(s){\text {d}}s\end{array}\right\} >1, \end{aligned}$$
(3.5)

such that

$$\begin{aligned} \left\{ \begin{array}{l}f_1(t,x_{1},x_{2},x_{3},x_{4})\ge a_{1}x_{1}+a_2x_2-a_{3},\\ f_2(t,x_{1},x_{2},x_{3},x_{4})\ge b_{1}x_{1}+b_2x_2-b_{3}\end{array}\right. \end{aligned}$$
(3.6)

for all \((t,x_{1},x_{2},x_{3},x_{4})\in [0,1]\times {\mathbb {R}}^4_{+};\)

\((F_{4})\) there exist constants \(c_1, d_1>0,\ c_i, d_i\ge 0 (i=2,3,4)\) and \(r>0\), such that

$$\begin{aligned} \left\{ \begin{array}{l}f_1(t,x_{1},x_{2},x_{3},x_{4})\le c_{1}x_{1}+c_2x_2+c_3x_3+c_4x_4,\\ f_2(t,x_{1},x_{2},x_{3},x_{4})\le d_{1}x_{1}+d_2x_2+d_3x_3+d_4x_4\end{array}\right. \end{aligned}$$
(3.7)

for all \((t,x_{1},x_{2},x_{3},x_{4})\in [0,1]\times [0,r]^{4}\), moreover the spectral radius \(r(L)<1\), where L is defined by (2.8).

If the following condition of Nagumo type is fulfilled, i.e.,

\((F_5)\) for any \(M>0\) there is a positive continuous function \(H_M(\rho )\) on \({\mathbb {R}}_+\) satisfying

$$\begin{aligned} \int _0^{+\infty }\frac{\rho d\rho }{H_M(\rho )+1}=+\infty , \end{aligned}$$
(3.8)

such that for all \((t,x_{1},x_{2},x_{3},x_{4})\in [0,1]\times [0,M]^2\times {\mathbb {R}}_+^2 (i=1,2)\),

$$\begin{aligned} f_i\left( t,x_{1},x_{2},x_{3},x_{4}\right) \le H_M\left( x_{3}+x_{4}\right) , \end{aligned}$$
(3.9)

then BVP (1.1) has at least one positive solution.

Proof

(i) First, we prove that \(\mu S(u,v)\ne (u,v)\) for \((u,v)\in K\cap \partial \Omega _{r}\) and \(\mu \in [0,1].\) In fact, if there exist \((u_1,v_1)\in K\cap \partial \Omega _{r}\) and \(\mu _0\in [0,1]\), such that \((u_1,v_1)=\mu _0S(u_1,v_1)\), then we deduce from (3.7) and \(0\le u_1(t)\le r,\ 0\le u_1'(t)\le r,\ \forall t\in [0,1]\) that

$$\begin{aligned} u_1(t)&=\mu _0(S_1(u_1,v_1))(t)\\&= \mu _0\Big (\int _{0}^{1}G_1(t,s)f_1(s,u_1(s),v_1(s),u_1'(s),v_1'(s)){\text {d}}s\\&\quad +\int _{0}^{1}H_1(t,s)f_2(s,u_1(s),v_1(s),u_1'(s),v_1'(s)){\text {d}}s\Big )\\&\le \int _{0}^{1}G_1(t,s)[c_1u_1(s)+c_2v_1(s)+c_3u'_1(s)+c_4v'_1(s)]{\text {d}}s\\&\quad +\int _{0}^{1}H_1(t,s)[d_1u_1(s)+d_2v_1(s)+d_3u'_1(s)+d_4v'_1(s)]{\text {d}}s\\&= (L_1(u_1,v_1))(t),\\ u'_1(t)&= \mu _0(S_{1}(u_1,v_1))'(t)\\&= \mu _0\left( \int _{0}^{1} \frac{1-\alpha [t]}{D} {\mathcal {K}}_{B}(s) f_{1}\left( s, u_1(s), v_1(s), u_1^{\prime }(s), v_1^{\prime }(s)\right) {\text {d}}s\right. \\&\quad +\int _{t}^{1} f_{1}\left( s, u_1(s), v_1(s), u_1^{\prime }(s), v_1^{\prime }(s)\right) {\text {d}}s\\&\quad \left. +\int _{0}^{1} \frac{\beta [1]}{D} {\mathcal {K}}_{A}(s) f_{2}\left( s, u_1(s), v_1(s), u_1^{\prime }(s), v_1^{\prime }(s)\right) {\text {d}}s\right) \\&\le \int _{0}^{1} \frac{1-\alpha [t]}{D} {\mathcal {K}}_{B}(s)\left[ c_1u_1(s)+c_2v_1(s)+c_3u_1^{\prime }(s)+c_4v_1^{\prime }(s)\right] {\text {d}}s\\&\quad +\int _{t}^{1} \left[ c_1u_1(s)+c_2v_1(s)+c_3u_1^{\prime }(s)+c_4v_1^{\prime }(s)\right] {\text {d}}s\\&\quad +\int _{0}^{1} \frac{\beta [1]}{D} {\mathcal {K}}_{A}(s) \left[ d_1u_1(s)+d_2v_1(s)+d_3u_1^{\prime }(s)+d_4v_1^{\prime }(s)\right] {\text {d}}s\\&= (L_1(u_1,v_1))'(t),\ \forall t\in [0,1]. \end{aligned}$$

Similarly, \(v_1(t)\le (L_2(u_1,v_1))(t)\) and \(v'_1(t)\le (L_2(u_1,v_1))'(t),\ \forall t\in [0,1]\), and thus, \((I-L)(u_1,v_1)\preceq (0,0)\). Because of the spectral radius \(r(L)<1\), we know that \(I-L\) has a bounded inverse operator \((I-L)^{-1}\) which can be written as

$$\begin{aligned} (I-L)^{-1}=I+L+L^{2}+\cdots +L^{n}+\cdots . \end{aligned}$$

Since \(L(P)\subset K\subset P\) by Lemma 2.3, we have \((I-L)^{-1}(P)\subset P\) which implies the inequality \((u_1,v_1)\preceq (I-L)^{-1}(0,0)=(0,0)\) which contradicts \((u_1,v_1)\in K\cap \partial \Omega _{r}\).

Therefore, \(i(S,K\cap \Omega _{r},K)=1\) follows from Lemma 2.4.

(ii) Let

$$\begin{aligned} M_1= & {} \frac{a_3\int _0^1\Phi _1(s){\text {d}}s+b_3\int _0^1\Phi _3(s){\text {d}}s}{a_1\int _0^1s\Phi _1(s){\text {d}}s+b_1\int _0^1s\Phi _3(s){\text {d}}s-1},\\ M_2= & {} \frac{a_3\int _0^1\Phi _4(s){\text {d}}s+b_3\int _0^1\Phi _2(s){\text {d}}s}{a_2\int _0^1s\Phi _4(s){\text {d}}s+b_2\int _0^1s\Phi _2(s){\text {d}}s-1}, \end{aligned}$$

and \(M_3=\max \{M_1,M_2\},\ c=\max \{a_3,b_3\}\). By (3.8), it is easy to see that

$$\begin{aligned} \int _{2M_3}^{+\infty }\frac{\rho d\rho }{H_{4M_3}(\rho )+c}=+\infty ; \end{aligned}$$

hence, there exists \(M_4>4M_3\), such that

$$\begin{aligned} \int _{2M_3}^{M_4}\frac{\rho d\rho }{H_{4M_3}(\rho )+c}>4M_3. \end{aligned}$$
(3.10)

(iii) For \((u,v)\in P\), define

$$\begin{aligned} {\widetilde{S}}(u,v)=({\widetilde{S}}_1(u,v),{\widetilde{S}}_2(u,v)), \end{aligned}$$
(3.11)

where

$$\begin{aligned} ({\widetilde{S}}_{1}(u, v))(t)= & {} \int _{0}^{1} G_{1}(t, s) \left[ f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) +a_3\right] {\text {d}}s\\&+\int _{0}^{1} H_{1}(t, s) \left[ f_{2}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) +b_3\right] {\text {d}}s, \\ ({\widetilde{S}}_{2}(u, v))(t)= & {} \int _{0}^{1} H_{2}(t, s) \left[ f_{1}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) +a_3\right] {\text {d}}s\\&+\int _{0}^{1} G_{2}(t, s) \left[ f_{2}\left( s, u(s), v(s), u^{\prime }(s), v^{\prime }(s)\right) +b_3\right] {\text {d}}s. \end{aligned}$$

Similar to Lemma 2.3, we know that \({\widetilde{S}}: P\rightarrow K\) is completely continuous.

Let \(R>\max \{r, M_4\}\) and we will prove that

$$\begin{aligned} (1-\lambda )S(u,v)+\lambda {\widetilde{S}}(u,v)\ne (u,v),\ \forall (u,v)\in K\cap \partial \Omega _R,\ \lambda \in [0,1]. \end{aligned}$$
(3.12)

If it does not hold, there exist \((u_2,v_2)\in K\cap \partial \Omega _R\) and \(\lambda _0\in [0,1]\), such that

$$\begin{aligned} (1-\lambda _0)S(u_2,v_2)+\lambda _0 {\widetilde{S}}(u_2,v_2)=(u_2,v_2); \end{aligned}$$
(3.13)

thus, by (3.5), (3.6), and Lemma 2.2, we obtain that

$$\begin{aligned} \Vert u_2\Vert _C&=\Vert (1-\lambda _0)S_1(u_2,v_2)+\lambda _0 {\widetilde{S}}_1(u_2,v_2)\Vert _C \\&= (1-\lambda _0)\int _{0}^{1} G_{1}(1, s)f_{1}\left( s, u_2(s), v_2(s), u_2^{\prime }(s), v_2^{\prime }(s)\right) {\text {d}}s\\&\quad +(1-\lambda _0)\int _{0}^{1} H_{1}(1, s)f_{2}\left( s, u_2(s), v_2(s), u_2^{\prime }(s), v_2^{\prime }(s)\right) {\text {d}}s\\&\quad +\lambda _0\int _{0}^{1} G_{1}(1, s)\left[ f_{1}\left( s, u_2(s), v_2(s), u_2^{\prime }(s), v_2^{\prime }(s)\right) +a_3\right] {\text {d}}s\\&\quad +\lambda _0\int _{0}^{1} H_{1}(1, s)\left[ f_{2}\left( s, u_2(s), v_2(s), u_2^{\prime }(s), v_2^{\prime }(s)\right) +b_3\right] {\text {d}}s\\&= \int _{0}^{1} G_{1}(1, s)\left[ f_{1}\left( s, u_2(s), v_2(s), u_2^{\prime }(s), v_2^{\prime }(s)\right) +\lambda _0 a_3\right] {\text {d}}s\\&\quad +\int _{0}^{1} H_{1}(1, s)\left[ f_{2}\left( s, u_2(s), v_2(s), u_2^{\prime }(s), v_2^{\prime }(s)\right) +\lambda _0 b_3\right] {\text {d}}s\\&\ge \int _0^1\Phi _1(s)(a_1u_2(s)+a_2v_2(s)-a_3){\text {d}}s\\&\quad +\int _0^1\Phi _3(s)(b_1u_2(s)+b_2v_2(s)-b_3){\text {d}}s\\&\ge a_1\Vert u_2\Vert _C\int _0^1s\Phi _1(s){\text {d}}s-a_3\int _0^1\Phi _1(s){\text {d}}s\\&\quad +b_1\Vert u_2\Vert _C\int _0^1s\Phi _3(s)-b_3\int _0^1\Phi _3(s){\text {d}}s, \end{aligned}$$

which implies that

$$\begin{aligned} \Vert u_2\Vert _C\le \frac{a_3\int _0^1\Phi _1(s){\text {d}}s+b_3\int _0^1\Phi _3(s){\text {d}}s}{a_1\int _0^1s\Phi _1(s){\text {d}}s+b_1\int _0^1s\Phi _3(s){\text {d}}s-1}=M_1<4M_3. \end{aligned}$$
(3.14)

Similarly, we also have

$$\begin{aligned} \Vert v_2\Vert _C\le \frac{a_3\int _0^1\Phi _4(s){\text {d}}s+b_3\int _0^1\Phi _2(s){\text {d}}s}{a_2\int _0^1s\Phi _4(s){\text {d}}s+b_2\int _0^1s\Phi _2(s){\text {d}}s-1}=M_2<4M_3. \end{aligned}$$
(3.15)

Now, we show that \(u'_2(t)+v'_2(t)\le M_4,\ \forall t\in [0,1]\). Otherwise, there exists \(t_0\in [0,1]\), such that \(u'_2(t_0)+v'_2(t_0)> M_4\). Since

$$\begin{aligned} u'_2(\xi )+v'_2(\xi )&=u_2(1)+v_2(1)-u_2(0)-v_2(0)\\&\le 2M_3<M_4<u'_2(t_0)+v'_2(t_0) \end{aligned}$$

for some \(\xi \in (0,1)\), it follows from intermediate value theorem that there exist \(t_1,t_2\in [0,1]\), such that \(u'_2(t_1)+v'_2(t_1)=M_4\) and \(u'_2(t_2)+v'_2(t_2)=2M_3\). Furthermore, \(u'_2(t)+v'_2(t)\) is decreasing by \(u_2''(t)+v_2''(t)\le 0\) for \(t\in [0,1]\) and it implies that \(t_1<t_2\). We can derive from (3.9), (3.13), (3.14), and (3.15) that

$$\begin{aligned} -u_2''(t)&= (1-\lambda _0)f_{1}\left( t, u_2(t), v_2(t), u_2^{\prime }(t), v_2^{\prime }(t)\right) \\&\quad +\lambda _0(f_{1}\left( t, u_2(t), v_2(t), u_2^{\prime }(t), v_2^{\prime }(t)\right) +a_3)\\&= f_{1}\left( t, u_2(t), v_2(t), u_2^{\prime }(t), v_2^{\prime }(t)\right) +\lambda _0a_3\\&\le f_{1}\left( t, u_2(t), v_2(t), u_2^{\prime }(t), v_2^{\prime }(t)\right) +a_3\\&\le H_{4M_3}(u'_2(t)+v'_2(t))+a_3\\&\le H_{4M_3}(u'_2(t)+v'_2(t))+c \end{aligned}$$

and

$$\begin{aligned} -v_2''(t)\le H_{4M_3}(u'_2(t)+v'_2(t))+b_3\le H_{4M_3}(u'_2(t)+v'_2(t))+c. \end{aligned}$$

Thus

$$\begin{aligned} -\frac{u_2''(t)+v_2''(t)}{H_{4M_3}(u'_2(t)+v'_2(t))+c}\le 2. \end{aligned}$$
(3.16)

Multiplying both sides of the inequality (3.16) by \(u'_2(t)+v'_2(t)\ge 0\), we have that

$$\begin{aligned} -\frac{(u'_2(t)+v'_2(t))(u_2''(t)+v_2''(t))}{H_{4M_3}(u'_2(t)+v'_2(t))+c}\le 2(u'_2(t)+v'_2(t)). \end{aligned}$$
(3.17)

Then, integrating the inequality (3.17) over \([t_1,t_2]\) and making the variable transformation \(\rho =u'_2(t)+v'_2(t)\), we obtain from (3.14) and (3.15) that

$$\begin{aligned} \int _{2M_3}^{M_4}\frac{\rho d\rho }{H_{4M_3}(\rho )+c}&=-\int _{u'_2(t_1)+v'_2(t_1)}^{u'_2(t_2)+v'_2(t_2)}\frac{\rho d\rho }{H_{4M_3}(\rho )+c}\\&\le 2(u_2(t_2)+v_2(t_2)-u_2(t_1)-v_2(t_1))\le 2(\Vert u_2\Vert _C+\Vert v_2\Vert _C)\\&\le 2(M_1+M_2)\le 4M_3, \end{aligned}$$

which is a contradiction to (3.10).

In summary, \(\Vert u'_2\Vert _C\le M_4\) and \(\Vert v'_2\Vert _{C}\le M_4\), and hence, \(\Vert (u_2,v_2)\Vert _{C^1}\le M_4\) by combining with (3.14) and (3.15). Therefore, a contradiction to \(\Vert (u_2,v_2)\Vert _{C^1}=R>M_4\) occurs.

From (3.12), it follows that:

$$\begin{aligned} i(S,K\cap \Omega _R,K)=i({\widetilde{S}},K\cap \Omega _R,K) \end{aligned}$$
(3.18)

by the homotopy invariance property of fixed point index.

(iv) Let \(h(t)=(h_1(t),h_2(t))=(t,t)\) and consider the linear operator on \(C[0,1]\times C[0,1]\) as follows:

$$\begin{aligned} {\widetilde{L}}(u, v)=\left( {\widetilde{L}}_{1}(u, v), {\widetilde{L}}_{2}(u, v)\right) , \end{aligned}$$

where

$$\begin{aligned}&({\widetilde{L}}_{1}(u, v))(t)\\&\quad = \int _{0}^{1} G_{1}(t, s)\left[ a_{1} u(s)+a_{2} v(s)\right] d s+\int _{0}^{1} H_{1}(t, s)\left[ b_{1} u(s)+b_{2} v(s)\right] {\text {d}}s,\\&({\widetilde{L}}_{2}(u, v))(t)\\&\quad = \int _{0}^{1} H_{2}(t, s)\left[ a_{1} u(s)+a_{2} v(s)\right] {\text {d}}s+\int _{0}^{1} G_{2}(t, s)\left[ b_{1} u(s)+b_{2} v(s)\right] {\text {d}}s; \end{aligned}$$

it is easy to see that \({\widetilde{L}}\) is a completely continuous operator with

$$\begin{aligned} {\widetilde{L}}(C^+[0,1]\times C^+[0,1])\subset C^+[0,1]\times C^+[0,1] \end{aligned}$$

and \(h\in C^+[0,1]\times C^+[0,1]{\setminus }\{(0,0)\}\), where

$$\begin{aligned} C^+[0,1]=\{u\in C[0,1]: u(t)\ge 0,\ \forall t\in [0,1]\} \end{aligned}$$

is the positive cone in C[0, 1]. We have from Lemma 2.2 that

$$\begin{aligned}&({\widetilde{L}}_{1}(h_1, h_2))(t)\\&\quad = \int _{0}^{1} G_{1}(t, s)(a_{1}s+a_{2}s){\text {d}}s+\int _{0}^{1} H_{1}(t, s)(b_{1}s+b_{2} s){\text {d}}s\\&\quad \ge \left( \int _{0}^{1} \Phi _{1}(s)(a_{1}s+a_{2}s)d s+\int _{0}^{1} \Phi _{3}(s)(b_{1}s+b_{2} s){\text {d}}s\right) h_1(t)\\&\quad \ge 2\left( \min \{a_1,a_2\}\int _0^1s\Phi _1(s){\text {d}}s+\min \{b_1,b_2\}\int _0^1s\Phi _3(s){\text {d}}s\right) h_1(t) \end{aligned}$$

and

$$\begin{aligned}&({\widetilde{L}}_2(h_1, h_2))(t)\\&\quad \ge 2\left( \min \{a_1,a_2\}\int _0^1s\Phi _4(s){\text {d}}s+\min \{b_1,b_2\}\int _0^1s\Phi _2(s){\text {d}}s\right) h_2(t), \end{aligned}$$

so by (3.5) and Lemma 3.2 in which \(P=C^+[0,1]\times C^+[0,1]\), there exist

$$\begin{aligned} \lambda _1\ge 2\min \left\{ \begin{array}{l} \min \{a_1,a_2\}\int _0^1s\Phi _1(s){\text {d}}s+\min \{b_1,b_2\}\int _0^1s\Phi _3(s){\text {d}}s,\\ \min \{a_1,a_2\}\int _0^1s\Phi _4(s){\text {d}}s+\min \{b_1,b_2\}\int _0^1s\Phi _2(s){\text {d}}s\end{array}\right\} >2 \end{aligned}$$

and \(\varphi _0=(\varphi _{01},\varphi _{02})\in C^+[0,1]\times C^+[0,1]{\setminus }\{(0,0)\}\) such that \(\varphi _0=\lambda _1^{-1}{\widetilde{L}}\varphi _0\). According to the definition of \({\widetilde{L}}\), similar to the proof of Lemma 2.3, we have that \(\varphi _0\in K\).

(v) In this step, we prove that \((u,v)-{\widetilde{S}}(u,v)\ne \tau (\varphi _{01},\varphi _{02})\) for \((u,v)\in K\cap \partial \Omega _{R}\) and \(\tau \ge 0\), and hence

$$\begin{aligned} i({\widetilde{S}},K\cap \Omega _R,K)=0 \end{aligned}$$
(3.19)

holds by Lemma 2.5.

If there exist \((u_0,v_0)\in K\cap \partial \Omega _{R}\) and \(\tau _{0}\ge 0\), such that \((u_0,v_0)-{\widetilde{S}}(u_0,v_0)=\tau _{0}(\varphi _{01},\varphi _{02})\). Obviously, \(\tau _{0}>0\) by (3.12) and

$$\begin{aligned} \left\{ \begin{array}{l}u_0(t)=({\widetilde{S}}_1(u_0,v_0))(t)+\tau _{0}\varphi _{01}(t)\ge \tau _{0}\varphi _{01}(t),\\ v_0(t)=({\widetilde{S}}_2(u_0,v_0))(t)+\tau _{0}\varphi _{02}(t)\ge \tau _{0}\varphi _{02}(t)\end{array}\right. \end{aligned}$$
(3.20)

for \(t\in [0,1]\). Set

$$\begin{aligned} \tau ^{*}=\sup \{\tau >0: u_0(t)\ge \tau \varphi _{01}(t),\ v_0(t)\ge \tau \varphi _{02}(t),\ \forall t\in [0,1]\}, \end{aligned}$$

then \(\tau _0\le \tau ^*<+\infty \), \(u_0(t)\ge \tau ^{*}\varphi _{01}(t)\) and \(v_0(t)\ge \tau ^{*}\varphi _{02}(t)\) for \(t\in [0,1]\). From (3.6) and (3.20), we have that for \(t\in [0,1]\)

$$\begin{aligned} u_0(t)&=({\widetilde{S}}_1(u_0,v_0))(t)+\tau _{0}\varphi _{01}(t)\ge ({\widetilde{L}}_1(u_0,v_0))(t)+\tau _{0}\varphi _{01}(t)\\&\ge \tau ^{*}({\widetilde{L}}_1(\varphi _{01},\varphi _{02}))(t)+\tau _{0}\varphi _{01}(t) =\lambda _1\tau ^{*}\varphi _{01}(t)+\tau _{0}\varphi _{01}(t)\\&= (\lambda _1\tau ^{*}+\tau _{0})\varphi _{01}(t) \end{aligned}$$

and

$$\begin{aligned} v_0(t)&=({\widetilde{S}}_2(u_0,v_0))(t)+\tau _{0}\varphi _{02}(t)\ge ({\widetilde{L}}_2(u_0,v_0))(t)+\tau _{0}\varphi _{02}(t)\\&\ge \tau ^{*}({\widetilde{L}}_2(\varphi _{01},\varphi _{02}))(t)+\tau _{0}\varphi _{02}(t) =\lambda _1\tau ^{*}\varphi _{02}(t)+\tau _{0}\varphi _{02}(t)\\&= (\lambda _1\tau ^{*}+\tau _{0})\varphi _{02}(t). \end{aligned}$$

Since \(\lambda _1>2\), we have \(\lambda _1\tau ^{*}+\tau _{0}>\tau ^*\) which contradicts the definition of \(\tau ^{*}.\)

(vi) From (3.18) and (3.19), it follows that \(i(S,K\cap \Omega _{R},K)=0\) and:

$$\begin{aligned} i(S,K\cap (\Omega _{R}{\setminus }{\overline{\Omega }}_{r}),K)=i(S,K\cap \Omega _{R},K)-i(S,K\cap \Omega _{r},K)=-1. \end{aligned}$$

Hence, S has at least one fixed solution and BVP (1.1) has at least one positive solution. \(\square \)

4 Examples

We consider second-order coupled system under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel

$$\begin{aligned} \left\{ \begin{array}{l}{-u^{\prime \prime }(t)=f_{1}\left( t, u(t), v(t), u^{\prime }(t), v^{\prime }(t)\right) , t \in [0,1]}, \\ {-v^{\prime \prime }(t)=f_{2}\left( t, u(t), v(t), u^{\prime }(t), v^{\prime }(t)\right) , t \in [0,1]}, \\ {u(0)=-\int _0^1v(t)\cos (\pi t){\text {d}}t, \quad u^{\prime }(1)=\frac{1}{2}u\left( \frac{1}{3}\right) -\frac{1}{10}u\left( \frac{2}{3}\right) }, \\ {v(0)=\frac{1}{2}u\left( \frac{1}{3}\right) -\frac{1}{10}u\left( \frac{2}{3}\right) , \quad v^{\prime }(1)=-\int _0^1v(t)\cos (\pi t){\text {d}}t},\end{array}\right. \end{aligned}$$
(4.1)

that is, \(\alpha [v]=-\int _0^1v(t)\cos (\pi t){\text {d}}t,\ \beta [u]=\frac{1}{2}u\left( \frac{1}{3}\right) -\frac{1}{10}u\left( \frac{2}{3}\right) \). For \(s\in [0,1]\)

$$\begin{aligned} {\mathcal {K}}_A(s)= & {} -\int _{0}^{1}k(t,s)\cos (\pi t){\text {d}}t =\frac{1-\cos (\pi s)}{\pi ^{2}},\\ {\mathcal {K}}_B(s)= & {} \frac{1}{2}k\Bigg (\frac{1}{3},s\Bigg )-\frac{1}{10}k\Bigg (\frac{2}{3},s\Bigg ) = \left\{ \begin{array}{ll}\displaystyle \frac{2 s}{5}, &{} \displaystyle 0\le s\le \frac{1}{3},\\ \displaystyle \frac{5-3s}{30}, &{} \displaystyle \frac{1}{3}<s\le \frac{2}{3},\\ \displaystyle \frac{1}{10}, &{} \displaystyle \frac{2}{3}<s\le 1,\end{array}\right. \end{aligned}$$

Since

$$\begin{aligned} 0\le & {} {\mathcal {K}}_A(s)\le \frac{2}{\pi ^2},\ 0\le {\mathcal {K}}_B(s)\le \frac{2}{15},\\ \alpha [1]= & {} 0,\ \beta [1]=\frac{2}{5},\ \alpha [t]=\frac{2}{\pi ^2},\ \beta [t]=\frac{1}{10} \end{aligned}$$

and

$$\begin{aligned} D=\left( 1-\alpha [t]\right) \left( 1-\beta [t]\right) -\alpha [1] \beta [1]=\frac{9\pi ^{2}-18}{10\pi ^{2}}>0, \end{aligned}$$

\((C_2)\) and \((C_3)\) are satisfied. Furthermore

$$\begin{aligned} G_{1}(t, s)= & {} \displaystyle \frac{10t}{9}{\mathcal {K}}_{B}(s)+k(t, s)\le \frac{31}{27},\\ H_{1}(t,s)= & {} \displaystyle \left( \frac{\pi ^2}{\pi ^{2}-2}+\frac{4\pi ^2}{9\pi ^{2}-18}t\right) {\mathcal {K}}_{A}(s)\le \displaystyle \frac{26}{9\pi ^{2}-18},\\ G_{2}(t, s)= & {} \displaystyle \left( \frac{4\pi ^2}{9\pi ^{2}-18}+\frac{\pi ^2}{\pi ^{2}-2}t\right) {\mathcal {K}}_{A}(s)+k(t, s)\le \frac{9\pi ^{2}+8}{9\pi ^{2}-18},\\ H_{2}(t, s)= & {} \displaystyle \frac{10}{9} {\mathcal {K}}_{B}(s)\le \displaystyle \frac{4}{27},\\ \Phi _1(s)= & {} \frac{10}{9}{\mathcal {K}}_{B}(s)+s,\ \ \Phi _2(s)=\frac{13\pi ^2}{9\pi ^{2}-18}{\mathcal {K}}_{A}(s)+s, \\ \Phi _3(s)= & {} \frac{13\pi ^2}{9\pi ^{2}-18}{\mathcal {K}}_{A}(s),\ \ \Phi _4(s)=\frac{10}{9}{\mathcal {K}}_{B}(s). \end{aligned}$$

Example 4.1

If

$$\begin{aligned} f_1\left( t,x_1,x_2,x_3,x_4\right)= & {} \frac{1}{3}(2+t)\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4},\\ f_2\left( t,x_1,x_2,x_3,x_4\right)= & {} \root 3 \of {x_1}+\root 3 \of {x_2}+\frac{1}{2}(1+t)\root 3 \of {x_3}+\root 3 \of {x_4},\\ a_{1}= & {} \frac{10}{49},\ b_{1}=\frac{2\pi ^2}{9\pi ^2+8} \end{aligned}$$

and thus

$$\begin{aligned}&a_{1} \displaystyle \int _{0}^{1}\left[ \Phi _{1}(s)+\Phi _{4}(s)\right] d s+b_{1} \int _{0}^{1}\left[ \Phi _{2}(s)+\Phi _{3}(s)\right] d s+\chi _{1}+\chi _{2}\\&\quad = \displaystyle \frac{10}{3}a_1\int _0^1{\mathcal {K}}_{B}(s){\text {d}}s+\frac{13\pi ^2}{3\pi ^2-6}b_1\int _0^1{\mathcal {K}}_{A}(s){\text {d}}s+\frac{3}{2}a_1+\frac{3}{2}b_1\\&\quad = \displaystyle \frac{49}{27}\times \frac{10}{49}+\frac{9\pi ^2+8}{6\pi ^2-12}\times \frac{2\pi ^2}{9\pi ^2+8}<1. \end{aligned}$$

Therefore, \((F_1)\) holds for \(\delta \) large enough. In addition, take

$$\begin{aligned} c _1=3,\ c_2=18,\ c_3=c_4=1,\ d_i=1 (i=1,2,3,4),\ r=\frac{1}{324}. \end{aligned}$$

From Lemmas 2.2 and 2.3, we have that for \((u,v)\in K{\setminus }\{(0,0)\}\) and \(t\in [0,1]\)

$$\begin{aligned} (L_1(u,v))(t)\ge & {} c_1t\int _0^{1}\Phi _1(s)u(s){\text {d}}s\ge c_1t\Vert {u}\Vert _{C}\int _0^{1}s\Phi _1(s){\text {d}}s,\\ \Vert L_1(u,v)\Vert _C= & {} (L_1(u,v))(1)\ge c_1\Vert u\Vert _{C}\int _0^1s\Phi _1(s){\text {d}}s,\\ (L_2(u,v))(t)\ge & {} c_2t\int _0^{1}\Phi _4(s)v(s){\text {d}}s\ge c_2t\Vert v\Vert _{C}\int _0^{1}s\Phi _4(s){\text {d}}s,\\ \Vert L_2(u,v)\Vert _C= & {} (L_2(u,v))(1)\ge c_2\Vert v\Vert _{C}\int _0^1s\Phi _4(s){\text {d}}s,\\ \Vert L(u,v)\Vert _C\ge & {} \max \left\{ c_1\Vert u\Vert _{C}\int _0^1s\Phi _1(s){\text {d}}s,\ c_2\Vert v\Vert _{C}\int _0^1s\Phi _4(s){\text {d}}s\right\} \\\ge & {} \min \left\{ c_1\int _0^1s\Phi _1(s){\text {d}}s,\ c_2\int _0^1s\Phi _4(s){\text {d}}s\right\} \Vert (u,v)\Vert _C, \end{aligned}$$

and

$$\begin{aligned} \left( L_1(L_1(u,v),L_2(u,v))\right) (t)&\ge c_1t\int _0^{1}\Phi _1(s)(L_1(u,v))(s){\text {d}}s\\&\ge c_1t\int _0^{1}s\Phi _1(s)\Vert L_1(u,v)\Vert _C{\text {d}}s\ge c_1^2t\Vert {u}\Vert _{C}\Big (\int _0^1s\Phi _1(s){\text {d}}s\Big )^2, \\ \Vert L_1(L_1(u,v),L_2(u,v))\Vert _C&= (L_1(L_1(u,v),L_2(u,v)))(1)\ge c_1^2\Vert {u}\Vert _{C}\Big (\int _0^1s\Phi _1(s){\text {d}}s\Big )^2,\\ (L_2(L_1(u,v),L_2(u,v)))(t)&\ge c_2t\int _0^{1}\Phi _4(s)(L_2(u,v))(s){\text {d}}s\\&\ge c_2t\int _0^{1}s\Phi _4(s)\Vert L_2(u,v)\Vert _C{\text {d}}s\ge c_2^2t\Vert v\Vert _{C}\Big (\int _0^1s\Phi _4(s){\text {d}}s\Big )^2,\\ \Vert L_2(L_1(u,v),L_2(u,v))\Vert _C&= (L_2(L_1(u,v),L_2(u,v)))(1)\ge c_2^2\Vert v\Vert _{C}\Big (\int _0^1s\Phi _4(s){\text {d}}s\Big )^2,\\ \Vert L^2(u,v)\Vert _C&\ge \max \left\{ c_1^2\Vert {u}\Vert _{C}\Big (\int _0^1s\Phi _1(s){\text {d}}s\Big )^2,\ c_2^2\Vert v\Vert _{C}\Big (\int _0^1s\Phi _4(s){\text {d}}s\Big )^2\right\} \\&\ge \min \left\{ c_1^2\Big (\int _0^1s\Phi _1(s){\text {d}}s\Big )^2,\ c_2^2\Big (\int _0^1s\Phi _4(s){\text {d}}s\Big )^2\right\} \Vert (u,v)\Vert _C, \end{aligned}$$

By induction

$$\begin{aligned}&\Vert L^n(u,v)\Vert _C\\&\quad \ge \min \left\{ c_1^n\Big (\int _0^1s\Phi _1(s){\text {d}}s\Big )^n,\ c_2^n\Big (\int _0^1s\Phi _4(s){\text {d}}s\Big )^n\right\} \Vert (u,v)\Vert _C. \end{aligned}$$

Consequently, for \((u,v)\in K{\setminus }\{(0,0)\}\)

$$\begin{aligned}&\Vert {L^{n}}\Vert \Vert (u,v)\Vert _{C^{1}}\ge \Vert {L^{n}}(u,v)\Vert _{C^1}\ge \Vert {L^{n}}(u,v)\Vert _{C}\\&\quad \ge \min \left\{ c_1^n\Big (\int _0^1s\Phi _1(s){\text {d}}s\Big )^n,\ c_2^n\Big (\int _0^1s\Phi _4(s){\text {d}}s\Big )^n\right\} \Vert (u,v)\Vert _C, \end{aligned}$$

and by virtue of Gelfand’s formula, the spectral radius

$$\begin{aligned} r(L)=&\lim _{n\rightarrow \infty }\Vert {L^{n}}\Vert ^{1/n}\\ \ge&\min \left\{ c_1\int _0^1s\Phi _1(s){\text {d}}s,\ c_2\int _0^1s\Phi _4(s){\text {d}}s\right\} \lim _{n\rightarrow \infty }\Big (\frac{\Vert (u,v)\Vert _C}{\Vert (u,v)\Vert _{C^1}}\Big )^{1/n}\\ =\,&\min \left\{ c_1\int _0^1s\Phi _1(s){\text {d}}s,\ c_2\int _0^1s\Phi _4(s){\text {d}}s\right\} \\ =\,&\min \left\{ \frac{95}{243}\times 3,\ \frac{14}{243}\times 18\right\} >1. \end{aligned}$$

Therefore, \((F_{2})\) holds, since (3.3) can be inferred easily. By Theorem 3.4, we know that the second-order coupled system (4.1) has at least one positive solution.

Example 4.2

If

$$\begin{aligned} f_1(t,x_{1},x_{2},x_3,x_4)= & {} \frac{(1+t){x_{1}}^{4}+4{x_{2}^{4}}+{x_3}^{4}+3{x_4^{4}}}{31(1+x_{1}^{2}+x_{2}^{2}+x_3^{2}+x_4^{2})},\\ f_2(t,x_{1},x_{2},x_3,x_4)= & {} \frac{\pi ^2({x_{1}}^{4}+{x_{2}^{4}}+(1+t)({x_3}^{4}+{x_4^{4}}))}{(18\pi ^2+16)(1+x_{1}^{2}+x_{2}^{2}+x_3^{2}+x_4^{2})}, \end{aligned}$$

take

$$\begin{aligned} a_{1}=18,\ a_2=19,\ b_1=\frac{18\pi ^4}{13\pi ^2+52},\ b_2=\frac{18\pi ^4+1}{13\pi ^2+52}, \end{aligned}$$

and thus

$$\begin{aligned}&\min \{a_1,a_2\}\int _0^1s\Phi _1(s){\text {d}}s+\min \{b_1,b_2\}\int _0^1s\Phi _3(s){\text {d}}s\\&\quad = 18\times \frac{95}{243}+\frac{18\pi ^4}{13\pi ^2+52}\times \frac{13\pi ^2+52}{18\pi ^4-36\pi ^2}>1,\\&\min \{a_1,a_2\}\int _0^1s\Phi _4(s){\text {d}}s+\min \{b_1,b_2\}\int _0^1s\Phi _2(s){\text {d}}s\\&\quad = 18\times \frac{14}{243}+\frac{18\pi ^4}{13\pi ^2+52}\times \frac{6\pi ^4+\pi ^2+52}{18\pi ^4-36\pi ^2}>1. \end{aligned}$$

Therefore, \((F_3)\) holds, since (3.6) is satisfied for \(a_3\) and \(b_3\) large enough. In addition, take

$$\begin{aligned} c_1= & {} \frac{2}{31},\ c_2=\frac{4}{31},\ c_3=\frac{1}{31},\ c_4=\frac{3}{31}, \\ d_1= & {} d_2=\frac{\pi ^2}{18\pi ^2+16},\ d_3=d_4=\frac{\pi ^2}{9\pi ^2+8}, \end{aligned}$$

and thus, for \((u,v)\in C^1[0,1]\times C^1[0,1]\)

$$\begin{aligned}&(L_1(u,v))(t)\\&\quad \le \frac{31}{27}\int _0^{1}\left[ c_{1} u(s)+c_{2} v(s)+c_{3} u^{\prime }(s)+c_{4} v^{\prime }(s)\right] {\text {d}}s\\&\qquad +\frac{26}{9\pi ^{2}-18}\int _{0}^{1} \left[ d_{1} u(s)+d_{2} v(s)+d_{3} u^{\prime }(s)+d_{4} v^{\prime }(s)\right] {\text {d}}s\\&\quad \le \left( \frac{31}{27}(c_{1}+c_{2}+c_{3}+c_{4})+\frac{26}{9\pi ^{2}-18}(d_{1}+d_{2}+d_{3}+d_{4})\right) \Vert (u,v)\Vert _{C^1}\\&\quad = \left( \frac{10}{27}+\frac{26}{9\pi ^{2}-18}\times \frac{3\pi ^2}{9\pi ^{2}+8}\right) \Vert (u,v)\Vert _{C^1}, \\&(L_1(u,v))'(t)\\&\quad \le \frac{4}{27}\int _0^{1}\left[ c_{1} u(s)+c_{2} v(s)+c_{3} u^{\prime }(s)+c_{4} v^{\prime }(s)\right] {\text {d}}s\\&\qquad +\int _0^{1}\left[ c_{1} u(s)+c_{2} v(s)+c_{3} u^{\prime }(s)+c_{4} v^{\prime }(s)\right] {\text {d}}s\\&\qquad +\frac{8}{9\pi ^{2}-18}\int _{0}^{1} \left[ d_{1} u(s)+d_{2} v(s)+d_{3} u^{\prime }(s)+d_{4} v^{\prime }(s)\right] {\text {d}}s\\&\quad \le \left( \frac{31}{27}(c_{1}+c_{2}+c_{3}+c_{4})+\frac{8}{9\pi ^{2}-18}(d_{1}+d_{2}+d_{3}+d_{4})\right) \Vert (u,v)\Vert _{C^1}\\&\quad = \left( \frac{10}{27}+\frac{8}{9\pi ^{2}-18}\times \frac{3\pi ^2}{9\pi ^{2}+8}\right) \Vert (u,v)\Vert _{C^1}; \end{aligned}$$

hence

$$\begin{aligned} \Vert L_1(u,v)\Vert _{C^1}\le \left( \frac{10}{27}+\frac{26}{9\pi ^{2}-18}\times \frac{3\pi ^2}{9\pi ^{2}+8}\right) \Vert (u,v)\Vert _{C^1}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \Vert L_2(u,v)\Vert _{C^1}\le \left( \frac{40}{837}+\frac{3\pi ^2}{9\pi ^{2}-18}\right) \Vert (u,v)\Vert _{C^1}, \end{aligned}$$

and thus

$$\begin{aligned} \Vert L(u,v)\Vert _{C^1}\le \left( \frac{10}{27}+\frac{26}{9\pi ^{2}-18}\times \frac{3\pi ^2}{9\pi ^{2}+8}\right) \Vert (u,v)\Vert _{C^1}. \end{aligned}$$

Therefore, the spectral radius

$$\begin{aligned} r(L)\le \Vert L\Vert \le \frac{10}{27}+\frac{26}{9\pi ^{2}-18}\times \frac{3\pi ^2}{9\pi ^{2}+8}<1. \end{aligned}$$

It is easy to see that (3.7) can be inferred easily for \(r<1\). As for \((F_3)\), one can let \(H_{M}(\rho )=M^2+\rho ^2\). By Theorem 3.5, we know that the second-order coupled system (4.1) has at least one positive solution.