1 Introduction

In the remarkable paper [1], Wang proved the existence of one positive solution of the following one-dimensional p-Laplacian equation:

$$ \bigl(\varphi_{p}\bigl(u^{\prime}\bigr) \bigr)'(t)+g(t)f\bigl(u(t)\bigr)=0, \quad t \in(0,1), $$
(1.1)

subject to one of the following three pairs of nonlinear boundary conditions (BCs)

$$\begin{aligned}& u'(0)=0,\quad\quad u(1)+B_{1} \bigl(u'(1) \bigr)=0, \\& u(0)=B_{2} \bigl(u'(0) \bigr), \quad\quad u'(1)=0, \\& u(0)=B_{2} \bigl(u'(0) \bigr),\quad\quad u(1)+B_{1} \bigl(u'(1) \bigr)=0, \end{aligned}$$

where \(B_{1},B_{2}:\mathbb{R}\to\mathbb{R}\) are continuous functions satisfying some suitable growth conditions. The results of [1] were extended by Karakostas [2] to the context of deviated arguments. In both cases, the existence results are obtained via a careful study of an associated integral operator combined with the use of the Krasnosel’skiĭ-Guo theorem on cone compressions and cone expansions.

The Krasnosel’skiĭ-Guo theorem and other topological methods are commonly used tools in the study of existence of positive solutions for the p-Laplacian equation (1.1) subject to different BCs. This is an active area of research, for example, homogeneous Dirichlet BCs were studied in [3–10], homogeneous Robin BCs in [7, 9, 10], nonlocal BCs of Dirichlet type in [1, 11–19] and nonlocal BCs of Robin type in [16, 19–23].

Here we study the one-dimensional \((p_{1},p_{2})\)-Laplacian system

$$ \begin{aligned} &\bigl(\varphi_{p_{1}} \bigl(u^{\prime}\bigr)\bigr)'(t)+g_{1}(t)f_{1} \bigl(t, u(t),v(t)\bigr)=0,\quad t \in (0,1), \\ &\bigl(\varphi_{p_{2}}\bigl(v^{\prime}\bigr)\bigr)'(t)+g_{2}(t)f_{2} \bigl(t, u(t),v(t)\bigr)=0,\quad t \in (0,1), \end{aligned} $$
(1.2)

with \(\varphi_{p_{i}}(w)=\vert w\vert^{p_{i}-2}w\), subject to the nonlinear boundary conditions

$$ u'(0)= 0,\quad\quad u(1)+B_{1} \bigl(u'(1) \bigr)=0,\quad\quad v(0)=B_{2} \bigl(v'(0) \bigr),\quad\quad v(1)=0. $$
(1.3)

The existence of positive solutions for systems of equations of the type (1.2) has been widely studied; see, for example, [24–27] under homogeneous Dirichlet BCs and [5, 28–31] with homogeneous Robin or Neumann BCs. For earlier contributions on problems with nonlinear BCs, we refer to [1, 2, 20, 32–39] and the references therein.

We improve and complement the previous results in several directions: we obtain multiplicity results for the \((p_{1}, p_{2})\)-Laplacian system subject to nonlinear BCs, we allow different growths in the nonlinearities \(f_{1}\) and \(f_{2}\), and also we discuss non-existence results. Finally we illustrate in an example that all the constants that occur in our results can be computed.

Our approach is to seek solutions of system (1.2)-(1.3) as fixed points of a suitable integral operator. We make use of the classical fixed point index theory and benefit from ideas of the papers [1, 2, 37, 40].

2 The system of integral equations

We recall that a cone K in a Banach space X is a closed convex set such that \(\lambda x\in K\) for \(x \in K\) and \(\lambda\geq0\) and \(K\cap(-K)=\{0\}\).

If Ω is an open bounded subset of a cone K (in the relative topology), we denote by Ω̅ and ∂Ω the closure and the boundary relative to K. When Ω is an open bounded subset of X, we write \(\Omega_{K}=\Omega\cap K\), an open subset of K.

The following lemma summarizes some classical results regarding the fixed point index; for more details, see [41, 42].

Lemma 2.1

Let Ω be an open bounded set with \(0\in\Omega_{K}\) and \(\overline{\Omega}_{K}\ne K\). Assume that \(F:\overline{\Omega}_{K}\to K\) is a compact map such that \(x\neq Fx\) for all \(x\in\partial\Omega_{K}\). Then the fixed point index \(i_{K}(F, \Omega_{K})\) has the following properties.

  1. (1)

    If there exists \(e\in K\setminus\{0\}\) such that \(x\neq Fx+\lambda e\) for all \(x\in\partial\Omega_{K}\) and all \(\lambda >0\), then \(i_{K}(F, \Omega_{K})=0\).

  2. (2)

    If \(\mu x \neq Fx\) for all \(x\in\partial\Omega_{K}\) and for every \(\mu\geq1\), then \(i_{K}(F, \Omega_{K})=1\).

  3. (3)

    If \(i_{K}(F,\Omega_{K})\ne0\), then F has a fixed point in \(\Omega_{K}\).

  4. (4)

    Let \(\Omega^{1}\) be open in X with \(\overline{\Omega ^{1}}\subset\Omega_{K}\). If \(i_{K}(F, \Omega_{K})=1\) and \(i_{K}(F, \Omega_{K}^{1})=0\), then F has a fixed point in \(\Omega_{K}\setminus\overline{\Omega_{K}^{1}}\). The same result holds if \(i_{K}(F, \Omega_{K})=0\) and \(i_{K}(F, \Omega_{K}^{1})=1\).

To system (1.2)-(1.3) we associate the following system of integral equations, which is constructed in a similar manner as in [1] where the case of a single equation is studied:

$$ \begin{aligned} &u(t)=\int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\hphantom{u(t)=}{} +B_{1} \biggl(\varphi_{p_{1}}^{-1} \biggl(\int _{0}^{1}g_{1}(\tau) f_{1} \bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr),\quad 0\le t\le1, \\ &v(t)= \textstyle\begin{cases} \int_{0}^{t} \varphi_{p_{2}}^{-1} (\int_{s}^{\sigma_{u,v}}g_{2}(\tau)f_{2}(\tau,u(\tau ),v(\tau))\,d\tau )\,ds\\ \quad{}+B_{2} (\varphi_{p_{2}}^{-1} (\int_{0}^{\sigma_{u,v}}g_{2}(\tau) f_{2}(\tau,u(\tau),v(\tau))\,d\tau ) ),& 0\le t\le\sigma_{u,v},\\ \int_{t}^{1}\varphi_{p_{2}}^{-1} (\int_{\sigma_{u,v}}^{s}g_{2}(\tau )f_{2}(\tau,u(\tau),v(\tau))\,d\tau )\,ds,& \sigma_{u,v}\leq t\leq1, \end{cases}\displaystyle \end{aligned} $$
(2.1)

where \(\varphi_{p_{i}}^{-1}(w)=\vert w\vert^{\frac{1}{p_{i}-1}}\) sgn w and \(\sigma_{u,v}\) is the smallest solution \(x\in[0,1]\) of the equation

$$\begin{aligned}& \int_{0}^{x}\varphi_{p_{2}}^{-1} \biggl(\int_{s}^{x}g_{2}(\tau) f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds +B_{2} \biggl(\varphi_{p_{2}}^{-1} \biggl(\int_{0}^{x}g_{2}(\tau) f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) \\& \quad= \int_{x}^{1} \varphi_{p_{2}}^{-1} \biggl(\int_{x}^{s} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds. \end{aligned}$$

By a solution of (1.2)-(1.3), we mean a solution of system (2.1).

In order to utilize the fixed point index theory, we state the following assumptions on the terms that occur in system (2.1):

  1. (C1)

    For every \(i=1,2\), \(f_{i}: [0,1]\times[0,\infty)\times [0,\infty) \to[0,\infty)\) satisfies Carathéodory conditions, that is, \(f_{i}(\cdot,u,v)\) is measurable for each fixed \((u,v)\) and \(f_{i}(t,\cdot,\cdot)\) is continuous for almost every (a.e.) \(t\in [0,1]\), and for each \(r>0\) there exists \(\phi_{i,r} \in L^{\infty }[0,1]\) such that

    $$ f_{i}(t,u,v)\le\phi_{i,r}(t) \quad\text{for } u,v\in[0,r] \text{ and a.e. } t\in[0,1]. $$
  2. (C2)

    \(g_{1}\in L^{1}[0,1]\), \(g_{1}\geq0\) and

    $$ 0< \int_{0}^{1}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)\,d\tau \biggr)\,ds < +\infty. $$
  3. (C3)

    \(g_{2}\in L^{1}[0,1]\), \(g_{2}\geq0\) and

    $$ 0< \int_{0}^{1/2} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{1/2}g_{2}( \tau)\,d\tau \biggr)\,ds +\int_{1/2}^{1} \varphi_{p_{2}}^{-1} \biggl(\int_{1/2}^{s} g_{2}(\tau)\,d\tau \biggr)\,ds< +\infty. $$
    (2.2)
  4. (C4)

    For every \(i=1,2\), \(B_{i}: \mathbb{R}\to\mathbb{R}\) is a continuous function, and there exist \(h_{i1}, h_{i2}\geq0\) such that

    $$ h_{i1} v\le B_{i}(v)\le h_{i2} v\quad \mbox{for any }v\geq0. $$

Remark 2.2

Condition (2.2) is weaker than the condition

$$ 0< \int_{0}^{1} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{1} g_{2}(\tau)\,d\tau \biggr)\,ds< +\infty. $$
(2.3)

In fact, for example, the function

$$g_{2}(t)= \textstyle\begin{cases} \frac{1}{(t-1)^{2}}, & t\in[0,1/2],\\\frac{1}{t^{2}},& t\in(1/2,1], \end{cases} $$

satisfies (2.2) but not (2.3).

Remark 2.3

From (C2) and (C3) it follows that there exists \([a_{1},b_{1}]\subset[0,1)\) such that \(\int_{a_{1}}^{b_{1}} g_{1}(s)\,ds >0\) and there exists \([a_{2},b_{2}]\subset(0,1)\) such that \(\int_{a_{2}}^{b_{2}} g_{2}(s)\,ds >0\).

We work in the space \(C[0,1]\times C[0,1]\) endowed with the norm

$$ \bigl\Vert (u,v)\bigr\Vert :=\max\bigl\{ \Vert u\Vert _{\infty},\| v\| _{\infty}\bigr\} , $$

where \(\| w\| _{\infty}:=\max\{| w(t)|,t\in[0,1] \}\).

Take the cones

$$\begin{aligned}& K_{1}:=\bigl\{ w\in C[0,1]: w\geq0, \mbox{ concave and nonincreasing} \bigr\} , \\& K_{2}:=\bigl\{ w\in C[0,1]:w\geq0, \mbox{ concave}\bigr\} . \end{aligned}$$

It is known (see, e.g., [1]) that

  • for \(w\in K_{1}\), we have \(w(t)\geq(1-t)\|w\|_{\infty}\) for \(t\in[0,1]\);

  • for \(w\in K_{2}\), we have \(w(t)\geq\min\{t,1-t\}\|w\|_{\infty}\) for \(t\in[0,1]\).

It follows that the functions in \(K_{i}\) are strictly positive on the sub-interval \([a_{i},b_{i}]\) and in particular

  • for \(w\in K_{1}\), we have \(\min_{t\in [a_{1},b_{1}]}w(t)\geq(1-b_{1})\|w\|_{\infty}\);

  • for \(w\in K_{2}\), we have \(\min_{t\in [a_{2},b_{2}]}w(t)\geq\min\{a_{2},1-b_{2}\}\|w\|_{\infty}\).

In the following we assume \(a_{1}=0\) and we make use of the notations

$$ c_{1}:=1-b_{1},\quad\quad c_{2}:=\min\{a_{2},1-b_{2} \}. $$

Consider now the cone K in \(C[0,1]\times C[0,1]\) defined by

$$ K:=\bigl\{ (u,v)\in K_{1}\times K_{2}\bigr\} .$$

For a positive solution of system (2.1) we mean a solution \((u,v)\in K\) of (2.1) such that \(\|(u,v)\|> 0\). We seek such solution as a fixed point of the following operator T.

Consider the integral operator

$$ T(u,v) (t):= \begin{pmatrix} T_{1}(u,v)(t) \\ T_{2}(u,v)(t)\end{pmatrix}, $$
(2.4)

where

$$\begin{aligned} T_{1}(u,v) (t) :=&\int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s}g_{1}( \tau) f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{}+B_{1} \biggl(\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}(\tau) f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) \end{aligned}$$

and

$$T_{2}(u,v) (t):= \textstyle\begin{cases} \int_{0}^{t} \varphi_{p_{2}}^{-1} (\int_{s}^{\sigma_{u,v}}g_{2}(\tau)f_{2}(\tau,u(\tau ),v(\tau))\,d\tau )\,ds\\ \quad{} +B_{2} (\varphi_{p_{2}}^{-1} (\int_{0}^{\sigma_{u,v}}g_{2}(\tau) f_{2}(\tau,u(\tau),v(\tau))\,d\tau ) ), & 0\le t\le\sigma_{u,v},\\ \int_{t}^{1}\varphi_{p_{2}}^{-1} (\int_{\sigma_{u,v}}^{s}g_{2}(\tau )f_{2}(\tau,u(\tau),v(\tau))\,d\tau )\,ds,& \sigma_{u,v}\leq t\leq1. \end{cases} $$

From the definitions, for every \((u,v)\in K\), we have

$$\max_{t\in[0,1]}T_{2}(u,v) (t)=T_{2}(u,v) ( \sigma_{u,v}). $$

Under our assumptions, we can show that the integral operator T leaves the cone K invariant and is compact.

Lemma 2.4

The operator (2.4) maps K into K and is compact.

Proof

Take \((u,v)\in K\). From the definition we have that the function \(T_{1}(u,v)\) is nonincreasing. The fact that \(T_{1}(u,v)\) and \(T_{2}(u,v)\) are convex functions is known, see Section 2, p.2279 of [1]. Thus \(T(u,v)\in K\). Now, we show that the map T is compact. Firstly, we show that T sends bounded sets into bounded sets. Take \((u,v)\in K\) such that \(\| (u,v)\| \leq r\). Then, for all \(t \in[0,1]\), we have

$$\begin{aligned} T_{1}(u,v) (t) =& \int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +B_{1} \biggl(\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}(\tau) f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) \\ \leq&\int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau) \phi_{1,r}(\tau)\,d\tau \biggr)\,ds \\ &{} +h_{12}\varphi_{p_{1}}^{-1} \biggl(\int _{0}^{1}g_{1}(\tau) f_{1} \bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \\ \leq&\int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1} g_{1}(\tau) \phi_{1,r}(\tau)\,d\tau \biggr)\,ds+h_{12}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}(\tau) \phi_{1,r}(\tau)\,d\tau \biggr) \\ \leq&\int_{0}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1} g_{1}(\tau) \phi_{1,r}(\tau)\,d\tau \biggr)\,ds+h_{12}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}(\tau) \phi_{1,r}(\tau)\,d\tau \biggr)< +\infty. \end{aligned}$$

We prove now that \(T_{1}\) sends bounded sets into equicontinuous sets. Let \(t_{1},t_{2}\in[0,1]\), \(t_{1}< t_{2}\), \((u,v)\in K\) such that \(\| (u,v)\| \leq r\). Then we have

$$\begin{aligned} \bigl\vert T_{1}(u,v) (t_{1})-T_{1}(u,v) (t_{2})\bigr\vert =& \biggl|\int_{t_{1}}^{t_{2}} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau )f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \biggr| \\ \leq& \biggl|\int_{t_{1}}^{t_{2}}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1} g_{1}(\tau) \phi_{1,r}(\tau)\,d\tau \biggr)\,ds \biggr|=C_{r}|t_{1}-t_{2}|. \end{aligned}$$

Therefore we obtain \(|T_{1}(u,v)(t_{1})-T_{1}(u,v)(t_{2})|\rightarrow0\) when \(t_{1}\rightarrow t_{2}\). By the Ascoli-Arzelà theorem we can conclude that \(T_{1}\) is a compact map.

For the sake of completeness, we sketch the proof of the fact that \(T_{2}\) sends bounded sets into equicontinuous sets. Let \(t_{1},t_{2}\in [0,1]\), \(t_{1}< t_{2}\), \((u,v)\in K\) such that \(\| (u,v)\| \leq r\). The cases \(0\leq t_{1}< t_{2}\leq\sigma_{u,v}\) or \(\sigma_{u,v}\leq t_{1}< t_{2}\leq 1\) can be handled as in the case of the operator \(T_{1}\). If \(0\leq t_{1}<\sigma_{u,v}<t_{2}\leq1\), we observe that

$$\begin{aligned}& \bigl\vert T_{2}(u,v) (t_{1})-T_{2}(u,v) (t_{2})\bigr\vert \\& \quad= \bigl\vert T_{2}(u,v) (t_{1})-T_{2}(u,v) ( \sigma_{u,v})+T_{2}(u,v) (\sigma _{u,v})-T_{2}(u,v) (t_{2})\bigr\vert \\& \quad\leq \bigl\vert T_{2}(u,v) (t_{1})-T_{2}(u,v) (\sigma_{u,v})\bigr\vert +\bigl\vert T_{2}(u,v) ( \sigma_{u,v})-T_{2}(u,v) (t_{2})\bigr\vert , \end{aligned}$$

and the proof follows as in previous cases.

Moreover, the map T is compact since the components \(T_{i}\) are compact maps. □

3 Existence results

For our index calculations, given \(\rho_{1}, \rho_{2}>0\) we use the following (relative) open bounded sets in K:

$$ K_{\rho_{1},\rho_{2}} = \bigl\{ (u,v) \in K : \|u\|_{\infty}< \rho_{1} \text{ and } \|v\|_{\infty}< \rho_{2}\bigr\} $$

and

$$ V_{\rho_{1},\rho_{2}} =\Bigl\{ (u,v) \in K: \min_{t\in [0,b_{1}]}u(t)< c_{1} \rho_{1} \text{ and } \min_{t\in [a_{2},b_{2}]}v(t)< c_{2}\rho_{2} \Bigr\} , $$

and if \(\rho_{1}=\rho_{2}=\rho\), we write simply \(K_{\rho}\) and \(V_{\rho}\). The set \(V_{\rho}\) was introduced in [43] as an extension to the case of systems of a set given by Lan [44]. The use of different radii, in the spirit of the paper [40], allows more freedom in the growth of the nonlinearities.

The following lemma is similar to Lemma 5 of [43] and therefore its proof is omitted.

Lemma 3.1

The sets defined above have the following properties:

  • \(K_{c_{1}\rho_{1},c_{2}\rho_{2}}\subset V_{\rho_{1},\rho_{2}}\subset K_{\rho _{1},\rho_{2}}\).

  • \((w_{1},w_{2}) \in\partial K_{\rho_{1},\rho_{2}}\) iff \((w_{1},w_{2})\in K\) and \(\| w_{i}\|_{\infty}= \rho_{i}\) for some \(i\in\{1,2\}\) and \(\| w_{j}\|_{\infty}\leq\rho_{j}\) for \(j\neq i\).

  • \((w_{1},w_{2}) \in\partial V_{\rho_{1},\rho_{2}}\) iff \((w_{1},w_{2})\in K\) and \(\min_{t\in[a_{i},b_{i}]} w_{i}(t)= c_{i}\rho_{i}\) for some \(i\in\{1,2\}\) and \(\min_{t\in[a_{j},b_{j}]}w_{j}(t) \le c_{j}\rho_{j}\) for \(j\neq i\).

  • If \((w_{1},w_{2}) \in\partial V_{\rho_{1},\rho_{2}}\), then for some \(i\in \{1,2\}\), \(c_{i}\rho_{i} \le w_{i}(t) \le\rho_{i}\) for each \(t \in[a_{i},b_{i}]\) and \(\|w_{i}\|_{\infty}\leq\rho_{i}\); moreover, for \(j\neq i\), we have \(\|w_{j}\|_{\infty}\leq\rho_{j}\).

We firstly prove that the fixed point index is 1 on the set \(K_{\rho_{1},\rho_{2}} \).

Lemma 3.2

Assume that

(\(\mathrm{I}_{\rho_{1},\rho_{2} }^{1}\)):

there exist \(\rho_{1},\rho_{2} >0\) such that for every \(i=1,2\)

$$ f_{i}^{\rho_{1},\rho_{2}} < \varphi_{p_{i}}(m_{i}), $$
(3.1)

where

$$\begin{aligned}& f_{i}^{\rho_{1},\rho_{2}}=\sup \biggl\{ \frac{f_{i}(t,u,v)}{\rho_{i}^{p_{i}-1}}: (t,u,v) \in[0,1]\times[ 0,\rho_{1} ]\times[0,\rho_{2} ] \biggr\} , \\& \frac{1}{m_{1}}=\int_{0}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)\,d\tau \biggr)\,ds+h_{12}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}(\tau)\,d\tau \biggr) \end{aligned}$$

and

$$\begin{aligned} \frac{1}{m_{2}} =& \max \biggl[\int_{0}^{\frac{1}{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\frac {1}{2}}g_{2}( \tau)\,d\tau \biggr)\,ds+h_{22}\varphi_{p_{2}}^{-1} \biggl( \int_{0}^{\frac{1}{2}}g_{2}(\tau)\,d\tau \biggr), \\ & \int_{\frac{1}{2}}^{1}\varphi _{p_{2}}^{-1} \biggl(\int_{\frac{1}{2}}^{s}g_{2}( \tau)\,d\tau \biggr)\,ds \biggr]. \end{aligned}$$

Then \(i_{K}(T,K_{\rho_{1},\rho_{2}})=1\).

Proof

We show that \(\lambda(u,v)\neq T(u,v)\) for every \((u,v)\in\partial K_{\rho_{1},\rho_{2} }\) and for every \(\lambda\geq1\); this ensures that the index is 1 on \(K_{\rho_{1},\rho_{2} }\). In fact, if this does not happen, there exist \(\lambda\geq1\) and \((u,v)\in\partial K_{\rho _{1},\rho_{2} }\) such that \(\lambda(u,v)=T(u,v)\).

Firstly we assume that \(\| u\| _{\infty}=\rho_{1} \) and \(\| v\| _{\infty }\leq\rho_{2} \).

Then we have

$$\begin{aligned} \lambda u(t) =& \int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}( \tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{}+B_{1} \biggl(\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}(\tau) f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) \\ \le&\int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}( \tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +h_{12}\varphi_{p_{1}}^{-1} \biggl(\int _{0}^{1}g_{1}(\tau) f_{1} \bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \\ =&\rho_{1}\int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)\frac{f_{1}(\tau,u(\tau),v(\tau))}{\rho^{p_{1}-1}_{1}}\,d\tau \biggr)\,ds \\ &{}+\rho_{1} h_{12}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}(\tau) \frac{f_{1}(\tau,u(\tau),v(\tau))}{\rho^{p_{1}-1}_{1}}\,d\tau \biggr). \end{aligned}$$

Taking \(t=0\) gives

$$\begin{aligned} \lambda u(0) =&\lambda{\rho_{1}} \\ \leq&\rho_{1} \int _{0}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}( \tau)f_{1}^{\rho_{1},\rho_{2}}\,d\tau \biggr)\,ds +\rho_{1} h_{12}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}( \tau)f_{1}^{\rho_{1},\rho_{2}}\,d\tau \biggr) \\ =& \rho_{1} \varphi_{p_{1}}^{-1}\bigl(f_{1}^{\rho_{1},\rho_{2}} \bigr) \biggl(\int_{0}^{1} \varphi_{p_{1}}^{-1} \biggl( \int_{0}^{s} g_{1}(\tau)\,d\tau \biggr)\,ds+h_{12}\varphi_{p_{1}}^{-1} \biggl(\int _{0}^{1}g_{1}(\tau)\,d\tau \biggr) \biggr) \\ =&\rho_{1}\frac{1}{m_{1}} \varphi_{p_{1}}^{-1} \bigl( f_{1}^{\rho_{1},\rho _{2}} \bigr). \end{aligned}$$

Using hypothesis (3.1) and the strict monotonicity of \(\varphi_{p_{1}}^{-1}\), we obtain \(\lambda\rho_{1} <\rho_{1} \). This contradicts the fact that \(\lambda\geq1\) and proves the result.

Now we assume \(\|v\|_{\infty}=\rho_{2}\) and \(\| u\| _{\infty}\leq\rho_{1} \).

Then we have

$$\lambda\rho_{2}= \bigl\Vert T_{2}(u,v)\bigr\Vert _{\infty}=T_{2}(u,v) (\sigma_{u,v}). $$

If \(\sigma_{u,v}\le \frac{1}{2} \), we have

$$\begin{aligned} \lambda\rho_{2} =& \bigl\Vert T_{2}(u,v)\bigr\Vert _{\infty}=T_{2}(u,v) (\sigma_{u,v}) \\ =& \int_{0}^{\sigma_{u,v}}\varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\sigma _{u,v}}g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +B_{2} \biggl(\varphi_{p_{2}}^{-1} \biggl(\int_{0}^{\sigma _{u,v}}g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) \\ \leq&\int_{0}^{\frac{1}{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\frac{1}{2}} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{}+h_{22} \varphi_{p_{2}}^{-1} \biggl(\int _{0}^{\sigma _{u,v}}g_{2}(\tau)f_{2}\bigl( \tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \\ \leq&\int_{0}^{\frac{1}{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\frac{1}{2}} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +h_{22} \varphi_{p_{2}}^{-1} \biggl(\int _{0}^{\frac {1}{2}}g_{2}(\tau)f_{2}\bigl( \tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \\ =&\rho_{2} \int_{0}^{\frac{1}{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\frac{1}{2}}g_{2}( \tau)\frac{f_{2}(\tau ,u(\tau),v(\tau))}{\rho_{2}^{p_{2}-1}}\,d\tau \biggr)\,ds \\ &{}+\rho_{2} h_{22} \varphi_{p_{2}}^{-1} \biggl(\int_{0}^{\frac{1}{2}}g_{2}(\tau) \frac{f_{2}(\tau,u(\tau),v(\tau))}{\rho _{2}^{p_{2}-1}}\,d\tau \biggr); \end{aligned}$$

thus we obtain

$$ \lambda\rho_{2}\leq \rho_{2} \varphi_{p_{2}}^{-1} \bigl(f_{2}^{\rho_{1},\rho_{2}}\bigr) \biggl(\int_{0}^{\frac {1}{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\frac{1}{2}}g_{2}( \tau)\,d\tau \biggr)\,ds+ h_{22} \varphi_{p_{2}}^{-1} \biggl(\int_{0}^{\frac{1}{2}}g_{2}(\tau)\,d\tau \biggr) \biggr). $$

If \(\sigma_{u,v}>\frac{1}{2} \), we have

$$\begin{aligned} \lambda\rho_{2} =&\bigl\Vert T_{2}(u,v)\bigr\Vert _{\infty}=T_{2}(u,v) (\sigma_{u,v}) \\ =&\int_{\sigma_{u,v}}^{1}\varphi_{p_{2}}^{-1} \biggl(\int_{\sigma_{u,v}}^{s}g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ \leq&\int_{\frac{1}{2}}^{1} \varphi_{p_{2}}^{-1} \biggl(\int_{\frac{1}{2}}^{s}g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ =&\rho_{2} \int_{\frac{1}{2}}^{1} \varphi _{p_{2}}^{-1} \biggl(\int_{\frac{1}{2}}^{s}g_{2}( \tau)\frac{f_{2}(\tau,u(\tau ),v(\tau))}{\rho_{2}^{p_{2}-1}}\,d\tau \biggr)\,ds \\ \leq&\rho_{2} \varphi_{p_{2}}^{-1} \bigl(f_{2}^{\rho_{1},\rho _{2}}\bigr)\int_{\frac{1}{2}}^{1} \varphi_{p_{2}}^{-1} \biggl(\int_{\frac {1}{2}}^{s}g_{2}( \tau)\,d\tau \biggr)\,ds. \end{aligned}$$

Then, in both cases, we have

$$\begin{aligned} \lambda\rho_{2} =&\bigl\Vert T_{2}(u,v)\bigr\Vert _{\infty}=T_{2}(u,v) (\sigma_{u,v}) \\ \leq&\rho_{2} \varphi_{p_{2}}^{-1} \bigl(f_{2}^{\rho_{1},\rho_{2}}\bigr) \max \biggl[\int_{0}^{\frac{1}{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\frac {1}{2}}g_{2}( \tau)\,d\tau \biggr)\,ds+h_{22} \varphi_{p_{2}}^{-1} \biggl( \int_{0}^{\frac{1}{2}}g_{2}(\tau)\,d\tau \biggr), \\ & \int_{\frac{1}{2}}^{1} \varphi_{p_{2}}^{-1} \biggl(\int_{\frac{1}{2}}^{s}g_{2}( \tau)\,d\tau \biggr)\,ds \biggr] \\ =&\rho_{2} \varphi_{p_{2}}^{-1}\bigl(f_{2}^{\rho_{1},\rho_{2}} \bigr)\frac{1}{m_{2}}. \end{aligned}$$

Using hypothesis (3.1) and the strict monotonicity of \(\varphi_{p_{2}}^{-1}\), we obtain \(\lambda\rho_{2} <\rho_{2} \). This contradicts the fact that \(\lambda\geq1\) and proves the result. □

We give a first lemma that shows that the index is 0 on a set \(V_{\rho_{1},\rho_{2} }\).

Lemma 3.3

Assume that:

(\(\mathrm{I}^{0}_{\rho_{1},\rho_{2}}\)):

there exist \(\rho_{1},\rho_{2}>0\) such that for every \(i=1,2\)

$$ f_{i,(\rho_{1}, \rho_{2})} > \varphi_{p_{i}}(M_{i}), $$
(3.2)

where

$$\begin{aligned}& f_{1,({\rho_{1},\rho_{2} })}= \inf \biggl\{ \frac{f_{1}(t,u,v)}{ \rho _{1}^{p_{1}-1}}: (t,u,v)\in[0,b_{1}] \times[c_{1}\rho_{1},\rho_{1}]\times[0, \rho _{2}] \biggr\} , \\& f_{2,({\rho_{1},\rho_{2} })}= \inf \biggl\{ \frac{f_{2}(t,u,v)}{ \rho _{2}^{p_{2}-1}}: (t,u,v)\in[a_{2},b_{2}] \times[0,\rho_{1}]\times[c_{2}\rho_{2}, \rho_{2}] \biggr\} , \\& \frac{1}{M_{1}}= \int_{0}^{b_{1}} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)\,d\tau \biggr)\,ds+h_{11} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{b_{1}}g_{1}(\tau)\,d\tau \biggr), \end{aligned}$$

and

$$\begin{aligned} \frac{1}{M_{2}} =&\frac{1}{2}\min_{a_{2}\leq\nu\leq b_{2}} \biggl[\int _{a_{2}}^{\nu}\varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\nu} g_{2}(\tau)\,d\tau \biggr)\,ds \\ &{} +\int_{\nu}^{b_{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{\nu}^{s} g_{2}(\tau)\,d\tau \biggr)\,ds+h_{21}\varphi_{p_{2}}^{-1} \biggl(\int_{a_{2}}^{\nu }g_{2}(\tau)\,d\tau \biggr) \biggr]. \end{aligned}$$

Then \(i_{K}(T,V_{\rho_{1},\rho_{2}})=0\).

Proof

Let \(e(t)\equiv1\) for \(t\in[0,1]\). Then \((e,e)\in K\). We prove that \((u,v)\ne T(u,v)+\lambda(e,e)\) for \((u,v)\in\partial V_{\rho_{1},\rho_{2}}\) and \(\lambda\geq0\). In fact, if this does not happen, there exist \((u,v)\in\partial V_{\rho_{1},\rho_{2} }\) and \(\lambda\geq0\) such that \((u,v)=T(u,v)+\lambda(e,e)\). We examine two cases.

Case (1): \(c_{1}\rho_{1} \le u(t) \le\rho_{1}\) for \(t\in[0,b_{1}]\) and \(0\leq v(t)\leq\rho_{2}\) for \(t\in[0,1]\).

Thus we have, for \(t\in[0,b_{1}]\),

$$\begin{aligned} \rho_{1} \geq& u(t) \\ =& \int_{t}^{1}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}( \tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +B_{1} \biggl(\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}(\tau) f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) +\lambda \\ \geq&\int_{t}^{b_{1}}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}( \tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +h_{11} \varphi_{p_{1}}^{-1} \biggl( \int_{0}^{1}g_{1}(\tau)f_{1} \bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) + \lambda \\ \geq&\int_{t}^{b_{1}}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}( \tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +h_{11} \varphi_{p_{1}}^{-1} \biggl( \int_{0}^{b_{1}}g_{1}(\tau) f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) + \lambda \\ =&\rho_{1}\int_{t}^{b_{1}} \varphi_{p_{1}}^{-1} \biggl(\int_{0} ^{s} g_{1}(\tau)\frac{f_{1}(\tau,u(\tau),v(\tau))}{\rho_{1}^{p_{1}-1}}\,d\tau \biggr)\,ds \\ &{} +\rho_{1} h_{11}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{b_{1}}g_{1}(\tau) \frac{f_{1}(\tau,u(\tau),v(\tau))}{\rho_{1}^{p_{1}-1}}\,d\tau \biggr) + \lambda. \end{aligned}$$

For \(t=0\), we obtain

$$\begin{aligned} \rho_{1} \geq&\rho_{1}\varphi_{p_{1}}^{-1}(f_{1},_{(\rho_{1},\rho_{2})}) \int_{0}^{b_{1}}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)\,d\tau \biggr)\,ds \\ &{} +\rho_{1}\varphi_{p_{1}}^{-1}(f_{1},_{(\rho_{1},\rho _{2})})h_{11} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{b_{1}}g_{1}( \tau)\,d\tau \biggr) + \lambda \\ >&{\rho_{1}}\varphi_{p_{1}}^{-1} (f_{1,(\rho_{1},\rho_{2})} ) \frac {1}{M_{1}}+{\lambda}. \end{aligned}$$

Using hypothesis (3.2) we obtain \(\rho_{1}>\rho_{1} +\lambda\), a contradiction.

Case (2): \(0\leq u(t)\leq\rho_{1}\) for \(t\in[0,1]\) and \(c_{2}\rho_{2} \le v(t) \le\rho_{2}\).

We distinguish three cases as follows.

Case (21): \(0<\sigma_{u,v}\leq a_{2}\).

Therefore we get

$$\begin{aligned} \rho_{2} \geq& v(\sigma_{u,v}) =T_{2}(u,v) ( \sigma_{u,v})+\lambda \\ =&\int_{\sigma_{u,v}}^{1}\varphi_{p_{2}}^{-1} \biggl(\int_{\sigma_{u,v}}^{s} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds+\lambda \\ \geq&\int_{a_{2}}^{b_{2}}\varphi_{p_{2}}^{-1} \biggl(\int_{a_{2}}^{s} g_{2}(\tau )f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds+\lambda \\ =&\rho_{2} \int_{a_{2}}^{b_{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{a_{2}}^{s} g_{2}(\tau )\frac{f_{2}(\tau,u(\tau),v(\tau))}{\rho_{2}^{p_{2}-1}}\,d\tau \biggr)\,ds+\lambda \\ \geq&\rho_{2}\varphi_{p_{2}}^{-1}(f_{2},_{(\rho_{1},\rho_{2})}) \int_{a_{2}}^{b_{2}}\varphi_{p_{2}}^{-1} \biggl(\int_{a_{2}}^{s} g_{2}(\tau)\,d\tau \biggr)\,ds+\lambda \\ \geq&\rho_{2}\varphi_{p_{2}}^{-1}(f_{2},_{(\rho_{1},\rho_{2})}) \frac {1}{M_{2}}+\lambda. \end{aligned}$$

Using hypothesis (3.2) we obtain \(\rho_{2}>\rho_{2} +\lambda\), a contradiction.

Case (22): \(\sigma_{u,v} \geq b_{2}\).

$$\begin{aligned} \rho_{2} \geq& v(\sigma_{u,v})=T_{2}(u,v) ( \sigma_{u,v})+\lambda \\ =&\int_{0}^{\sigma_{u,v}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\sigma _{u,v}} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +B_{2} \biggl(\varphi_{p_{2}}^{-1} \biggl(\int_{0}^{\sigma_{u,v}}g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) +\lambda \\ \geq&\int_{a_{2}}^{b_{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{b_{2}} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +h_{21} \varphi_{p_{2}}^{-1} \biggl( \int_{a_{2}}^{b_{2}}g_{2}(\tau)f_{2} \bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) +\lambda \\ =&\rho_{2} \int_{a_{2}}^{b_{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{b_{2}} g_{2}(\tau)\frac{f_{2}(\tau,u(\tau),v(\tau))}{\rho_{2}^{p_{1}-1}}\,d\tau \biggr)\,ds \\ &{} +\rho_{2} h_{21} \varphi_{p_{2}}^{-1} \biggl(\int_{a_{2}}^{b_{2}}g_{2}(\tau) \frac{f_{2}(\tau,u(\tau),v(\tau))}{\rho_{2}^{p_{2}-1}}\,d\tau \biggr) +\lambda \\ \geq&\rho_{2}\varphi_{p_{2}}^{-1}(f_{2},_{(\rho_{1},\rho_{2})}) \int_{a_{2}}^{b_{2}}\varphi_{p_{2}}^{-1} \biggl(\int_{s}^{b_{2}} g_{2}(\tau)\,d\tau \biggr)\,ds \\ &{} +\rho_{2}\varphi_{p_{2}}^{-1}(f_{2},_{(\rho_{1},\rho _{2})})h_{21} \varphi_{p_{2}}^{-1} \biggl(\int_{a_{2}}^{b_{2}}g_{2}( \tau)\,d\tau \biggr) +\lambda \\ \geq&\rho_{2}\varphi_{p_{2}}^{-1}(f_{2},_{(\rho_{1},\rho_{2})}) \frac {1}{M_{2}}+\lambda. \end{aligned}$$

Using hypothesis (3.2) we obtain \(\rho_{2}>\rho_{2} +\lambda\), a contradiction.

Case (23): \(a_{2}<\sigma_{u,v}<b_{2}\).

$$\begin{aligned} 2\rho_{2} \geq&2v(\sigma_{u,v})=2\lambda+2T_{2}(u,v) (\sigma_{u,v}) \\ =& 2\lambda+\int_{0}^{\sigma_{u,v}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\sigma_{u,v}} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +B_{2} \biggl(\varphi_{p_{2}}^{-1} \biggl(\int_{0}^{\sigma_{u,v}}g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) \\ &{} +\int_{\sigma_{u,v}}^{1} \varphi_{p_{2}}^{-1} \biggl(\int_{\sigma_{u,v}}^{s} g_{2}(\tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ \geq&2\lambda+\int_{a_{2}}^{\sigma_{u,v}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\sigma_{u,v}} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +h_{21} \varphi_{p_{2}}^{-1} \biggl( \int_{a_{2}}^{\sigma_{u,v}}g_{2}(\tau)f_{2} \bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \\ &{} + \int_{\sigma_{u,v}}^{b_{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{\sigma _{u,v}}^{s} g_{2}(\tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ =&2\lambda+\rho_{2} \int_{a_{2}}^{\sigma_{u,v}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\sigma_{u,v}} g_{2}(\tau)\frac{f_{2}(\tau,u(\tau),v(\tau))}{\rho_{2}^{p_{2}-1}}\,d\tau \biggr)\,ds \\ & {}+\rho_{2} h_{21} \varphi_{p_{2}}^{-1} \biggl(\int_{a_{2}}^{\sigma_{u,v}}g_{2}(\tau) \frac{f_{2}(\tau,u(\tau),v(\tau))}{\rho _{2}^{p_{2}-1}}\,d\tau \biggr) \\ &{} + \rho_{2}\int_{\sigma_{u,v}}^{b_{2}} \varphi _{p_{2}}^{-1} \biggl(\int_{\sigma_{u,v}}^{s}g_{2}( \tau)\frac{f_{2}(\tau,u(\tau ),v(\tau))}{\rho_{2}^{p_{2}-1}}\,d\tau \biggr)\,ds \\ \geq& 2\lambda+\rho_{2}\varphi_{p_{2}}^{-1}(f_{2},_{(\rho_{1},\rho_{2})}) \biggl[\int_{a_{2}}^{\sigma_{u,v}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\sigma_{u,v}} g_{2}(\tau)\,d\tau \biggr)\,ds \\ &{} +h_{21} \varphi_{p_{2}}^{-1} \biggl( \int_{a_{2}}^{\sigma _{u,v}}g_{2}(\tau)\,d\tau \biggr) + \int_{\sigma_{u,v}}^{b_{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{\sigma _{u,v}}^{s}g_{2}(\tau)\,d\tau \biggr)\,ds \biggr] \\ \geq&2\lambda+ 2\rho_{2}\varphi_{p_{2}}^{-1}(f_{2},_{(\rho_{1},\rho_{2})}) \frac{1}{M_{2}}. \end{aligned}$$

Using hypothesis (3.2) we obtain \(\rho_{2}>\lambda+\rho_{2} \), a contradiction. □

Remark 3.4

We point out that a stronger, but easier to check, hypothesis than (3.2) is

$$ f_{i,(\rho_{1}, \rho_{2})} > \varphi_{p_{i}}(\tilde{M}_{i}), $$

where

$$\frac{1}{\tilde{M}_{1}}= \int_{0}^{b_{1}} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)\,d\tau \biggr)\,ds $$

and

$$\frac{1}{\tilde{M}_{2}}=\frac{1}{2}\min_{a_{2}\leq\nu\leq b_{2}} \biggl\{ \int _{a_{2}}^{\nu}\varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\nu} g_{2}(\tau)\,d\tau \biggr)\,ds +\int_{\nu}^{b_{2}}\varphi_{p_{2}}^{-1} \biggl(\int_{\nu}^{s} g_{2}(\tau)\,d\tau \biggr)\,ds \biggr\} . $$

In the following lemma we exploit an idea that was used in [37, 38, 40], and we provide a result of index 0 controlling the growth of just one nonlinearity \(f_{i}\), at the cost of having a larger domain. Nonlinearities with different growths were considered, for example, in [45–47].

Lemma 3.5

Assume that

\((\mathrm{I}^{0}_{\rho_{1},\rho_{2}})^{\star}\) :

there exist \(\rho _{1},\rho_{2}>0\) such that for some \(i\in\{1,2\}\) we have

$$ f^{*}_{i,(\rho_{1}, \rho_{2})}>\varphi_{p_{i}}(M_{i}), $$
(3.3)

where

$$ f^{*}_{i,(\rho_{1},{\rho_{2}})}=\inf \biggl\{ \frac{f_{i}(t,u,v)}{ \rho_{i}^{p_{i}-1}}: (t,u,v) \in[a_{i},b_{i}]\times[0,\rho_{1}]\times[0, \rho_{2}] \biggr\} . $$

Then \(i_{K}(T,V_{\rho_{1},\rho_{2}})=0\).

Proof

Suppose that condition (3.3) holds for \(i=1\). Let \((u,v)\in\partial V_{\rho_{1},\rho_{2} }\) and \(\lambda\geq0\) such that \((u,v)=T(u,v)+\lambda(e,e)\). Thus we proceed as in the proof of Lemma 3.3. □

The proof of the next result regarding the existence of at least one, two or three positive solutions follows by the properties of fixed point index and is omitted. It is possible to state results for four or more positive solutions, in a similar way as in [48], by expanding the lists in conditions (S5), (S6).

Theorem 3.6

System (2.1) has at least one positive solution in K if one of the following conditions holds.

(S1):

For \(i=1,2\), there exist \(\rho_{i},r _{i}\in (0,\infty)\) with \(\rho_{i}< r _{i}\) such that (\(\mathrm{I}_{\rho _{1},\rho_{2}}^{0}\)) [or \((\mathrm{I}_{\rho_{1},\rho_{2}}^{0})^{\star}\)], (\(\mathrm {I}_{r _{1},r_{2}}^{1}\)) hold.

(S2):

For \(i=1,2\), there exist \(\rho_{i},r _{i}\in (0,\infty)\) with \(\rho_{i}< c_{i} r _{i}\) such that (\(\mathrm{I}_{\rho _{1},\rho_{2}}^{1}\)), (\(\mathrm{I}_{r _{1},r_{2}}^{0}\)) hold.

System (2.1) has at least two positive solutions in K if one of the following conditions holds.

(S3):

For \(i=1,2\), there exist \(\rho_{i},r _{i},s_{i}\in (0,\infty)\) with \(\rho_{i}< r_{i} < c_{i} s _{i}\) such that (\(\mathrm {I}_{\rho_{1},\rho_{2}}^{0}\)) [or \((\mathrm{I}_{\rho_{1},\rho_{2}}^{0})^{\star }\)], (\(\mathrm{I}_{r _{1},r_{2}}^{1}\)) and (\(\mathrm{I}_{s _{1},s_{2}}^{0}\)) hold.

(S4):

For \(i=1,2\), there exist \(\rho_{i},r _{i},s_{i}\in (0,\infty)\) with \(\rho_{i}< c_{i} r _{i}\) and \(r _{i}< s _{i}\) such that (\(\mathrm{I}_{\rho_{1},\rho_{2}}^{1}\)), (\(\mathrm{I}_{r _{1},r_{2}}^{0}\)) and (\(\mathrm{I}_{s _{1},s_{2}}^{1}\)) hold.

System (2.1) has at least three positive solutions in K if one of the following conditions holds.

(S5):

For \(i=1,2\), there exist \(\rho_{i},r _{i},s_{i},\delta _{i}\in(0,\infty)\) with \(\rho_{i}< r _{i}< c_{i} s _{i}\) and \(s _{i}<\delta _{i}\) such that (\(\mathrm{I}_{\rho _{1},\rho_{2}}^{0}\)) [or \((\mathrm{I}_{\rho _{1},\rho_{2}}^{0})^{\star}\)], (\(\mathrm{I}_{r _{1},r_{2}}^{1}\)), (\(\mathrm{I}_{s_{1},s_{2}}^{0}\)) and (\(\mathrm{I}_{\delta _{1},\delta_{2}}^{1}\)) hold.

(S6):

For \(i=1,2\), there exist \(\rho_{i},r _{i},s_{i},\delta _{i}\in(0,\infty)\) with \(\rho_{i}< c_{i} r _{i}\) and \(r _{i}< s _{i}< c_{i} \delta_{i}\) such that (\(\mathrm{I}_{\rho _{1},\rho_{2}}^{1}\)), (\(\mathrm{I}_{r_{1},r_{2}}^{0}\)), (\(\mathrm{I}_{s _{1},s_{2}}^{1}\)) and (\(\mathrm{I}_{\delta _{1},\delta_{2}}^{0}\)) hold.

4 Non-existence results

We now provide some non-existence results for system (2.1). We use an argument similar to the ones of [40, 49–53].

Theorem 4.1

Assume that one of the following conditions holds.

  1. 1.

    For \(i=1,2\),

    $$ f_{i}(t,u_{1},u_{2})< \varphi_{p_{i}}(m_{i} u_{i}) \quad\textit{for every } t\in [0,1] \textit{ and } u_{i}>0, $$
    (4.1)

    where \(m_{i}\) is defined in Lemma 3.2.

  2. 2.

    For \(i=1,2\),

    $$ f_{i}(t,u_{1},u_{2})> \varphi_{p_{i}} \biggl(\frac{M_{i}}{c_{i}} u_{i} \biggr) \quad\textit{for every } t\in[a_{i},b_{i}] \textit{ and } u_{i}>0, $$
    (4.2)

    where \(M_{i}\) is defined in Lemma 3.3.

  3. 3.

    There exists \(k\in\{1,2\}\) such that (4.1) is verified for \(f_{k}\) and for \(j\neq k\) condition (4.2) is verified for \(f_{j}\).

Then there is no positive solution of system (2.1) in K.

Proof

(1) Assume, on the contrary, that there exists \((u,v)\in K\) such that \((u,v)=T(u,v)\) and \((u,v)\neq(0,0)\). We distinguish two cases.

  • Let \(\|u\|_{\infty}\neq0\). Then we have

    $$\begin{aligned} u(t) = &\int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}( \tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +B_{1} \biggl(\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}(\tau) f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) \\ < & m_{1} \int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)\varphi_{p_{1}}\bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds + m_{1} h_{12} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1} g_{1}(\tau)\varphi _{p_{1}}\bigl(u(\tau)\bigr)\,d\tau \biggr) \\ \leq& m_{1} \|u\|_{\infty}\biggl(\int_{t}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)\,d\tau \biggr)\,ds+h_{12} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1} g_{1}(\tau)\,d\tau \biggr) \biggr). \end{aligned}$$

    Taking \(t=0\) gives

    $$\begin{aligned} \|u\|_{\infty} =& u(0)< m_{1} \|u\|_{\infty}\int _{0}^{1} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}(\tau)\,d\tau \biggr)\,ds +m_{1} \|u\|_{\infty}h_{12} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1} g_{1}(\tau)\,d\tau \biggr)\\ =& m_{1} \|u\|_{\infty}\frac{1}{m_{1}}, \end{aligned}$$

    a contradiction.

  • Let \(\|v\|_{\infty} \neq0\).

    Reasoning as in Lemma 3.2 we distinguish the cases \(\sigma _{u,v} \leq1/2\) and \(\sigma_{u,v} >1/2\).

    In the first case we have

    $$\begin{aligned} \|v\|_{\infty} =& \bigl\Vert T_{2}(u,v)\bigr\Vert _{\infty}\\ =&T_{2}(u,v) (\sigma_{u,v}) \\ =& \int_{0}^{\sigma_{u,v}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\sigma _{u,v}} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +B_{2} \biggl(\varphi_{p_{2}}^{-1} \biggl(\int_{0}^{\sigma_{u,v}}g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) \\ < & m_{2} \|v\|_{\infty}\int_{0}^{\sigma_{u,v}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\sigma_{u,v}} g_{2}(\tau)\,d\tau \biggr)\,ds \\ &{} + h_{22}m_{2} \|v\|_{\infty}\varphi_{p_{2}}^{-1} \biggl(\int_{0}^{\sigma_{u,v}}g_{2}( \tau)\,d\tau \biggr) \\ \le& m_{2} \|v\|_{\infty}\biggl(\int_{0}^{\frac{1}{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\frac{1}{2}} g_{2}(\tau)\,d\tau \biggr)\,ds+ h_{22} \varphi_{p_{2}}^{-1} \biggl(\int_{0}^{\frac{1}{2}}g_{2}(\tau)\,d\tau \biggr) \biggr) \\ \le& m_{2} \|v\|_{\infty}\frac{1}{m_{2}} , \end{aligned}$$

    a contradiction.

    In a similar manner we proceed in the case \(\sigma_{u,v} >1/2\).

(2) Assume, on the contrary, that there exists \((u,v)\in K\) such that \((u,v)=T(u,v)\) and \((u,v)\neq(0,0)\). We distinguish two cases.

  • Let \(\|u\|_{\infty}\neq0\). Then, for \(t\in[a_{1},b_{1}]= [0,b_{1}]\), we have

    $$\begin{aligned} u(t) =& \int_{t}^{1}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}( \tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +B_{1} \biggl(\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{1}g_{1}( \tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) \\ \geq&\int_{t}^{b_{1}}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}( \tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +h_{11}\varphi_{p_{1}}^{-1} \biggl(\int _{0}^{1}g_{1}(\tau) f_{1} \bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \\ \geq&\int_{t}^{b_{1}}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{s} g_{1}( \tau)f_{1}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +h_{11}\varphi_{p_{1}}^{-1} \biggl(\int _{0}^{b_{1}}g_{1}(\tau) f_{1} \bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \\ >&\frac{M_{1}}{c_{1}}\int_{t}^{b_{1}} \varphi_{p_{1}}^{-1} \biggl(\int_{0} ^{s} g_{1}(\tau)\varphi_{p_{1}}\bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +\frac{M_{1}}{c_{1}} h_{11}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{b_{1}}g_{1}(\tau)\varphi _{p_{1}}\bigl(u(\tau)\bigr)\,d\tau \biggr) \\ >&\frac{M_{1}}{c_{1}}\int_{t}^{b_{1}} \varphi_{p_{1}}^{-1} \biggl(\int_{0} ^{s} g_{1}(\tau)\varphi_{p_{1}}\bigl(c_{1} \Vert u\Vert _{\infty}\bigr)\,d\tau \biggr)\,ds \\ &{} +\frac{M_{1}}{c_{1}}h_{11}\varphi_{p_{1}}^{-1} \biggl(\int_{0}^{b_{1}}g_{1}(\tau) \varphi_{p_{1}}(c_{1}\|u\|_{\infty})\,d\tau \biggr) \\ =&M_{1}\|u\|_{\infty} \biggl(\int_{t}^{b_{1}} \varphi_{p_{1}}^{-1} \biggl(\int_{0} ^{s} g_{1}(\tau)\,d\tau \biggr)\,ds+ h_{11} \varphi_{p_{1}}^{-1} \biggl(\int_{0}^{b_{1}}g_{1}( \tau)\,d\tau \biggr) \biggr). \end{aligned}$$

    For \(t=0\) we obtain

    $$ u(0)= \|u\|_{\infty}>M_{1} \|u\|_{\infty} \frac{1}{M_{1}}, $$

    a contradiction.

  • Let \(\|v\|_{\infty} \neq0\). We examine the case \(\sigma_{u,v} \geq b_{2}\). We have

    $$\begin{aligned} \|v\|_{\infty} =&v(\sigma_{u,v})=T_{2}(u,v) ( \sigma_{u,v}) \\ =& \int_{0}^{\sigma_{u,v}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{\sigma _{u,v}} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} +B_{2} \biggl(\varphi_{p_{2}}^{-1} \biggl(\int_{0}^{\sigma _{u,v}}g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr) \biggr) \\ \geq&\int_{a_{2}}^{b_{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{b_{2}} g_{2}( \tau)f_{2}\bigl(\tau,u(\tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{} + h_{21} \varphi_{p_{2}}^{-1} \biggl( \int_{a_{2}}^{b_{2}}g_{2}(\tau)f_{2} \bigl(\tau,u(\tau ),v(\tau)\bigr)\,d\tau \biggr) \\ >& M_{2}\|v\|_{\infty} \biggl(\int_{a_{2}}^{b_{2}} \varphi_{p_{2}}^{-1} \biggl(\int_{s}^{b_{2}} g_{2}(\tau)\,d\tau \biggr)\,ds +h_{21} \varphi_{p_{2}}^{-1} \biggl(\int_{a_{2}}^{b_{2}}g_{2}(\tau)\,d\tau \biggr) \biggr) \\ \geq& M_{2}\|v\|_{\infty}\frac{1}{M_{2}}, \end{aligned}$$

    a contradiction. By similar proofs, the cases \(0<\sigma_{u,v}\leq a_{2}\) and \(a_{2}<\sigma_{u,v}< b_{2}\) can be examined.

(3) Assume, on the contrary, that there exists \((u,v)\in K\) such that \((u,v)=T(u,v)\) and \((u,v)\neq(0,0)\). If \(\|u\|_{\infty}\neq 0\), then the function \(f_{1}\) satisfies either (4.1) or (4.2), and the proof follows as in the previous cases. If \(\|v\|_{\infty}\neq0\), then the function \(f_{2}\) satisfies either (4.1) or (4.2), and the proof follows as in the previous cases. □

5 An example

We illustrate in the following example that all the constants that occur in Theorem 3.6 can be computed.

Consider the system

$$ \begin{aligned} &\bigl(\varphi_{p_{1}} \bigl(u^{\prime}\bigr)\bigr)'(t)+g_{1}(t)f_{1} \bigl(t, u(t),v(t)\bigr)=0,\quad t \in (0,1), \\ &\bigl(\varphi_{p_{2}}\bigl(v^{\prime}\bigr)\bigr)'(t)+g_{2}(t)f_{2} \bigl(t, u(t),v(t)\bigr)=0,\quad t \in (0,1), \end{aligned} $$
(5.1)

subject to the boundary conditions

$$ u'(0)= 0,\quad\quad u(1)+B_{1} \bigl(u'(1) \bigr)=0,\quad\quad v(0)=B_{2} \bigl(v'(0) \bigr),\quad\quad v(1)=0, $$
(5.2)

where \(B_{1}\) and \(B_{2}\) are defined by

$$B_{1}(w)= \textstyle\begin{cases} w, & w\le0, \\ \frac{w}{2}, & 0\leq w \leq1,\\ \frac{w}{6} +\frac{1}{3},& w\geq1, \end{cases} $$

and

$$B_{2}(w)= \textstyle\begin{cases} \frac{w}{3}, & 0\leq w \leq1,\\ \frac{w}{9} +\frac{2}{9},& w\geq1. \end{cases} $$

Now we assume \(g_{1} =g_{2}\equiv1\). Thus we have

$$\begin{aligned}& \frac{1}{m_{1}}=\frac{p_{1}-1}{p_{1}} +h_{12}, \\& \frac{1}{m_{2}}=\frac{p_{2}-1}{p_{2}} \biggl(\frac{1}{2} \biggr)^{\frac {p_{2}}{p_{2}-1}}+h_{22} \biggl(\frac{1}{2} \biggr)^{\frac{1}{p_{2}-1}}, \\& \frac{1}{M_{1}}=\frac{1}{M_{1}[0,b_{1}]}=\frac{p_{1}-1}{p_{1}} b_{1}^{\frac {p_{1}}{p_{1}-1}}+h_{11} b_{1}^{\frac{1}{p_{1}-1}} \end{aligned}$$

and

$$\begin{aligned} \frac{1}{M_{2}} =&\frac{1}{M_{2}[a_{2},b_{2}]} \\ =&\frac{1}{2}\min_{a_{2}\le\nu\le b_{2}} \biggl(\frac{p_{2}-1}{p_{2}} \bigl((\nu-a_{2})^{\frac{p_{2}}{p_{2}-1}}+(b_{2}-\nu)^{\frac{p_{2}}{p_{2}-1}} \bigr) +h_{21}(\nu-a_{2})^{\frac{1}{p_{2}-1}} \biggr). \end{aligned}$$

The choice \(p_{1}=\frac{3}{2}\), \(p_{2}=3\), \(b_{1}=\frac{2}{3}\), \(a_{2}=\frac{1}{4}\), \(b_{2}=\frac{3}{4}\), \(h_{11}=\frac{1}{6}\), \(h_{12}=\frac{1}{2}\), \(h_{21}=\frac{1}{9}\) and \(h_{22}=\frac{1}{3}\) gives, by direct computation,

$$c_{1}=\frac{1}{3};\quad\quad c_{2}=\frac{1}{4};\quad\quad m_{1}=1.2;\quad\quad M_{1}=5.785;\quad\quad m_{2}=2.121;\quad\quad M_{2}=9.145. $$

Let us now consider

$$f_{1}(t, u,v)=\frac{1}{16}\bigl(u^{4}+t^{3}v^{3} \bigr)+\frac{27}{50},\quad\quad f_{2}(t,u,v)=\sqrt{tu}+10v^{9}. $$

Then, with the choice of \(\rho_{1}=\rho_{2}=\frac{1}{20}\), \(r_{1}=1\), \(r_{2}=\frac{2}{3}\), \(s_{1}=s_{2}=9\), we obtain

$$\begin{aligned}& \inf \biggl\{ f_{1}(t,u,v): (t,u,v)\in\biggl[0,\frac{2}{3} \biggr]\times[0,\rho _{1}]\times[0,\rho_{2}] \biggr\} \\& \quad= f_{1}(0,0,0)=0.54>\sqrt{M_{1}\rho_{1}}=0.538, \\& \sup \bigl\{ f_{1}(t,u,v): (t,u,v)\in[0,1]\times[0,r_{1}] \times[0, r_{2}] \bigr\} \\& \quad=f_{1}(1,r_{1},r_{2})=0.62< \sqrt{m_{1} r_{1}}=1.095, \\& \sup \bigl\{ f_{2}(t,u,v): (t,u,v)\in[0,1]\times[0,r_{1}] \times[0, r_{2}] \bigr\} \\& \quad=f_{2}(1,r_{1},r_{2})=1.260< (m_{2} r_{2})^{2}=2, \\& \inf \biggl\{ f_{1}(t,u,v): (t,u,v)\in \biggl[0,\frac{2}{3} \biggr]\times[c_{1}s_{1},s_{1}] \times[0,s_{2}] \biggr\} \\& \quad=f_{1}(0,c_{1}s_{1},0)=5.602 > \sqrt{M_{1} s_{1}}=1.247, \\& \inf \biggl\{ f_{2}(t,u,v): (t,u,v)\in \biggl[\frac{1}{4}, \frac{3}{4}\biggr]\times[0,s_{1}]\times[c_{2}s_{2},s_{2}] \biggr\} \\& \quad=f_{2}(t,0,c_{2}s_{2})=14778.9 >M^{2}_{2} s_{2}^{2}=6774.07. \end{aligned}$$

Thus the conditions \((\mathrm{I}^{0}_{\frac{1}{20},\frac{1}{20}})^{\star}\), (\(\mathrm{I}^{1}_{1,\frac{2}{3}}\)) and (\(\mathrm{I}^{0}_{9,9}\)) are satisfied; therefore system (5.1)-(5.2) has at least two positive solutions \((u_{1},v_{1})\) and \((u_{2},v_{2})\) such that \(\frac{1}{20}<\|(u_{1},v_{1})\|\leq1\) and \(1<\|(u_{2},v_{2})\|\leq 9\).