We consider the equation

where $L\colon X\to X$ is a linear operator and $F\colon K\to K$ is a (nonlinear) continuous mapping. In the subsequent we denote

Our first condition is related to the linear operator $L$.

(h1)

The cone $K$ is invariant under the linear operator $L$, i.e., $L(K)\subset K$. Moreover, $L$ admits a positive eigenvalue $\lambda_{1}>0$ with the corresponding eigenfunction $\varphi_{1}$ from the cone $K$ ($\varphi_{1}\in K$).

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From (h1) it follows that $L$ is increasing (order-preserving), that is, for all $u,v\in K$, one has

Indeed, if $0\leq u\leq v$, then $v-u\in K$. Since $L(v-u)\in K$, i.e., $L(v-u)\geq 0$, it follows that $Lv-Lu\geq 0$, which proves the claim.

Under certain conditions on $L$ and $F$, we show that there exists a lower solution $\underline{u}$ , i.e.,

and an upper solution

for the equation (2).

Before stating our assumptions, let $\Phi\colon K\to\mathbb{R}_{+}$ be a positively homogeneous mapping, that is,

and with the additional property that $\Phi(u)=0$ if and only if $u=0$.

The following conditions are assumed to be satisfied:

(h2)

The operator $F$ is increasing (order-preserving), that is, for $u,v\in K$ one has

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$$0\leq u\leq v\quad\text{implies}\quad F(u)\leq F(v).$$

(h3)

There exists a constant $r>0$ such that

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$$F(\lambda_{1}u)\geq u\quad\text{for all }u\in K\text{ with }\Phi(u)=r.$$

(h4)

There exists $\alpha>0$ and $\mu\in K\setminus\{0\}$ such that

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$$L\mu\leq\alpha\mu$$

and

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$$F(\alpha\mu)\leq\mu.$$

Denote

We now show that $\underline{u}$ and $\overline{u}$ are a lower and an upper solution, respectively, for the equation (2).

Further, assume that

(h5)

The lower and upper solutions $\underline{u}$ and $\overline{u}$ satisfy (are comparable)

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$$\underline{u}\leq\overline{u}.$$

Then the following invariance result holds.

Under an additional compactness condition on $L$, we obtain the following fixed-point result.

If, instead of conditions (h3) and (h4), we consider

(h3)’

There exists $R>0$ such that

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$$F(\lambda_{1}u)\leq u\quad\text{for all }u\in K\text{ with }\Phi(u)=R,$$

(h4)’

There exist $\alpha>0$ and $\mu\in K\setminus\{0\}$ such that

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$$L\mu\geq\alpha\mu$$

and

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$$F(\alpha\mu)\geq\mu,$$

an analogue of Theorem 3 can be established.

In the next section, we present another method for constructing a lower and an upper solution for a fixed point equation, applicable when the nonlinear operator satisfies an abstract Harnack inequality.

We consider the fixed point problem

where $N\colon X\to X$ is a (nonlinear) increasing operator, i.e., for $u,v\in K$ one has

On $X$, we consider a seminorm $\|\cdot\|$, which is assumed to be increasing, i.e., for $u,v\in K$ one has

Moreover, we assume that there exists $\psi\in K\setminus\{0\}$ such that

The following condition plays a key role in the subsequent analysis and can be regarded as a weak type Harnack inequality [13].

(a1)

There exists $\chi\in K\setminus\{0\}$ such that

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$$N(u)\geq\|N(u)\|\chi\quad\text{for all }u\in K.$$

(7)

The next two additional conditions are required.

(a2)

There exists $r>0$ such that

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$$\|N(r\chi)\|\geq r.$$

(a3)

There exists $R>0$ such that

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$$|N(R\psi)|\leq R.$$

Now, we are ready to present the main result of this subsection.

In this subsection, we consider the fixed point problem

for which we show how Theorem 5 can be applied. Here, $k$ is a symmetric ($k(t,s)=k(s,t)$ for all $t,s\in[0,1]$), nonnegative continuous function on $[0,1]^{2}$, and $f\in C(\mathbb{R},\mathbb{R}_{+})$ is nondecreasing on $\mathbb{R}_{+}$.

Let $X=C[0,1]$ be endowed with the supremum norm $|\cdot|_{\infty}$, and let $K$ denote the cone of continuous nonnegative functions. Also, let $L\colon X\to X$ be the linear Hammerstein operator

and $F\colon K\to K$ the Nemytskii operator, which assigns to each $u\in K$ the function $Fu\colon[0,1]\to\mathbb{R}_{+}$ given by

Note that $F$ is well defined since $f$ is continuous and nonnegative on $\mathbb{R}$.

The complete continuity of $L$ follows from standard arguments based on the Arzelà–Ascoli theorem (see, e.g., [7, 10, 3, 11]). To guarantee that $L$ has an eigenvalue, we need the following additional condition on the kernel $k$ (see [3]).

(H1)

The function $k$ satisfies a Green like inequality, that is, there exists a continuous function $\theta\colon[0,1]\to\mathbb{R}_{+}$ such that

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$$k(t,s)\geq\theta(t)k(q,s)\quad\text{for all}\quad t,s,q\in[0,1],$$

and

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$$|L\theta|_{\infty}>0.$$

The following result is of great importance for the subsequent analysis.

By Lemma 1, the operator $L$ has a positive eigenvalue $\lambda_{1}=r(L)>0$ with the corresponding eigenfunction $\varphi_{1}\in K$, hence condition (h1) is fulfilled. Moreover, since the function $f$ is continuous and nondecreasing, the Nemytskii operator $F$ is continuous and increasing, therefore, condition (h2) is also satisfied.

To ensure that (h3) and (h4) are valid as well, the following asymptotic conditions on $f$ are required:

(H2)

One has,

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$$\lim_{t\searrow 0}\frac{f(t)}{t}>\frac{1}{\lambda_{1}}\quad\text{ and }\quad\lim_{t\to\infty}\frac{f(t)}{t}<\frac{1}{|L1|_{\infty}},$$

where

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$$(L1)(t)=\int_{0}^{1}k(t,s)ds.$$

Note that, by (H2), there exists $0<r<R$ such that

and

Now, the following result holds.

In this subsection, inspired by [13], we apply Theorem 7 for the $p$-Laplace problem

where $p>1$, $\Omega$ is a smooth domain in $\mathbb{R}^{n}$ and $f\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a nondecreasing continuous function. Following [1, Lemma 1.1] (see also [6]), for each $h\in L^{\infty}(\Omega)$, there exists a unique (weak) solution $S(h)\in C_{0}^{1}(\overline{\Omega})$ to the problem

Moreover, the operator $S\colon L^{\infty}(\Omega)\to C_{0}^{1}(\overline{\Omega})$ is completely continuous, order preserving (increasing), and invariant with respect to the cone of positive functions, i.e., $S(h)\geq 0$ whenever $h\geq 0$. Thus, letting $X=L^{\infty}(\Omega)$ be endowed with the supremum norm $|\cdot|_{\infty}$ (hence increasing), and denoting by $K$ the cone of positive functions, the problem (16) allows for the fixed point formulation

where $N=SF$, and $F$ is the Nemytskii operator that associates to each function $u\in L^{\infty}(\Omega)$ the function

Note that, since $f$ is continuous and nonnegative, the operator $F$ is continuous and $F(K)\subset K$.

We immediately observe that relation (6) holds with $\psi\equiv 1$. Moreover, given that the function $f$ is nondecreasing, the operator $T$ is increasing, and because $S$ is order-preserving, it follows that $N$ is also increasing.

By Theorem 2, for some fixed compact set $D\subset\Omega$, there exists $M>0$ such that for every $p$-superharmonic function $u\in K$, one has

Let us consider the seminorm $\|\cdot\|$ be given by

Then, since $S(h)$ is nonnegative and $p$-superharmonic for each $h\in K$, condition (a1) (inequality  (7)) holds with

Note for any $q>0$, one has (see [13])

Thus, it follows that

and

If

(A1)

There exists $0<r\leq R$ such that

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$$f(r)\geq\frac{r^{p-1}}{\|S1\|^{p-1}},$$

and

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$$f(R)\leq\frac{r^{p-1}}{|S1|_{\infty}^{p-1}},$$

then conditions (a2) and (a3) are verified. In addition, since $r\leq R$ then relation (8) is also valid, where

We easily see that, since $F$ is continuous, the operator $N$ is also continuous. Moreover, as $F$ is increasing, the set $F([\underline{u},\overline{u}])$ is bounded. Hence, the complete continuity of $S$ implies that the set $N([\underline{u},\overline{u}])$ is relatively compact.

Therefore, all the requirements of Theorem 7 are satisfied, so the following result holds.

Acknowledgements The author would like to thank Prof. Radu Precup for insightful discussions on the subject and verification of the paper.

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

The author have no relevant financial or non-financial interests to disclose.

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