Existence and uniqueness of fixed points of an integral operator of Hammerstein type

Abstract

We study the existence and uniqueness of positive fixed points of an integral operator of the Hammerstein type in the space of continuous functions. Under certain conditions on the integral operator, we establish a sufficient condition for the the integral operator to have exactly one positive fixed point.

1. Introduction

The Hammerstein equation covers a large variety of areas and is of great interest to a wide audience because of its applications in numerous areas. Several problems that arise in differential equations (ordinary and partial), for instance, elliptic boundary value problems whose linear parts have a Green’s function, can be transformed into the Hammerstein integral equations. Equations of the Hammerstein type play a crucial role in the theory of optimal control systems and in automation and network theory (see, e.g., Doležal [1]).

The problem of the existence of solutions arises naturally in different areas of life. There are methods that help one to ensure the existence of a solution to a particular problem. In general fixed-point theorems, the Banach contraction mapping principle and the Schauder–Tychonov fixed point theorem are used. However, none of these theorems is applicable here because our operator is not compact or contracting.

The concept of monotone operators, introduced in the 1960s, has proved very effective in obtaining existence results in nonlinear problems. One of the reasons is certainly the absence of compactness among the basic requirements. Also, compactness is not always easy to check and it does represent a rather severe restriction on the operator. Many researchers have successfully applied monotonicity concepts to the Hammerstein equations.

Also, there are some works devoted to fixed points of the Hammerstein operator on cones. The main results on the existence and multiplicity of fixed points of Hammerstein equations can be found, e.g., in [1]–[6]. On the other hand, new results on the uniqueness of fixed points of Hammerstein equations on cones are needed. For instance, an increasing attention was recently given to models with an uncountably many spin values on a Cayley tree. Splitting Gibbs measures on Cayley trees are described by positive fixed points of a Hammerstein integral operator. But the known results on the existence and uniqueness of fixed points of a Hammerstein integral operator on cones cannot be used directly. In this paper, we give new conditions for the existence and uniqueness of fixed points of a Hammerstein integral operator, with regard to problems in the theory of Gibbs measures.

2. Existence and uniqueness of fixed points of the Hammerstein operator

2.1. Existence of fixed points of the Hammerstein operator

Definition.

Let $(\mathcal N,\|\,{\cdot}\,\|)$ be a real normed space. A cone $\mathcal K$ in $\mathcal N$ is a closed set such that

• • $u+v\in\mathcal K$ for all $u,v\in\mathcal K$,

• • $\lambda u\in\mathcal K$ for all $u\in\mathcal K$, $\lambda\in[0,+\infty)$,

• • $\mathcal K\cap(-\mathcal K)=\{0\}$.

The cone $\mathcal K$ defines a partial ordering in $\mathcal N$ such that $x\preceq y$ if and only if $y-x\in\mathcal K$. For $x,y\in\mathcal N$, with $x\preceq y$, we define the ordered interval

$$[x,y]=\{z\in X\colon x\preceq z\preceq y\}.$$

The cone $\mathcal K$ is normal if there exists $d>0$ such that for all $x,y\in\mathcal N$ with $0\preceq x\preceq y$, we have $\|x\|\leqslant d\|y\|$.

We let

$$B[x_0,r]=\{x\in\mathcal N\colon\|x-x_0\|\leqslant r\}$$

denote the closed ball with radius $r>0$ centered at $x_0\in\mathcal N$. We also let $\mathcal K_r=\mathcal K\cap\{x\in\mathcal N\colon \|x\|<r\}$ be the intersection of the cone with the open ball with radius $r>0$ centered at the origin.

We next recall some well-known results (see, e.g., [4], [7]–[9]).

Theorem 1.

Let $\mathcal N$ be a real Banach space with a normal order cone $\mathcal K$ . Suppose that there exist $\alpha\preceq\beta$ $($ where $[\alpha,\beta]\subset\mathcal N$ $)$ such that $T\colon [\alpha,\beta]\subset\mathcal N$ is a completely continuous monotone nondecreasing operator with $\alpha\preceq T\alpha$ and $T\beta\preceq\beta$ . Then $T$ has a fixed point and the iterative sequence $\alpha_{n+1}=T\alpha_n$ , with $\alpha_0=\alpha$ , converges to the greatest fixed point of $T$ in $[\alpha,\beta]$ , and the sequence $\beta_{n+1}=T\beta_n$ , with $\beta_0=\beta$ , converges to the smallest fixed point of $T$ in $[\alpha,\beta]$ .

In what follows, the closure and the boundary of subsets of $\mathcal K$ are understood relative to $\mathcal K$.

Proposition 1.

Let $D$ be an open bounded set of $\mathcal N$ with $0\in D_{\mathcal K}$ and $\kern1.8pt\overline{\vphantom{D}\kern5.8pt}\kern-7.6pt D _\mathcal K\neq \mathcal{\mathcal K}$ , where $D_\mathcal{\mathcal K}=D\cap\mathcal K$ . Assume that $T\colon \kern1.8pt\overline{\vphantom{D}\kern5.8pt}\kern-7.6pt D _{\mathcal K}\to\mathcal K$ is a completely continuous operator such that $x\neq Tx$ for $x\in\partial D_{\mathcal K}$ . Then the fixed point index $i_{\mathcal K}(T,D_{\mathcal K})$ has the following properties.

• 1. If there exists $e\in\mathcal K\backslash\{0\}$ such that $x\neq Tx+\lambda e$ for all $x\in\partial D_{\mathcal K}$ and all $\lambda>0$ , then $i_{\mathcal K}(T,D_{\mathcal K})=0$ . For example, this condition holds if $Tx\npreceq x$ for $x\in\partial D_{\mathcal K}$ .

• 2. If $\|Tx\|\geqslant\|x\|$ for $x\in\partial D_{\mathcal K}$ , then $i_{\mathcal K}(T,D_{\mathcal K})=1$ .

• 3. If $Tx\neq\lambda x$ for all $x\in\partial D_{\mathcal K}$ and all $\lambda>1$ , then $i_{\mathcal K}(T,D_{\mathcal K})=1$ . For example, this condition holds if either $Tx\nsucceq x$ or $\|Tx\|\leqslant\|x\|$ for all $x\in\partial D_{\mathcal K}$ .

• 4. Let $D^1$ be open in $\mathcal N$ such that $\overline{D^1}\subset D_{\mathcal K}$ . If $i_{\mathcal K}(T,D_\mathcal K)=1$ and $i_{\mathcal K}(T,D_{\mathcal K}^1)=0$ , then $T$ has a fixed point in $D_{\mathcal K}^{}\backslash\overline{D_{\mathcal K}^1}$ . The same holds if $i_{\mathcal K}(T,D_{\mathcal K})=0$ and $i_{\mathcal K}(T,D_{\mathcal K}^1)=1$ .

The next theorem is obtained by using Theorem 1 and Proposition 1.

Theorem 2.

Let $\mathcal N$ be a real Banach space, $\mathcal K$ be a normal cone with the normal constant $d\geqslant 1$ and a nonempty interior $($ i.e., solid $)$ , and $T\colon\mathcal K\to\mathcal K$ be a completely continuous operator. Assume that

• 1) there exists $\beta\in\mathcal K$ , with $T\beta\preceq\beta$ , and $R>0$ such that $B[\beta, R]\subset\mathcal K$ ;

• 2) the map $T$ is nondecreasing in the set

  $$\mathcal P=\biggl\{x\in\mathcal K\colon x\preceq\beta\;\,\textit{and}\;\, \frac{R}{d}\leqslant\|x\|\biggr\};$$

• 3) there exists a (relatively) open bounded set $V\subset\mathcal K$ such that $i_\mathcal K(T,V)=0$ and either $\overline{\mathcal K_R}\subset V$ or $\kern0.5pt\overline{\vphantom{V}\kern6.9pt}\kern-7.4pt V \subset\mathcal K_R$ .

Then the map $T$ has at least one nonzero fixed point $x_1$ in $\mathcal K$ that either belongs to $\mathcal P$ or belongs to $V\backslash \kern1.1pt\overline{\vphantom{\mathcal K}\kern6.0pt}\kern-7.1pt\mathcal K _R$ if $\kern1.1pt\overline{\vphantom{\mathcal K}\kern6.0pt}\kern-7.1pt\mathcal K _R\subset V$ or $\kern1.1pt\overline{\vphantom{\mathcal K}\kern6.0pt}\kern-7.1pt\mathcal K _R \backslash \kern0.5pt\overline{\vphantom{V}\kern6.9pt}\kern-7.4pt V$ if $\kern0.5pt\overline{\vphantom{V}\kern6.9pt}\kern-7.4pt V \subset\mathcal K_R$ .

Let $\Omega$ be a set. A nonlinear integral equation of the Hammerstein type on $\Omega$ has the form

$$ u(x)+\int_{\Omega}K(x,y)f(u(y))\,dy=h(x)$$

(2.1)

where $dy$ stands for a $\sigma$-finite measure on the measure space $\Omega$; the kernel $K$ is defined on $\Omega\times\Omega$, $f$ is a real-valued function defined on $\Omega\times\mathbb{R}$ (and is nonlinear in general), and $h$ is a given function on $\Omega$. If we define the operator $A$ by

$$ Av(x)=\int_{\Omega}K(x,y)v(y)\,dy,$$

(2.2)

and let $N_f$ be the Nemystkii operator associated with $f$,

$$ \widetilde N_fu(x)=f(u(x))$$

(2.3)

then integral equation (2.1) can be expressed in the form of a functional equation,

$$ u+A\widetilde N_fu=0,$$

(2.4)

where we take $h\equiv 0$ without loss of generality. For $h\neq 0$, we have

$$u+A\widetilde N_fu=h\quad\Longrightarrow\quad u-h+A\widetilde N_f u=0\quad\Longrightarrow\quad\omega+A\widetilde N_f(\omega+h)=0,$$

where $\omega=u-h$. Thus, $\omega+A\widetilde N_f\omega=0$, where $\widetilde N_f\omega=\widetilde N_f(\omega+h)$. Hence, it suffices to solve the following equation instead of (2.1):

$$ u(x)+\int_{\Omega}K(x,y)f(u(y))\,dy=0.$$

(2.5)

Let $\Omega=[a,b]$. We consider a special Hammerstein-type operator $A\widetilde N_f$ and study positive fixed points, i.e.,

$$ \int_{a}^b K(x,y)f(u(y))\,dy=u(x).$$

(2.6)

Here and hereafter, $K\colon[a,b]^2\to(0,+\infty)$ and $f\colon[0,+\infty)\to[0,+\infty)$ are continuous functions; moreover, $f$ is nonlinear. We note that the operator $A\widetilde N_f$ is completely continuous.

Condition C1.

There exists a continuous function $\mathfrak{R}\colon[a,b]\to[0,+\infty)$ and constants $c\in(0,1)$ and $\alpha,\beta\in[a,b]$ ($\alpha<\beta$) such that $K(x,y)\leqslant\mathfrak{R}(y)$ for $x,y\in[a,b]$ and $c\,\mathfrak{R}(y)\leqslant K(x,y)$ for $x\in[\alpha,\beta]$ and $y\in[a,b]$.

Monotonicity with respect to one projection

We say that $\mathcal D\colon(x,y)\in\mathbb{R}^2\to\mathcal D(x,y)$ is monotone (nondecreasing) with respect to $x$ if for a fixed $y_0\in\mathbb{R}$, the function $\mathcal D_{y_0}\colon x\in\mathbb{R}\to\mathcal D(x,y_0)$ is monotone (nondecreasing).

Example 1.

Let $K\colon[a,b]^2\to(0,+\infty)$ be a nonconstant continuous function. If $K(x,y)$ is monotone (nondecreasing) with respect to $x$, then the function satisfies condition C1. Indeed, ${K(x,y)\,{\leqslant}\, K(b, y)\!:=\!\mathfrak{R}(y)}$.

If we set

$$M=\max_{y\in[a,b]}K(b,y),\qquad m=\min_{y\in[a,b]}K(a,y)$$

and write $\alpha=a$ and $\beta=b$, then $c=m/M<1$, i.e.,

$$\frac{m}{M}K(b,y)\leqslant K(x,y)\quad\Longrightarrow\quad\frac{m}{M}\mathfrak{R}(y)\leqslant K(x,y),\qquad (x,y)\in[a,b]^2.$$

We work in the space $C[a,b]$ endowed with the usual norm

$$\|\kern1pt\omega\|=\max_{t\in[a,b]}|\omega(t)|.$$

We set

$$\begin{aligned} \, &C^{+}[a,b]=\bigl\{\omega\in C[a,b]\colon\omega(x)\geqslant 0\;\,\text{for all}\;\,x\in[a,b]\bigr\}, \\ &\widehat{\mathcal K}:=\bigl\{\omega\in C^{+}[a,b]\colon\min_{x\in[\alpha,\beta]}\omega(x)\geqslant c\|\kern1pt\omega\|\bigr\}, \end{aligned}$$

where $\alpha$, $\beta$, and $c$ are defined in condition C1.

Property 1.

The following assertions hold:

• 1) $\widehat{\mathcal K}$ is a normal cone with $d=1$ ;

• 2) If the kernel $K(x,y)$ of Eq. (2.6) satisfies condition C1, then $A\widetilde N_f(C^{+}[a,b])\subset\widehat{\mathcal K}$.

Proof.

1. Clearly, $\widehat{\mathcal K}\cap(-\widehat{\mathcal K}\,)=\{\mathcal O\}$ and $\lambda\omega\in\widehat{\mathcal K}$ for all $\omega\in\widehat{\mathcal K}$, $\lambda\in[0,+\infty)$, where $\mathcal O$ is the identically zero function. Let $\omega_1,\omega_2\in\widehat{\mathcal K}$; then

$$\min_{x\in[\alpha,\beta]}(\omega_1(x)+\omega_2(x))\geqslant\min_{x\in[\alpha,\beta]}\omega_1(x)+\min_{x\in[\alpha,\beta]}\omega_2(x)\geqslant c(\|\kern1pt\omega_1\|+\|\kern1pt\omega_2\|)\geqslant c\|\kern1pt\omega_1+\omega_2\|,$$

i.e., $\omega_1+\omega_2\in\widehat{\mathcal K}$. We verify that $\widehat{\mathcal K}$ is normal with $d=1$, i.e.,

$$\mathcal O\preceq\omega_1\preceq\omega_2\quad\Longrightarrow\quad\omega_1-\mathcal O,\;\omega_2-\omega_1\in\widehat{\mathcal K}\subset C^{+}[a,b].$$

Therefore, $\omega_1(x)\geqslant 0$, $\omega_2(x)-\omega_1(x)\geqslant 0$ for all $x\in[a,b]$ and $\|\kern1pt\omega_1\|\leqslant \|\kern1pt\omega_2\|$.

2. Let $\omega\in A\widetilde N_f(C^{+}[a,b])$; then there exists $\nu\in C^{+}[a,b]$ such that $\omega=A\widetilde N_f\nu$. We then have

$$\begin{aligned} \, \min_{x\in[\alpha,\beta]}\omega(x)=\int_a^b K(x,y)f(\nu(y))\,dy& \geqslant c\int_a^b \mathfrak{R}(y)f(\nu(y))\,dy\geqslant \\ &\geqslant c\int_a^b K(t,y)f(\nu(y))\,dy=c\,\omega(t),\qquad t\in[a,b]. \end{aligned}$$

Hence, $\min_{x\in[\alpha,\beta]}\omega(x)\geqslant c\|\kern1pt\omega\|$, and therefore $\omega\in\widehat{\mathcal K}$.

With the notation used in condition C1, we give one more condition.

Condition C2.

For every $\varepsilon>0$, there exists $\delta(\varepsilon)>0$ such that

$$\inf_{t\in[1,c^{-1}]} f(\delta t)>\varepsilon\delta.$$

Example 2.

Let $\alpha>1$ and $\beta\in\mathbb{R}_{+}:=(0, +\infty)$. If a function $f\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ satisfies the condition $f(x)>\beta x^\alpha$ for all $x\in\mathbb{R}_{+}$, then the function satisfies condition C2. Indeed, for every $\varepsilon>0$, we choose $\delta(\varepsilon)=\sqrt[\alpha-1]{\varepsilon\beta^{-1}}$. Then

$$\inf_{t\in[1,c^{-1}]} f(\delta(\varepsilon)t)>\inf_{t\in[1,c^{-1}]} \beta(\delta(\varepsilon)t)^\alpha =\beta\delta(\varepsilon)^\alpha=\varepsilon\delta(\varepsilon).$$

Proposition 2.

Assume that conditions C1 and C2 hold and, moreover,

• 1) there exists $\tau>0$ such that $f(\,{\cdot}\,)$ is nondecreasing on $[0,\tau];$

• 2) the following inequality holds:

  $$ \sup_{s\in (0,\tau)}\frac{(1-c)s}{f(s)\big\|\int_a^b K(x,y)\,dy\big\|}>1.$$

  (2.7)

Then the operator $A\widetilde N_f$ has at least one positive fixed point in $\widehat{\mathcal K}$ .

Proof.

Let $\theta\in(0,\tau)$ satisfy the inequality

$$ \theta-f(\theta)\bigg\|\int_a^b K(x,y)\,dy\bigg\|>c\,\theta.$$

(2.8)

Also, for $\varepsilon>\|\int_a^b K(x,y)\,dy\|^{-1}$, let $\delta=\delta(\varepsilon)>0$ as in C2 and $\mathcal R$ be a fixed number such that

$$\mathcal R<\min\biggl\{\frac{(1-c)\theta}{1+c},\,\delta \biggr\}.$$

We set $\varphi(x)=\theta$ for all $x\in[a,b]$ and

$$\mathcal V:=\bigl\{\omega\in\widehat{\mathcal K}\colon\min_{x\in[\alpha,\beta]}\omega(x)<\delta\bigr\}.$$

We now verify the conditions of Theorem 2.

1. $B[\theta,\mathcal R]\subset\widehat{\mathcal K}$ and $A\widetilde N_f \theta\preceq\theta$.

Because $\theta$ is constant and $\mathcal R<\min\frac{(1-c)\theta}{1+c}$, direct computation shows that $B[\theta,\mathcal R]\subset\widehat{\mathcal K}$. For every $x\in[a,b]$, Eq. (2.8) implies that

$$[A\widetilde N_f\varphi](x)=\int_a^b K(x,y)f(\theta)\,dy\leqslant\bigg\|\int_a^bK(x,y)\,dy\bigg\|f(\theta)<\theta.$$

Because $\|\varphi-A\widetilde N_f\theta\|\leqslant \varphi$ and (2.8), for $x\in[\alpha,\beta]$ we have

$$\theta-[A\widetilde N_f\varphi](x)=\theta-\int_a^bK(x,y)f(\theta)\,dy\geqslant \theta-\bigg\|\int_a^bK(x,y)\,dy\bigg\|f(\theta)>c\theta\geqslant c\|\theta-A\widetilde N_f\varphi\|.$$

Hence, $A\widetilde N_f\theta\preceq\theta$ and claim 1 of Theorem 2 is proved.

2. $A\widetilde N_f$ is nondecreasing on the set $\{s\in\widehat{\mathcal K}\colon s\preceq \varphi\}$.

Let $\omega_1, \omega_2\in\widehat{\mathcal K}$ and $\omega_1\preceq\omega_2\preceq\varphi$. Then we obtain

$$0\leqslant \omega_1(x)\leqslant \omega_2(x)\leqslant \theta,\qquad x\in[a,b].$$

Also, because $f$ is nondecreasing in $[\mathcal O,\varphi]$ ($\mathcal O$-zero function), we have

$$f(\mathcal O)\preceq f(\omega_1)\preceq f(\omega_2)\preceq f(\varphi)\quad\Longrightarrow\quad 0\leqslant f(\omega_1(x))\leqslant f(\omega_2(x))\leqslant \theta.$$

Therefore,

$$[A\widetilde N_f\omega_2](x)-[A\widetilde N_f\omega_1](x)=\int_a^bK(x,y)[f(\omega_2(y))-f(\omega_1(y))]\,dy\geqslant 0.$$

In addition, for all $x\in[\alpha,\beta]$, $s\in[a,b]$,

$$\begin{aligned} \, [A\widetilde N_f\omega_2](x)-[A\widetilde N_f\omega_1](x)&=\int_a^b K(x,y)[f(\omega_2(y))-f(\omega_1(y))]\,dy\geqslant \\ &\geqslant c\int_a^b \mathfrak{R}(y)[f(\omega_2(y))-f(\omega_1(y))]\,dy\geqslant \\ &\geqslant c\int_a^b K(s,y)[f(\omega_2(y))-f(\omega_1(y))]\,dy= \\ &=c([A\widetilde N_f\omega_2](s)-[A\widetilde N_f\omega_1](s)). \end{aligned}$$

Consequently,

$$\min_{x\in[\alpha,\beta]}([A\widetilde N_f\omega_2](x)-[A\widetilde N_f\omega_1](x))\geqslant c\|A\widetilde N_f\omega_2-A\widetilde N_f\omega_1\|.$$

Therefore, $A\widetilde N_f\omega_1\preceq A\widetilde N_f\omega_2$.

3. Let $\mathcal K_{\mathcal R}:=\widehat{\mathcal K}\cap\{x\in\mathcal N\colon\|x\|<\mathcal R\}$; then $\kern1.1pt\overline{\vphantom{\mathcal K}\kern6.0pt}\kern-7.1pt\mathcal K _{\mathcal R}\subset\mathcal V$ and the fixed-point index (see, e.g., [10]) is zero, i.e., $i_{\widehat{\mathcal K}}(A\widetilde N_f, \mathcal V)=0$. From $\mathcal R<\delta$, we obtain $\kern1.1pt\overline{\vphantom{\mathcal K}\kern6.0pt}\kern-7.1pt\mathcal K _{\mathcal R}\subset\mathcal K_{\delta}\subset\mathcal V$.

Let $e(x)=1$ for all $x\in[a,b]$. Then $e\in\mathcal K$ and we show that $\omega\neq A\widetilde N_f(\omega)+\mu e$ for $\omega\in\partial (\mathcal V)$ and $\mu\geqslant 0$. Conversely, suppose that there exists $\omega\in\partial(\mathcal V)$ and $\mu\geqslant 0$ such that $\omega=A\widetilde N_f(\omega)+\mu e$. For all $x\in[\alpha,\beta]$, we then have

$$\omega(x)=\int_a^b K(x,y)f(\omega(y))\,dy+\mu e(x).$$

It follows from C2 that

$$\omega(x)\geqslant\int^{b}_{a}K(x,y)f(\omega(y))\,dy+\mu\geqslant\min_{x\in[\alpha,\beta]}\omega(x)+\mu.$$

Consequently, we obtain

$$\delta=\min_{x\in[\alpha,\beta]}\omega(x)>\delta+\mu\geqslant\delta.$$

This contradicts our assumption.

Thus, by Theorem 2, we can conclude that the operator $A\widetilde N_f$ has at least one positive fixed point in $\widehat{\mathcal K}$.

2.2. Uniqueness of a fixed point of the Hammerstein operator

Here, we study the problem of uniqueness of a positive fixed point of $A\widetilde N_f$.

Let $M$ and $m$ be the respective maximum and minimum values of the kernel $K(x,y)$, $(x,y)\in[a,b]\times [a,b]$. We set

$$C'_{+}[a,b]=\{f(x)\in C^{+}[a,b]\colon\ \text{there exists}\;\,f'(x),\,x\in[a,b],\;\;\textrm{and}\;\,f'(x)\in C^{+}[a,b]\}$$

and

$$\max_{x\in[a,b]}f(x)=f_{\max},\qquad \min\limits_{x\in[a,b]}f(x)=f_{\min}.$$

Theorem 3.

Assume that the assumptions of Proposition 2 hold and, moreover,

• 1) let $\nu\in[1,+\infty)$ and $\omega_i(x)\in C^{+}[a,b]$ , $i=1,2$ , then

  $$ \omega_1(x)\geqslant\nu\omega_2(x)\quad\Longrightarrow\quad f(\omega_1(x))\geqslant\nu f(\omega_2(x));$$

  (2.9)

• 2) $f$ satisfies the condition

  $$ \gamma_1x^{\sigma_1}\leqslant f(x)\leqslant\gamma_2 x^{\sigma_2}, \qquad \gamma_i>0,\quad \sigma_i>1,\quad i=1,2,$$

  (2.10)

  and $f(x)-\gamma_1x^{\sigma_1}$ , $\gamma_2x^{\sigma_2}-f(x)$ are monotone nondecreasing functions;

• 3) For the maximum $M$ and minimum $m$ values of the kernel of $A\widetilde N_f$ , the following inequality holds:

  $$ \mathfrak{I}_2(M,m)-\mathfrak{I}_1(m,M)<\frac{1}{b-a},$$

  (2.11)

  where

  $$\mathfrak{I}_i(x,y)=x\sigma_i\gamma_i\biggl(\frac{x}{y}(m\gamma_{3-i}(b-a))^{1/(1-\sigma_{3-i})}\biggr)^{\!\sigma_i-1}.$$

Then the operator $A\widetilde N_f$ , $f\in C'_{+}[a,b]$ has exactly one positive fixed point in $\widehat{\mathcal K}$ .

Proof.

By Proposition 2, the operator $A\widetilde N_f$ has at least one positive fixed point. It therefore suffices to prove that it is unique. We assume that the operator $A\widetilde N_f$ has two distinct positive fixed points $\omega_1$ and $\omega_2$. Let $\omega(x)=\omega_1(x)-\omega_2(x)$; we then prove that $\omega(x)$ changes sign on $[a,b]$.

We set

$$\delta_{\mathrm s}:=\delta_{\sup}(\omega_1,\omega_2)= \sup\bigl\{\delta\in[0,\infty)\colon\omega_1(x)-\delta \omega_2(x)>0\;\,\text{for all}\;\,x\in[a,b]\bigr\}.$$

Then

$$\begin{aligned} \, \omega_1(x)-\delta_{\mathrm s}\omega_2(x)&=A\widetilde N_f(\omega_1)(x)-\delta_{\mathrm s} A\widetilde N_f(\omega_2)(x)= \nonumber\\ &=\int_a^b K(x,y)[f(\omega_1(y))-\delta_{\mathrm s}f(\omega_2(y))]\,dy. \end{aligned}$$

(2.12)

We suppose that $\delta_{\mathrm s}\geqslant 1$; because $\omega_1(y)\neq\omega_2(y)$ for some $y\in[a,b]$, we then have

$$\omega_1(y)-\delta_{\mathrm s}\omega_2(y)\geqslant 0\quad \text{for all}\quad y\in[a,b].$$

Now, we show that

$$\int_a^b K(x,y)[f(\omega_1(y))-\delta_{\mathrm s}f(\omega_2(y)]\,dy>0.$$

Indeed, if

$$\int_a^b K(x,y)[f(\omega_1(y))-\delta_{\mathrm s}f(\omega_2(y)]\,dy=0$$

then, by the definition of $\delta_{\mathrm s}$ and (2.9), we have $f(\omega_1(y))=\delta_{\mathrm s}f(\omega_2(y))$ for all $y\in[a,b]$. Because $\omega_1$ and $\omega_2$ are fixed points, we obtain $\omega_1(y)=\delta_{\mathrm s}\omega_2(y)$ for all $y\in[a,b]$. The last equality contradicts the assumption that $\omega_1$ and $\omega_2$ are two distinct positive fixed points. Hence,

$$ \omega_1(x)-\delta_{\mathrm s}\omega_2(x)=\int_a^b K(x,y)[f(\omega_1(y))-\delta_{\mathrm s}f(\omega_2(y))]\,dy>0.$$

(2.13)

On the other hand, by the definition of $\delta_{\mathrm s}$, there is a $y_0\in[0,1]$ such that $\omega_1(y_0)-\delta_{\mathrm s}\omega_2(y_0)=0$. But Eq. (2.12) contradicts inequality (2.13).

Hence, $\delta_{\mathrm s}<1$, i.e., $\omega(x)$ changes sign on $[a,b]$. We can say without loss of generality that the maximum value $\omega_{\max}$ of $\omega(x)=\omega_1(x)-\omega_2(x)$ is less than or equal to the absolute value of $\omega_{\min}$, i.e., $\|\kern1pt\omega\|\leqslant |\omega_{\min}|$ (otherwise, we choose $-\omega(x)=\omega_2(x)-\omega_1(x)$).

Now, subtracting

$$ \omega_1(x)- \omega_2(x)=\int_a^bK(x,y)[f(\omega_1(y))-f(\omega_2(y))]\,dy,$$

(2.14)

we can use Cauchy’s mean value theorem to conclude that

$$\omega(x)=\int^b_a K(x,y)f'(\xi(y))\omega(y)\,dy,$$

where

$$ \min\{\omega_1(y), \omega_2(y)\}\leqslant\xi(y)\leqslant \max\{\omega_1(y),\omega_2(y)\},\qquad y\in[a,b].$$

(2.15)

We set $\varrho_1=\mathfrak{I}_1(m,M)$ and $\varrho_2=\mathfrak{I}_2(M,m)$. Let the image (range) of $\xi$ be denoted by $\operatorname{Im}(\xi)$. We now show that

$$\operatorname{Im}(\xi)\subset\mathcal Q:=[\varrho_1, \varrho_2].$$

If $\psi\in A\widetilde N_f(C[a,b])$, then the inequality $\psi_{\min}\geqslant(m/M)\|\psi\|$ holds. Indeed, there exists a continuous function $\psi_1$ such that $\psi=A\widetilde N_f \psi_1$. Then

$$\begin{aligned} \, &\psi_{\min}\geqslant m\int^b_a f(\psi_1(y))\,dy\geqslant m\int_a^b\frac{K(x_1, y)}{M}f(\psi_1(y))\,dy\geqslant\frac{m}{M}\|\psi_1\|, \\ &\max_{x\in[a,b]}\psi(x)=\psi(x_1), \end{aligned}$$

i.e.,

$$\psi\in\mathcal B:=\biggl\{\phi\in C[0,1]\colon\phi_{\min}\geqslant\frac{m}{M}\,\|\phi\|\biggr\}.$$

By (2.15), it is sufficient to prove that any positive fixed point of $A\widetilde N_f$ belongs to the set $\mathcal Q$. Let $\mu$ be a positive fixed point of $A\widetilde N_f$; then

$$\|\mu\|\leqslant M\|f(\mu)\|(b-a)\leqslant\gamma_2 M(b-a)\|\mu\|^{\sigma_2}\quad\Longrightarrow\quad (\gamma_2 M(b-a))^{1/(1-\sigma_2)}\leqslant\|\mu\|.$$

Because $\mu\in\mathcal B$, we obtain

$$\mu(t)\geqslant\mu_{\min}\geqslant\frac{m}{M}\|\mu\|\geqslant\frac{m}{M}(\gamma_2 M(b-a))^{1/(1-\sigma_2)}.$$

On the other hand, we can estimate $\mu(x)$ from above:

$$\mu(x)=(A\widetilde N_f \mu)(x)\geqslant m\int^b_af(\mu(y))\,dy\geqslant m\gamma_1\mu_{\min}^{\sigma_1}(b-a).$$

Therefore,

$$\mu_{\min}\geqslant m\gamma_1\mu_{\min}^{\sigma_1}(b-a)\quad\Longrightarrow\quad\omega_{\min}\leqslant (m\gamma_1(b-a))^{1/(1-\sigma_1)}.$$

Because $\mu\in\mathcal B$, we obtain

$$\mu(t)\leqslant\mu_{\max}\leqslant\frac{M}{m}\mu_{\min}\leqslant \frac{M}{m}(m\gamma_1(b-a))^{1/(1-\sigma_1)}.$$

Hence,

$$\operatorname{Im}(\omega)\subset\mathcal Q\quad\Longrightarrow\quad \operatorname{Im}(\xi)\subset\mathcal Q.$$

Consequently, for all $x,y\in[a,b]$ we have

$$K(x,y)f'_{}(\xi(y))\in[\mathcal M_1^{}, \mathcal M_2^{}],\quad\text{where}\quad \mathcal M_1^{}=m\sigma_1^{}\gamma_1^{}\varrho_1^{\sigma_1-1},\quad \mathcal M_2^{}=M\sigma_2^{}\gamma_2^{}\varrho_2^{\sigma_2-1}.$$

Thus, the following inequality holds:

$$|2 K(x,y)f'(\xi(y))-(\mathcal M_1+\mathcal M_2)|\leqslant \mathcal M_2-\mathcal M_1.$$

We multiply both sides by $|\omega(y)|$:

$$|2 K(x,y)f'(\xi(y))\omega(y)-(\mathcal M_1+\mathcal M_2)\omega(y)|\leqslant (\mathcal M_2-\mathcal M_1)|\omega(y)|.$$

After integrating both sides, we have

$$ \biggl|\omega(x)-\frac{\mathcal M_1+\mathcal M_2}{2}\int_a^{b}\omega(y)\,dy\biggr|\leqslant\frac{\mathcal M_1-\mathcal M_2}{2}(b-a)\|\kern1pt\omega\|.$$

(2.16)

Now, we use Lemma 3.9 in [11] i.e., assume that the function $\zeta\in C[a,b]$ changes sign on $[a,b]$. For every $r\in\mathbb{R}$, we then have the inequality $2\|\zeta-r\|\geqslant\|\zeta\|$. We choose

$$r=\frac{\mathcal M_1+\mathcal M_2}{2}\int_a^{b}\omega(y)\,dy,$$

and it then follows from Lemma 3.9 that

$$ \max_{x\in[0,1]}\biggl|\kern1pt\omega(x)- \frac{\mathcal M_1+\mathcal M_2}{2}\int_a^{b}\omega(y)\,dy\biggr|\geqslant\frac{1}{2}\|\kern1pt\omega\|.$$

(2.17)

From (2.16) and (2.17), we derive the inequality

$$\frac{1}{2}\|\kern1pt\omega\|\leqslant \max_{x\in[0,1]}\,\biggl|\kern1pt\omega(x)-\frac{\mathcal M_1+\mathcal M_2}{2}\int_a^{b}\omega(y)\,dy\kern1pt\biggr|\leqslant \frac{\mathcal M_1-\mathcal M_2}{2}(b-a)\|\kern1pt\omega\|.$$

Now, if $\mathcal M_2-\mathcal M_1<1/(b-a)$, then the operator $A\widetilde N_f$ has exactly one positive fixed point. The last inequality is equivalent to condition 3 of the theorem.

Example 3.

For every $k\in\mathbb{N}$, we consider the integral operator $H_{k}$ acting in the cone $C^{+}[0,1]$ as

$$ (H_{k}\varphi)(t)=\int^1_{0} K(t,u)\varphi^{k}(u)du,\qquad k\in\mathbb{N}.$$

(2.18)

From [11]–[13], we know that translation-invariant Gibbs measures of models with an uncountable set of spin values on Cayley trees can be described by positive fixed points of $H_k$. Also, $H_k$ has at least one positive fixed point (see [12]).

We verify the conditions of Theorem 3 for the operator $H_k$.

• 1. Let $\nu\in[1,+\infty)$ and $\omega_i(x)\in C^{+}[a,b]$, $i=1,2$. Then

  $$\omega_1(x)\geqslant\nu\omega_2(x)\quad\Longrightarrow\quad\omega^k_1(x)\geqslant\nu\omega^k_2(x).$$

• 2. $f(x)=x^k$ satisfies the condition

  $$\gamma_1 x^{\sigma_1}\leqslant x^k\leqslant\gamma_2 x^{\sigma_2},\qquad \gamma_i>0,\quad \sigma_i>1,\quad i=1,2,$$

  and $x^k-\gamma_1 x^{\sigma_1}$, $\gamma_2 x^{\sigma_2}-x^k$ are monotone nondecreasing functions if we choose $\gamma_1=\gamma_2=1$ and $\sigma_1=\sigma_2=k$.

• 3. For the maximum $M$ and minimum $m$ values of the kernel of $H_k$, the following inequality holds:

  $$ \mathfrak{I}_2(M,m)-\mathfrak{I}_1(m,M)<1,$$

  (2.19)

  where

  $$\mathfrak{I}_i(x,y)=x\sigma_i\gamma_i\biggl(\frac{x}{y}(m\gamma_{3-i}(b-a))^{1/(1-\sigma_{3-i})}\biggr)^{\!\sigma_i-1}=\frac{x^k}{y^k}.$$

Hence, we arrive at the following result (the main theorem in [11]).

Corollary 1.

Let $k\geqslant 2$ . If the kernel $K(t,u)$ satisfies the condition

$$ \biggl(\frac{M}{m}\biggr)^{\!k}-\biggl(\frac{m}{M}\biggr)^{\!k}<\frac{1}{k},$$

(2.20)

then the operator $H_k$ has a unique fixed point in $C^{+}[0,1]$ .

Example 4.

In [11]–[13], the Ising model with nearest-neighbor interactions with the spin space $[0,1]$ on the Cayley tree of order $k\in\mathbb{N}$ was considered. It was proved that the splitting Gibbs measures of the model can be described by positive fixed points of a Hammerstein integral operator, i.e.,

$$ (H_k\omega)(x):=\int_0^1 e^{\vartheta xy}\omega^k(y)\,dy=\omega(x),\qquad \vartheta\in\mathbb{R}.$$

(2.21)

It is easy verify using Proposition 2 that Eq. (2.21) has at least one positive fixed point. Because ${f(\omega(x))=\omega^{k}(x)}$, the first and second conditions of Theorem 3 are automatically satisfied. In this case, the third condition of Theorem 3 is equivalent to the condition

$$\vartheta\in(-\infty,\vartheta_{\mathrm{cr}}),\qquad \vartheta_{\mathrm{cr}}=\frac{1}{k}\ln\frac{1+\sqrt{4k^2+1}}{2k}.$$

In the language of the theory of Gibbs measures, $\vartheta:=J/T$, where $T>0$ is the temperature and $J\in\mathbb{R}$. If $J>0 (J<0)$, then the Ising model is called ferromagnetic (antiferromagnetic). We can deduce the following result from Theorem 3.

Corollary 2.

For the Ising model with spin values in $[0,1]$ on the Cayley tree of order $k$ , the following statements hold:

• 1) there is a unique translation-invariant splitting Gibbs measure for the antiferromagnetic Ising model;

• 2) if the temperature $T$ satisfies the condition $T\geqslant J/\vartheta_{\mathrm{cr}}$ , then there is a unique translation-invariant splitting Gibbs measure for the ferromagnetic Ising model.

We note that we consider one of the classical models on Cayley trees, but Theorem 3. can be used for other model Hamiltonians on Cayley trees.

Conflict of interest

The author declares no conflicts of interest.