A fixed point theorem for non-negative functions

1.   Introduction

The theory of fixed points constitutes one of the important topics in pure and applied mathematics. Indeed, the most results related to the existence of solutions for nonlinear problems arising in physics and engineering are based on the use of certain fixed point theorems. For some contributions related to the applications of the theory of fixed points, we refer to the series of papers ^[1,2,3].

One of the most important fixed point results is the Banach fixed point theorem ^[4], which states that; if $(M, d)$ is a complete metric space and $f: M\to M$ is a mapping satisfying the inequality

for every $u, v\in M$, where $k\in (0, 1)$ is a constant, then $f$ possesses a unique fixed point, and for every $u_0\in M$, the Picard sequence $u_{n+1} = f(u_n)$ converges to this fixed point. A mapping $f$ satisfying (1.1) is called a contraction on $M$. The literature includes several generalizations and extensions of Banach's fixed point theorem. Some of them are concerned with the study of fixed points for mappings satisfying various kinds of contractions. For instance, Boyd and Wong ^[5] considered nonlinear contractions involving a function $\varphi: [0, \infty)\to [0, \infty)$ satisfying certain conditions. Reich ^[6,7] studied a contraction of the form

where $k_1, k_2, k_3\geq 0$ and $k_1+k_2+k_3 < 1$. Ćirić ^[8] proposed a contraction of the form

where $k\in (0, 1)$. Further contributions for other kinds of contractions can be found in ^[9,10,11]. Other fixed point results were obtained when the underlying set is equipped with a generalized metric such as $b$-metric spaces ^[12], rectangular metric spaces ^[13], $G$-metric spaces ^[14], partial metric spaces ^[15], $JS$-metric spaces ^[16], supra-metric spaces ^[17], Hemi metric spaces ^[18], and fractional metric spaces ^[19].

Another category of fixed point results is concerned with the study of fixed points for mappings satisfying functional inequalities. For instance, in the monograph ^[20], Guo et al. considered the class of mappings $f: {\rm{Int}}(P)\to {\rm{Int}}(P)$ satisfying the inequality

where $\xi\in (0, 1)$ is a constant. Here, $P$ is a normal solid cone of a Banach space $E$ and ${\rm{Int}}(P)$ is the interior of $P$. Namely, it was proved that, if $f$ is an increasing operator (with respect to the partial order induced by the cone $P$), then $f$ possesses a unique fixed point. Moreover, for any $x_0\in {\rm{Int}}(P)$, the sequence $x_{n+1} = f(x_{n})$, $n\geq 0$, converges to this fixed point. Other results that belong to the same category can be found in ^[21,22,23]. In the mentioned contributions, the mapping $f$ is always supposed to be monotone or mixed monotone with respect to the partial order induced by the cone.

The present contribution belongs to the category of fixed point results for mappings satisfying functional inequalities. Our main idea is motivated by the following example. Let us consider the function $f: [0, 1]\to [0, 1]$ defined by

Let $d$ be the standard metric on $[0, 1]$, that is,

It is clear that $0$ is the unique fixed point of $f$. However, there is no $k\in [0, 1)$ such that $f$ satisfies (1.1) for every $u, v\in [0, 1]$. This can be easily seen by remarking that $f$ is not continuous at $1$ (with respect to the metric $d$). Then, the Banach fixed theorem is not applicable in this case. On the other hand, the function $f$ satisfies an interesting property. Indeed, if $\alpha\in (0, 1)$, then for every $t, s\in [0, 1]$, we have

So, a natural question arises: Is it possible to obtain suitable conditions under which a function $f$ satisfying inequalities of type (1.2) possesses a unique fixed point? The aim of this work is to investigate this question. Namely, we are concerned with the class of real-valued functions $f: C\to C$ satisfying the inequality

for every $t, s\in C$ with $f(t)\neq f(s)$. Here, $C\subset [0, \infty)$, $\alpha, \sigma\in (0, 1)$ are constants, and

is a function satisfying $\inf_{t > 0} \frac{\ell(t)}{t^\rho} > 0$ for some constant $\rho > 0$. Notice that in the special case $\ell(t) = t$, the above inequality reduces to (1.2) with $\sigma = \frac{1}{2}$. First, we establish a fixed point theorem for the above class of functions. Next, we discuss the particular cases when $C$ is an interval and $\ell$ is a convex or concave function. Namely, making use of Hermite-Hadamard inequalities, we deduce several new fixed point theorems. We also provide an example where our approach can be used while the Banach fixed point theorem is not applicable.

We point out that unlike the most contributions related to the study of fixed points for mappings satisfying functional inequalities, in this paper, no monotony condition is imposed on the function $f$.

Our main result is stated and proved in Section 2. Some particular cases of our main result are studied in Section 3.

We end this section by fixing some notations that will be used throughout this paper. Let $f: C\to C$. We denote by ${\rm{Fix}}(f)$ the set of fixed points of $f$, that is,

For $t\in C$, we denote by $\{f^n(t)\}$ the sequence in $C$ defined by

2.   Main result

For $\rho > 0$, let us denote by $L_\rho$ the set of functions $\ell:[0, \infty)\to [0, \infty)$ satisfying the condition

For a nonempty subset $C\subset [0, \infty)$, $\rho > 0$, $\ell\in L_\rho$, and $\alpha, \sigma\in (0, 1)$, let $\mathbb{F}_C(\rho, \ell, \sigma, \alpha)$ be the set of functions $f: C\to C$ satisfying the inequality

for every $t, s\in C$ with $f(t)\neq f(s)$.

Our main result is the following fixed point theorem.

Theorem 2.1. Let $C\subset [0, \infty)$, $C\neq \emptyset$, and $f: C\to C$. Assume that the following conditions hold:

(ⅰ) $C$ is a closed subset;

(ⅱ) $f\in \mathbb{F}_C(\rho, \ell, \sigma, \alpha)$ for some $\rho > 0$, $\ell\in L_\rho$, and $\alpha, \sigma\in (0, 1)$;

(ⅲ) For every $t, s\in C$, if $\lim\limits_{n\to \infty}f^n(t) = s$, then $\{f^n(t)\}$ admits a subsequence $\{f^{n_q}(t)\}$ such that $\lim\limits_{q\to \infty}f(f^{n_q}(t)) = f(s)$.

Then, $f$ possesses a unique fixed point. Moreover, for every $t_0\in C$, the sequence $\{f^n(t_0)\}$ converges to this fixed point.

Proof. The first step of the proof is to show that ${\rm{Fix}}(f)\neq \emptyset$. Indeed, for an arbitrary $t_0\in C$, let $\{t_n\}\subset C$ be the sequence defined by

that is,

If for some $k\geq 0$, we have $t_k = t_{k+1}$, then $t_k\in {\rm{Fix}}(f)$ and ${\rm{Fix}}(f)\neq \emptyset$. So, we may assume that

which implies that

Then, by (ⅱ), taking $(t, s) = (t_0, t_1)$ in (2.2), we obtain

that is,

Similarly, (2.2) with $(t, s) = (t_1, t_2)$ yields

that is,

Then, from (2.3) and (2.4), we deduce that

Repeating the same argument as above, we obtain by induction that

On the other hand, by (2.1), for all $n\geq 0$, we have

which implies by (2.5) that

We now discuss two cases.

Case 1: If $t_n > t_{n+1}$ for some $n\geq 0$.

In this case, we obtain

which implies that

Case 2: If $t_n < t_{n+1}$ for some $n\geq 0$.

In this case, we obtain

which implies that

Consequently, in both cases, we have

where

Thus, by (2.6), we obtain

that is,

where $\sigma_\rho = \sigma^{\frac{1}{\rho}}\in (0, 1)$. Hence, for all $n\geq 0$ and $m\geq 1$, using (2.7), we obtain

Since $\sigma_\rho^n\to 0$ as $n\to \infty$, it holds that $\{t_n\}$ is a Cauchy sequence. Thus, there exists $t^*\geq 0$ such that

Furthermore, since $C$ is closed and $\{t_n\}\subset C$, then

Next, by (ⅲ), (2.8), and (2.9), we deduce that $\{f^n(t_0)\}$ admits a subsequence $\{f^{n_q}(t_0)\}$ such that

In view of (2.8) and (2.10), we obtain $t^* = f(t^*)$, that is, $t^*\in {\rm{Fix}}(f)$.

The second step is to show that $t^*$ is the unique fixed point of $f$. We use the contradiction assuming that there exists $s^*\in {\rm{Fix}}(f)$ such that $t^*\neq s^*$ (or, equivalently, $f(t^*)\neq f(s^*)$). Then, using (2.2) with $(t, s) = (t^*, s^*)$, we obtain

that is,

On the other hand, if

then $t^* = s^* = 0$, which is impossible, since $t^*\neq s^*$. Then,

which implies by (2.1) that

Hence, dividing (2.11) by $\ell\left(\alpha t^*+(1-\alpha)s^*\right)$, we reach a contradiction with $\sigma < 1$. Then, $t^*$ is the unique fixed point of $f$. The proof of Theorem 2.1 is completed. □

3.   Some special cases

It can be easily seen that, if $f$ is continuous on $C$, then condition (ⅲ) of Theorem 2.1 is satisfied. Then, from Theorem 2.1, we deduce the following result.

Corollary 3.1. Let $C\subset [0, \infty)$, $C\neq \emptyset$, and $f: C\to C$. Assume that the following conditions hold:

(ⅰ) $C$ is a closed subset;

(ⅱ) $f\in \mathbb{F}_C(\rho, \ell, \sigma, \alpha)$ for some $\rho > 0$, $\ell\in L_\rho$, and $\alpha, \sigma\in (0, 1)$;

(ⅲ) $f$ is continuous on $C$.

Then, $f$ possesses a unique fixed point. Moreover, for every $t_0\in C$, the sequence $\{f^n(t_0)\}$ converges to this fixed point.

Next, we consider the case when $C\subset [0, \infty)$ is a closed interval and $\ell$ is a convex or concave function on $C$. Just before, we recall the following lemma (see, e.g., ^[24]).

Lemma 3.1. (Hermite-Hadamard inequalities) Let $C\subset [0, \infty)$ be an interval.

($\iota$) If $\ell: C\to \mathbb{R}$ is a convex function, then for all $a,b\in C$ with $a<b$, we have

($\iota\iota$) $ If $\ell: C\to \mathbb{R}$ is a concave function, then for all $a,b\in C$ with $a<b$, we have

Corollary 3.2. Let $C\subset [0, \infty)$, $C\neq \emptyset$, and $f: C\to C$. Assume that the following conditions hold:

(ⅰ) $C$ is a closed interval;

(ⅱ) There exist $\rho > 0$ and $\ell\in L_\rho$ such that $\ell$ is convex on $C$;

(ⅲ) There exist $\alpha, \sigma\in (0, 1)$ such that the inequality

holds for every $t, s\in C$ with $f(t)\neq f(s)$;

(ⅳ) $f$ is continuous on $C$.

Then, $f$ possesses a unique fixed point. Moreover, for every $t_0\in C$, the sequence $\{f^n(t_0)\}$ converges to this fixed point.

Proof. By the convexity of $\ell$ on $C$, for every $t, s\in C$ with $f(t)\neq f(s)$, we obtain

which implies by (3.1) that

Consequently, $f\in \mathbb{F}_C(\rho, \ell, \sigma, \alpha)$. Then, applying Corollary 3.1, we obtain the desired result. □

Corollary 3.3. Let $C\subset [0, \infty)$, $C\neq \emptyset$, and $f: C\to C$. Assume that the following conditions hold:

(ⅰ) $C$ is a closed interval;

(ⅱ) There exist $\rho > 0$ and $\ell\in L_\rho$ such that $\ell$ is convex on $C$;

(ⅲ) There exist $\alpha, \sigma\in (0, 1)$ such that the inequality

holds for every $t, s\in C$ with $f(t)\neq f(s)$;

(ⅳ) $f$ is continuous on $C$.

Then, $f$ possesses a unique fixed point. Moreover, for every $t_0\in C$, the sequence $\{f^n(t_0)\}$ converges to this fixed point.

Proof. Since $\ell$ is convex on $C$, by Lemma 3.1 ($\iota$), for every $t, s\in C$ with $f(t)\neq f(s)$, we have

which implies by (3.2) that

This shows that $f$ satisfies (2.2) with $\alpha = \frac{1}{2}$, that is, $f\in \mathbb{F}_C(\rho, \ell, \sigma, \frac{1}{2})$. Then, applying Corollary 3.1, we obtain the desired result. □

Corollary 3.4. Let $C\subset [0, \infty)$, $C\neq \emptyset$, and $f: C\to C$. Assume that the following conditions hold:

(ⅰ) $C$ is a closed interval;

(ⅱ) There exist $\rho > 0$ and $\ell\in L_\rho$ such that $\ell$ is concave on $C$;

(ⅲ) There exist $\alpha, \sigma\in (0, 1)$ such that the inequality

holds for every $t, s\in C$ with $f(t)\neq f(s)$;

(ⅳ) $f$ is continuous on $C$.

Then, $f$ possesses a unique fixed point. Moreover, for every $t_0\in C$, the sequence $\{f^n(t_0)\}$ converges to this fixed point.

Proof. By the concavity of $\ell$ on $C$, for every $t, s\in C$ with $f(t)\neq f(s)$, we obtain

which implies by (3.3) that

This shows that $f\in \mathbb{F}_C(\rho, \ell, \sigma, \alpha)$. Then, applying Corollary 3.1, we obtain the desired result. □

Corollary 3.5. Let $C\subset [0, \infty)$, $C\neq \emptyset$, and $f: C\to C$. Assume that the following conditions hold:

(ⅰ) $C$ is a closed interval;

(ⅱ) There exist $\rho > 0$ and $\ell\in L_\rho$ such that $\ell$ is concave on $C$;

(ⅲ) There exist $\alpha, \sigma\in (0, 1)$ such that the inequality

holds for every $t, s\in C$ with $f(t)\neq f(s)$;

(ⅳ) $f$ is continuous on $C$.

Then, $f$ possesses a unique fixed point. Moreover, for every $t_0\in C$, the sequence $\{f^n(t_0)\}$ converges to this fixed point.

Proof. Since $\ell$ is concave on $C$, by Lemma 3.1 ($\iota\iota$), for every $t, s\in C$ with $f(t)\neq f(s)$, we have

which implies by (3.4) that

This shows that $f\in \mathbb{F}_C(\rho, \ell, \sigma, \frac{1}{2})$. Then, applying Corollary 3.1, we obtain the desired result. □

We now give an example to illustrate Theorem 2.1.

Example 3.1. Let $C = \{0,1,3,5\}$ and $f: C\to C$ be the function defined by

Observe that

which shows that there is no $k\in (0, 1)$ such that

for every $t, s\in C$. Then, the Banach fixed point theorem is not applicable in this example.

We now introduce the function $\ell: [0, \infty)\to [0, \infty)$ defined by

It can be easily seen that

which shows that $\ell\in L_1$. On the other hand, for all $t, s\in C$, we have

where

and

Let $(t, s)\in U$. If $(t, s) = (0, 1)$, then

If $(t, s) = (0, 5)$, then

If $(t, s) = (1, 3)$, then

If $(t, s) = (1, 5)$, then

If $(t, s) = (3, 5)$, then

The above calculations show that for all $(t, s)\in U$, we have

Notice that by symmetry, the above inequality holds also for all $(t, s)\in U'$. Consequently, $f$ satisfies (2.2) for every $t, s\in C$ with $f(t)\neq f(s)$, where $\alpha = \frac{1}{2}$, $\sigma = \frac{7}{9}$, and $\ell$ is the mapping defined above, that is, $f\in \mathbb{F}_C\left(1, \ell, \frac{7}{9}, \frac{1}{2}\right)$. Furthermore, for all $n\geq 3$, we have

This shows that condition (ⅲ) of Theorem 2.1 is satisfied. Thus, Theorem 2.1 applies. Notice that ${\rm{Fix}}(f) = \{0\}$, which confirms Theorem 2.1.

4.   Conclusions

We introduced the new class of functions $f: C\to C$ satisfying the functional inequality

for every $t, s\in C$ with $f(t)\neq f(s)$, where $C$ is a closed subset of $[0, \infty)$, $\alpha, \sigma\in (0, 1)$ are constants, and $\ell: [0, \infty)\to [0, \infty)$ is a function satisfying the condition $\inf_{t > 0} \frac{\ell(t)}{t^\rho} > 0$ for some constant $\rho > 0$. We proved that, if $f$ is continuous (or, more generally, $f$ satisfies condition (ⅲ) of Theorem 2.1), then $f$ possesses a unique fixed point. Moreover, for any $t_0\in C$, the Picard sequence $\{f^n(t_0)\}$ converges to this fixed point. Next, making use of the Hermite-Hadamard inequalities, we deduced from our main result new fixed point theorems in the special cases when $f$ is a convex or concave function.

The proposed approach needs to be more developed. For instance, the following issues deserve to be studied:

Ⅰ. The study of fixed points for mappings $f: C\to C$ satisfying new functional inequalities of type (2.2). For instance, one can study the possibility of extending the obtained results to functions $f$ satisfying the inequality

for every $t, s\in C$ with $f(t)\neq f(s)$, where $\sigma_1+\sigma_2+\sigma_3 > 0$.

Ⅱ. The extension of the obtained results from $\mathbb R$ to a Banach space $E$ partially ordered by a cone $P$. Indeed, one can study the class of mappings $f: C\to C$ satisfying the functional inequality

for every $u, v\in C$ with $f(u)\neq f(v)$, where $C\subset P$, $\sigma\in (0, 1)$, $\ell: P\to P$, and $\preceq$ is the partial order induced by the cone $P$.

Ⅲ. It would be interesting to study the possible applications of the obtained results.

Author contributions

Hassen Aydi: Conceptualization, methodology, investigation, formal analysis, writing review and editing; Bessem Samet: Conceptualization, methodology, validation, investigation, writing original draft preparation; Manuel De La Sen: Methodology, validation, formal analysis, investigation, writing review and editing. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

B. Samet is supported by Researchers Supporting Project number (RSP2024R4), King Saud University, Riyadh, Saudi Arabia. M. De La Sen is supported by the project: Basque Government IT1555-22.

Conflict of interest

The authors declare no conflict of interest.