On Fixed Point for Derivative of Set-Valued Functions

Abstract

In this article, we showed the existence of a fixed point for the derivative of interval-valued functions. The investigation of the existence of such fixed points utilizes the common fixed point concepts for two mappings. Under the condition of compatibility of the hybrid composite mappings in the sense of the Pompei-Hausdorff metric the existence of a fixed point for the derivative is shown. Some examples to support the usability of the result of this study are also given.

1. Introduction

E. Dyer in [1] conjectured that $f$ and $g$ must have a common fixed point in $\left[0,1\right]$ if $f\left(g\left(t\right)\right)=g\left(f\left(t\right)\right)$ for each $t\in \left[0,1\right]$. In 1967, W.M. Boyce [2] replied in his paper that Dyer’s question is negative as well as an answer from Husein [3] and Singh [4]. However, many researchers are curious about conjecture. In 1976, G Jungck [5] shows the existence of the common fixed point for two mappings by the commuting mapping method in general metric spaces. Since then the common fixed point research had quickly grown. In development, some of the researchers not only involved two mappings (single-valued mappings) but also they are more than it is [6]. In fact, some involve the set-valued mapping forms [7, 8].

In progress, the composition mappings are discussed not only between fellow of single-valued mappings or set-valued mappings but also its combination (mixed compositions between of single-valued and set-valued mappings). Since then several authors have studied common fixed point theorems for such mapping in different ways ([9, 10, 11] and references therein).

Itoh et al. [12] introduce “commute” term of hybrid composite functions in 1977. By this properties, they have proven common fixed point theorems in topological vector spaces. In 1982, Fisher [13] has introduced common fixed point theorems for commuting mappings in the sense of the other in metric spaces. Then Imdad [14] mentioned the properties $\mathit{fFx}\subseteq \mathit{Ffx}$ as “quasi-commute” to distinguish with the latter term. Whereas two commuting mappings $F$ and $f$ are weakly commuting, but in general two weakly commuting mappings do not commute as it is shown in Example 1 of [15].

In 1989, Kaneko [16] introduced the concept of “compatible” by using the Hausdorff metric and proved the existence of a common fixed point theorem by the concept. In 1993, Jungck [17] introduced the same things but used the concept of “$\delta$-compatible” mappings in metric spaces and proved some common fixed point theorems for $\delta$-compatible mappings.

Regarding the fixed point for derivatives has been observed by M. Elekes at all in [18]. In his paper, he shows the compositions of two functions derivatives have fixed points. This result is an affirmative answer to a question of K. Ciesielski, whether the composition of two derivatives on interval closed has a fixed point? The fixed point for a function is usual but for its derivatives is another something. Here have we the quadruplets $\left(X,x,f,f^{\prime }\right)$. How do these problems? By the device of commutativity and compatibility between the function and its derivatives, the author shows that the function derivatives of the real-valued function have a fixed point [19].

Motivated by the results mentioned above, in this article, we introduced the existence theorem of a fixed point for $\mathit{gh}$-derivative of the interval-valued function. To this work, we used hybrid composite mappings involving $\mathit{gh}$-derivative under the compatibility conditions.

2. gh-Differences

Suppose $\left(X,d\right)$ is a metric spaces. The collection of all non-empty subsets of $X$ is denoted by ${\mathcal{P}}_0\left(X\right)$. Whereas, the notation $\mathcal{B}\left(X\right)$ (resp. $\mathcal{C}\mathcal{B}\left(X\right)$, $\mathcal{K}\left(X\right)$ and $\mathcal{KC}\left(X\right)$) is the collection of all non-empty bounded (resp. closed-bounded, compact and compact-convex) subsets of $X$.

In 1905, In his PhD thesis [20], Pompeiu defined the notions of $e^{\prime }\mathit{\text{cart}}$ between two sets. Hausdorff [21] studies the notion of set distance in the natural setting of metric spaces and with a small modification (the the sum is replaced by the maximum).

Let $\left(X,d\right)$ be a metric spaces and $A,B\subset X$. The Hausdorff distance between $A$ and $B$ is a distance function $H:{\mathcal{P}}_0\left(X\right)\times {\mathcal{P}}_0\left(X\right)\to {\mathbb{R}}^{+}$ which is defined as

$$H\left(A,B\right)=sup\left\{d\left(A,B\right),d(B,A)\right\},$$E1

where $d\left(A,B\right)={sup}_{a\in A}d\left(a,B\right)$. Certainly value that $d\left(A,B\right)\ne d\left(B,A\right)$. The distance functions $H$ to be a metric on the collection of all non-empty closed-bounded subset of $X$, $\mathcal{C}\mathcal{B}\left(X\right)$. The metric spaces $\left(\mathcal{C}\mathcal{B}\left(X\right),H\right)$ is called a complete metric spaces if the metric space $X$ is a complete.

Suppose $\mathcal{I}\left(\mathbb{R}\right)=\left\{I=\left[a^{-},a^{+}\right]|a^{-},a^{+}\in \mathbb{R},a^{-}<a^{+}\right\}$. In [22], R.E. Moore et al. introduced an absolute value of the interval $J=\left[x^{-},x^{+}\right]$ asfollows.

$$\Vert J\Vert =max\left\{|x^{-}|,|x^{+}|\right\}.$$E2

For a given interval $I=\left[a^{-},a^{+}\right]$ define the width, midpoint and radius of $I$, respectively, by

$$w\left(I\right)=a^{+}-a^{-},m\left(I\right)=\frac{1}{2}\left(a^{-}+a^{+}\right),\mathit{\text{and}}r\left(I\right)=\frac{1}{2}\left(a^{+}-a^{-}\right)\ge 0,$$E3

so that $a^{-}=m\left(I\right)-r\left(I\right)$ and $a^{+}=m\left(I\right)+r\left(I\right)$. Thus the interval notasion $I=\left[a^{-},a^{+}\right]$ can be written as the pair $I=\left(m\left(I\right);r\left(I\right)\right).$.

The Pompeiu-Hausdorff distance on $\mathcal{I}\left(\mathbb{R}\right)$ defined as

$$H\left(I,J\right)=max\left\{|a^{-}-b^{-}|,|a^{+}-b^{+}|\right\},$$E4

where $I=\left[a^{-},a^{+}\right]$ and $J=\left[b^{-},b^{+}\right]$. The pair $\left(\mathcal{I}\left(\mathbb{R}\right),H\right)$ is a complete and separable metric space.

In 1967, M. Hukuhara [23] introduced the difference ($h$-difference) between $U$ and $V$ defined as $U\stackrel{h}{-}V=W$ if and only if $U=V+W$ for each $U,V,W\in \mathcal{KC}\left({\mathbb{R}}^k\right)$. An important properties of the Hukuhara difference is that $U\stackrel{h}{-}U=\left\{\mathrm{\Theta }\right\}$ and $\left(U+V\right)\stackrel{h}{-}V=U$. The Hukuhara difference is unique, but it does not always exists.

The Hukuhara difference had generalized by Markov in [24]. He defined is following as

$$U\stackrel{\mathit{gh}}{-}V=W\iff \left(\mathbf{a}\right)U=V+W\mathit{\text{or}}\left(\mathbf{b}\right)V=U+\left(-1\right)W.$$E5

Furthermore, Hukuhara difference generalized is called the $\mathit{gh}$-difference.

Both the equation $U=V+W$ and the equation $V=U+\left(-1\right)W$ can simultaneously holds. It is clear that $h$-difference is part of $\mathit{gh}$-difference. Therefore, the $\mathit{gh}$-difference is often said to be a generalization of the $h$-difference. The $\mathit{gh}$-difference of two intervals in $\mathcal{I}\left(\mathbb{R}\right)$ always exists.

Proposition 1. Suppose $I=\left[a^{-},a^{+}\right]$ and $J=\left[b^{-},b^{+}\right]$ are intervals in $\mathcal{I}\left(\mathbb{R}\right)$. The $\mathit{gh}$-difference of two intervals $I$ and $J$ always exists and

$$I\stackrel{\mathit{gh}}{-}J=\left[a^{-},a^{+}\right]\stackrel{\mathit{gh}}{-}\left[b^{-},b^{+}\right]=\left[c^{-},c^{+}\right]$$E6

where $c^{-}=min\left\{\left(a^{-}-b^{-}\right),\left(a^{+}-b^{+}\right)\right\}$ and $c^{+}=max\left\{\left(a^{-}-b^{-}\right),\left(a^{+}-b^{+}\right)\right\}$.

In [25, 26] defined $H\left(I,J\right)=\Vert I\stackrel{\mathit{gh}}{-}J\Vert$ for each $I,J\in I\left(\mathbb{R}\right)$. An immediate property of the $\mathit{gh}$-difference for $I,J\in I\left(\mathbb{R}\right))$ is

$$H\left(I,I\right)=0\iff I\stackrel{\mathit{gh}}{-}J=0\iff I=J$$E7

It is also well known that $\left(I\left(\mathbb{R}\right),H\right)$is complete metric space.

2.1 gh-Derivative of set-valued functions

The mapping $F:X\to {\mathcal{P}}_0\left(Y\right)$ is called set-valued functions where the maps $F\left(x\right)\in {\mathcal{P}}_0\left(Y\right)$ for each $x\in X$. The function $f:X\to Y$ is said to be selection of $F$ if $f\left(x\right)\in F\left(x\right)$ for all $x\in X$. We say that a point $z\in X$ is a fixed point of $F$ if $z\in F\left(z\right)$.

The $\mathit{gh}$-derivative for an interval-valued function, expressed in terms of the difference quotient by $\mathit{gh}$-difference, has been first introduced in 1979 by S. Markov. A very recent and complete description of the algebraic properties of $\mathit{gh}$-derivative can be found in [27].

Definition 1 Let $F:\left[a,b\right]\to \mathcal{I}\left(\mathbb{R}\right)$ be an interval-valued function and suppose $t_0,t_0+h\in \left(a,b\right)$. The $\mathit{gh}$-derivative $F_{\mathit{gh}}^{\prime }\left(t_0\right)\in \mathcal{I}\left(\mathbb{R}\right)$ defined as

$$F_{\mathit{gh}}^{\prime }\left(t_0\right)=\underset{h\to 0}{lim}\frac{F\left(t_0+h\right)\stackrel{\mathit{gh}}{-}F\left(t_0\right)}{h}.$$E8

If the limit, ${lim}_{h\to 0}\frac{F\left(t_0+h\right)\stackrel{\mathit{gh}}{-}F\left(t_0\right)}{h}$ exists and satisfies Eq. (6), then $F$ is said differentiable in the sense of generalized Hukuhara difference or $\mathit{gh}$-differentiable at a point $t_0\in \left(a,b\right)$. The set-valued $F_{\mathit{gh}}^{\prime }$ is called a generalized Hukuhara derivative.

Theorem 1.1 If interval-valued functions $F:\left[a,b\right]\to \mathcal{I}\left(\mathbb{R}\right)$ is a $\mathit{gh}$-differentiable at a point $p\in \left(a,b\right)$ then $F$ is continuous at $p$.

Proof:

$$\begin{array}{c}\underset{x\to p}{lim}F\left(x\right)\stackrel{\mathit{gh}}{-}F\left(p\right)=\underset{x\to p}{lim}\left[\frac{F\left(x\right)\stackrel{\mathit{gh}}{-}F\left(p\right)}{\left(x-p\right)}\left(x-p\right)\right]\\ =\left[\underset{x\to p}{lim}\frac{F\left(x\right)\stackrel{\mathit{gh}}{-}F\left(p\right)}{\left(x-p\right)}\right]\left[\underset{x\to p}{lim}\left(x-p\right)\right]\\ =F_{\mathit{gh}}^{\prime }\left(p\right)\cdot 0=0.\end{array}$$

So $F$ is continuous at the point $p\in \left[a,b\right]$.

Theorem 1.2 [26] Let $F:\left[a,b\right]\to \mathcal{I}\left(\mathbb{R}\right)$ be an interval-valued functions and $F\left(x\right)=\left[f\left(x\right),g\left(x\right)\right]$, where $f,g:\left[a,b\right]\to \mathbb{R}$. $F$ is $\mathit{gh}$-differentiable on $(a,b$) if and only if $f$ and $g$ are differentiable on $\left(a,b\right)$ and

$$F_{\mathit{gh}}^{\prime }\left(x\right)=\left[min\left\{f^{\prime }\left(x\right),g^{\prime }\left(x\right)\right\},max\left\{f^{\prime }\left(x\right),g^{\prime }\left(x\right)\right\}\right],$$

for all $x\in \left(a,b\right)$.

This means that

$$F_{\mathit{gh}}^{\prime }\left(x\right)=\left\{\begin{array}{ll}\left[f^{\prime }\left(x\right),g^{\prime }\left(x\right)\right]& \text{if}{f}^{\prime }\left(x\right)<g^{\prime }\left(x\right),\\ \left[g^{\prime }\left(x\right),f^{\prime }\left(x\right)\right]& \text{if}{f}^{\prime }\left(x\right)<g^{\prime }\left(x\right)\end{array}\right.$$

for all $x\in \left(a,b\right)$.

3. Common fixed point

Definition 2 Suppose $\left(X,d\right)$ is a metric space, $E\subset X$, $F:E\to \mathcal{B}\left(X\right)$ is a set-valued mapping and $f:E\to X$ is single-valued mapping.

1. $F$ and $f$ are said to quasi commute if $\mathit{fFx}\subseteq \mathit{Ffx}$ for each $x\in E$

2. $F$ and $f$ are said to commute if $\mathit{fFx}=\mathit{Ffx}$ for each $x\in E$

3. $F$ and $f$ are said to slightly commute if $\mathit{fFx}\in \mathcal{B}\left(X\right)$ for each $x\in E$ and $\delta \left(\mathit{fFx},\mathit{Ffx}\right)\le max\left\{\delta \left(\mathit{fx},\mathit{Fx}\right),\mathit{\text{diam}}\left(\mathit{Fx}\right)\right\}$

4. $F$ and $f$ are said to weakly commute if $\mathit{fFx}\in \mathcal{B}\left(X\right)$ for each $x\in E$ and $\delta \left(\mathit{fFx},\mathit{Ffx}\right)\le max\left\{\delta \left(\mathit{fx},\mathit{Fx}\right),\mathit{\text{diam}}\left(\mathit{fFx}\right)\right\}$.

Suppose $\left(X,d\right)$ is a metric space. The mapping $f,g:X\to X$ is a single-valued function or function and $F,G:X\to \mathcal{B}\left(X\right)$ is a set-valued function. For each $x,y\in X$, we used the notation as follows.

$$\mathcal{M}\left(F,f\right)=max\left\{d\left(\mathit{fx},\mathit{Fx}\right),d\left(\mathit{fy},\mathit{Fy}\right),d\left(\mathit{fx},\mathit{Fy}\right),d\left(\mathit{fy},\mathit{Fx}\right),d\left(\mathit{fx},\mathit{fy}\right)\right\}.$$E9

and

$$\mathcal{N}\left(F,f\right)=max\left\{d\left(\mathit{fx},\mathit{fy}\right),d\left(\mathit{fx},\mathit{Fx}\right),d\left(\mathit{fy},\mathit{Fy}\right),\frac{1}{2}\left[d\left(\mathit{fx},\mathit{Fy}\right)+d\left(\mathit{fy},\mathit{Fx}\right)\right]\right\}.$$E10

and

$$\mathcal{M}\left(F,G,f,g\right)=max\left\{d\left(\mathit{fx},\mathit{gy}\right),\delta \left(\mathit{fx},\mathit{Gy}\right),\delta \left(\mathit{gy},\mathit{Fx}\right)\right\}.$$E11

The following is the existence of common fixed point theorem that result by B. Fisher [13].

Theorem 1.3 Suppose $\left(X,d\right)$ is a complete metric space, $F:X\to \mathcal{B}\left(X\right)$ is a set-valued mapping and $f:X\to X$ is a single-valued mapping satisfying the inequality

$$\delta \left(\mathit{Fx},\mathit{Fy}\right)\le c\mathcal{M}\left(F,f\right)$$E12

for all $x,y\in X$, where $0\le c<1$. If.

1. $f$ is continuous,

2. $F\left(X\right)\subseteq f\left(X\right)$, and

3. $F$ and $f$ are commute,

then $F$ and $f$ have a unique common fixed point.

B. Fisher also shown the same with assumes the continuity of $F$ in $X$ instead of the continuity of $f$ [28].

The following theorem is generalization of Theorem 1.3 that has been resulted by M Imdad et al. [14].

Theorem 1.4 Suppose $\left(X,d\right)$ is a complete metric space, $F:X\to \mathcal{B}\left(X\right)$ is a set-valued mapping and $f:X\to X$ is a single-valued mapping satisfying the inequality

$$\delta \left(\mathit{Fx},\mathit{Fy}\right)\le \psi \mathcal{M}\left(F,f\right)$$

for all $x,y\in X$, where $\psi :\left[0,\infty \right)\to \left[0,\infty \right)$ is a nondecreasing, right continuous and $\psi \left(t\right)<t$, for all $t>0$. If this following is satisfied

1. the function $f$ is continuous,

2. the image of $F\left(X\right)$ is a subset of $f\left(X\right)$,

3. the set-valued $F$ and single-valued $f$ are weakly commute, and

4. $\exists$$x_0\in X$ such that $sup\left\{\delta \left({\mathit{Fx}}_n,{\mathit{Fx}}_1\right):n=0,1\cdots \right\}<+\infty$,

then $F$ and $f$ have a unique common fixed point on $X$.

Theorem 1.5 Suppose $\left(X,d\right)$ is a complete metric space, $F:X\to \mathcal{B}\left(X\right)$ is a set-valued mapping and $f:X\to X$ is a single-valued mapping satisfying the inequality

$$\delta \left(\mathit{Fx},\mathit{Fy}\right)\le \psi \mathcal{M}\left(F,f\right)$$E13

for all $x,y\in X$, where $\psi :\left[0,\infty \right)\to \left[0,\infty \right)$ is a non-decreasing, right continuous and $\psi \left(t\right)<t$, for all $t>0$. If this following is satisfied

1. the set-valued mapping $F$ or the single-valued mapping $f$ are continuous,

2. the image $F\left(X\right)$ is a subset of the image $f\left(X\right)$,

3. the set-valued $F$ and the singel valued $f$ are slightly commute, and

4. $\exists$$x_0\in X$ such that $sup\left\{\delta \left({\mathit{Fx}}_n,{\mathit{Fx}}_1\right):n=0,1\cdots \right\}<+\infty$,

then $F$ and $f$ have a unique common fixed point on $X$.

In the other context, Kaneko and Sessa in [16] introduce the “compatibility” term for the set-valued mapping $F$ and the single-valued mapping $f$ defined as follows:

Definition 3 Let $\left(X,d\right)$ be a metric spaces. Suppose that $F:X\to \mathcal{C}\mathcal{B}\left(X\right)$ is a set-valued mapping and $f:X\to X$ is a single-valued mapping. The mappings $F$ and $f$ is called compatible if the composition $\mathit{fFx}\in \mathcal{C}\mathcal{B}\left(X\right)$ and the sequence $H\left({\mathit{Ffx}}_n,{\mathit{fFx}}_n\right)\to 0$ whenever $\left\{x_n\right\}$ is sequence in $X$ such that ${\mathit{fx}}_n\to t\in B\in \mathcal{C}\mathcal{B}\left(X\right)$ and ${\mathit{Fx}}_n\to B\in \mathcal{C}\mathcal{B}\left(X\right)$.

By using such the notion obtained the following theorem and lemma [16].

Theorem 1.6 Suppose $\left(X,d\right)$ is a complete metric space, $F:X\to \mathcal{C}\mathcal{B}\left(X\right)$ is a set-valued mapping, and $f:X\to X$ is a single-valued mapping satisfying the inequality

$$H\left(\mathit{Fx},\mathit{Fy}\right)\le c\mathcal{N}\left(F,f\right)$$E14

for all $x,y\in X$, where $0\le c<1$. If this following is satisfied.

1. the set-valued mapping $F$ and the single-valued mapping $f$ are continuous,

2. the image $F\left(X\right)$ is a subset of the image $f\left(X\right)$, and

3. the set-valued $F$ and the single-valued $f$ are compatible,

then there exists a point $z\in X$ such that $f\left(z\right)\in F\left(z\right)$.

Lemma 1 Let $\left(X,d\right)$ be a metric spaces. Suppose that $F:X\to \mathcal{C}\mathcal{B}\left(X\right)$ and $f:X\to X$ are a compatible. If $\mathit{fw}\in \mathit{Fw}$ for some $w\in X$, then $\mathit{Ffw}=\mathit{fFw}$.

4. Fixed point for derivative

In this discussion, we shall make frequent use of the following Lemmas.

Lemma 2 [29] Let $\left(X,d\right)$ be a metric spaces. If $C,D\in \mathcal{K}\left(X\right)$ and $c\in C$, then there exists the points $d\in D$ such that $d\left(c,d\right)\le H\left(C,D\right)$.

Lemma 3 (Lemma 1 [4]) Let $\psi :\left[0,\infty \right)\to \left[0,\infty \right)$ be a real function such that non-decreasing, right continuous on $\left[0,\infty \right)$.

$$\underset{n\to \infty }{lim}{\psi }^n\left(t\right)=0.$$

if and only if for every $t>0$ and $\psi \left(t\right)<t$.

In this result, we found that Lemma 1 also conversely holds provided its values of mapping are compact sets.

Lemma 4 Let $\left(X,d\right)$ be a metric spaces and the set-valued $F:X\to \mathcal{K}\left(X\right)$ is a continuous on $X$. If there exists single-valued $f:X\to X$ is continuous on $X$ such that $\mathit{fw}\in \mathit{Fw}$ for some $w\in X$, then the mappings $F$ and $f$ are compatible.

Proof: Since $\mathit{Fx}\in \mathcal{K}\left(X\right)$ for each $x\in X$ and $f$ is continuous, the composition $\mathit{fFx}\in \mathcal{K}\left(X\right)$ for all $x\in X$. Suppose that the sequence $\left\{x_n\right\}$ on $X$ such that the sequence of sets ${\mathit{Fx}}_n$ converges to $K\in \mathcal{K}\left(X\right)$ and the sequence function ${\mathit{fx}}_n$ converges to $z\in K$. In this case, we choose $z\in X$ such that $\mathit{fz}\in \mathit{Fz}$. Since $F$ and $f$ are continuous, we obtained

$$\begin{array}{c}\underset{n\to \infty }{lim}H\left({\mathit{Ffx}}_n,{\mathit{fFx}}_n\right)\le \underset{n\to \infty }{lim}\left[H\left({\mathit{Ffx}}_n,\mathit{Fz}\right)+H\left(\mathit{Fz},\left\{\mathit{fz}\right\}\right)+H\left(\left\{\mathit{fz}\right\},{\mathit{fFx}}_n\right)\right]\\ =H\left(\mathit{Fz},\mathit{Fz}\right)+H\left(\mathit{Fz},\left\{\mathit{fz}\right\}\right)+H\left(\left\{\mathit{fz}\right\},\mathit{fK}\right)\\ =0.\end{array}$$

The pairs $F$ and $f$ are proved as compatible by Definition 3.

By using the Lemma 4 we obtain the theorem as follows:

Theorem 1.7 Let $\left(X,d\right)$ be a complete metric space, $F:X\to \mathcal{K}\left(X\right)$ be a continuous. Suppose there exists $f:X\to X$ is continuous on $X$ such that $F\left(X\right)\subseteq f\left(X\right)$ and for all $x,y\in X$ satisfying the inequality

$$H\left(\mathit{Fx},\mathit{Fy}\right)\le c\mathcal{N}\left(F,f\right),$$E15

where $0\le c<1$. Then $\mathit{fz}\in \mathit{Fz}$ for some $z\in X$ if and only if the pairs $F$ and $f$ are compatible.

Proof: Let $x_0\in X$ be an arbitrary. Since $F\left(X\right)\subseteq f\left(X\right)$, we choose the point $x_1\in X$ such that ${\mathit{fx}}_1\in {\mathit{Fx}}_0$. If $c=0$, then

$$d\left({\mathit{fx}}_1,{\mathit{Fx}}_1\right)\le H\left({\mathit{Fx}}_0,{\mathit{Fx}}_1\right)=0.$$

Since ${\mathit{Fx}}_1$ is compact (hence closed), we obtain ${\mathit{fx}}_1\in {\mathit{Fx}}_1$.

Now we assume $c\ne 0$. By Lemma 2 for each $\epsilon =\frac{1}{\sqrt{c}}>1$ there exists a point $y_1\in {\mathit{Fx}}_1$ such that

$$d\left(y_1,{\mathit{Fx}}_1\right)\le H\left({\mathit{Fx}}_1,{\mathit{Fx}}_0\right)<\mathit{\epsilon H}\left({\mathit{Fx}}_1,{\mathit{Fx}}_0\right).$$

Choose $x_2\in X$ such that $y_1={\mathit{fx}}_2\in {\mathit{Fx}}_1$ and so on. In general, if $x_n\in X$ there exists $x_{n+1}\in X$ such that $y_n={\mathit{fx}}_{n+1}\in {\mathit{Fx}}_n$ and

$$d\left(y_n,{\mathit{fx}}_n\right)<\mathit{\epsilon H}\left({\mathit{Fx}}_n,{\mathit{Fx}}_{n-1}\right)$$

for each $n\ge 1$. By the inequality (10) for each $n\in \mathrm{\mathbb{N}}$ we have

$$\begin{array}{c}d\left({\mathit{fx}}_{n+1},{\mathit{fx}}_n\right)<\mathit{\epsilon H}\left({\mathit{Fx}}_n,{\mathit{Fx}}_{n-1}\right)\le \frac{c}{\sqrt{c}}\mathcal{N}\left(F,f\right)=\sqrt{c}\mathcal{N}\left(F,f\right)\\ <\sqrt{c}max\{d\left({\mathit{fx}}_n,{\mathit{fx}}_{n-1}\right),d\left({\mathit{fx}}_n,{\mathit{Fx}}_n\right),d\left({\mathit{fx}}_{n-1},{\mathit{Fx}}_{n-1}\right),\\ \frac{1}{2}\left[d\left({\mathit{fx}}_n,{\mathit{Fx}}_{n-1}\right)+d\left({\mathit{fx}}_{n-1},{\mathit{Fx}}_n\right)\right]\}.\\ <\sqrt{c}max\{d\left({\mathit{fx}}_n,{\mathit{fx}}_{n-1}\right),d\left({\mathit{fx}}_n,{\mathit{fx}}_{n+1}\right),d\left({\mathit{fx}}_{n-1},{\mathit{fx}}_n\right),\\ \frac{1}{2}d\left({\mathit{fx}}_{n-1},{\mathit{fx}}_{n+1}\right)\left]\right\}.\\ <\sqrt{c}max\{d\left({\mathit{fx}}_n,{\mathit{fx}}_{n-1}\right),d\left({\mathit{fx}}_n,{\mathit{fx}}_{n+1}\right),d\left({\mathit{fx}}_{n-1},{\mathit{fx}}_n\right),\\ \frac{1}{2}\left[d\left({\mathit{fx}}_{n-1},{\mathit{fx}}_n\right)+d\left({\mathit{fx}}_n,{\mathit{fx}}_{n+1}\right)\right]\}.\\ =\sqrt{c}max\left\{d\left({\mathit{fx}}_{n-1},{\mathit{fx}}_n\right),d\left({\mathit{fx}}_n,{\mathit{fx}}_{n+1}\right)\right\}.\\ =\sqrt{c}d\left({\mathit{fx}}_{n-1},{\mathit{fx}}_n\right)\end{array}$$

Since $\sqrt{c}<1$, the sequence $\left\{{\mathit{fx}}_n\right\}$ is a Cauchy sequence on the complete metric space $X$. Therefore, it converges to a point $z\in X$. Likewise $\left\{{\mathit{Fx}}_n\right\}$ is a Cauchy sequence on the complete metric space ($\mathcal{K}\left(X\right),H$), hence it converges to a set $K\in \mathcal{K}\left(X\right)$. As a result

$$d\left(z,K\right)\le d\left(z,{\mathit{fx}}_n\right)+d\left({\mathit{fx}}_n,K\right)\le d\left(z,{\mathit{fx}}_n\right)+H\left({\mathit{Fx}}_{n-1},K\right).$$

Certainly that $d\left(z,K\right)=0$ by $d\left(z,{\mathit{fx}}_n\right)\to 0$ and $H\left({\mathit{Fx}}_{n-1},K\right)\to 0$ as $n\to \infty$. This implies $z\in K$ since $K$ is a compact set. Since $F$ and $f$ are compatible, we have

$$\begin{array}{c}d\left(\mathit{fz},\mathit{Fz}\right)=\underset{n\to \infty }{lim}d\left(\mathit{fz},\mathit{Fz}\right)\le \underset{n\to \infty }{lim}\left[d\left(\mathit{fz},{\mathit{ffx}}_n\right)+d\left({\mathit{ffx}}_n,\mathit{Fz}\right)\right]\\ \le \underset{n\to \infty }{lim}\left[d\left(\mathit{fz},{\mathit{ffx}}_n\right)+H\left({\mathit{fFx}}_n,\mathit{Fz}\right)\right]\\ \le \underset{n\to \infty }{lim}\left[d\left(\mathit{fz},{\mathit{ffx}}_n\right)+H\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+H\left({\mathit{Ffx}}_n,\mathit{Fz}\right)\right]\\ =d\left(\mathit{fz},\mathit{fz}\right)+H\left(\mathit{Fz},\mathit{Fz}\right)\\ =0.\end{array}$$

So $\mathit{fz}\in \mathit{Fz}$. Conversely, it’s clear by Lemma.

Remark 1 Theorem 1.7 is a special occurrence of results obtained by H Kaneko and S Sessa [16]. Certainly the provisioning should be satisfied as in Theorem 3.

This result modify of Theorem 1.5 by substituting compatibility with respect to Hausdorff metric on $\mathcal{K}\left(X\right)$ for slight commutativity at once improvement Theorem 1.6 in finding common fixed point for the mapping of the hybrid composite.

Theorem 1.8 Let $\left(X,d\right)$ be a complete metric space, $F:X\to \mathcal{K}\left(X\right)$ be a set-valued mapping and $f:X\to X$ be a single-valued mapping satisfying the inequality

$$H\left(\mathit{Fx},\mathit{Fy}\right)\le \psi \mathcal{N}\left(F,f\right)$$E16

for all $x,y\in X$, where $\psi :\left[0,\infty \right)\to \left[0,\infty \right)$ is a nondecreasing, right continuous and $\psi \left(t\right)<t$, for all $t>0$. If this following is satisfied

1. the set-valued mapping $F$ and the single-valued mapping $f$ are continuous,

2. the image $F\left(X\right)$ is a subset of the image $f\left(X\right)$,

3. the pairs $F$ and $f$ are compatible, and

4. $\exists$$x_0\in X$ such that $sup\left\{H\left({\mathit{Fx}}_n,{\mathit{Fx}}_1\right):n=0,1\cdots \right\}<+\infty$,

then $F$ and $f$ have a unique common fixed point on $X$.

Proof: This proof is the same as Theorem 5 in [14]

Example 1 Let $X=\left[0,3\right]$ with usual metric. Let $F\left(x\right)=\left[0,x^2\right]$ and $f\left(x\right)=2x^2-1$ for each $x\in \left[0,3\right]$. Its clear that the image $F\left(X\right)=F\left(\left[0,3\right]\right)=\left[0,9\right]\subset f\left(\left[0,3\right]\right)=\left[-1,17\right]=f\left(X\right)$ and the both $F$ and $f$ are continuous on $\left[0,3\right]$. If the sequence $x_n\to 1$, then ${\mathit{Fx}}_n\to \left[0,1\right]=K$ and ${\mathit{fx}}_n\to 1\in K$. We know that ${\mathit{Ffx}}_n=\left[0,{\left(2x_n^2-1\right)}^2\right]$ and ${\mathit{fFx}}_n=\left[-1,2x_n^4-1\right]$ so that we obtained

$$H\left({\mathit{Ffx}}_n,{\mathit{fFx}}_n\right)=\mid 4x_n^4-6x_n^2+2\mid \to 0$$

since $x_n\to 1$. It is clear $sup\left\{H\right({\mathit{Fx}}_n,{\mathit{Fx}}_1:n=0,1,\cdots \}=9<+\infty$. Since $F$ and $f$ are continuous, we have

$$\underset{x_n\to 1}{lim}{\mathit{Ffx}}_n=F\left(1\right)=\left[0,1\right]=K,\mathit{\text{and}}\underset{x_n\to 1}{lim}{\mathit{fx}}_n=1=f\left(1\right).$$

This means $1=f\left(1\right)\in F\left(1\right)=K$.

Remark 2 Simple examples above prove that the condition of the continuity of the both mappings $F$ and $f$ is important in Theorem 4 other than the other requirements. However, in general the common fixed point theorems for hybrid composite mappings only required one of the mappings $F$ or $f$ is continuous. In our opinion, such case it can be used if the set $K$ is a singleton.

The following main result is a discussion of the existence of a fixed point for the derivative of an interval-valued function.

Theorem 1.9 Suppose that $F:\left[a,b\right]\to \mathcal{I}\left(\mathbb{R}\right)$ is a continuously $\mathit{gh}$-differentiable on $\left(a,b\right)$ such that there exists $f:\left[a,b\right]\to \mathbb{R}$ and $\mathit{fx}\in F_{\mathit{gh}}^{\prime }\left(x\right)$ for all $x\in \left[a,b\right]$ satisfying the inequality

$$H\left(\mathit{Fx},\mathit{Fy}\right)\le \psi \mathcal{N}\left(F,f\right)$$E17

for all $x,y\in \left[a,b\right]$, where $\psi :\left[0,\infty \right)\to \left[0,\infty \right)$ is a non-decreasing, right continuous, and $\psi \left(t\right)<t$, for all $t>0$. If this following is satisfied.

1. the image $F\left(\left[a,b\right]\right)$ is subsets of the image $f\left(\left[a,b\right]\right)$,

2. the pairs $F$ and $f$ are compatible, and

3. $\exists$$x_0\in \left[a,b\right]$ such that $sup\left\{H\left({\mathit{Fx}}_n,{\mathit{Fx}}_1\right):n=0,1\cdots \right\}<+\infty$,

then the $\mathit{gh}$-derivative $F_{\mathit{gh}}^{\prime }$ has a unique fixed point.

Proof: From hypothesis (C), suppose $H\left({\mathit{Fx}}_s,{\mathit{Fx}}_t\right)\le H\left({\mathit{Fx}}_s,{\mathit{Fx}}_1\right)+H\left({\mathit{Fx}}_t,{\mathit{Fx}}_1\right)\le M$ so that

$$sup\left\{H\left({\mathit{Fx}}_s,{\mathit{Fx}}_t\right):s,t=\mathrm{0,1,2}\cdots \right\}=M<+\infty .$$E18

Suppose that $N\in \mathrm{\mathbb{N}}$ such that for each $\epsilon >0$

$${\psi }^{N}L<\epsilon$$E19

by Lemma 3.

Let $x_0\in \left[a,b\right]$ be an arbitrary. Since $F\left(\left[a,b\right]\right)\subseteq f\left(\left[a,b\right]\right)$, we choose the point $x_1\in \left[a,b\right]$ such that $y_1={\mathit{fx}}_1\in {\mathit{Fx}}_0$. In general, if $x_n\in X$ there exists $x_{n+1}\in X$ such that $y_n={\mathit{fx}}_n\in {\mathit{Fx}}_{n-1}$. By applying inequality (11) to term $H\left({\mathit{Fx}}_m,{\mathit{Fx}}_n\right)$ we have for $m,n\ge N$:

$$\begin{array}{c}H\left({\mathit{Fx}}_m,{\mathit{Fx}}_n\right)\le \psi max\{d\left({\mathit{fx}}_m,{\mathit{fx}}_n\right),d\left({\mathit{fx}}_n,{\mathit{Fx}}_n\right),d\left({\mathit{fx}}_m,{\mathit{Fx}}_m\right),\backslash \\ \frac{1}{2}\left[d\left({\mathit{fx}}_n,{\mathit{Fx}}_m\right)+d\left({\mathit{fx}}_m,{\mathit{Fx}}_n\right)\right]\}\\ \le \psi max\{H\left({\mathit{Fx}}_{m-1},{\mathit{Fx}}_{n-1}\right),H\left({\mathit{Fx}}_{n-1},{\mathit{Fx}}_n\right),H\left({\mathit{Fx}}_{m-1},{\mathit{Fx}}_m\right),\\ \frac{1}{2}\left[H\left({\mathit{Fx}}_{n-1},{\mathit{Fx}}_m\right)+H\left({\mathit{Fx}}_{m-1},{\mathit{Fx}}_n\right)\right]\}\\ \le \psi max\{H\left({\mathit{Fx}}_{m-1},{\mathit{Fx}}_{n-1}\right),H\left({\mathit{Fx}}_{n-1},{\mathit{Fx}}_n\right),H\left({\mathit{Fx}}_{m-1},{\mathit{Fx}}_m\right),\\ \frac{1}{2}\left[H\left({\mathit{Fx}}_{n-1},{\mathit{Fx}}_{m-1}\right)+H\left({\mathit{Fx}}_{m-1},{\mathit{Fx}}_m\right)\right],\\ \frac{1}{2}\left[H\left({\mathit{Fx}}_{m-1},{\mathit{Fx}}_{n-1}\right)+H\left({\mathit{Fx}}_{n-1},{\mathit{Fx}}_n\right)\right]\}\\ =\psi max\left\{H\left({\mathit{Fx}}_{m-1},{\mathit{Fx}}_{n-1}\right),H\left({\mathit{Fx}}_{n-1},{\mathit{Fx}}_n\right),H\left({\mathit{Fx}}_{m-1},{\mathit{Fx}}_m\right)\right\}\end{array}$$E20

By iterating (14) above as much as $N$ times, we deduce for each $m,n>N$ as follows:

$$\begin{array}{rcl}H\left({\mathit{Fx}}_m,{\mathit{Fx}}_n\right)& \le & \psi max\{H\left({\mathit{Fx}}_r,{\mathit{Fx}}_s\right),H\left({\mathit{Fx}}_r,{\mathit{Fx}}_t\right),\\ & & H\left({\mathit{Fx}}_s,{\mathit{Fx}}_k\right):m-1\le r;t\le n;n-1\le s;k\le m\}\\ & \le & {\psi }^{2}max\{H\left({\mathit{Fx}}_r,{\mathit{Fx}}_s\right),H\left({\mathit{Fx}}_r,{\mathit{Fx}}_t\right),\\ & & H\left({\mathit{Fx}}_s,{\mathit{Fx}}_k\right):m-2\\ & \le & r;t\le n;n-2\le s;k\le m\}\le \cdots \\ & \le & {\psi }^{N}max\{H\left({\mathit{Fx}}_r,{\mathit{Fx}}_s\right),H\left({\mathit{Fx}}_r,{\mathit{Fx}}_t\right),\\ & & H\left({\mathit{Fx}}_s,{\mathit{Fx}}_k\right):4m-N\\ & \le & r;t\le n;n-N\le s;k\le m\}\le {\psi }^{N}M<\epsilon ,\end{array}$$E21

by inequality (13).

Accordingly the sequence $\left\{{\mathit{Fx}}_n\right\}$ is a Cauchy sequence on the complete metric spaces $\left(\mathcal{I}\left(\mathbb{R}\right),H\right)$ so that converges to an interval $J\in \mathcal{I}\left(\mathbb{R}\right)$. The sequence of single-valued functions $\left\{{\mathit{fx}}_n\right\}$ is also a Cauchy sequence on $\mathbb{R}$ hence it converges to a point $z\in \mathbb{R}$. We have

$$\mid z-J\mid \le \mid z-{\mathit{fx}}_n\mid +\mid {\mathit{fx}}_n-J\mid \le \mid z-{\mathit{fx}}_n\mid +H\left({\mathit{Fx}}_{n-1},J\right),$$E22

as $n\to \infty$, $\mid z-J\mid =0$. This means, $z\in J$ since $J\in \mathcal{I}\left(\mathbb{R}\right)$. By compatibility of $F$ and $f$, we obtain

$$\underset{n\to \infty }{lim}H\left({\mathit{Ffx}}_n,{\mathit{fFx}}_n\right)=0.$$E23

By using inequality (11), we have

$$\begin{array}{c}H\left({\mathit{Ffx}}_{n+1},{\mathit{Fx}}_n\right)\le \psi max\{d\left(f^2{x}_{n+1},{\mathit{fx}}_n\right),d\left(f^2{x}_{n+1},{\mathit{Ffx}}_{n+1}\right),d\left({\mathit{fx}}_n,{\mathit{Fx}}_n\right),\\ \frac{1}{2}\left[d\left(f^2{x}_{n+1},{\mathit{Fx}}_n\right)+d\left({\mathit{fx}}_n,{\mathit{Ffx}}_{n+1}\right)\right\}\\ \le \psi max\{d\left({\mathit{fFx}}_n,{\mathit{fx}}_n\right),d\left({\mathit{fFx}}_n,{\mathit{Ffx}}_{n+1}\right),d\left({\mathit{fx}}_n,{\mathit{Fx}}_n\right),\\ \frac{1}{2}\left[d\left({\mathit{fFx}}_n,{\mathit{Fx}}_n\right)+d\left({\mathit{fx}}_n,{\mathit{Ffx}}_{n+1}\right)\right\}\\ \le \psi max\{d\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+d\left({\mathit{Ffx}}_n,{\mathit{Fx}}_n\right)+d\left({\mathit{Fx}}_n,{\mathit{fx}}_n\right),\\ d\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+d\left({\mathit{Ffx}}_n\right),{\mathit{fx}}_n)+d\left({\mathit{fx}}_n,{\mathit{Ffx}}_{n+1}\right),\\ d\left({\mathit{fx}}_n,{\mathit{Fx}}_n\right),\frac{1}{2}\left[d\left({\mathit{fFx}}_n,{\mathit{Fx}}_n\right)+d\left({\mathit{fx}}_n,{\mathit{Ffx}}_{n+1}\right)\right\}\\ \le \psi max\{d\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+d\left({\mathit{Ffx}}_n,{\mathit{Fx}}_n\right)+d\left({\mathit{Fx}}_n,{\mathit{fx}}_n\right),\\ d\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+d\left({\mathit{Ffx}}_n,{\mathit{fx}}_n\right)+d\left({\mathit{fx}}_n,{\mathit{Ffx}}_{n+1}\right)\}\\ \le \psi max\{H\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+H\left({\mathit{Ffx}}_n,{\mathit{Fx}}_n\right)+d\left({\mathit{Fx}}_n,{\mathit{fx}}_n\right),\\ H\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+d\left({\mathit{Ffx}}_n,{\mathit{fx}}_n\right)+d\left({\mathit{fx}}_n,{\mathit{Ffx}}_{n+1}\right)\}\\ \le \psi max\{H\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+H\left({\mathit{Ffx}}_n,{\mathit{Fx}}_n\right)+d\left({\mathit{Fx}}_n,{\mathit{fx}}_n\right),\\ H\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+H\left({\mathit{Ffx}}_n,{\mathit{Fx}}_{n-1}\right)+H\left({\mathit{Fx}}_{n-1},{\mathit{Ffx}}_{n+1}\right)\}\end{array}$$

since $f^2{x}_{n+1}\in {\mathit{fFx}}_n$ and $\psi$ are non-decreasing. Since the pairs $F$ and $f$ are compatible, we obtain

$$\begin{array}{c}H\left(\mathit{Fz},J\right)\le \psi max\left\{0+H\left(\mathit{Fz},J\right)+d\left(J,z\right),0+2H\left(\mathit{Fz},J\right)\right\}\\ \le \psi max\left\{H\left(\mathit{Fz},J\right),2H\left(\mathit{Fz},J\right)\right\}\\ \le 2\mathit{\psi H}\left(\mathit{Fz},J\right).\end{array}$$

Since $\psi \left(t\right)<t$ for all $t>0$, we have $H\left(\mathit{Fz},J\right)=0$. This means $\mathit{Fz}=J$. Since the pairs $F$ and $f$ are compatible and $F$ is continuously differentiable on $\left[a,b\right]$ (hence continuous), we have

$$\underset{n\to \infty }{lim}H\left(\mathit{Fz},\mathit{fJ}\right)=\underset{n\to \infty }{lim}H\left({\mathit{Ffx}}_n,{\mathit{fFx}}_n\right)=0.$$E24

So $\mathit{Fz}=\mathit{fJ}$. Since $z\in J$, $f\left(z\right)\in f\left(J\right)$, consequently

$$f\left(z\right)\in F\left(z\right)=f\left(J\right)=J.$$E25

Since $f\in F_{\mathit{gh}}^{\prime }$ and $F_{\mathit{gh}}^{\prime }$ is continuous, the function $f$ is continuous. Of course, the sequence $\left\{f^2{x}_n\right\}$ converges to the point $\mathit{fz}$ and the sequence of set $\left\{{\mathit{fFx}}_n\right\}$ converges to a set $\mathit{fJ}$. Since the limit

$$\underset{n\to \infty }{lim}H\left({\mathit{Ffx}}_n,\mathit{fJ}\right)\le \underset{n\to \infty }{lim}\left[H\left({\mathit{Ffx}}_n,{\mathit{fFx}}_n\right)+H\left({\mathit{fFx}}_n,\mathit{fJ}\right)\right]=0,$$E26

the sequence of set $\left\{{\mathit{Ffx}}_n\right\}$ also converges to a set $\mathit{fJ}$.

Since $f^2{x}_{n+1}\in {\mathit{fFx}}_n$ and using inequality (11), we get

$$\begin{array}{l}\mid f^2{x}_{n+1}-{\mathit{fx}}_{n+1})\mid \le H\left({\mathit{fFx}}_n,{\mathit{Fx}}_n\right)\le H\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+H\left({\mathit{Ffx}}_n,{\mathit{Fx}}_n\right)\\ \le H\left({\mathit{fFx}}_n,{\mathit{Ffx}}_n\right)+\psi max\{\mid f^2{x}_n-{\mathit{fx}}_n\mid ,\mid f^2{x}_n-{\mathit{Ffx}}_n\mid ,\\ \mid {\mathit{fx}}_n-{\mathit{Fx}}_n\mid ,\frac{1}{2}\left[|f^2{x}_n-{\mathit{Fx}}_n|+|{\mathit{fx}}_n,{\mathit{Ffx}}_n|\right]\}.\end{array}$$

For $n\to \infty$, it allows from hypothesis the part $B$ (compatibility) and the Eq. (19) we obtain

$$\begin{array}{l}\mid \mathit{fz}-z\mid \le 0+\psi max\left\{|\mathit{fz}-z|,|\mathit{fz}-\mathit{fJ}|,|z-J|,\frac{1}{2}\left[|\mathit{fz}-J|+|z-\mathit{Fz}|\right]\right\}\\ \le \psi max\left\{|\mathit{fz}-z|,0,0,\frac{1}{2}\left[0+0\right]\right\}\\ \le \psi \mid \mathit{fz}-z\mid .\end{array}$$

It implies $z=\mathit{fz}$. Meaning the point $z$ is a fixed point of $f$. This allows $z=\mathit{fz}\in F\left(z\right)$ since the Eq. (19) and hence $z$ is also a fixed point of $F_{\mathit{gh}}^{\prime }$ by $z=\mathit{fz}\in F_{\mathit{gh}}^{\prime }\left(z\right)$.

Let $u$ is another common fixed point of $F$ and $f$. By inequality (11), we have that

$$\begin{array}{c}H\left(\mathit{Fz},\mathit{Fu}\right)\le \mathit{\text{ψmax}}\left\{|\mathit{fz}-\mathit{fu}|,|\mathit{fz}-\mathit{Fz}|,|\mathit{fu}-\mathit{Fu}|,\frac{1}{2}\left[|\mathit{fz}-\mathit{Fu}|+|\mathit{fu}-\mathit{Fz}|\right]\right\}.\\ \le \mathit{\text{ψmax}}\left\{H\left(\mathit{Fz},\mathit{Fu}\right),0,0,\frac{1}{2}\left[H\left(\mathit{Fz},\mathit{Fu}\right)+H\left(\mathit{Fu},\mathit{Fz}\right)\right]\right\}.\\ =\mathit{\text{ψmax}}\left\{H\left(\mathit{Fz},\mathit{Fu}\right)\right\}.\\ =\mathit{\psi H}\left(\mathit{Fz},\mathit{Fu}\right).\end{array}$$

It implies that $H\left(\mathit{Fz},\mathit{Fu}\right)=0$. Since $d\left(z,u\right)\le H\left(\mathit{Fz},\mathit{Fu}\right)=0$, we have $z=u$. Thus the fixed point $z$ is unique. This completes the proof.

Remark 3 To get a common fixed point through the hybrid composite mapping usually contains at least two mappings in its hypothesis. This study shows enough one mapping in its hypothesis. In this case, the mapping given must be differentiable (Theorem 1.9). In addition, the continuity of function is not needed explicitly stated in its hypothesis. Thus this result is more simple than the results reached by past researchers.

Example 2 Let $F\left(x\right)=\left[\left(x^2-x\right),x\right]$, be an interval-valued function for all $x\in \left[0,2\right]$. It is clear $F$ is $\mathit{gh}$-differentiable on $\left(0,2\right)$ with derivative

$$F_{\mathit{gh}}^{\prime }\left(x\right)=\left\{\begin{array}{ll}\left[\left(2x-1\right),x\right]& \text{if}0\le x\le 1,\\ \left[x,\left(2x-1\right)\right]& \text{if}1\le x\le 2.\end{array}\right.$$

In this case, we can take the selector $f\left(x\right)=\left(2x-1\right)\in F_{\mathit{gh}}\left(x\right)$ for all $x\in \left[0,2\right]$. We obtain the image

$$F\left(\left[0,2\right]\right)=\left[-\frac{1}{4},1\right]\cup \left[1,2\right]=\left[-\frac{1}{4},2\right]\subset \left[-1,2\right]=f\left(\left[0,2\right]\right).$$

This means that the condition in Theorem 4 part (A) is satisfied.

If the sequence $x_n\to 1$, then ${\mathit{Fx}}_n\to \left[0,1\right]=K$ and ${\mathit{fx}}_n\to 1\in K$. First, we start with the formula ${\mathit{Ffx}}_n=\left[{\left(2x_n-1\right)}^2-\left(2x_n-1\right),\left(2x_n-1\right)\right]$ and ${\mathit{fFx}}_n=\left[2\left(x_n^2-x_n\right)-1,\left(2x_n-1\right)\right]$, we obtain

$$H\left({\mathit{Ffx}}_n,{\mathit{fFx}}_n\right)=\mid 2x_n^2-4x_n+2\mid \to 0$$

since $x_n\to 1$. Thus $F$ and $f$ are compatible. It is clear that $sup\left\{H\left({\mathit{Fx}}_n,{\mathit{Fx}}_1\right):n=0,1,\cdots \right\}=3<+\infty$. Since $F$ is continuously $\mathit{gh}$-differentiable on $\left(0,2\right)$, then implies that $F$ and $f$ are continuous on $\left(0,2\right)$ (see Theorem 1.1). Hence we have

$$\underset{x_n\to 1}{lim}{\mathit{Ffx}}_n=F\left(1\right)=\left[0,1\right]=K,\mathit{\text{and}}\underset{x_n\to 1}{lim}{\mathit{fx}}_n=1=f\left(1\right).$$

Certainly $1=f\left(1\right)\in F\left(1\right)=K$. Since $f\left(x\right)\in F_{\mathit{gh}}^{\prime }\left(x\right)$ for all $x\in \left[0,2\right]$, we obtain $1=f\left(1\right)\in F_{\mathit{gh}}^{\prime }\left(1\right)=1$. Thus the point $z=1$ is a unique fixed point of $F_{\mathit{gh}}^{\prime }$.

Furthermore, if $f\in F$, then we have the following.

Corollary 1 Suppose that $F:\left[a,b\right]\to \mathcal{I}\left(\mathbb{R}\right)$ is a continuously $\mathit{gh}$-differentiable on $\left(a,b\right)$ such that there exists $f:\left[a,b\right]\to \mathbb{R}$ and $\mathit{fx}\in F\left(x\right)$ for all $x\in \left[a,b\right]$. If the function $f$ and the derivative $F_{\mathit{gh}}^{\prime }$ satisfies the inequality

$$H\left(F_{\mathit{gh}}^{\prime }x,F_{\mathit{gh}}^{\prime }y\right)\le \psi \mathcal{N}\left(F_{\mathit{gh}}^{\prime },f\right)$$

for all $x,y\in \left[a,b\right]$, where $\psi :\left[0,\infty \right)\to \left[0,\infty \right)$ is a nondecreasing, right continuous, and $\psi \left(t\right)<t$, for all $t>0$ and satisfies the condition.

1. the image $F^{\prime }\left(\left[a,b\right]\right)$ is subsets of the image $f\left(\left[a,b\right]\right)$,

2. the pairs $F^{\prime }$ and $f$ are compatible, and

3. $\exists$$x_0\in \left[a,b\right]$ such that $sup\left\{H\left({\mathit{Fx}}_n,{\mathit{Fx}}_1\right):n=0,1\cdots \right\}<+\infty$,

then $F_{\mathit{gh}}^{\prime }$ has a unique fixed point on $X$.

Example 3 Let $X=\left[-2,2\right]$ with usual metric. Let $F:\left[-2,2\right]\to \mathcal{I}\left(\mathbb{R}\right)$ with the formula

$$F\left(x\right)=\left\{\begin{array}{ll}\left[\left(x+sin\left(x+\frac{1}{2}\right)\right),x\right]& \text{if}-2\le x\le -\frac{1}{2},\\ \left[x,\left(x+sin\left(x+\frac{1}{2}\right)\right)\right]& \text{if}-\frac{1}{2}\le x\le -2.\end{array}\right.$$

It is clear that $F$ is $\mathit{gh}$-differentiable on $\left(-2,2\right)$ by Theorem 1.2 with derivative

$$F_{\mathit{gh}}^{\prime }\left(x\right)=\left\{\begin{array}{ll}\left[1,\left(1+cos\left(x+\frac{1}{2}\right)\right)\right]& \text{if}-2\le x\le 1,\\ \left[\left(1+cos\left(x+\frac{1}{2}\right)\right),1\right]& \text{if}1\le x\le 2.\end{array}\right.$$

If we choose $f\left(x\right)=x\in F\left(x\right)$ for all $x\in \left[-2,2\right]$, then we obtain

$$F_{\mathit{gh}}^{\prime }X=F_{\mathit{gh}}^{\prime }\left(\left[-2,2\right]\right)=\left[1,2\right]\cup \left[\mathrm{0.198,1}\right]=\left[\mathrm{0.198,2}\right]\subset \left[-2,2\right]=f\left(\left[-2,2\right]\right)=\mathit{fX}.$$

This means the condition in Corollary 1 part (A) is satisfied. If the sequence $x_n\to 1$, then $F_{\mathit{gh}}^{\prime }\left(x_n\right)\to \left\{1\right\}=K$ and ${\mathit{fx}}_n\to 1\in K$. First, we start with the formula $F_{\mathit{gh}}^{\prime }{\mathit{fx}}_n=F_{\mathit{gh}}^{\prime }\left(x_n\right)=\left[\left(1+cos\left(x_n+\frac{1}{2}\right)\right),1\right]\cup [1,(1+cos\left(x_n+\frac{1}{2}\right)]$ and $f{F}_{\mathit{gh}}^{\prime }\left(x_n\right)=F_{\mathit{gh}}^{\prime }\left(x_n\right)=\left[\left(1+cos\left(x_n+\frac{1}{2}\right)\right),1\right]\cup [1,(1+cos\left(x_n+\frac{1}{2}\right)]$, we obtain

$$H\left(F_{\mathit{gh}}^{\prime }{\mathit{fx}}_n,f{F}_{\mathit{gh}}^{\prime }{x}_n\right)=0.$$

Thus $F$ and $f$ are compatible. Since $F$ is continuously $\mathit{gh}$-differentiable on $\left(-2,2\right)$, this implies that $F$ and $f$ are continuous on $\left(-2,2\right)$ (see Theorem 1.1). Hence we have

$$\underset{x_n\to 1}{lim}{F}_{\mathit{gh}}^{\prime }{\mathit{fx}}_n=F_{\mathit{gh}}^{\prime }\left(1\right)=\left\{1\right\},\mathit{\text{and}}\underset{x_n\to 1}{lim}{\mathit{fx}}_n=1=f\left(1\right).$$

Consequently, $1=f\left(1\right)\in F_{\mathit{gh}}^{\prime }\left(1\right)=\left\{1\right\}$.

5. Conclusions

The existence of a fixed point for the derivative of set-valued mappings can be obtained by using the method of the compatibility of the hybrid composite mappings in the sense of the Pompei-Hausdorff metric.

Conflict of interest

The authors declare no conflict of interest.