Fixed Point of Modified F-Contraction with an Application

Abstract

In this paper, we prove the existence of fixed points for every 2-rotative continuous mapping in Banach spaces to answer an open question raised by Goebel and Koter. Further, we modify F-contraction by developing F-rotative mapping and establish some fixed-point theorems. Finally, we apply our results to prove the existence of a solution of a non-linear fractional differential equation.

1. Introduction

In 1981, Goebel and Koter [1] proved that every non-expansive rotative mapping has a fixed point for nonempty, closed convex subsets of Banach spaces. We use the rotativeness of a mapping in $\mathbb{R}$ to weaken certain mapping conditions and prove that if a continuous mapping is not non-expansive then it will have a fixed point. This result proves the existence of fixed points for the whole class of continuous mappings that are not non-expansive mappings, such as mean non-expansive, mean Lipschitzian, $(\alpha ,p)$ non-expansive mappings, etc. Further, Wardowski [2] obtained some results regarding the existence of fixed points of F-contraction under certain conditions. Due to its importance and simplicity, the researchers obtained many significant extensions and interesting generalizations of this new modification of the F-contraction (for details, see [3,4,5,6]). We modify the idea of F-contraction by introducing F-rotative mapping and establishing some fixed point results. Finally, we apply our main result to find the existence a solution to a nonlinear fractional differential equation.

2. Preliminaries

Let C be a nonempty closed convex subset of a Banach space X. Let us denote the set of real numbers by $\mathbb{R}$ and the set of natural numbers by $\mathbb{N}.$ Moreover, we denote the image of any element c under the function g by $g_c$ and the image of $g_c$ under g by $g_c^2$.

Definition 1.

Consider a mapping $g:C\to C,$$k>0,$$n\ge 2$ and $a\in [0,n).$

(i) 

If for all $m,l\in C,$ we have $∥g_m-g_l∥\le k∥m-l∥,$ then g is a k-Lipschitzian mapping. For some $k>0,$ g is Lipschitzian. Therefore, 1-Lipschitzian mapping is non-expansive, whereas, a contraction is also a k-Lipschitzian with $k<1$.

(ii) 

If $g^n$ is an identity mapping, then g is called periodic with period n.

(iii) 

If for all $m\in C,$ we have $∥g_m^n-m∥\le a∥g_m-m∥,$ then g is $(n,a)$-rotative. For some $a\in [0,n),$g is n-rotative and for some $n\ge 2$ and $a\in [0,n),$g is rotative.

Using $\phi \left(C,n,a,k\right),$ we denote the class of all $(n,a)$-rotative k-Lipschitzian mappings g from C into itself.

Theorem 1

([1]). For each $n\ge 2$ and $a\in \left[0,n\right)$, there is a real number $\gamma >1$, such that if $\gamma >k$ and $g\in \phi \left(C,n,a,k\right),$ then g has a fixed point in X.

It is easy to observe that $\gamma$ in Theorem (1) depends on $X,n,a$. Therefore, we can define

$$\gamma (X,n,a)=inf\left\{\begin{array}{c}k>1:g\in \phi \left(C,n,a,k\right)\mathrm{and}fix\left(g\right)=\varnothing \}.\end{array}\right.$$

It is clear from Theorem (1) that, for the existence of a fixed point in Banach space X with $n\ge 2$ and $a\in [0,n),$ we have $\gamma (X,n,a)>1.$ In the literature, we can find certain results about some upper and lower bounds of $\gamma \left(X,n,a\right)$ (for more details, we refer readers to [7,8,9,10,11,12,13,14,15]). In this research article, we restrict the space to $\mathbb{R}$ and use $2-$rotative mapping to find the precise values of $\gamma \left(X,n,a\right)$.

The following results give the bounds of $\gamma \left(X,n,a\right)$ and $\gamma \left(H,n,a\right)$ for a Banach space X and a Hilbert space H.

• (Goebel and Koter) [1]

  $$\gamma \left(X,n,0\right)\ge \left\{\begin{array}{cc}2& \mathrm{for}n=2\\ \sqrt[n-1]{\left(-1+\sqrt{\frac{n^3-2n^2+n-1}{n-1}}\right)\left(\frac{1}{n-2}\right)}& \mathrm{for}n>2\end{array}\right..$$
• (Koter) [8]

  $$\gamma \left(H,n,0\right)\ge \sqrt{{\pi }^2-3}.$$
• (Goebel) [13]

  $$\gamma \left(C\left[0,1\right],2,a\right)\le \frac{1}{a-1}.$$

The obtained bounds of $\gamma \left(X,n,a\right)$ and $\gamma \left(H,n,a\right)$ show that the greatest lower bound is no greater than $3.$ This immediately raises a question: Is it possible to find the exact value of $\gamma \left(X,n,a\right)$ for some $X,n,$ and $a?$ Goebel and Koter [1] proved some fixed point results by obtaining the upper and lower bounds of $\gamma \left(X,n,a\right)$ but remained unable to find its precise value. In this paper, we find the exact value of $\gamma$ for certain values of $X,n,$ and $a.$

Definition 2.

Let $\left(X,d\right)$ be a metric space. A mapping $g:X\to X$ is said to be a F-contraction if $t>0$, such that $\forall m,l\in X$,

$$d\left(g_m,g_l\right)>0\Rightarrow t+F\left(d\left(g_m,g_l\right)\right)\le F\left(d\left(m,l\right)\right),$$

where $F:{\mathbb{R}}_{+}\to \mathbb{R}$ is a mapping that satisfies the following conditions:

(F1) 

For all $m,l\in {\mathbb{R}}_{+}$ such that $m<l,$ we have $F\left(m\right)<F\left(l\right).$

(F2) 

For every positive sequence {${\eta }_1,{\eta }_2,....\},{lim}_{n\to \infty }{\eta }_n=0$ if, and only if, ${lim}_{n\to \infty }F\left({\eta }_n\right)=-\infty .$

(F3) 

${lim}_{\eta \to 0^{+}}{\eta }^{k}F\left(\eta \right)=0$ for some $k\in \left(0,1\right).$

Let $\mathcal{F}$ be the set of all functions that satisfy these three conditions (F1)–(F3).

Example 1.

Let $F_i:{\mathbb{R}}^{+}\mapsto \mathbb{R}$, where $i=1,2,3,4$ defined as:

1. 

$F_1\left(m\right)=lnm.$

2. 

$F_2\left(m\right)=m+lnm.$

3. 

$F_3\left(m\right)=-\frac{1}{\sqrt{m}}.$

4. 

$F_4\left(m\right)=ln\left(m^2+m\right).$

Then, $F_1,F_2,F_3,F_4\in \mathcal{F}.$

Wardowski [2] established the following fixed point theorem, which is regarded as a generalization of the Banach contraction principle.

Theorem 2.

Let $g:X\to X$ be a F-contraction in a metric space $\left(X,d\right)$. Then for every $x\in X$ the sequence ${\left\{g_x^n\right\}}_{n\in N}$ converges to a unique fixed point in X.

Enthused by Wardowski’s results, there many authors are attempting to extend and refine this idea by relaxing some of its conditions, generalizing the definition space or modifying the respective contraction type mapping. In this paper, we introduce a significant modification to the F-contraction, named F-rotative mapping. It is important to note that a contraction may or may not be a rotative mapping.

3. Main Results

In this section, we first define a control function F with less restrictive conditions than the conditions used by Wardowski [2].

Consider a strictly increasing function $F:\left(0,\infty \right)\mapsto \mathbb{R}$ that satisfies the following conditions:

$\left(F{1}^{\prime }\right)$         $F\left(\eta \zeta \right)\ge F\left(\eta \right)+F\left(\zeta \right).$

$\left(F{2}^{\prime }\right)$         F is continuous on $\left(0,\infty \right).$

In the set of all functions F satisfying $\left(F{1}^{\prime }\right),\left(F{2}^{\prime }\right)$ is represented by $\mathfrak{F}.$

Example 2.

Let $F_i:\left(0,\infty \right)\mapsto \mathbb{R},i=1,2,3,4$ defined as:

1. 

$F_1\left(c\right)=lnc.$

2. 

$F_2\left(c\right)=e^c.$

3. 

$F_3\left(c\right)=-\frac{1}{c}.$

4. 

$F_4\left(c\right)=lnc-\frac{1}{c}+\sqrt{c}.$

Then, $F_1,F_2,F_3,F_4\in \mathfrak{F}.$

Remark 1.

In the comparison of conditions (F1)–(F3) with the conditions (F1’), (F3’), together with the given examples, one can easily observe that by relaxing some conditions of control function F, we can find a large number of functions that will be helpful for weakening certain conditions on the existence theorem applications of a kind of differential equation.

Definition 3.

Let $\left(X,d\right)$ be a metric space. A mapping $g:X\to X$ is said to be F-rotative if there is a function $F\in \mathfrak{F}$ with $t+F\left(a\right)\ge 0,a\in \left(0,n\right),n\ge 2$ such that, for all $c\in X$, and $t\in \mathbb{R}$, we have

$$d\left(c,g_c^n\right)>0\Rightarrow t+F\left(d\left(c,g_c^n\right)\right)\le F\left(d\left(c,g_c\right)\right).$$

It is interesting to note that t can take negative values as well, which means that, for some $c\in X$, we may have $F\left(d\left(c,g_c^n\right)\right)>F\left(d\left(c,g_c\right)\right)$, or, equivalently, $d\left(c,g_c^n\right)>d\left(c,g_c\right)$. This means that F-rotative mappings belong to a larger class than F-contractions.

Theorem 3.

Let $g:I\to I$ be a 2-rotative continuous mapping, where I is a non-empty, closed interval of $\mathbb{R}$. If, for some $c\in I$,

$$\left|g_c^2-g_c\right|>\left|g_c-c\right|,$$

(1)

then g admits a fixed point.

Proof. 

Using the trichotomy property of a real number, for some $c\in I$, we have

(A) 

$g_c=c.$

(B) 

$g_c>c.$

(C) 

$g_c<c.$

If $\left(A\right)$ is true, then c is the fixed point of g.

Let $\left(B\right)$ is true. We will prove that $g_c^2<c<g_c$.

Suppose, however, that

$$c<g_c<g_c^2.$$

(2)

Then,

$$\left|g_c^2-c\right|=\left|g_c^2-g_c\right|+\left|g_c-c\right|.$$

Using the inequality (1), we have

$$\left|g_c^2-c\right|>\left|g_c-c\right|+\left|g_c-c\right|$$

$$>2\left|g_c-c\right|,$$

which contradicts g is $2-$rotative. Therefore, inequality (2) is not true.

Now if,

$$c<g_c^2<g_c.$$

(3)

Then

$$\left|g_c-c\right|=\left|g_c^2-g_c\right|+\left|g_c^2-c\right|.$$

Inequality $\left(1\right)$ yields,

$$\left|g_c-c\right|>\left|g_c-c\right|+\left|g_c^2-c\right|,$$

or,

$$0>\left|g_c^2-c\right|.$$

which is a contradiction. Therefore, the inequality (3) is also not true.

Therefore, the falsity of inequalities (2) and (3) implies the truth of the following inequality

$$g_c^2<c<g_c.$$

(4)

We can define $\mathcal{F}\left(c\right)=g_c-c$. Since $g_c$ is a continuous function, so is $\mathcal{F}\left(c\right)$.

For $g_c>c,\mathrm{we}\mathrm{have}{g}_c-c>0,$ so that

$$\mathcal{F}\left(c\right)>0.$$

Likewise, if

$$g_c^2<g_c,$$

we can write

$$g\left(g_c\right)<g_c.$$

Let $g_c=r,$ so that

$$g_r<r.$$

This yields

$$\mathcal{F}\left(r\right)<0.$$

Since $c<r,\mathcal{F}\left(c\right)>0$ and $\mathcal{F}\left(r\right)<0$, it follows from Bolzano Theorem that there exists a number $\beta \in [c,r]$ such that $\mathcal{F}\left(\beta \right)=0.$

That is

$$g_{\beta }-\beta =0.$$

Hence, we have $g_{\beta }=\beta$; that is, g has a fixed point.

Now, if $\left(C\right)$ is true, that is, $g_c<c$, we will prove that

$$g_c<c<g_c^2.$$

Suppose, however, that we have

$$g_c^2<g_c<c.$$

(5)

Meaning that

$$\left|g_c^2-c\right|=\left|g_c^2-g_c\right|+\left|g_c-c\right|.$$

Using (1), we can write

$$\left|g_c^2-c\right|>\left|g_c-c\right|+\left|g_c-c\right|,$$

which is a contradiction, as g is $2-$rotative. Therefore, (5) is not true.

Similarly, if we suppose that

$$g_c<g_c^2<c.$$

(6)

We can write

$$\left|g_c-c\right|=\left|g_c^2-g_c\right|+\left|g_c^2-c\right|.$$

From inequality (1), it follows that

$$\left|g_c-c\right|>\left|g_c-c\right|+\left|g_c^2-c\right|,$$

or,

$$0>\left|g_c^2-c\right|,$$

which is a contradiction. Therefore, the inequality (6) is also not true.

Therefore, the falsity of inequalities (5) and (6) implies the truth of the following inequality

$$g_c<c<g_c^2.$$

(7)

Let $\mathcal{F}\left(c\right)=g_c-c.$ Since $g_c$ is a continuous function, then $\mathcal{F}\left(c\right)$ is also continuous.

If $g_c<c,$ we have

$$\mathcal{F}\left(c\right)<0.$$

Likewise, if

$$g_c^2>g_c,$$

we can write

$$g\left(g_c\right)>g_c.$$

Let $g_c=r,$ so that

$$g_r>r.$$

That yields

$$\mathcal{F}\left(r\right)>0.$$

Therefore, $g_c$ guarantees the existence of a fixed point for all three possibilities (A), (B), (C). □

Theorem 4.

Every 2-rotative continuous mapping has a fixed point in a non-empty closed interval of $\mathbb{R}.$

Proof. 

Let $g:I\to I$ be a 2-rotative continuous mapping, where I is non-empty closed interval of $\mathbb{R}$. If g is nonexpansive, then by Goeble and Koter the mapping g admits the fixed point (see, [1]). Suppose that g is not nonexpansive mapping; then, the distance between some or all of its iterates must expand. Suppose that for some $c\in I$,

$$\left|g_c^{n+2}-g_c^{n+1}\right|>\left|g_c^{n+1}-g_c^n\right|.$$

Then, there exists $g_c^n=u\in I$, such that

$$\left|g_u^2-g_u\right|>\left|g_u-u\right|.$$

Therefore, Theorem (3) guarantees the existence of the fixed point of g. □

Since we have proved that every $2-$rotative continuous mapping in $\mathbb{R}$ has a fixed point, it is easy to observe that

$$\varnothing =\left\{\begin{array}{c}k>1:g\in \phi \left(C,n,a,k\right)\mathrm{and}fix\left(g\right)=\varnothing \}.\end{array}\right.$$

Therefore, $\gamma =inf\varnothing =\infty .$ That provides an answer to the questions raised by Goebel and Koter [1], and Górnicki [16] that, under certain conditions on $X,n,a,$ the number $\gamma$ attains the value $\infty .$

Corollary 1.

Let $\left(X,{∥.∥}_{\infty }\right)$ be a complete metric space, with $X=C\left([0,1],\mathbb{R}\right)$, such that ${∥x∥}_{\infty }={\displaystyle \underset{t\in [0,1]}{sup}}\left|x\left(t\right)\right|.$ Let $g:X\to X$ be a 2-rotative continuous mapping, such that, for each $c\in X$, one of the following condition holds:

1. 

$g_c$ is an increasing function.

2. 

$g_c$ is a decreasing function.

Then, the arguments of Theorems 3 and 4 ensure the existence of the fixed point of g in X.

Theorem 5.

Let I be a non-empty closed interval in $\mathbb{R}$. Then, a F-rotative continuous mapping $g:I\to I$ with $n=2$ has a fixed point for all $F\in \mathfrak{F}.$

Proof. 

Using the definition of F-rotative mapping, we can write

$$t+F∥g_c^2-c∥\le F∥g_c-c∥.$$

The condition $t+F\left(a\right)\ge 0$ in the definition yields

$$-F\left(a\right)+F∥g_c^2-c∥\le F∥g_c-c∥.$$

Therefore, we can write

$$F∥g_c^2-c∥\le F∥g_c-c∥+F\left(a\right).$$

Now, the condition $\left(F1\right)$ will imply

$$F∥g_c^2-c∥\le F\left(a∥g_c-c∥\right),$$

or

$$∥g_c^2-c∥\le a∥g_c-c∥,a\in \left(0,2\right),$$

which shows that g is $2-$rotative continuous mapping in I. Therefore, Theorem 4 guarantees the existence of a fixed point. □

Theorem 6.

Let C be a nonempty closed convex subset of a Banach space X. If a k-Lipschitzian mapping $g:X\to X$ represents a F-rotative mapping with $F\in \mathfrak{F}$ and $\gamma >1$ such that $k<\gamma$, then g has a fixed point in X.

Proof. 

Since g is a k-Lipschitzian mapping, we can write

$$∥g_c-g_r∥\le k∥c-r∥,$$

for all $c,r\in C$ and $k>0$.

If $k\le 1$, then g, being a non-expansive mapping, admits a fixed point.

Next, for some $k>1$, along with F-rotative mapping, we will prove that $fix\left(g\right)\ne \varnothing .$

Using the definition of F-rotative mapping, we can write

$$t+F∥g_c^n-c∥\le F∥g_c-c∥.$$

The condition $t+F\left(a\right)\ge 0$ yields

$$-F\left(a\right)+F∥g_c^n-c∥\le F∥g_c-c∥.$$

Therefore, we can write

$$F∥g_c^n-c∥\le F∥g_c-c∥+F\left(a\right).$$

Moreover, the condition $\left(F{1}^{\prime }\right)$ also implies

$$F∥g_c^n-c∥\le F\left(a∥g_c-c∥\right),$$

or,

$$∥g_c^n-c∥\le a∥g_c-c∥,a\in \left(0,n\right),$$

which shows that g is $n-$rotative mapping in X. Then, $\gamma >1$ such that $k<\gamma$ and $g\in \phi \left(C,n,a,k\right)$. Therefore, Theorem 1 guarantees that $fix\left(g\right)\ne \varphi$. □

Remark 2.

It is important to note that the condition we used to prove the existence of fixed point result of k-Lipschitzian mapping is weaker than the condition used in Theorem 1 (see [1]) as it does not restrict k to be slightly greater than 1.

Lemma 1.

Consider a real valued function g: $\mathbb{R}\to \mathbb{R},$ defined by $g\left(x\right)=cx+d.$

$Itiseasytoseethatifc=1,theng\mathrm{is}2-rotativeif,andonlyif,d=0.$

$Let,c\ne 1$ we can write

$$\left|g_x-x\right|=\left|\left(c-1\right)x+d\right|,$$

$$g_x^2=g\left(g_x\right)=c(cx+d)+d=c^{2}x+cd+d,$$

or,

$$g_x^2-x=\left(c^2-1\right)x+cd+d,$$

or,

$$\left|g_x^2-x\right|=\left|\frac{c^2-1}{c-1}\right|\left|g_x-x\right|.$$

Therefore, for g to be 2–rotative, we must have

$$\left|g_x^2-x\right|\le \left|\frac{c^2-1}{c-1}\right|\left|g_x-x\right|,-3<c<1.$$

Example 3.

Let $g:I\to I$ be a F-rotative mapping with $F\left(\alpha \right)=ln\alpha$, such that

$$\tau +F\left|g_x^2-x\right|\le F\left|g_x-x\right|,$$

where $\tau =F\left(\left|\frac{1}{c+1}\right|\right),-3<c<1c\ne -1.$

$Itiseasytoseethat,for1<\left|c+1\right|<2,wehave\tau <0whichmeansthatgisnotF$-contraction mapping. We will prove that a F-rotative mapping has fixed point.

Since,

$$F\left(\left|\frac{1}{c+1}\right|\right)+F\left|g_x^2-x\right|\le F\left|g_x-x\right|.$$

By applying the logarithm, we have

$$ln\left(\left|\frac{1}{c+1}\right|\right)+ln\left|g_x^2-x\right|\le ln\left|g_x-x\right|,$$

or,

$$ln\left|\left(\frac{c-1}{c^2-1}\right)\left(g_x^2-x\right)\right|\le ln\left|g_x-x\right|.$$

Therefore,

$$\left|g_x^2-x\right|\le \left|\frac{c^2-1}{c-1}\right|\left|g_x-x\right|,$$

which is a $2-$rotative mapping and guarantees fixed point of F-rotative mapping in $\mathbb{R}$.

4. Application

Next, we apply our work to establish the existence of solutions for a nonlinear fractional differential equation considered in [3,17].

Let ${}^C{D}^{\mathfrak{k}}$ represents the Caputo fractional derivative of order $\mathfrak{k}.$

Consider the following problem:

$${}^C{D}^{\mathfrak{k}}(u\left(t\right)=h(t,u\left(t\right)),(0<t<1,1<\mathfrak{k}\le 2),$$

(8)

via the integral boundary conditions

$$u\left(0\right)=0,u\left(1\right)={\int }_0^{\delta }u\left(s\right)ds(0<\delta <1).$$

Consider a continuous function $h:[0,1]\times \mathbb{R}\to \mathbb{R}$ and a Banach space $(X,\parallel .{\parallel }_{\infty })$ of continuous functions $C\left(\right[0,1],\mathbb{R})$ from $[0,1]$ into $\mathbb{R}$. Define a Caputo derivative of fractional order $\mathfrak{k}$ for a continuous function $T:(0,\infty )\to \mathbb{R}$ as follows:

$${}^C{D}^{\mathfrak{k}}T\left(t\right)=\frac{1}{\mathrm{\Gamma }(n-\mathfrak{k})}{\int }_0^t{(t-s)}^{n-\mathfrak{k}-1}{T}^n\left(s\right)ds(n-1<\mathfrak{k}<n,n=\left[\mathfrak{k}\right]+1),$$

where $\left[\mathfrak{k}\right]$ denotes the greatest integer, which is no greater than $\mathfrak{k}$. Next, for a continuous function $T:{\mathbb{R}}^{+}\to \mathbb{R},$ we define the Riemann–Liouville fractional derivatives of order $\mathfrak{k}$ as follows:

$$D^{\mathfrak{k}}T\left(t\right)=\frac{1}{\mathrm{\Gamma }(n-\mathfrak{k})}\frac{d^n}{d{t}^n}{\int }_0^t\frac{T\left(s\right)}{{(t-s)}^{\mathfrak{k}-n+1}}(n=\left[\mathfrak{k}\right]+1),$$

such that the function of t on the right side is point-wise defined on $(0,+\infty )$, see [18].

Next, we establish the existence theorem.

Theorem 7.

1.   For all $t\in [0,1]$ and $a,b\in \mathbb{R}$ with $\xi (a,b)>0$, consider a function $\xi :{\mathbb{R}}^2\to \mathbb{R}$ and $\tau >0$ such that

$$|h(t,a)-h(t,b)|\le \frac{\mathrm{\Gamma }(\mathfrak{k}+1)}{5}{e}^{-\tau }|a-b|.$$

2. 

For some $u_0\in X$ such that $\xi (u_0\left(t\right),\mathfrak{M}{u}_0\left(t\right))>0,t\in [0,1]$, define $\mathfrak{M}:X\to X$ as:

$$g_{u\left(t\right)}=\frac{1}{\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^t{(t-s)}^{\mathfrak{k}-1}h(s,u\left(s\right))\mathrm{d}s-\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^1{(1-s)}^{\mathfrak{k}-1}h(s,u\left(s\right))\mathrm{d}s$$

$$+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^{\delta }\left({\int }_0^s{(s-\kappa )}^{\mathfrak{k}-1}h(\kappa ,u\left(\kappa \right))\mathrm{d}\kappa \right)\mathrm{d}s(t\in [0,1]),$$

and

$$g_{u\left(t\right)}^2=\frac{1}{\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^t{(t-s)}^{\mathfrak{k}-1}h(s,g_{u\left(s\right)})\mathrm{d}s-\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^1{(1-s)}^{\mathfrak{k}-1}h(s,g_{u\left(s\right)})\mathrm{d}s$$

$$+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^{\delta }\left({\int }_0^s{(s-\kappa )}^{\mathfrak{k}-1}h(\kappa ,g_{u\left(\kappa \right)})\mathrm{d}\kappa \right)\mathrm{d}s(t\in [0,1]),$$

3. 

for each $t\in [0,1]$ and $u,v\in X,\xi \left(u\right(t),v(t\left)\right)>0$ implies $\xi (g_{u\left(t\right)},\mathfrak{M}v\left(t\right))>0$,

4. 

if $\left\{u_n\right\}$ is a sequence in X such that $u_n\to u$ in X and $\xi (u_n,u_{n+1})>0$ for all $n\in \mathbb{N}$, then $\xi (u_n,u)>0$ for all $n\in \mathbb{N}.$

Then, problem (8) has at least one solution.

Proof. 

Since $u\in X$ represents the solution to (8) if, and only if, it satisfies the following integral equation

$$u\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^t{(t-s)}^{\mathfrak{k}-1}h(s,u\left(s\right))\mathrm{d}s-\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^1{(1-s)}^{\mathfrak{k}-1}h(s,\mathfrak{k}\left(s\right))\mathrm{d}s$$

$$+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^{\delta }\left({\int }_0^s{(s-\kappa )}^{\mathfrak{k}-1}h(\kappa ,u\left(\kappa \right))\mathrm{d}\kappa \right)\mathrm{d}s,t\in [0,1].$$

The problem (8) is equivalent to finding $u^{*}\in X$, which is a fixed point of $\mathfrak{M}$.

Now, let $u,v\in X$ such that $\xi \left(u\right(t),v(t\left)\right)>0$ for all $t\in [0,1]$. From $\left(i\right)$, we have

$$|g_{u\left(t\right)}^2-u\left(t\right)|=\left|\frac{1}{\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^t{(t-s)}^{\mathfrak{k}-1}h(s,g_{u\left(s\right)})ds-\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^1{(1-s)}^{\mathfrak{k}-1}h(s,g_{u\left(s\right)})ds\right.$$

$$+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^{\delta }\left({\int }_0^s{(s-\kappa )}^{\mathfrak{k}-1}h(\kappa ,g_{u\left(\kappa \right)})d\kappa \right)ds$$

$$-\frac{1}{\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^t{(t-s)}^{\mathfrak{k}-1}h(s,u\left(s\right))ds+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^1{(1-s)}^{\mathfrak{k}-1}h(s,\mathfrak{k}\left(s\right))ds$$

$$\left.-\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^{\delta }\left({\int }_0^s{(s-\kappa )}^{\mathfrak{k}-1}h(\kappa ,u\left(\kappa \right))d\kappa \right)ds\right|$$

$$\le \frac{1}{\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^t|t-s{|}^{\mathfrak{k}-1}|h(s,g_{u\left(s\right)})-h(s,u\left(s\right))|ds$$

$$+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^1|{(1-s)}^{\mathfrak{k}-1}|h(s,g_{u\left(s\right)})-h(s,\mathfrak{k}\left(s\right))|ds$$

$$+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^{\delta }\left|\left.{\int }_0^s{(s-\kappa )}^{\mathfrak{k}-1}(h(\kappa ,u\left(\kappa \right))-h(\kappa ,g_{u\left(\kappa \right)}))d\kappa \right|ds\right.$$

$$\le \frac{1}{\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^t|t-s{|}^{\mathfrak{k}-1}\frac{\mathrm{\Gamma }(\mathfrak{k}+1)}{5}{e}^{-\tau }|u\left(t\right)-g_{u\left(t\right)}|ds$$

$$+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^1{(1-s)}^{\mathfrak{k}-1}\frac{\mathrm{\Gamma }(\mathfrak{k}+1)}{5}{e}^{-\tau }|\mathfrak{k}\left(s\right)-g_{u\left(s\right)}|ds$$

$$+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^{\delta }\left({\int }_0^s|s-\kappa {|}^{\mathfrak{k}-1}\frac{\mathrm{\Gamma }(\mathfrak{k}+1)}{5}{e}^{-\tau }|u\left(\kappa \right)-g_{u\left(\kappa \right)}|d\kappa \right)ds$$

$$\le \frac{\mathrm{\Gamma }(\mathfrak{k}+1)}{5}{e}^{-\tau }{\parallel \mathfrak{M}u-u\parallel }_{\infty }\underset{t\in (0,1)}{sup}\left(\frac{1}{\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^1|\mathfrak{t}-s{|}^{\mathfrak{k}-1}ds\right.$$

$$\left.+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^1{(1-s)}^{\mathfrak{k}-1}ds+\frac{2t}{(2-{\delta }^2)\mathrm{\Gamma }\left(\mathfrak{k}\right)}{\int }_0^{\delta }{\int }_0^s|s-\kappa {|}^{\mathfrak{k}-1}d\kappa ds\right)$$

$$\le e^{-\tau }{\parallel g_u-u\parallel }_{\infty }.$$

Thus, for each $u,g_u\in X$, we have

$$\parallel u-{g^2}_u{\parallel }_{\infty }\le e^{-\tau }{\parallel u-g_u\parallel }_{\infty },$$

or,

$$d(u,{g^2}_u)\le e^{-\tau }d(u,g_u).$$

By applying logarithm, we can write

$$ln\left(d(u,{g^2}_u)\right)\le ln(e^{-\tau }d(u,g_u),$$

and hence

$$\tau +ln\left(d(u,{g^2}_u)\right)\le lnd(u,g_u).$$

which proves that $\mathfrak{M}$ is a F-rotative in X and the existence of $u^{*}\in X$ such that $u^{*}\in g_u^{*}.$ Hence, $u^{*}$ is a solution to problem (8). □

Remark 3.

Several results in the literature provide various applications to differential and integral equations concerning F-contraction mapping (For details, see [3,19,20,21,22] etc.). F-contraction represents a generalization of the famous Banach contraction Principle [23]. Since the relation $\left|{g^2}_x-g_x\right|<\left|g_x-x\right|$indicates that the mapping g is a contraction and the relation $\left|{g^2}_x-x\right|<\left|g_x-x\right|$ indicates that the distance between consecutive iterates of g for some x may also expand. Therefore, the relation $\tau +\left|{g^2}_x-x\right|\le \left|g_x-x\right|$ is a significant modification of $\tau +\left|{g^2}_x-g_x\right|\le \left|g_x-x\right|$. Moreover, the provided applications of our obtained result concerning F-rotative mapping has weakened the main condition of F-contraction by allowing for some of its iterates to expand the distance between them.

5. Questions

In this paper, we have presented a positive answer to the open question by Goebel and Koter [1] and found the precise value of $\gamma$ for $2-$rotative continuous mapping in $\mathbb{R}$. Next, it may be interesting to find the precise values of $\gamma$ for $3-$rotative mappings or their higher orders. Another potential attractive work is to find fixed points for $n-$rotative continuous mappings in Banach spaces. Further, we also need to illustrate some generalizations of the introduced rotative contraction mappings for an $n-$rotative with constant a.