Some new generalizations of $F-$contraction type mappings that weaken certain conditions on Caputo fractional type differential equations

1.   Introduction

The Banach contraction principle generally known as Banach fixed-point theorem emerged in 1922 ^[1], and due to its coherence and effectiveness, it has turned out to be a very popular tool in several branches of mathematical analysis for solving the existence problems. Numerous researchers studied the Banach fixed point theorem in different directions and established the extensions, generalizations and the applications of their findings. Among them, Wardowski ^[2] provided very interesting extension of Banach's fixed point theorem.

Definition 1.1. ^[2] Let a function $F:(0, \infty) \mapsto \mathbb{R}$ satisfy the following conditions:

$(F1)$ $F$ is strictly increasing, i.e. for all $\alpha, \beta\in(0, \infty)$ such that $\alpha < \beta\Rightarrow F(\alpha) < F(\beta)$.

$(F2)$ For each sequence $\{\alpha_n\}$ of positive numbers $\underset{n\rightarrow \infty}\lim\alpha_n = 0 \Leftrightarrow \underset{n\rightarrow \infty}\lim F(\alpha_n) = -\infty.$

$(F3)$ There exists $k\in(0, 1)$ such that $\underset{\alpha\rightarrow0^+}\lim\alpha^k F(\alpha) = 0$.

The set of all functions $F$ satisfying $(F1)-(F3)$ will be denoted by $\mathcal{F}$.

Example 1.1. ^[2] Suppose that the functions $F_i: (0, \infty)\mapsto \mathbb{R}$, $i = 1, 2, 3, 4$ are defined by

$(1)$ ${{F}_{1}}:t \to \ln t$.

$(2)$ ${{F}_{2}}:t \to t +\ln t$.

$(3)$ ${{F}_{3}}:t \to \frac{-1}{\sqrt{t }}$.

$(4)$ ${{F}_{4}}:t \to \ln ({{t }^{2}}+t)$.

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

Definition 1.2. ^[2] Let $(X, \mathcal{D})$ be a metric space. A mapping $T: X \mapsto X$ is said to be a $F-$contraction on $(X, \mathcal{D})$ if there exists $F\in \mathcal{F}$ and $\tau > 0$ such that for all $u, v\in X$,

Note that we have $\mathcal{D}(Tu, Tv) < \mathcal{D}(u, v)$ for all $u, v\in X$ with $Tu\neq Tv$ concluding that every $F-$contraction is a contractive mapping.

Wardowski ^[2] proved the existence of a unique fixed point in a complete metric space for every $F-$contraction mapping $T$. He also showed that $F-$contractions are the generalizations of Banach contractions.

Theorem 1.1. ^[2] Let $T$ be a self-mapping on a complete metric space $(X, \mathcal{D}).$ If $T$ forms a $F-$contraction, then it possesses a unique fixed point $u^*$. Moreover, for any $u \in X$ the sequence $\{T^{n}u\}$ is convergent to $u^*$.

After the development of $F-$contractions, several authors looked into the necessity of the conditions $(F1)$ – $(F3)$ and presented some weaken conditions by replacing or removing some of them.

We briefly present some existing cases. Secelean ^[3] suggested that the condition $(F2)$ can be replaced by a simple one:

$(F2^{'}) \inf F = -\infty$, or, also by

$(F2^{''})$ there exists a sequence $\{\alpha_{n}\}$ of positive real numbers such that $\underset{n \rightarrow \infty}\lim F(\alpha_{n}) = -\infty.$

Secelean ^[4] also removed condition $(F3)$ on the operator $T$ by assuming some boundedness condition. Moreover he also proved that $(F3)$ can be dropped without any additional supposition on $T$. Piri and Kumam ^[5] replaced condition $(F3)$ by the continuity of $F$. Vetro ^[6] replaced the constant $\tau$ with a function and generalized the $F-$contraction. Secelean and Wardowski ^[7] introduced $\psi F-$contraction and weak $\psi F-$contraction by weakening condition $(F1)$ and introducing the class of increasing functions $\psi$. Further, Lukács and Kajántó ^[8] found some results in $b-$metric spaces of $F-$contraction by omitting condition $(F2)$. Alsulami ^[9] introduced generalized $F-$Suzuki contraction in $b-$metric spaces and established the existence of fixed points by using the conditions $(F1)$ and $(F2)$ only.

Definition 1.3. Let $(X, \mathcal{D})$ be a metric space. A mapping $T :X\mapsto X$ is said to be a $F$-Suzuki contraction if there exist a real number $\tau > 0$ such that for all $u, v\in X$,

where, $F\in\mathcal{F}.$

On the other hand, in 2017, Gornicki ^[10] introduced a new type of mappings called $F$-expanding mappings and proved some new fixed point results for this new kind of mapping, especially on a complete $G-$metric space.

Definition 1.4. ^[10] Let $(X, \mathcal{D})$ be a metric space. A mapping $T: X \mapsto X$ is called $F-$expanding if there exist $F\in \mathcal{F}$ and $\tau > 0$ such that for all $u, v \in X$

Theorem 1.2. ^[10] Let $(X, \mathcal{D})$ be a complete metric space and $T: X \mapsto X$ be surjective and $F-$expanding. Then $T$ has a unique fixed point.

In 2001, James Merryfield ^[11] established a generalization of Banach contraction principal by following conjecture named as generalized Banach contraction conjecture.

Theorem 1.3. ^[11] Let $(X, \mathcal{D})$ be a complete metric space. Suppose $T:X\to X$ satisfies the following condition:

There exists an integer $p$ and a number $k\in [0, 1)$ such that for all $u, v\in X$ we have

Then, $T$ has exactly one fixed point.

Following the Wardowski's idea along with the conjecture presented by James Merryfield ^[11], in this paper, we introduce some new generalizations of $F-$contraction, $F$-Suzuki contraction and $F-$expanding mappings and prove the existence of their unique fixed points. Moreover, as an application of our main result, we investigate the existence of unique solution of the nonlinear Caputo fractional differential equations.

Recently Proinov ^[12] proved some fixed point theorems that extend earlier results of Moradi ^[13], Geraghty ^[14], Amini- Harandi and Petruşel ^[15], Li and Jiang ^[16], Jleli and Samet ^[17], Wardowski ^[2], Wardowski and Van Dung ^[18], Piri and Kumam ^[5], Secelean ^[7], Lukács and Ka$\mathop {\rm{j}}\limits^{'}$anto ^[8] and others. He also proved that the fixed point theorems of Wardowski ^[2] and Jleli and Samet ^[17] are equivalent to a special case of the well-known fixed point theorem of Skof ^[19].

Proinov established the following fixed point theorem for a self-mapping $T$ on a complete metric space $(X, \mathcal{D}).$

Theorem 1.4. ^[12] Let $(X, \mathcal{D})$ be a metric space and $T : X \mapsto X$ be a mapping such that ${{F}_{1}}\left(\mathcal{D}\left(Tu, Tv \right) \right)\le F\left(\mathcal{D}\left(u, v \right) \right)$ for all $u, v\in X$,

where, the functions ${{F}_{1}}, F:\left(0, \infty \right)\to \mathbb{R}$ satisfy the following conditions.

i) $F_{1}$ is nondecreasing;

ii) $F(t) < {{F}_{1}}(t)$ for $t > 0$;

iii) $\lim {{\sup }_{t\to \varepsilon +}}F(t) < {{F}_{1}}(\varepsilon +)$ for any $\varepsilon > 0$.

Then $T$ has a unique fixed point $u^* \in X$ and the iterative sequence $\left\{ {{T}^{n}}u \right\}$ converges to $u^{*}$ for every $u \in X$.

Proinov also obtained the following improvement of Wardowski's fixed point theorem.

Theorem 1.5. Let $(X, \mathcal{D})$ be a metric space and $T: X \mapsto X$ be a mapping such that

where, $\tau > 0$ and the function $F:\left(0, \infty\right)\to \mathbb{R}$ is nondecreasing. Then $T$ has a unique fixed point $u^* \in X$ and the sequence $\left\{T^nu\right\}$ converges to $u^{*}$ for every $u \in X$.

In this paper, we establish a fixed point theorem using a certain condition that generalizes the main contractive-type conditions used by Wardowski and Proinov.

2.   Main results

We start this section by introducing some new types of generalized $F-$contraction and generalized $F-$Suzuki contraction mappings.

Let $F:(0, \infty) \mapsto \mathbb{R}$ satisfy the following conditions:

$(F^*_1)$ For some $t > 0$ and $0 < F(t) < t$, if $0 < \alpha < \beta$, we have $F(\alpha) < F(\beta)$ and $F(\beta)\leq F(\alpha)+F(t)$.

$(F^*_{1'})$ For some $t > 0$ and $0 < F(t) < t$, if $0 < \alpha < \beta$, we have $F(\alpha) < F(\beta)$.

$(F^*_2)$ $\underset{n\to \infty}\lim F(\alpha_{n}) = -\infty \Leftrightarrow \underset{n\to\infty}\lim \alpha_{n} = 0$, for each positive number sequence $\{\alpha _n\}$.

$(F^*_3)$ $F$ is continuous on $(0, \infty)$.

Let us denote by $\mathfrak{F}$ the set of all functions $F$ satisfying the conditions $(F^*_1), (F^*_2), (F^*_3)$ and denote by $\mathfrak{T}$ the set of all functions $F$ satisfying the conditions $(F^*_{1'}), (F^*_2), (F^*_3)$.

Example 2.1. Define $F:(0, \infty) \mapsto \mathbb{R}$ by

$F(u) = \begin{cases} & \ln u, \quad 0 < u\leq 1\\ & 1-\frac{1}{u}, \quad u > 1 \end{cases}.$

Note that $F$ is strictly increasing with an upper bound $F(t) = 1$. Therefore, for some $t > 0$, $0 < F(t) < t,$ and $0 < \alpha < \beta,$ we have $F(\alpha) < F(\beta)\leq F(\alpha)+F(t)$ and $F \in \mathfrak{F}$. Further, we can easily check that the conditions $(F^*_{1'})$, $(F^*_2)$ and $(F^*_3)$ are also satisfied, hence $F \in \mathfrak{T}$.

Definition 2.1. Let $(X, \mathcal{D})$ be a metric space. A mapping $T: X\to X$ is said to be a generalized $F$-contraction if there exist $\tau > 0$ and an integer $p > 1$ such that for all $u, v\in X$,

where $F\in\mathfrak{F}$.

Definition 2.2. Let $(X, \mathcal{D})$ be a metric space. A mapping $T :X\mapsto X$ is said to be a generalized $F$-Suzuki contraction if there exist $\tau > 0$ and an integer $p > 1$ such that for all $u, v\in X$, with $u\neq v$

where, $F\in\mathfrak{T}$.

Theorem 2.1. Let $(X, \mathcal{D})$ be a complete metric space and $T:X\to X$ be continuous generalized $F$-contraction mapping with $p = 2$. Then $T$ has a unique fixed point $u^* \in X$ and for every $u_0\in X$ the sequence $\{T^nu_0\}_{n = 1}^{\infty}$ converges to the fixed point.

Proof. Let $u_0\in X$ be an arbitrary point and define a sequence $\{u_n\}\subseteq X$ by $u_{n+1} = Tu_n = T^{n+1}u_0$, for all $n\in\mathbb{N}$.

Now, we will prove that $\underset{n\rightarrow \infty}\lim \mathcal{D}(u_n, Tu_n) = 0$.

If $u_{n_0} = Tu_{n_0}$ for some $n_0\in \mathbb{N}$, then $\mathcal{D}(u_n, Tu_n) = \mathcal{D}(u_{n+1}, Tu_{n+1}) = \cdots = 0$, for all $n\geq n_0$ and so $\mathcal{D}(u_n, Tu_n)$ converges to 0, as $n\rightarrow \infty$.

Assume that $u_n\neq u_{n+1}$, for all $n\in\mathbb{N}$.

By the contraction assumption on $T$, there exits $\tau > 0$ such that

or

If $\mathcal{D}(u_{n-1}, u_{n}) = \min\{\mathcal{D}(Tu_{n-1}, Tu_{n}), \mathcal{D}(u_{n-1}, u_{n})\}$, the inequality (2.2) will take the form as follows

which also can be written as

If $\mathcal{D}(Tu_{n-1}, Tu_{n}) = \min\{\mathcal{D}(Tu_{n-1}, Tu_{n}), \mathcal{D}(u_{n-1}, u_{n})\}$.

Then, by condition $(F^*_1)$, we have

The above inequality together with inequality (2.1) yields that

which is equivalent to,

Combining (2.3) and (2.5) we have,

where, $\delta_{n} = \begin{cases} & 1, \quad \mathcal{D}(u_{n-1}, u_{n}) > \mathcal{D}(Tu_{n-1}, Tu_{n}) = \mathcal{D}(u_{n}, u_{n+1})\\ & 0, \quad \mathcal{D}(u_{n-1}, u_{n}) < \mathcal{D}(Tu_{n-1}, Tu_{n}) = \mathcal{D}(u_{n}, u_{n+1}) \end{cases}.$

Repeating this process we get,

where, $0\leq(\sum\limits_{k = 0}^{n}{\delta_k}) < \frac{n+1}{2}$.

Therefore,

So that,

Therefore, the results of (2.6) and condition $(F^*_2)$ implies,

or,

If $\underset{n\rightarrow \infty}\lim \min\{\mathcal{D}(Tu_n, T^{2}u _{n}), \mathcal{D}(u_n, Tu_{n})\} = \underset{n\rightarrow \infty}\lim \mathcal{D}(u_n, Tu_n) = 0$, then

If $\underset{n\rightarrow \infty}\lim \min\{\mathcal{D}(Tu_n, T^{2}u _{n}), \mathcal{D}(u_n, Tu_{n})\} = \underset{n\rightarrow \infty}\lim \mathcal{D}(Tu_n, T^2u_n) = 0$, then

Therefore, Eq (2.7) implies

Next, we claim that $\{u_n\}$ is a Cauchy sequence. Arguing by contradiction, we assume that there exist $\epsilon > 0$ and sequence $\{n(k)\}_{k = 1}^{\infty}$ and $\{m(k)\}_{k = 1}^{\infty}$ of natural numbers such that

So we have

It follows from (2.8) and the above inequality that

On the other hand, from (2.8) there exits $N\in\mathbb{N}$, such that

Next, we claim that

Arguing by contradiction, there exists $l\geq N$ such that

It follows from (2.9), (2.11) and (2.13) that

which is a contradiction.

Therefore, it follows from (2.12) and the assumptions of the theorem that

From condition $(F^*_3)$, (2.10) and (2.14), we get

which shows that $\{u_n\}$ is a Cauchy sequence. From the completeness of $(X, \mathcal{D})$, $\{u_n\}$ converges to some point $u^*\in X$.

Finally, the continuity of $T$ yields that

For uniqueness, we assume that $u'$ is another fixed point such that $Tu^* = u^*\neq u' = Tu'$. Then we have

which is a contradiction.

Therefore, $T$ has a unique fixed point in $X$.

Theorem 2.2. Let $(X, \mathcal{D})$ be a complete metric space and $T:X\to X$ be a generalized $F$-Suzuki contraction mapping with $p = 2$. Then $T$ has a unique fixed point in $X$ and for every $u_0\in X$ the sequence $\{T^nu_0\}_{n = 1}^{\infty}$ converges to the fixed point.

Proof. Let $u_0\in X$ be an arbitrary point and define a sequence $\{u_n\}\subseteq X$ by $u_{n+1} = Tu_n = T^{n+1}u_0$, for all $n\in\mathbb{N}$.

Now, we will prove that $\underset{n\rightarrow \infty}\lim \mathcal{D}(u_n, Tu_n) = 0$.

If $u_{n_0} = Tu_{n_0}$ for some $n_0\in \mathbb{N}$, then $\mathcal{D}(u_n, Tu_n) = \mathcal{D}(u_{n+1}, Tu_{n+1}) = \cdots = 0$, for all $n\geq n_0$ and so $\mathcal{D}(u_n, Tu_n)$ converges to 0, as $n\rightarrow \infty$.

Assume that $u_n\neq u_{n+1}$, for all $n\in\mathbb{N}$.

Since, ${\rm{ for \;all }}\, n\in \mathbb{N}$,

and

so, from the contraction assumption on $T$, there exits $\tau > 0$ such that

or

As in the proof of Theorem 2.1, the above inequality gives that

Moreover, analysis similar to that in the proof of Theorem 2.1 shows the sequence $\{u_n\}_{n = 1}^{\infty}$ is a Cauchy sequence.

Since, $(X, \mathcal{D})$ is complete, then the sequence $\{u_n\}_{n = 1}^{\infty}$ converges to some point $u^*\in X$, that is,

Now, we will claim that

Suppose, on the contrary, that there exists a $m\in\mathbb{N}$ satisfying the following three inequalities,

Now, (2.17) along with triangular inequality gives

which implies that

Also, (2.20) along with (2.18) gives

Similarly, (2.18) yields

which implies

From (2.19) and (2.22), we have

Now, by the contraction assumption on $T$, there exists $\tau > 0$ such that

If $\min\{\mathcal{D}(Tu_m, T^2u_m), \mathcal{D}(T^2u_m, T^3u_m)\} = \mathcal{D}(Tu_m, T^2u_m)$, we have

which yields,

From condition $(F^*_{1'})$, we have

which is a contradiction.

Likewise, if $\min\{\mathcal{D}(Tu_m, T^2u_m), \mathcal{D}(T^2u_m, T^3u_m)\} = \mathcal{D}(T^2u_m, T^3u_m)$, we have

which is a contradiction. Hence, (2.16) holds true.

Again, from (2.16), we have

Using (2.15) and condition $(F^*_2)$, we can write

and

Hence, $u^*$ is a fixed point of $T$.

For uniqueness, let us suppose that $T$ has another fixed point $u'$, such that $Tu^* = u^*\neq u' = Tu'$.

Since, $0 = \frac{1}{2}\mathcal{D}(u', Tu') < \mathcal{D}(u', u^*)$ and $0 = \frac{1}{2}\mathcal{D}(Tu', T^2u') < \mathcal{D}(Tu', Tu^*)$, from the contraction assumption on $T$, there exists $\tau > 0$ such that,

which is a contradiction. Therefore, $T$ has a unique fixed point.

In the next definition, we introduce some new notions of generalized $F-$expanding mappings.

Definition 2.3. Let $(X, \mathcal{D})$ be a metric space. A mapping $T: X\mapsto X$ is said to be a generalized $F-$expanding mapping of type $(A)$, if there exists $\tau > 0$ such that for all $u, v\in X$,

where $F\in\mathfrak{F}$.

Definition 2.4. Let $(X, \mathcal{D})$ be a metric space. A mapping $T: X\mapsto X$ is said to be a generalized $F-$expanding mapping of type $(B)$, if there exists $\tau\geq0$ such that for all $u, v\in X$,

where $F\in\mathfrak{F}$.

Theorem 2.3. Let $(X, \mathcal{D})$ be a complete metric space and let $T: X\mapsto X$ be continuous surjective and generalized $F-$expanding of type $(A)$. Then $T$ has a unique fixed point in $X$ and for every $u_0\in X$ the sequence $\{T^nu_0\}_{n = 1}^{\infty}$ converges to the fixed point.

Proof. Firstly, we will show that $T$ is bijective, which only needs to show that

If $\mathcal{D}(u, v) > 0$, from the assumption on $T$, there exits $\tau > 0$ such that

or

The second above inequality together with condition $(F^*_1)$ implies,

which yields that $0\geq \mathcal{D}(u, v)$, a contradiction.

So, we have $Tu\neq Tv$ or $\mathcal{D}(Tu, Tv) > 0$, hence $T$ is injective and then bijective. Consider a mapping $S$ such that $TS = ST = I_u$, where $I_u$ is identity mapping on $X$.

Let $u = S^2u, v = S^2v$, so that $Tu = Su, Tv = Sv$ and $T^2u = u, T^2v = v$.

If $\min\{\mathcal{D}(T^2u, T^2v), \mathcal{D}(Tu, Tv)\} = \mathcal{D}(Tu, Tv)$, there exits $\tau > 0$ such that

Condition $(F^*_1)$ yields

which together with (2.25) implies that

Using the inverse mapping $S$, the above inequality takes the form

Moreover, we have

Combining (2.27) and (2.28), we have

Again, if $\min\{ \mathcal{D}(T^2u, T^2v), \mathcal{D}(Tu, Tv)\} = \mathcal{D}(T^2u, T^2v)$, there exists $\tau > 0$ such that

and

From condition $(F^*_1)$ we have,

Combining (2.29) and (2.30), together with the assumption on $T$, we have

which shows that $S$ is the generalized $F-$contraction defined in Theorem 2.1. From the conclusion of Theorem 2.1, $S$ has a unique fixed point, so does $T$.

Theorem 2.4. Let $(X, \mathcal{D})$ be a complete metric space and let $T: X\mapsto X$ be continuous surjective and generalized $F-$expanding of type $(B)$. Then $T$ has a unique fixed point in $X$ and for every $u_0\in X$ the sequence $\{T^nu_0\}_{n = 1}^{\infty}$ converges to the fixed point.

Proof. Firstly, we will show that $T$ is bijective, which only needs to show that

If $\mathcal{D}(u, v) > 0$, from the assumption on $T$, there exits $\tau\geq0$ such that

or

The second inequality together with condition $(F^*_1)$ implies,

which yields that $0\geq \mathcal{D}(u, v)$, a contradiction.

So, we have $Tu\neq Tv$ or $\mathcal{D}(Tu, Tv) > 0$, hence $T$ is injective and then bijective.

Consider a mapping $S$ such that $TS = ST = I_u$, where $I_u$ is identity mapping on $X$.

Let $u = S^2u, v = S^2v$, so that $Tu = Su, Tv = Sv$ and $T^2u = u, T^2v = v$.

If $\min\{\mathcal{D}(T^2u, T^2v), \mathcal{D}(Tu, Tv)\} = \mathcal{D}(Tu, Tv)$, there exists $\tau\geq0$ such that

so that

From condition $(F^*_1)$, we have

Again, if $\min\{\mathcal{D}(T^2u, T^2v), \mathcal{D}(Tu, Tv)\} = \mathcal{D}(T^2u, T^2v)$, there exists $\tau\geq0$ such that

and

From condition $(F^*_1)$, we can write

Since $T$ is bijective, we have

Combining (2.32) and (2.33), together with the assumption on $T$, we have

which is equivalently stated as there exists $\tau' > 0$ such that

which shows that $S$ is the generalized $F-$contraction defined in Theorem 2.1. From the conclusion of Theorem 2.1, $S$ has a unique fixed point, so does $T$.

Theorem 2.5. Let $(X, \mathcal{D})$ be a complete metric space. Suppose a continuous mapping $T:X\mapsto X$ satisfy

where, non-decreasing functions $F, F_1 \in \mathfrak{F}$ and for all $t, t_1\in\mathbb{R_+}$, there exist $\upsilon > 0, \tau > 2\upsilon,$ such that

Then $T$ has a unique fixed point in $X$ and for every $u_0\in X,$ the sequence $\{ T^mu_0\}_{m = 1}^{+\infty }$ converges to the fixed point.

Proof. As, $F, F_1: (0, \infty) \to \mathbb{R_{+}}$ are non-decreasing functions, so that we can write

If, $F\left(\mathcal{D}\left({{u}_{m-1}}, {{u}_{m}} \right) \right)\le {{F}_{1}}\left(\mathcal{D}\left({{u}_{m-1}}, {{u}_{m}} \right) \right),$ we have

If, $F\left(\mathcal{D}\left({{u}_{m-1}}, {{u}_{m}} \right) \right) > {{F}_{1}}\left(\mathcal{D}\left({{u}_{m-1}}, {{u}_{m}} \right) \right),$

Using condition $(2.35)$, we can write (2.36) as follows

Then, we have either

or

If inequality (2.38) holds true, the inequality (2.37) will take the form

If inequality (2.39) is true, we have $F_1(\mathcal{D}(Tu_{m-1}, Tu_m)) < F_1(\mathcal{D}(u_{m-1}, u_m)).$

From condition $(A)$, we have

Using inequality (2.41) in (2.37), we can write

Moreover, from (2.39), we have

Combining both inequalities (2.40) and (2.42), we have

where $\delta_m = \begin{cases}1\quad & {\rm{if}}\quad F_1(t_2) > F_1(t_1)\\ 2\quad & {\rm{if}}\quad F_1(t_2) < F_1(t_1) \end{cases}$, $t_1, t_2\in \mathbb{R_+}, t_1\neq t_2$.

The above inequality can be written as

Repeating this process, we have

So that

Since, $\tau > 2\upsilon$ and $\sum\limits_{j = 1}^{m}\delta_j < m+1$, we have

So that we can write

Now, Eq (2.44) further has two possible cases.

Condition $(F^*_2)$ among case $(G)$ yields

or equivalently,

Condition $(F^*_2)$ among case $(H)$ yields

Therefore, from (2.44), we get

Now, we will prove that the sequence $\{u_m\}_{m = 1}^{+\infty }$ is a Cauchy sequence.

Suppose, on the contrary, that there exist $\varepsilon > 0$ and sequences $\{ \mathfrak{g}(m)\}_{m = 1}^{+\infty }$ and $\{\mathfrak{h}(m)\}_{m = 1}^{+\infty }$ of natural numbers such that for all $m\in\mathbb{N}$,

So that we can write

That is,

Inequalities (2.45) and (2.47) yield

Further, from (2.45) there exists $N\in \mathbb{N}$ such that for all $m\geq N$,

Next we claim that for all $m\geq N$,

Suppose, on the contrary, that there exist $r\geq N,$ such that

It follows from (2.45), (2.46) and (2.50) that

Which is a contradiction. Therefore, (2.49) together with the assumption of the theorem gives

From $(F^*_3)$, (2.45), (2.51) and the assumption of $F$-contraction, we get $\tau+\min\{F_1(\varepsilon), F_1(\varepsilon)\}\leq F(\varepsilon),$ which yields $\tau +F_1(\varepsilon)\leq F(\varepsilon)$. Then the condition $(2.35)$ allows us to write $\tau +{{F}_{1}}\left(\varepsilon \right)\le {{F}_{1}}\left(\varepsilon \right)+\upsilon.$ That yields a contradiction as $\tau > 2 \upsilon.$ The completeness of $(X, \mathcal{D})$ proves that $\{u_m\}_{m = 1}^{+\infty}$ converges to some point $u^*$ in $X$. Now, the continuity of $T$ implies

Therefore, $T$ has a unique fixed point $u^*$.

Here is an example to show the validity of Theorem 2.1.

Example 2.2. ^[20] Let $B$ be closed unit ball in $l_1$ space of all absolutely summable sequence $u = (u_1, u_2, \cdots)$ with a metric inherited from the standard norm $\|u\| = \sum\limits_{i = 1}^{\infty }\|u_i\|$.

Consider a function $h:[-1, 1 ]\mapsto [-1, 1 ]$ given by

$h(w) = \begin{cases} & 1+2w, \quad -1\leq w\leq -1/2\\ & 0, \quad -1/2\leq w\leq 1/2\\ & -1+2w, \quad 1/2\leq w\leq 1 \end{cases}.$

It is easy to observe that, for all $w_1, w_2\in[-1, 1],$ we have

and

Further, let us define a surjective mapping $T:B\mapsto B$ by

Then for $i\geq 2$, we have

For each $u = (u_1, u_2, \cdots), v = (v_1, v_2, \cdots)\in B$, we have

and for $i\geq 2$,

Then, we have for all $u = (u_1, u_2, \cdots), v = (v_1, v_2, \cdots)\in B$,

which implies that,

If $\|Tu-Tv\|\neq\|T^2u-T^2v\|$, there exists $\tau > 0$, such that

or

Therefore, $F(\alpha) = \ln\alpha$ represents generalized $F$-contraction mapping, hence Theorem 2.1 guarantees the existence of a unique fixed point of $T$.

Note that, $F(\alpha) = \ln\alpha$ does not contract, whenever $\max\{\|Tu-Tv\|, \|T^2u-T^2v\|) > \|u-v\|$.

That is, $\tau+\ln(\|Tu-Tv\|)\geq\ln(\|u-v\|)$, for all $u, v\in B$.

Therefore, $T$ does not represent $F$-contraction mapping defined in ^[2].

Hence Theorem 1.1 does not guarantee the existence of a fixed point.

Similarly, for $F'(\alpha) = \ln \alpha +\alpha$, we can write

so,

This can be written as,

Therefore, for

inequality (2.52) shows that $T$ is a generalized $F$-contraction mapping.

3.   Applications to Caputo fractional differential equations

As an application of our work, we will study the existence of solutions to Caputo fractional differential equations of the fractional order in (1, 2) and the integral boundary condition. The main condition in the problems studied in ^[21,22] is associated with sufficient small Lipschitz constant. We will use a less restrictive condition than the Lipschitz condition by applying our obtained fixed point theorems.

For, $1 < l < 2$, and a Caputo fractional derivative ${}_{\vartheta_{1}}^{C}D_{t}^{l}z(t) = \frac{1}{\Gamma (2-l)}\int\limits_{\vartheta_{1}}^{t}{{{(t-s)}^{1-l}}}{z}''(s)ds$, consider a nonlinear Caputo fractional differential equation:

with an integral boundary condition:

where $z\in \mathbb{R}$, $\vartheta_{1}, \vartheta_{2}$ are the given real numbers such that $0 \le \vartheta_{1} < \vartheta_{2}$.

Let $\Omega = C([\vartheta_{1}, \vartheta_{2}], \mathbb{R})$ with a norm ${{\left\| z \right\|}_{[\vartheta_{1}, \vartheta_{2}]}} = {{\sup }_{s\in [\vartheta_{1}, \vartheta_{2}]}}|z(s)|$.

For any $z, v\in \Omega,$ we define $\mathcal{D} (z, v) = {{\left\| z-v \right\|}_{[\vartheta_{1}, \vartheta_{2}]}}$.

Consider the linear fractional differential equation:

with the integral boundary condition (3.2) where $g\in \Omega$.

Lemma 3.1. For $g\in \Omega$, following function represents the solution of boundary value problem $(3.1), (3.2)$.

The proof of Lemma 3.1 is based on the presentation of the solution

given in ^[23].

Next, we will define a mild solution of (3.1) and (3.2).

Definition 3.1. The function $z \in \Omega$ is a mild solution of the boundary value problem (3.1) and (3.2) if it satisfies:

For any function $u\in \Omega$, we define a mapping $\Upsilon :\Omega \to \Omega$ by

for $t\in [\vartheta_{1}, \vartheta_{2}]$. Now, we establish the existence result as follows.

Theorem 3.1. Suppose that,

(i) There exists a constant $K > 0$, such that,

and a function $u\in C([\vartheta_{1}, \vartheta_{2}]\times \mathbb{R}, \mathbb{R})$ such that

Where, $r\in(0, 1],$

(ii) There exists a function ${{z}_{0}} \in \Omega$ such that $\mathcal{D} ({{z}_{0}}, \Upsilon ({{z}_{0}})) > 0$, where the operator $\Upsilon$ is defined by (3.6);

(iii) For any two functions $u, \, z\in \Omega,$ such that $\, \mathcal{D} (u, \, z) > 0,$ the inequality $\mathcal{D} (\Upsilon (u), \Upsilon (z)) > 0$ holds.

Then the boundary value problem (3.1), (3.2) has a mild solution.

Proof. Note that any fixed point of the mapping $\Upsilon$ is a mild solution of the boundary value problem (3.1) and (3.2). Now, let $z, v \in \Omega$ be such that $\mathcal{D}(z, v) > 0$. By condition (i) of the theorem, we obtain

where,

Therefore,

Further, relation (3.8) yields

As, $\left\| z-v \right\|_{\infty }^{{{r}^{2}}}\le \left\| z-v \right\|_{\infty }^{r}, \, \, \, \, \, \, r\in \left(0, 1 \right].$

The inequality (3.9) can be written as

Relations (3.8), (3.10) can be combined in one of the following forms

If relation (3.12) holds, we can write

Define, $F(t) = r\ln t-\ln p\in \mathfrak{F}$, and ${{F}_{1}}(t) = \ln t\in \mathfrak{F}$, where $p\in (0, 1)$, $p\Lambda < 1$ and $p\Lambda^{r+1} < 1$, so that the relation (3.13) can be written as

or,

If relation (3.11) holds, we can write

The relation (3.14) can be written as

So that,

Therefore, $\Upsilon :\Omega \to \Omega$ is a generalized $F$-contraction mapping and the operator $\Upsilon$ has a fixed point in $\Omega$. That is, there exists a function ${{z}^{*}}\in C([\vartheta_{1}, \vartheta_{2}], \mathbb{R})$ such that ${{z}^{*}} = \Upsilon ({{z}^{*}})$. The function ${{z}^{*}}$ is a mild solution of the boundary value problem for (3.1) and (3.2).

Remark 3.1. In comparison with the result of ^[24], we used a weaker condition $\Lambda \in (0, \infty)$ to prove the existence of solution to (3.1) and (3.2) instead of $\Lambda \in (0, 1)$.

Moreover, one can easily observe that the use of multiple functions in the generalized $F$-contraction also allows us to define a function $u\in C([\vartheta_{1}, \vartheta_{2}]\times \mathbb{R}, \mathbb{R})$ in Theorem 3.1 with a weaker condition $|u(\tau, z)-u(\tau, v){{|}^{m}}\le K|z-v{{|}^{r}}, z, v\in \mathbb{R}, \tau\in [{{\vartheta }_{1}}, {{\vartheta }_{2}}]$, where $m, r \in (0, \infty).$

Example 3.1. Consider the nonlinear Caputo fractional differential equation

with the integral boundary condition:

In this case,

and

where,

Therefore, Theorem 3.1 guarantees the solution of boundary value problem (3.15) and (3.16).

Remark 3.2. Note that the boundary value problem (3.15) and (3.16) are also studied in ^[22] (see Example 5 therein) and ^[24] (see Example 3.3 therein). Based on the obtained fixed points theorems we used the weaker conditions for the right hand side part of the equation and found the existence of fixed point for $K > 0$ and $\Lambda > 1.$

Remark 3.3. Wardowski obtained some fixed point theorems (see; Theorem 1.1) assuming that $T$ satisfies the following contractive-type condition

where, $F:(0, \infty)\mapsto \mathbb{R}$ is nondecreasing. Whereas, the condition that we used in Theorem 2.1 is of the following form

One can easily observe that relation (3.18) represents a generalization of (3.17).

Moreover, the following Proinov's condition represents a generalization of Wardowski's contraction condition.

The main condition we used in Theorem 2.5 is of the form

where, $F(\mathcal{D}(x, y)) = {F}'(\mathcal{D}(x, y))-\tau.$ In (3.19), function ${{F}_{1}}$ cannot exceed $F$. Whereas, the condition in (3.20) allows ${{F}_{1}}$ to exceed $F$ for different iterates.

4.   Conclusions

Although, Proinov ^[12] claimed that being a special case of Skof's result ^[19], the $F-$ contraction type mappings and their generalizations do not add a valuable work in the literature anymore, we found some new generalizations that extend Wardowski ^[2], Skof ^[19] as well as Proinov's idea ^[12] of $F-$contraction type mappings. Moreover, with the use of multiple functions and the idea of generalized Banach contraction principal ^[11], we applied less restrictive conditions on Caputo fractional differential equations than the sufficient small Lipschitz constant studied by Mehmood ^[22] and Hanadi ^[24]. The new generalizations of $F$-contraction, $F$-expanding type mappings and the corresponding results will break open new grounds for the research workers as they will be able to find the existence of solution to an extensive range of differential equations (see ^{[25,26,27,28,29,30,31]}) with some weaker conditions.

5.   Questions

In this research, the new generalizations of $F$-contraction mapping, $F$-Suzuki contraction mapping, $F$-expanding mapping and the corresponding results will provide a new direction of metric fixed point theory for the research workers. They may try to find the existence of fixed point for the further extensions of certain generalized mappings.

$1)$ One may find the above results with $p > 2$, for the generalizations of $F$-contraction, $F$-Suzuki contraction and $F$-expanding mappings.

$2)$ One may work on the idea of introducing new generalizations of $F$-contraction, $F$-Suzuki contraction and $F$-expanding mappings.

$3)$ There may exist the possibility of finding fixed points for these generalized mappings in other generalized metric spaces.

Acknowledgements

Both authors thank all the anonymous reviewers for their helpful and valuable comments, which definitely helped to improve the manuscript.

Conflicts of interest

The authors declare that they have no competing interests.