1 Introduction

In 2015, Khojasteh et al. [1] initiated the concept of simulation functions.

Definition 1.1

([1])

A mapping \(\zeta :[0,\infty )\times {}[ 0,\infty )\rightarrow \mathbb{R}\) is called a simulation function if the following conditions hold:

\((\zeta _{1})\):

for all ;

\((\zeta _{2})\):

if , are sequences in \((0,\infty )\) such that , then

(1.1)

We denote by \(\mathcal{Z}\) the family of all above simulation functions.

Let be a metric space and \(\alpha :\mathcal{X}\times \mathcal{X}\rightarrow [0,\infty )\) be a function. A mapping is called α-orbital admissible if the following condition holds:

(1.2)

for all \(\nu \in \mathcal{X}\). Moreover, an α-orbital admissible mapping is called triangular α-orbital admissible if for all \(\nu , \omega \in \mathcal{X}\), we have

(1.3)

Definition 1.2

A set \(\mathcal{X}\) is said to be regular with respect to a given function \(\alpha :\mathcal{X}\times \mathcal{X} \to [0,\infty )\) if for each sequence \(\{\nu _{n}\}\) in \(\mathcal{X}\) such that \(\alpha (\nu _{n},\nu _{n+1})\geq 1\) for all n and \(\nu _{n} \rightarrow \nu \in \mathcal{X}\) as \(n\rightarrow \infty \), then \(\alpha (\nu _{n},\nu )\geq 1\) for all n.

The notion of α-admissible \(\mathcal{Z}\)-contractions with respect to a given simulation function was merged and used by Karapinar in [2]. Using this new type of contractive mappings, he investigated the existence and uniqueness of a fixed point in standard metric spaces.

Definition 1.3

([2])

Let T be a self-mapping defined on a metric space \((\mathcal{X}, d)\). If there exist a function \(\zeta \in \mathcal{Z}\) and \(\alpha :\mathcal{X}\times \mathcal{X} \to [0,\infty )\) such that

$$ \zeta \bigl(\alpha (\nu , \omega )d(T\nu , T\omega ), d(\nu , \omega ) \bigr)\geq 0 \quad \text{for all }\nu , \omega \in \mathcal{X}, $$
(1.4)

then we say that T is an α-admissible \(\mathcal{Z}\)-contraction with respect to ζ.

Theorem 1.4

([2])

Let \((\mathcal{X},d)\)be a complete metric space and let \(T:\mathcal{X}\rightarrow \mathcal{X}\)be an α-admissible \(\mathcal{Z}\)-contraction with respect to ζ. Suppose that:

  1. (a)

    T is triangular α-orbital admissible;

  2. (b)

    there exists \(\nu _{0}\in \mathcal{X}\)such that \(\alpha (\nu _{0}, T\nu _{0})\geq 1\);

  3. (c)

    T is continuous.

Then there is \(\nu _{*}\in \mathcal{X}\)such that \(T\nu _{*}=\nu _{*}\).

Remark 1.5

The continuity condition in Theorem 1.4 can be replaced by the “regularity” condition, which is considered in Definition 1.2.

We will consider the following set of functions:

and we denote

Several interesting extensions and generalizations of the Banach contraction principle [3] appeared in the literature. For instance, see [4–10]. Among these generalizations, we cite the paper of Pata [11]. Since then, much work appeared in the same direction; see [12–15].

Theorem 1.6

([11])

Let be a complete metric space and let \(\Lambda \geq 0\), \(\lambda \geq 1\), \(\beta \in [0,\lambda ]\)be fixed constants. The mapping has a fixed point in \(\mathcal{X}\)if the inequality

(1.5)

is satisfied for every \(\varepsilon \in [0,1]\)and .

Definition 1.7

Let be a metric space. We say that is a Pata type Zamfirescu mapping if for all \(\nu , \omega \in \mathcal{X}\), \(\psi \in \mathcal{Z}\) and for every \(\varepsilon \in [0,1]\), , it satisfies the following inequality:

(1.6)

where

and \(\Lambda \geq 0\), \(\lambda \geq 1\) and \(\beta \in [0,\lambda ]\) are constants.

Theorem 1.8

([16])

Let be a complete metric space and let be a Pata type Zamfirescu mapping. Then has a unique fixed point in \(\mathcal{X}\).

We state the following useful known lemma.

Lemma 1.9

Let be a complete metric space and be a sequence in \(\mathcal{X}\)such that . If the sequence is not Cauchy, then there exist and subsequences and of such that

(1.7)

and

(1.8)

In this paper, we combine the concepts of simulation functions and α-admissibility to give a generalized Pata type fixed point result. At the end, we present an application on fractional calculus.

2 Main results

We denote by \(\tilde{\mathcal{Z}}\) the set of all functions \(\tilde{\zeta }:[0,\infty )\times {}[ 0,\infty )\rightarrow \mathbb{R}\) satisfying the following condition:

\((\tilde{\zeta }_{1})\):

for all .

Definition 2.1

Let be a metric space and \(\phi \in \Phi \). Let \(\Lambda \geq 0\), \(\lambda \geq 1\) and \(\beta \in [0,\lambda ]\) be fixed constants. A triangular α-orbital admissible mapping is called an α-ζ̃-- Pata contraction if there exists a function \(\tilde{\zeta }\in \tilde{\mathcal{Z}}\) such that, for every \(\varepsilon \in [0,1]\), the following condition is satisfied:

(2.1)

for all \(\nu ,\omega \in \mathcal{X}\), where

(2.2)

and

(2.3)

Remark 2.2

It is clear that any Pata type Zamfirescu mapping is also an α-ζ̃-- Pata mapping. Indeed, letting \(\alpha (\nu , \omega )=1\) and , the inequality (2.1) becomes

Moreover, note that for all \(\nu , \omega \in \mathcal{X}\).

Theorem 2.3

Every α–ζ̃––Pata contraction on a complete metric space possesses a fixed point if

(i):

there exists such that ;

(ii):

is triangular α-orbital admissible;

(iii):

either is continuous, or the set \(\mathcal{X}\)is regular.

If in addition we assume that the following condition is satisfied:

(iv):

for all ,

then such a fixed point of is unique.

Proof

Let be a point such that . On account of the assumption that is a triangular α-orbital admissible mapping, we derive that

and iteratively we find

(2.4)

Moreover, by (2.4) together with (1.3), we have

Again, iteratively, one writes

(2.5)

Starting from this point , we build an iterative sequence where for \(n=1,2,3, \ldots \). We can presume that any two consequent terms of this sequence are distinct. Indeed, if, on the contrary, there exists \(i_{0} \in \mathbb{N}\) such that

then is a fixed point. To avoid this, we will assume in the following that for all \(n\in \mathbb{N}\)

We mention that (2.4) can be rewritten as

(2.6)

respectively,

(2.7)

for any \(n \in \mathbb{N}\). In the sequel, we will denote for all \(\nu \in X\).

Since is an α-ζ̃--Pata contraction, we have

Thus, taking into account \((\tilde{\zeta }_{1})\), together with (2.6) we get

(2.8)

where

and

Denoting by , we have

Thus, (2.8) becomes

(2.9)

We claim that the sequence \(\{ \gamma _{n} \} \) is non-increasing. Indeed, if we suppose the contrary that, for some p, \(\gamma _{p}<\gamma _{p+1}\), and so \(\max \{ \gamma _{p}, \gamma _{p+1} \} =\gamma _{p+1}\), then we have \(\vert \gamma _{p}-\gamma _{p+1} \vert =\gamma _{p+1}-\gamma _{p}\).

(2.10)

Consequently, from (2.9), we get, for such an integer p,

(2.11)

The above inequality is true for all \(\varepsilon \in [0,1]\). In particular, for \(\varepsilon =0\), we get \(\gamma _{p+1}\leq \gamma _{p+1}\), which clearly is a contradiction. In this case, we find that the sequence \(\{ \gamma _{n} \} \) is non-increasing. So we can find a non-negative real number γ such that

We claim that \(\gamma =0\). In order to prove this, we have to show that the sequence \(\{ \kappa _{n} \} \) is bounded, where . Since the sequence is non-increasing, we have

By the triangle inequality, we get

(2.12)

On account of (2.5), regarding that is an α-ζ̃-Pata- contraction, we have

Taking into account (2.7), this is equivalent to

Using (2.12) and the above inequality, we get

Moreover, since \(\beta \leq \lambda \), we have

$$ \begin{aligned} \varepsilon \kappa _{n} &\leq (3- \varepsilon ) \kappa _{1}+\Lambda \varepsilon ^{\lambda }\psi ( \varepsilon ) [1+2 \kappa _{n}+2\kappa _{1} ]^{\beta } \\ &\leq (3-\varepsilon )\kappa _{1}+ \Lambda \varepsilon ^{\lambda } \psi ( \varepsilon ) [1+2\kappa _{n}+2\gamma _{1} ]^{\lambda } \\ & = (3-\varepsilon )\kappa _{1}+\Lambda \varepsilon ^{\lambda } \psi ( \varepsilon ) (1+2\kappa _{n})^{\lambda } \biggl[1+ \frac{2\kappa _{1}}{1+2\kappa _{n}} \biggr]^{\lambda } \\ & \leq 3\kappa _{1}+\Lambda \varepsilon ^{\lambda }\psi ( \varepsilon )2^{\lambda }\kappa _{n}^{\lambda }\biggl(1+ \frac{1}{2\kappa _{n}}\biggr)^{\lambda }(1+2 \kappa _{1})^{\lambda }. \end{aligned} $$

Now, supposing that the sequence \(\{ \kappa _{n} \} \) is not bounded, there exists a subsequence \(\{ \kappa _{n_{l}} \} \) of \(\{ \kappa _{n} \} \) such that \(\kappa _{n_{l}}\rightarrow \infty \) as \(l\rightarrow \infty \). In this case, letting \(\varepsilon =\varepsilon _{l}=\frac{1+3\kappa _{1}}{\kappa _{n_{l}}} ( \in [0,1])\), the above inequality yields

$$ \begin{aligned} 1& \leq \Lambda 2^{\lambda }\bigl[\varepsilon ^{\lambda }\kappa _{n}^{\lambda }\bigr](1+2 \kappa _{1})^{\lambda }\biggl(1+\frac{1}{2\kappa _{n_{l}}}\biggr)^{\lambda } \psi ( \varepsilon _{l}) \\ &\leq \Lambda 2^{\lambda }(1+3\kappa _{1})^{\lambda }(1+2\kappa _{1})^{ \lambda }\biggl(1+\frac{1}{2\kappa _{n_{l}}}\biggr)^{\lambda } \psi (\varepsilon _{l}) \\ &\leq \Lambda 2^{\lambda }(1+3\kappa _{1})^{2\lambda }\biggl(1+ \frac{1}{2\kappa _{n_{l}}}\biggr)^{\lambda }\psi (\varepsilon _{l}) \rightarrow 0 \quad \text{as }l\rightarrow \infty . \end{aligned} $$

This is a contradiction. Thus, we conclude that our presumption is false and then the sequence \(\{ \kappa _{n} \} \) is bounded. Furthermore, there exists such that for all \(n\in \mathbb{N}\).

Let us go back now and prove that \(\gamma =0\) (where \(\gamma =\lim_{n\rightarrow \infty }\gamma _{n}\)). In view of (2.10) and the fact that the sequence \(\{ \gamma _{n} \} \) is non-increasing, one writes

Recall that

Taking into account that is an α-ζ̃-\({E_{*}}\) contraction, keeping in mind (2.6) and using \((\tilde{\zeta }_{1})\), we have

We have

(2.13)

Letting \(n \to \infty \) in the previous inequality, we obtain

which is equivalent to

When \(\varepsilon \rightarrow 0\), we get \(\gamma \leq 0\). Therefore,

(2.14)

As a next step, we claim that is a Cauchy sequence. On the contrary, assuming that the sequence is not Cauchy, it follows from Lemma 1.9 that there exist and subsequences and such that (1.7) and (1.8) hold. Replacing and in (2.1), we have

(2.15)

where

The triangular α-orbital admissibility of shows that . Thus,

(2.16)

Letting \(l\rightarrow \infty \) and taking into account (2.14) and Lemma 1.9, we have

(2.17)

At the same time, one writes

Denoting by and , by Lemma 1.9, it follows that

Thus, passing to the limit as \(l\rightarrow \infty \) in (2.16), we get

Furthermore,

i.e.,

That is, . Therefore, is a Cauchy sequence in the complete metric space. For this reason, there exists \(\nu ^{*}\in \mathcal{X}\) such that , as \(n\rightarrow \infty \).

Furthermore, in the case that is a continuous mapping, we get , that is, is a fixed point of .

Now, suppose that \(\mathcal{X}\) is regular. From (2.1), one writes

(2.18)

Using the regularity of \(\mathcal{X}\) and \((\tilde{\zeta }_{1})\), we get

(2.19)

where

and

Taking into account the boundedness of the sequence \(\{ \kappa _{n} \} \), we have

On the other hand,

Letting \(n\rightarrow \infty \) in the inequality (2.19), we find

which is equivalent to

Obviously, we obtain for \(\varepsilon =0\) that , so . Thus, \(\nu ^{*}\) is a fixed point of . Finally, to prove the uniqueness of the fixed point, we suppose that there exist two fixed points such that \(\nu ^{*}\neq \omega ^{*}\). We have

Taking into account \((\mathit{iv})\), we obtain

which leads to

In the limit \(\varepsilon \rightarrow 0\), we get , that is, \(\nu ^{*}=\omega ^{*}\), which is a contradiction. Therefore, the fixed point of is unique. □

In the following, we present an example that supports our statement, that is, Theorem 2.3 is a generalization of Theorem 1.8.

Example 2.4

Take \(\mathcal{X}=A \times A\), where \(A=[0,11]\) and is the usual distance. Define the mapping by

where . For \(\nu _{1}=(11,0)\) and \(\nu _{2}=(2,0)\), we have

and

Thus,

so that the inequality (1.6) does not hold for \(\varepsilon =0\). That is, is not a Pata type Zamfirescu mapping.

Consider the function \(\alpha :\mathcal{X}\times \mathcal{X} \to [0,\infty )\) given as

$$\alpha (\nu , \omega )= \textstyle\begin{cases} 2, &\text{if }\nu , \omega \in B, \\ 1, &\text{if }\nu =(11,0), \omega =(2,0), \\ 0, &\text{otherwise.} \end{cases} $$

Since the assumptions \((i)\)–\((\mathit{iv})\) are obviously satisfied, we have to prove that is an α-ζ̃-- Pata contraction. Take \(\alpha =\beta =1\), \(\Lambda =6\) and the functions \(\Psi (t)=\frac{t}{2}\), .

For \(\nu , \omega \in B\), we have , so that (2.1) holds.

For \(\nu =(11,0)\) and \(\omega =(2,0)\) we have

Due to the way the function α was defined, we omit the other cases.

3 An application on a fractional boundary value problem

In this section, we ensure the existence of a solution of a nonlinear fractional differential equation (for more related details, see [17–23]). Denote by \(\mathcal{X}= C[0,1]\) the set of all continuous functions defined on \([0,1]\). We endow \(\mathcal{X}\) with the metric given as

$$ d(\rho ,\omega ) = \Vert \rho -\omega \Vert _{\infty }=\max _{s\in [0,1]} \bigl\vert \rho (s)- \omega (s) \bigr\vert . $$

Consider the fractional differential equation

$$ {}^{c}D^{\mu }\rho (t)=f\bigl(t,\rho (t) \bigr),\quad 0< t< 1,1< \mu \leq 2, $$
(3.1)

with boundary conditions

$$ \textstyle\begin{cases} \rho (0)=0, \\ I\rho (1)=\rho '(0). \end{cases} $$
(3.2)

Here, \({}^{c}D^{\mu }\) corresponds for the Caputo fractional derivative of order μ, given as

$$ D^{\mu }f(t)=\frac{1}{\Gamma (n-\mu )} \int _{0}^{1}(t-s)^{n-\mu -1}f^{n}(s) \,ds, $$
(3.3)

where \(n-1<\mu <n\) and \(n=[\mu ]+1\), and \(I^{\mu }f\) is the Riemann–Liouville fractional integral of order μ of a continuous function f, defined by

$$ I^{\mu }f(t)=\frac{1}{\Gamma (\mu )} \int _{0}^{t}(t-s)^{\mu -1}f(s) \,ds, \quad \mu >0. $$
(3.4)

In [24], it is showed that the problem (3.1) and (3.2) can be written in the following integral form:

$$ \rho (t)=\frac{1}{\Gamma (\mu )} \int _{0}^{t}(t-s)^{\mu -1}f\bigl(s,\rho (s) \bigr) \,ds+\frac{2t}{\Gamma (\mu )} \int _{0}^{1} \int _{0}^{s} (s-r)^{\mu -1}f\bigl(r, \rho (r)\bigr)\,dr \,ds. $$
(3.5)

Theorem 3.1

Assume that

  1. 1.

    \(f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}\)is continuous;

  2. 2.

    for all \(\rho ,\omega \in \mathcal{X}\), we have

    $$ \bigl\vert f\bigl(s,\rho (s)\bigr)-f\bigl(s,\omega (s)\bigr) \bigr\vert \leq \frac{\varepsilon ^{2}}{4}\Gamma ( \mu +1) \bigl\vert \rho (s)-\omega (s) \bigr\vert , $$
    (3.6)

    for each \(s\in [0,1]\), where \(\varepsilon \in [0,1]\).

Then the problem 3.1and 3.2possesses a unique solution.

Proof

Consider the functional

$$ T\rho (t)=\frac{1}{\Gamma (\mu )} \int _{0}^{t}(t-s)^{\mu -1}f\bigl(s,\rho (s) \bigr) \,ds+\frac{2t}{\Gamma (\mu )} \int _{0}^{1} \int _{0}^{s} (s-r)^{\mu -1}f\bigl(r, \rho (r)\bigr)\,dr \,ds. $$
(3.7)

Note that a solution of (3.5) is also a fixed point of T. We mention that T is well posed. For all \(\rho ,\omega \in \mathcal{X}\) and \(s\in [0,1]\), we have

$$\begin{aligned} &\bigl|T\rho (t)-T(\omega (t)\bigr|\\ &\quad = \biggl\vert \frac{1}{\Gamma (\mu )} \int _{0}^{t}(t-s)^{ \mu -1}f\bigl(s,\rho (s) \bigr) \,ds+\frac{2t}{\Gamma (\mu )} \int _{0}^{1} \int _{0}^{s} (s-r)^{\mu -1}f\bigl(r,\rho (r) \bigr)\,dr \,ds \\ &\qquad{}-\frac{1}{\Gamma (\mu )} \int _{0}^{t}(t-s)^{\mu -1}f\bigl(s,\omega (s)\bigr) \,ds- \frac{2t}{\Gamma (\mu )} \int _{0}^{1} \int _{0}^{s} (s-r)^{\mu -1}f\bigl(r, \omega (r)\bigr)\,dr \,ds \biggr\vert \\ &\quad \leq \biggl\vert \frac{1}{\Gamma (\mu )} \int _{0}^{t}(t-s)^{\mu -1}f\bigl(s, \rho (s) \bigr) \,ds-\frac{1}{\Gamma (\mu )} \int _{0}^{t}(t-s)^{\mu -1}f\bigl(s, \omega (s)\bigr) \,ds \biggr\vert \\ &\qquad{}+ \biggl\vert \frac{2t}{\Gamma (\mu )} \int _{0}^{1} \int _{0}^{s} (s-r)^{ \mu -1}f\bigl(r,\rho (r) \bigr)\,dr \,ds-\frac{2t}{\Gamma (\mu )} \int _{0}^{1} \int _{0}^{s} (s-r)^{\mu -1}f\bigl(r,\omega (r)\bigr)\,dr \,ds \biggr\vert \\ &\quad \leq \frac{1}{\Gamma (\mu )} \biggl\vert \int _{0}^{t}(t-s)^{\mu -1}f\bigl(s, \rho (s) \bigr) \,ds- \int _{0}^{t}(t-s)^{\mu -1}f\bigl(s,\omega (s)\bigr) \,ds \biggr\vert \\ &\qquad{}+ \frac{2}{\Gamma (\mu )} \biggl\vert \int _{0}^{1} \int _{0}^{s} (s-r)^{ \mu -1}f\bigl(r,\rho (r) \bigr)\,dr \,ds- \int _{0}^{1} \int _{0}^{s} (s-r)^{\mu -1}f\bigl(r, \omega (r)\bigr)\,dr \,ds \biggr\vert \\ &\quad \leq \frac{\varepsilon ^{2}\Gamma (\mu +1)}{4\Gamma (\mu )} \int _{0}^{t} (t-s)^{\mu -1} \bigl\vert \rho (s)-\omega (s) \bigr\vert \,ds \\ &\qquad{}+ \frac{2 \varepsilon ^{2}\Gamma (\mu +1)}{4\Gamma (\mu )} \int _{0}^{1} \int _{0}^{s} (s-r)^{\mu -1} \bigl\vert \rho (r)-\omega (r) \bigr\vert \,dr \,ds \\ &\quad \leq \frac{\varepsilon ^{2}\Gamma (\mu +1)}{4\Gamma (\mu )}d(x,y) \int _{0}^{t} (t-s)^{\mu -1} \,ds \\ &\qquad{}+ \frac{2 \varepsilon ^{2}\Gamma (\mu +1)}{4\Gamma (\mu )}d(x,y) \int _{0}^{1} \int _{0}^{s} (s-r)^{\mu -1} \,dr \,ds \\ &\quad \leq \frac{\varepsilon ^{2}\Gamma (\mu )\Gamma (\mu +1)}{4\Gamma (\mu )\Gamma (\mu +1)}d( \rho ,\omega ) \\ &\qquad{}+ 2 \varepsilon ^{2}B(\mu +1,1) \frac{\Gamma (\mu )\Gamma (\mu +1)}{4\Gamma (\mu )\Gamma (\mu +1)}d( \rho ,\omega ) \\ &\quad \leq \frac{\varepsilon ^{2}}{4}d(\rho ,\omega )+ \frac{\varepsilon ^{2}}{2}d(\rho ,\omega ) \\ &\quad \leq \varepsilon ^{2} d(\rho ,\omega ), \end{aligned}$$

where B is the beta function. Consequently, one has

where \(\psi (\varepsilon )=\varepsilon \), \(\beta =\lambda =1\) and \(\Lambda =2\). Applying Theorem 2.3, the functional T admits a unique fixed point, that is, the problem (3.1) and (3.2) possesses a unique solution. □

4 Conclusion and remarks

Our results merged from and generalized several existing results in the related literature. First of all, as underlined in Remark 2.2, the main result of [16] is a consequence of our given theorem. On the other hand, by choosing the auxiliary functions in a proper way, we may state a long list of corollaries. More precisely, by choosing the mapping α in a proper way, we can get the analogue of our result in the setting of partially ordered metric spaces, or in the set-up of cyclic mappings. Note that, if we take \(\alpha (x,y)=1\) for all x, y, we get the standard fixed point theorems in the context of complete metric spaces; see [25–29]. In addition, by choosing the appropriate simulation function, one can get several more results; see [30–35].