Fixed point theory in the setting of $(\alpha,\beta,\psi,\phi)$-interpolative contractions

Abstract

In this manuscript we introduce the notion of $(\alpha,\beta,\psi,\phi)$-interpolative contraction that unifies and generalizes significant concepts: Proinov type contractions, interpolative contractions, and ample spectrum contraction. We investigate the necessary and sufficient conditions to guarantee existence and uniqueness of the fixed point of such mappings.

1 Introduction

For many years researchers in the field of fixed point theory have known and employed contractivity conditions of the type:

where is a metric space, is a self-mapping, and $\psi ,\phi : [ 0,+ \infty ) \rightarrow [ 0,+ \infty )$ are two auxiliary functions satisfying certain general conditions whose main aim is to help in the task of proving existence and, in most of cases, uniqueness of fixed points of the mapping (Boyd and Wong [1], Geraghty [2], Amini-Harandi and Petruşel [3], Jleli and Samet [4], Wardowski [5], etc.) This is only a simple, but interesting, way to generalize the original Banach’s contractivity condition that firstly appeared in [6]. After that, many other ways to generate the Banach contractive mapping principle have appeared [6], based mainly on three distinct directions: on the abstract metric structure of the underlying space, on the formulation of novel increasingly general contractive conditions, and also on the introduction of algebraic (no necessarily metric) structures (binary relations, partial orders [7], fuzzy metrics [8, 9], etc.)

In the line of research regarding the generalization of contractive conditions, it is worth highlighting the evolution of two distinct models. On the one hand, there are models that are only based on the usage of two main terms: the distance between any two points in space and the distance between their respective images. Along this line, it is worth highlighting the efforts due to Boyd and Wong [1], etc. More recently, Khojasteh et al. [10] introduced the notion of simulation function, which was later subtly modified by Roldán López de Hierro et al. in [11]. After a series of successive research papers deepening this area (see [12–14]), this last author and Shahzad presented the notion of ample spectrum contraction in [15] with a double aim: to analyze the essential properties, from an abstract point of view, that any contraction that may arise in the future must satisfy and generalize as many results as possible in the field of fixed point theory.

On the other hand, the efforts made in the line of research making new terms intervene in the contractive condition (for example, the distance between any point and its image) are also noteworthy. In this line we can highlight the results given by Ćirić, Kannan [16, 17], Hardy-Rogers [18], etc. Inspired by Kannan contractions and the notion of interpolation triple of type γ described in [19], Karapınar introduced in [20] the interpolative Kannan type contractions. Such advance was later extended in [21] with the help of simulation functions (see [10, 11]). Very recently, Aydi et al. [22] have presented the ω-interpolative Ćirić–Reich–Rus-type contractions by including in the contractivity condition some new terms and algebraic structures. This last class of contractive applications inspires some of the content that we will present in this paper.

Returning to the first commented line of research (employing only two terms in the contractive condition), last year Proinov [23] published a manuscript in which new results about existence and uniqueness of fixed points were demonstrated. This paper attracted much attention among researchers in this area, essentially due to the weak hypotheses that were used to develop the main results. For example, it was demonstrated that the involved Picard sequence was a Cauchy sequence without the need to establish monotonic conditions on the functions ψ and ϕ that appeared in the contractive condition. Furthermore, it was shown that a large number of contractions introduced by other authors (Amini-Harandi and Petrusel, Moradi, Geragthy, Jleli and Samet, Wardowski) could be seen as Proinov contractions.

In this paper, drawing inspiration from the above-mentioned works, we introduce a new class of contractions that generalize the Proinov contractions. Namely, we use a Ćirić–Reich–Rus type contractive condition in which new terms appear (distinct to the two used in the first discussed line of research). For this class of operators, we introduce new results about existence and uniqueness of fixed points. Furthermore, before that, we point out that this class properly extends the family of Proinov contractions since, specifically, we proved that all Proinov contractions are particular cases of the Roldán López de Hierro and Shahzad ample spectrum contractions.

2 Preliminaries

In this section we introduce the notation we are going to use, and we present the three classes of contractions in which this paper is inspired.

2.1 Algebraic background

The framework in which we will introduce our main results from now on is a metric space endowed with a self-mapping . An element such that is a fixed point of . The set of all fixed points of is denoted by .

Henceforth, let $\mathbb{N}= \{ 1,2,3,\ldots \}$ stand for the set of all positive integer numbers. Following [24], a sequence in satisfying that for all $k,p\in \mathbb{N}$, $k\neq p$, is called an infinite sequence. If is given by for all $k\in \mathbb{N}$, then that sequence is known as a Picard sequence of .

Proposition 1

([18], Proposition 2)

Let be a Picard sequence in a metric space such that . If there are $k_{1},k_{2}\in \mathbb{N}$ such that $k_{1}< k_{2}$ and , then there are $k_{0}\in \mathbb{N}$ and such that for all $k\geq k_{0}$ (that is, is constant from a term onwards). In such a case, is a fixed point of the self-mapping for which is a Picard sequence.

Lemma 1

([23, 25, 26])

Let be a sequence in a metric space such that as $k\rightarrow +\infty$. If the sequence is not -Cauchy, then there exist and two partial subsequences and of such that

2.2 Roldán–Shahzad ample spectrum contractions

In [15], Roldán López de Hierro and Shahzad introduced a great class of contractions that generalized many previous kinds of contractions with a particular property: they only used the terms and on their contractivity conditions. In the following definitions, $\mathcal{S}$ represents a binary relation on and $\mathcal{S}^{\ast }$ is given by when and .

Definition 1

Let $\{a_{k}\}$ and $\{b_{k}\}$ be two sequences of real numbers. We say that $\{ ( a_{k},b_{k} ) \}$ is a -sequence if there exist two sequences such that

Definition 2

([15], Definition 4)

We will say that is an ample spectrum contraction if there exists a function such that and ϱ satisfy the following four conditions:

$( \mathcal{B}_{1} )$:

is nonempty and .

$( \mathcal{B}_{2} )$:

If is a Picard $\mathcal{S}$-nondecreasing sequence of such that

then .

$( \mathcal{B}_{3} )$:

If is a $(T,\mathcal{S}^{\ast })$-sequence such that $\{a_{k}\}$ and $\{b_{k}\}$ converge to the same limit and verifying that and $\varrho (a_{k},b_{k})\geq 0$ for all $k\in \mathbb{N}$, then .

$( \mathcal{B}_{4} )$:

for all such that and .

In such a case, we will say that is an ample spectrum contraction with respect to $\mathcal{S}$ and ϱ.

Some additional properties were taken into account in order to introduce very general results in the field of fixed point theory.

$( \mathcal{B}_{2}^{\prime } )$:

If are two points such that

then .

$( \mathcal{B}_{5} )$:

If $\{ ( a_{k},b_{k} ) \}$ is a $(T,\mathcal{S}^{\ast })$-sequence such that $\{b_{k}\}\rightarrow 0$ and $\varrho (a_{k},b_{k})\geq 0$ for all $k\in \mathbb{N}$, then $\{a_{k}\}\rightarrow 0$.

Although it was not explicitly enunciated in [15], we state here the following result which is a simple consequence of the main theorems given in [15]. The trivial binary relation $\mathcal{S}_{X}$ on X is given by $x\mathcal{S}_{X}y$ for all $x,y\in X$.

Theorem 1

([15])

Each ample spectrum contraction w.r.t. $\mathcal{S}_{X}$ from a complete metric space into itself satisfying $( \mathcal{B}_{2}^{\prime } )$ and $( \mathcal{B}_{5} )$ has a unique fixed point. In fact, each Picard sequence converges to such a fixed point.

Proof

The existence of fixed points follows from Theorem 2 in [15] using item $(b)$ and the uniqueness follows from Theorem 3 in [15] because we assume condition $( \mathcal{B}_{2}^{\prime } )$. □

2.3 ω-Interpolative Ćirić–Reich–Rus-type contractions

The notion of ω-interpolative Ćirić–Reich–Rus-type contraction was firstly inspired by the celebrated Kannan type contractions. A self-mapping is a Kannan contraction if there is such that

where is a metric space. The reason why Kannan contractions were considered as an outstanding generalization of the Banach contractions lies in the fact that Kannan contractions are not necessarily continuous. Having in mind such kind of contractions and the concept of interpolation triple of type γ described in [19], Karapınar introduced in [20] the notion of interpolative Kannan type contraction, which corresponds to mappings , from a metric space into itself, such that the following condition is fulfilled:

where α is a constant belonging to the interval $( 0,1 )$. This author was able to prove that each interpolative Kannan type contraction defined on a complete metric space has a unique fixed point.

Other significant advance in this line of study was the so-called α-admissible interpolative Rus–Reich–Ćirić type $\mathcal{Z}$-contractions which involve a contractivity condition as follows:

where

$\zeta \in \mathcal{Z}$ is a simulation function (see [10, 11]), ψ and α are appropriate functions, and β and γ are positive constants such that $\gamma +\beta <1$ (see [21] for more details).

Finally, it is worth mentioning the class of contractions that inspire this work. Following Definition 2 in [22], given a metric space , a function , two positive real $\beta ,\gamma >0$ such that $\beta +\gamma <1$, a function $\psi \in \Psi$, we say that a mapping is an ω-interpolative Ćirić–Reich–Rus-type contraction if it verifies

for all . For the sake of completeness, we clarify that here Ψ represents the family of all nondecreasing functions $\psi : [ 0, + \infty ) \rightarrow [ 0, + \infty )$ satisfying the following condition:

$$ \sum_{p=1}^{ + \infty } \psi ^{p}(s)< + \infty \quad \text{for all }s>0. $$

Theorem 2

([22, Theorem 3])

Suppose that a continuous self-mapping is ω-orbital admissible also an ω-interpolative Ćirić–Reich–Rus-type contraction on a complete metric space . If there is a point such that , then has a fixed point in .

The notion of ω-orbital admissible mapping can be found in [22, 27].

2.4 Proinov contractions

Very recently, Proinov announced some results which unify many known results.

Theorem 3

(Proinov [23], Theorem 3.6)

Let be a complete metric space and be a mapping such that

(2.1)

where the functions $\psi ,\phi :(0, + \infty )\rightarrow \mathbb{R}$ satisfy the following conditions:

($a_{1}$):

ψ is nondecreasing;

($a_{2}$):

$\phi (s)<\psi (s)$ for any $s>0$;

($a_{3}$):

for any .

Then has a unique fixed point and the iterative sequence converges to for every .

3 $(\alpha ,\beta ,\psi ,\phi )$-Interpolative contractions

As we have commented in the introduction, Theorem 3 has attracted the attention of many researchers in the field of fixed point theory due to the very weak hypotheses that are assumed in the statement in order to guarantee the existence and uniqueness of fixed points of the involved operator . Furthermore, this author also showed in [23] that many fixed point results introduced along the last years in nonlinear analysis can be easily deduced from Theorem 3. Due to its great impact, we will say that a mapping from a metric space into itself is a Proinov contraction if there are two functions $\psi ,\phi :(0, + \infty )\rightarrow \mathbb{R}$, satisfying the conditions ($a_{1}$), ($a_{2}$), and ($a_{3}$), such that contractivity condition (2.1) holds. Theorem 3 guarantees that each Proinov contraction from a complete metric space into itself has a unique fixed point.¹ Inspired by this class of contractions and also by ω-interpolative Ćirić–Reich–Rus-type contractions [22], in this paper we are going to study the existence and uniqueness of fixed points of self-mapping satisfying a more general contractivity condition.

Definition 3

Let be a metric space. A mapping is said to be an $(\alpha ,\beta ,\psi ,\phi )$-interpolative contraction if there exist $\alpha ,\beta \in [ 0,1 )$ such that $\alpha +\beta <1$ and a pair of functions $\psi ,\phi :(0, + \infty )\rightarrow \mathbb{R}$ such that:

(3.1)

Although at a first sight contractivity condition (3.1) seems to be similar to (2.1), we would like to highlight that they are definitively distinct in nature. The main difference lies in the fact that contractive condition (2.1) only uses the terms and , while in condition (3.1) the terms and also appear. This fact forces us to assume that the initial points and that are used in contractivity condition (3.1) are not fixed points of the mapping , since the function ϕ is not defined for $t=0$. Furthermore, in such a case, if $\alpha =0$ and is a fixed point of , the argument of ϕ in the right-hand side of (3.1) could contain an algebraic indetermination of type 0⁰. To show that this difference is very important, we start our main results by showing that each Proinov contraction is an ample spectrum contraction.

Theorem 4

Let be a Proinov contraction from a metric space into itself associated with the functions $\psi ,\phi :(0, + \infty )\rightarrow \mathbb{R}$. Let us define:

• the binary relation $\mathcal{S}_{X}$ on X given by for all ,

• $A=(0, + \infty )$,

• $\varrho _{\psi ,\phi }:(0, + \infty )\times (0, + \infty )\rightarrow \mathbb{R}$,

  $$\begin{aligned}& \varrho _{\psi ,\phi }(t,s)=\phi ( s ) - \psi ( t ) \quad \textit{for all }t,s\in (0, + \infty ). \end{aligned}$$

Then is an ample spectrum contraction with respect to $\mathcal{S}$ and $\varrho _{\psi ,\phi }$ satisfying additional properties $( \mathcal{B}_{2}^{\prime } )$ and $( \mathcal{B}_{5} )$.

Proof

We check all the assumptions given in Definition 2.

$( \mathcal{B}_{1} )$:

It is trivial because $A=(0, + \infty )$.

$( \mathcal{B}_{2} )$:

Let be a Picard $\mathcal{S}$-nondecreasing sequence of such that

Then Lemma 2 implies that .

$( \mathcal{B}_{2}^{\prime } )$:

Let be two initial points such that the Picard sequences and , defined by and for all $k\in \mathbb{N}$, satisfy

Therefore, for all $k\in \mathbb{N}$,

Taking into account condition $(a_{2})$ and the fact that , the previous inequality means that

(3.2)

Since the function ψ is nondecreasing, then for all $k\in \mathbb{N}$. Let . Notice that for all $k\in \mathbb{N}$. To prove that , assume that . In this case the following limits are equal:

Then (3.2) leads to

which contradicts ($a_{3}$) because

As a consequence, , that is, .

$( \mathcal{B}_{2} )$:

It directly follows from $( \mathcal{B}_{2}^{\prime } )$ by choosing an arbitrary point and using in $( \mathcal{B}_{2}^{\prime } )$.

$( \mathcal{B}_{3} )$:

Let $\{ ( a_{k},b_{k} ) \}\subseteq A\times A$ be a -sequence such that $\{a_{k}\}$ and $\{b_{k}\}$ converge to the same limit and verifying that and $\varrho _{\psi ,\phi }(a_{k},b_{k})\geq 0$ for all $k\in \mathbb{N}$. Therefore, for all $k\in \mathbb{N}$, $0\leq \varrho _{\psi ,\phi }(a_{k},b_{k})=\phi ( b_{k} ) - \psi ( a_{k} )$, so $\psi ( a_{k} ) \leq \phi ( b_{k} )$. Since $a_{k}>0$ and $b_{k}>0$, then ($a_{2}$) implies that

$$ \psi ( a_{k} ) \leq \phi ( b_{k} ) < \psi ( b_{k} ) \quad \text{for all }k\in \mathbb{N}. $$

(3.3)

As ψ is nondecreasing, $a_{k}< b_{k}$, so for all $k\in \mathbb{N}$. Taking into account that

property (3.3) leads to

However, if , we deduce a contradiction with ($a_{3}$) because

Therefore, .

$( \mathcal{B}_{4} )$:

Let be such that and , that is, and . Therefore contractivity condition (2.1) implies that

$( \mathcal{B}_{5} )$:

Let $\{ ( a_{k},b_{k} ) \}$ be a $(T,\mathcal{S}^{\ast })$-sequence such that $\{b_{k}\}\rightarrow 0$ and $\varrho _{\psi ,\phi }(a_{k},b_{k})\geq 0$ for all $k\in \mathbb{N}$. We can repeat the argument of (3.3), so we deduce that $a_{k}< b_{k}$ for all $k\in \mathbb{N}$. Hence $\{a_{k}\}\rightarrow 0$. □

Corollary 1

Each Proinov contraction is an ample spectrum contraction.

Corollary 2

Theorem 3directly follows from Theorem 1.

Theorem 4 implies that $(\alpha ,\beta ,\psi ,\phi )$-interpolative contractions belong to a category of contractions that are very distinct in nature to the class of Proinov contractions: each Proinov contraction is a particular case of an ample spectrum contraction; however, for the moment, it is an open problem to study the possible relationship between $(\alpha ,\beta ,\psi ,\phi )$-interpolative contractions and ample spectrum contractions.

4 Fixed point theorems for $(\alpha ,\beta ,\psi ,\phi )$-interpolative contractions

In this section we introduce some fixed point results for $(\alpha ,\beta ,\psi ,\phi )$-interpolative contractions. The following lemma is important in order to show later that a Picard sequence is Cauchy.

Lemma 2

Let be an $(\alpha ,\beta ,\psi ,\phi )$-interpolative contraction from a metric space into itself, and suppose that the functions ϕ and ψ satisfy, at least, one of the following conditions:

($\mathbb{A}$):

It holds, at the same time,

($a_{1}$):

ψ is nondecreasing,

($a_{2}$):

$\phi (s)<\psi (s)$ for all $s>0$,

($a_{3}$):

for all .

($\mathbb{B}$):

If $\{s_{n}\}\subset ( 0, + \infty )$ is a sequence such that $\psi ( s_{k+1} ) \leq \phi (s_{k}^{1-\beta }s_{k+1}^{ \beta })$ for all $k\in \mathbb{N}$, then $\{s_{k}\}\rightarrow 0$.

Then is an asymptotically regular mapping.

Proof

Let be arbitrary, and let us define and for all $k\in \mathbb{N}$. If there is $k_{0}\in \mathbb{N}$ such that , then is a fixed point of . In such a case, . On the contrary, suppose that for all $k\in \mathbb{N}$. Then each is not a fixed point of and also

Applying contractivity condition (3.1), we deduce that, for all $k\in \mathbb{N}$,

If we define for all $k\in \mathbb{N}$, the previous inequality means that the sequence $\{s_{k}\}$ satisfies $\psi ( s_{k+1} ) \leq \phi (s_{k}^{1-\beta }s_{k+1}^{ \beta })$ for all $k\in \mathbb{N}$. Under condition ($\mathbb{B}$), we deduce that , and this completes the proof.

Next, suppose that ($\mathbb{A}$) holds. Since and , property ($a_{2}$) guarantees that, for all $k\in \mathbb{N}$,

(4.1)

Taking into account that the function ψ is nondecreasing, it follows that

which immediately leads to

As the sequence of positive real numbers is strictly decreasing, it is convergent. Then there is such that

To prove that , we assume, by contradiction, that . In this case, both sequences,

are strictly decreasing and converging to (notice that if $\beta =0$, then for all $k\in \mathbb{N}$). As ψ is nondecreasing, the following limits exist and they are equal:

As a consequence, letting $k\rightarrow + \infty$ in (4.1), the following limit exists and it is finite:

However, this fact contradicts the property ($a_{3}$) because

As a consequence, and is an asymptotically regular mapping. □

Remark 1

• 1 Notice that the previous result shows that condition (ii) is not necessary in [23, Lemma 3.2] when we suppose that (iii) holds.

• 2 We include assumption ($\mathbb{B}$) in the previous result because it can be easy to check in practical examples. For instance, if there is $\lambda \in [ 0,1 )$ such that $\psi (s)=s$ and $\phi (s)=\lambda s$ for all $s\in ( 0, + \infty )$, then ($\mathbb{B}$) trivially holds and we do not need to check the properties given in ($\mathbb{A}$).

Our first main theorem in this paper is the following result.

Theorem 5

Let be an $(\alpha ,\beta ,\psi ,\phi )$-interpolative contraction from a complete metric space into itself, and suppose that the functions ϕ and ψ satisfy the following properties:

($a_{1}$):

ψ is nondecreasing,

($a_{2}$):

$\phi (s)<\psi (s)$ for all $s>0$,

($a_{3}$):

for all .

Then has a fixed point and the Picard sequence converges to a fixed point of whatever the initial point .

Proof

First of all, let us prove that is not fixed-points free. Let be an arbitrary initial point, and let us consider the Picard sequence of starting from , that is, define and for all $k\in \mathbb{N}$. If there is $k_{0}\in \mathbb{N}$ such that , then is a fixed point of . In this case, the first part of the proof is finished. Next suppose that for all $k\in \mathbb{N}$. Hence Lemma 2 implies that

(4.2)

Furthermore, Proposition 1 guarantees that each two terms of the Picard sequence are distinct, that is, for any $k_{1},k_{2}\in \mathbb{N}$ such that $k_{1}\neq k_{2}$.

In order to prove that is a -Cauchy sequence, we reason by contradiction. Suppose that is not -Cauchy. In such a case, Lemma 1 states that there exist and two partial subsequences and of such that

(4.3)

(4.4)

Applying contractivity condition (3.1) and property ($a_{2}$), we deduce that, for all $k\in \mathbb{N}$,

Since ψ is nondecreasing, then, for all $k\in \mathbb{N}$,

which, in particular, means that, for all $k\in \mathbb{N}$,

(4.5)

Taking into account (4.2)–(4.4), notice that

If $\alpha >0$ or $\beta >0$, then, letting $k\rightarrow + \infty$ in (4.5), we observe that

which contradicts the fact that . To avoid this contradiction, assume that $\alpha =\beta =0$. In such a case, contractivity condition (3.1) is given by

Although this condition is more general than condition (3.1) given in [23], the same arguments of the proof of Lemma 3.3 in that paper show that is a Cauchy sequence in . Hence, we have proved that, in any case, the sequence is -Cauchy in the complete metric space . As a consequence, there is such that , that is,

We claim that . To prove it, assume by contradiction that , that is, . As the sequence is infinite, then there is $k_{0}\in \mathbb{N}$ such that and for all $k\geq k_{0}$. Using contractivity condition (3.1) and property ($a_{2}$), we deduce that, for all $k\geq k_{0}$,

Since ψ is nondecreasing, we deduce that

(4.6)

Since $1-\alpha -\beta >0$, then , so . Hence the uniqueness of the limit of the sequence in the metric space guarantees that , which is a contradiction because we have assumed that is not a fixed point of .

This contradiction finally proves that is a fixed point of and, also, that the Picard sequence converges to such a fixed point whatever the initial point . □

In general, the uniqueness of the fixed point is not guaranteed. In fact, contractivity condition (3.1) that an $(\alpha ,\beta ,\psi ,\phi )$-interpolative mapping satisfies cannot be employed to establish a relationship between any two distinct fixed points. The problem of condition

is that the functions $\psi ,\phi :(0, + \infty )\rightarrow \mathbb{R}$ are not necessarily defined for $t=0$. Therefore, if (or ) is a fixed point of , then the right-hand side of the previous inequality is not well defined. On the one hand, when ψ is nondecreasing, the limit

$$ \lim_{s\rightarrow 0^{+}}\psi ( s ) $$

exists but it could take the value −∞ when the function ψ is not bounded from below. Avoiding this last case, it seems reasonable to assume that ψ is well defined on $t=0$. On the other hand, there is not a clear constraint about the possible value of $\phi ( 0 )$ because the condition $\phi <\psi$ does not lead to an additional condition unless we assume that both functions ϕ and ψ would be continuous at $t=0$. In the following result we illustrate a simple way to deduce the uniqueness of the fixed point by assuming an additional condition.

Theorem 6

Given a mapping from a complete metric space into itself, suppose that there are two real numbers $\alpha ,\beta \in [ 0,1 )$ such that $\alpha +\beta <1$ and a pair of functions $\psi ,\phi : [ 0, + \infty ) \rightarrow \mathbb{R}$ satisfying

for any with (we agree here that if $\alpha =0$ and is a fixed point of , and similarly with ). Additionally, assume that the following conditions are fulfilled:

($a_{1}$):

ψ is nondecreasing on $( 0, + \infty )$;

($a_{2}$):

$\phi (s)<\psi (s)$ for all $s>0$;

($a_{3}$):

for all ;

($a_{4}$):

if $s_{0}\in [ 0, + \infty )$ verifies that $\psi ( s_{0} ) \leq \phi ( 0 )$, then $s_{0}=0$.

Then has a unique fixed point and the Picard sequence converges to such a fixed point of whatever the initial point .

Proof

Let us consider the restricted functions $\psi ^{\prime }=\psi| _{ ( 0, + \infty ) }$, $\phi ^{\prime }=\phi| _{ ( 0, + \infty ) }: ( 0, + \infty ) \rightarrow \mathbb{R}$. Then is an $(\alpha ,\beta ,\psi ^{\prime },\phi ^{\prime })$-interpolative mapping. Since $\psi ^{\prime }$ and $\phi ^{\prime }$ satisfy the auxiliary conditions ($a_{1}$), ($a_{2}$), and ($a_{3}$), Theorem 5 guarantees that the mapping has a fixed point and the Picard sequence converges to a fixed point of whatever the initial point . To prove the uniqueness of the fixed point, suppose that there exist two distinct fixed points of . Then . In this case, the contractivity condition implies that

(4.7)

We can reason as follows to obtain a contradiction.

• If $\alpha >0$, then , and if $\beta >0$, then . In both cases, . However, assumption ($a_{4}$) would imply that , which contradicts the fact that .

• If $\alpha =\beta =0$, then (4.7) means that , which contradicts ($a_{2}$) because .

In any case, the obtained contradiction leads to the uniqueness of the fixed point of . □

Next we show some consequences of the previous main results.

Corollary 3

Let be a complete metric space, and let be a mapping. Suppose that there exist a constant $\alpha \in ( 0,1 )$ and two functions $\psi ,\phi :(0, + \infty )\rightarrow \mathbb{R}$, satisfying ($a_{1}$), ($a_{2}$), and ($a_{3}$), such that

Then has a fixed point and the Picard sequence converges to a fixed point of whatever the initial point .

Proof

It follows from Theorem 5 applied to the case $\beta =0$. □

We can easily deduce the Proinov theorem even if we assume that .

Corollary 4

Let be a complete metric space, and let be a mapping. Suppose that there exist two functions $\psi ,\phi :(0, + \infty )\rightarrow \mathbb{R}$, satisfying ($a_{1}$), ($a_{2}$), and ($a_{3}$), such that

Then has a fixed point and the Picard sequence converges to a fixed point of whatever the initial point .

Proof

It follows from Theorem 5 applied to the case $\alpha =\beta =0$. □

Corollary 5

Let be a complete metric space, and let be a mapping. Suppose that there exist a constant $\alpha \in ( 0,0.5 )$ and two functions $\psi ,\phi :(0, + \infty )\rightarrow \mathbb{R}$, satisfying ($a_{1}$), ($a_{2}$), and ($a_{3}$), such that

Then has a fixed point and the Picard sequence converges to a fixed point of whatever the initial point .

Proof

It corresponds to the case $\alpha =\beta$ in Theorem 5. □

Corollary 6

Let be a complete metric space, let be a mapping, and let $\phi :(0, + \infty )\rightarrow \mathbb{R}$ be a function satisfying the following conditions:

($a_{2}$):

$\phi (s)< s$ for all $s>0$;

($a_{3}$):

for all .

If we assume that there exist two constants $\alpha ,\beta \in ( 0,1 )$ such that $\alpha +\beta <1$ and

then has a fixed point and the Picard sequence converges to a fixed point of whatever the initial point .

Proof

It is only necessary to apply Theorem 5 when $\psi (s)=s$ for all $s\in (0, + \infty )$. □

Corollary 7

Let be a complete metric space, let be a mapping, and let $\psi :(0, + \infty )\rightarrow \mathbb{R}$ be a function satisfying the following conditions:

($a_{1}$):

ψ is nondecreasing on $( 0, + \infty )$;

($a_{2}$):

$s<\psi (s)$ for all $s>0$.

If we assume that there exist two constants $\alpha ,\beta \in ( 0,1 )$ such that $\alpha +\beta <1$ and

then has a fixed point and the Picard sequence converges to a fixed point of whatever the initial point .

Proof

We use the function $\phi :(0, + \infty )\rightarrow \mathbb{R}$ defined by $\phi (s)=s$ for all $s\in (0, + \infty )$. Notice that, in this case, for all ,

so assumption ($a_{3}$) can be deduced from ($a_{1}$) and ($a_{2}$). Therefore, it is only necessary to apply Theorem 5 when $\phi (s)=s$ for all $s\in (0, + \infty )$. □

Corollary 8

Let be a complete metric space, and let be a mapping. Assume that there exist three constants such that $\alpha +\beta <1$ and

then has a fixed point and the Picard sequence converges to a fixed point of whatever the initial point .

Proof

This result follows from Theorem 5 by using $\psi (s)=s$ and for all $s\in (0, + \infty )$. Such functions clearly satisfy conditions ($a_{1}$), ($a_{2}$), and ($a_{3}$). □

5 Conclusions and prospect work

Inspired by two well-known families of contractions, namely, Proinov contractions and ω-interpolative Ćirić–Reich–Rus-type contractions, in this paper we have introduced the notion of $(\alpha ,\beta ,\psi ,\phi )$-interpolative contraction as a way to take advantage of the best properties of both classes of contractive mappings. Furthermore, we have shown that this new kind of contractions is distinct in nature to the previous definitions, which provides substantial added value to our proposal compared to the above-mentioned families.

In prospect work we will study the possible relationships between this novel class of contractions and other families of contractive mappings, mainly in order to search for new applications, and we will also try to generalize them by employing more general conditions, auxiliary functions, and algebraic structures.