In this section we shall present some useful lemmas and definitions regarding rectangular and b-rectangular metric spaces. Also, we shall present some recent results in the field of fixed point theory concerning expansive operators and some generalized contraction mappings.
In [6], A. Branciari introduced a new metric-type space, when triangle inequality is replaced by an inequality which involves four different elements. This is called a rectangular metric space or a generalized metric space (g.m.s.)

Definition 1.1. Let $X\neq\emptyset,d:X\times X\rightarrow[0,\infty)$, such that for each $x,y\in X$ and $u,v\in X$ (each distinct from $x$ and $y$ ), we have that
(1) $d(x,y)=0\Longleftrightarrow x=y$,
(2) $d(x,y)=d(y,x)$,
(3) $d(x,y)\leq d(x,u)+d(u,v)+d(v,y)$.

Furthermore, from [10] we mention that convergent sequences and Cauchy sequences can be introduced in a similar manner as in metric spaces.
Also, from the same paper, we know that if ( $X,d$ ) is a rectangular metric space and if ( $x_{n}$ ) is a b-rectangular Cauchy sequence with the property that $x_{n}\neq x_{m}$, for each $n\neq m$, then $\left(x_{n}\right)$ converge to at most one point, i.e. the property that ( $X,d$ ) is

Haussdorf becomes superfluous.
Moreover, from [8], [9], [22], we recall the definition of b-rectangular metric spaces (or b-generalized metric spaces), briefly b-g.m.s.

Definition 1.2. Let $X\neq\emptyset,s\geq 1$ be a given real number and $d:X\times X\rightarrow[0,\infty)$, such that for each $x,y\in X$ and $u,v\in X$ (each distinct from $x$ and $y$ ), we have that
(1) $d(x,y)=0\Longleftrightarrow x=y$,
(2) $d(x,y)=d(y,x)$,
(3) $d(x,y)\leq s[d(x,u)+d(u,v)+d(v,y)]$.

As in metric spaces, we recall the basic notions regarding sequences in b-g.m.s:
Definition 1.3. Let ( $X,d$ ) be a b-g.m.s, $x\in X$ and $\left(x_{n}\right)\subset X$ be a given sequence. Then
(a) ( $x_{n}$ ) is convergent in ( $X,d$ ) to an element $x\in X$, if for each $\varepsilon>0$, there exists $n_{0}\in\mathbb{N}$, such that $d\left(x_{n},x\right)<\varepsilon$, for each $n>n_{0}$. We denote this by $\lim_{n\rightarrow\infty}x_{n}=x$.
(b) ( $x_{n}$ ) is Cauchy in ( $X,d$ ) (or b-rectangular Cauchy, briefly b-g.m.s.), if for each $\varepsilon>0$, there exists $n_{0}\in\mathbb{N}$, such that $d\left(x_{n},x_{n+p}\right)<\varepsilon$, for each $n>n_{0}$ and for each $p>0$. We denote this by $\lim_{n\rightarrow\infty}d\left(x_{n},x_{n+p}\right)=0$, for each $p>0$.
(c) ( $X,d$ ) is said to be complete b-g.m.s, if every Cauchy sequence in X converges to some $x\in X$.

We recall the following important remark from [8]:
Remark 1.4. (1) Every metric space and every rectangular metric space (g.m.s) is b-g.m.s.
(2) The limit of a sequence in a b-rectangular metric space is not unique.
(3) Every convergent sequence in a b-g.m.s is not necessarily a b-g.m.s Cauchy.

For this, we recall a crucial lemma from [8], i.e. (Lemma 1.5), that specify when a b-rectangular Cauchy sequence can’t have two limits in a b-g.m.s.

Lemma 1.5. Let $(X,d)$ be a $b$-rectangular metric space, with the coefficient $s\geq 1$. Let $\left(x_{n}\right)$ be a b-rectangular Cauchy sequence in $X$, such that $x_{n}\neq x_{m}$, for each $n\neq m$. Then ( $x_{n}$ ) can converge to at most one point.

Also, we recall from [12] and [8] the following crucial lemma.
Lemma 1.6. Let $(X,d)$ be a $b$-rectangular metric space, with the coefficient $s\geq 1$. Also, let $\left(x_{n}\right)$ be a sequence for which $x_{n}\neq x_{m}$, for every $n\neq m$, with $\lim_{n\rightarrow\infty}d\left(x_{n},x_{n+1}\right)=0$. If $\left(x_{n}\right)$ is not a b-rectangular Cauchy sequence, then there exists $\varepsilon>0$, such that for each $k\in\mathbb{N}$, there exists ( $m(k)$ ) and ( $n(k)$ ) two sequences of positive integers, such that

In [22], another crucial lemma regarding sequences in b-rectangular metric spaces was presented. For convenience, we remind it below.

Lemma 1.7. Let $(X,d)$ be a b-g.m.s., with coefficient $s\geq 1$.
(a) Consider two sequences $\left(x_{n}\right)$ and $\left(y_{n}\right)$, such that $x_{n}$ converges to $x\in X$ and $y_{n}$ converges to $y\in X$, with $x\neq y$. Also, suppose that for each $n\in\mathbb{N},x_{n}\neq x$ and $y_{n}\neq y$. Then

(b) Consider an element $y\in X$ and a b-rectangular Cauchy sequence ( $x_{n}$ ), such that $x_{n}\neq x_{m}$, for each $n\neq m$. Moreover, suppose that the sequence ( $x_{n}$ ) converges to an element $x\neq y$. Then

Finally, for the convenience of the reader, we recall some important results in brectangular metric spaces. In [9], George et.al.studied basic contraction-type mappings in b-rectangular metric spaces, like Kannan operators, i.e.

In [8], Radenovic et.al. extended the results to mappings satisfying

for each $x,y\in X$ and studied unique coincidence and common fixed points for the pair of operators ( $f,g$ ) that satisfies some additional assumptions.
Also, for more results in b-rectangular metric spaces and for a consistent survey on different generalized metric-type spaces, we recommend [11] and [12].
Now, regarding generalized contraction mappings we recall some recent advances in this subfield of fixed point theory.
In [13], Karapinar studied unique fixed points for some generalized contractions on cone Banach spaces satisfying the following contractive-type conditions

and

Moreover, in 2009, Kumar [14] presented some theorems for two maps satisfying the following

where $f$ is onto and $g$ is one-to-one.
Moosaei, Azizi, Asadi and Wang generalized the results of Karapinar as follows In [15], Moosaei used Krasnoselskii’s iteration defined in convex metric spaces, for the following mappings, that satisfy

respectively

In [17], Moosaei and Azizi extended the results to generalized contraction-type operators, studying coincidence points for various mappings, such as

where $T(K)\subset S(K),K$ and $S(K)$ are closed and convex subsets of a convex metric space and the coefficients satisfy

Nevertheless, in 2014, Moosaei [16] studied a more generalized pair of contractions $(S,T)$, where

with some assumptions on contractive-coefficients, i.e.

Asadi in [3], using the same iteration (Krasnoselskii) on convex metric spaces, studied fixed points for generalized Hardy-Rogers type-mappings, as follows

where

and $\lambda\in[0,1]$ is the coefficient of Krasnoselskii’s iteration.
Furthermore, Wang and Zhang, in [23] extended the above results for pairs of generalized Hardy-Rogers type contractions.
Now, expansive and expansive-type mappings can be considered a particular case of generalized contractions. Regarding the former ones, we recall some recent development into the study of this type of operators.
In 2011, Aage [1] considered expansive mappings in cone metric spaces. The more general form of these mappings, with some underlying assumptions, are

where $T$ satisfies $K\geq-1,p<1,l>1$ and $k+l+p>1$.
Aydi et.al. studied in [4] some interesting fixed point theorems for pairs of expansive mappings for spaces endowed with c-distances. We recall them using the standard notations for metric spaces, i.e.

with $b<1,a\neq 0,f(X)\subseteq T(X)$ and $(T(X),d)\subset(X,d)$ complete.
Also, in cone rectangular metric spaces, some fixed point theorems were developed. For example, in [20], pair of mappings satisfying

were studied, with some assumptions on the coefficients $\alpha,\beta$ and $\gamma$ and on the range of $g$ and $f$.
These pairs of generalized mappings were extended by Olaoluwa and Olaleru in [18], but in the framework of b-metric spaces and for a pair of four mappings, as follows

Also, for the sake of convenience, we recall other studies in metric-type spaces and for expansive-type mappings, as follows: in [24] generalized mappings were studied on cone rectangular metric spaces using the technique of scalarizing, in [21] mappings that satisfy

were studied on cone rectangular metric spaces and in [19], fixed point theorems for a general type of expansive mappings were developed, satisfying

Also, in the context of dislocated metric spaces, Daheriya et.al. [7] studied rationaltype expansive mappings, and in [2] Alghamdi studied fixed points for generalized expansive mappings in b-metric like spaces.
The purpose of this work is to extend some fixed results for a hybrid class of generalized contractive-type mappings and for some expansive-type operators in the context of b-rectangular metric spaces. Moreover, at the end of the second section, we shall let and open problem.

Moosaei in [15] used Krasnoselskii iteration to develop fixed point theorems for generalized contractions on convex metric spaces. It is easily seen that we can use Picard instead of Krasnoselkii sequences in metric spaces.
In this section, our aim is to extend the results of Moosaei [15] for generalized contraction mappings from metric spaces to b-rectangular metric spaces. Also, we extend and develop the fixed point results of Aage [1] from cone metric spaces to b-g.m.s. Furthermore, we extend results from [20] of Patil, from rectangular metric spaces to b-rectangular ones (b-g.m.s).
Also, examples similar to those in [1], [12] and [20] justifying our theorems are given. Now, let’s consider generalized contractions $f:X\rightarrow X$ on a b-g.m.s. $X$, satisfying the following condition:

We will analyze two separate cases: when $c>0$ and $c<0$. Also, for expansive-type mappings, i.e. when $c<0$, we consider two types of sequence, namely the classical Picard iteration $x_{n+1}=fx_{n}$, for each $n\in\mathbb{N}$ and the ’inverse’ Picard iteration, i.e. $x_{n}=fx_{n+1}$, for each $n\in\mathbb{N}$, for which we require that the operator $f$ is onto.
Our first result is a theorem for the existence and uniqueness of the fixed point of a mapping satisfying the contractive condition from above. The technique we will use is based on the (Lemma 1.6).

Theorem 2.1. Let ( $X,d$ ) be a complete $b$-rectangular metric space (b-gms), with coefficient $s>1$. Consider a mapping $f:X\rightarrow X$, satisfying the following contractive condition

Also, suppose the following assumptions are satisfied
(A) If $c>0$ and $k\geq 0$, then $\frac{k}{c}<\frac{1}{s}$,
(B) If $c>0$ and $k\leq 0$, then we have no additional conditions,
(C) If $c<0$ and $k<0$, then $\frac{k}{c}>s^{2}$.

Then, the Picard sequence ( $x_{n}$ ), defined as $x_{n+1}=fx_{n}$, for each $n\in\mathbb{N}$ converges to a fixed point of the mapping $f$.

Proof. We consider the Picard iterative process ( $x_{n}$ ), defined as $x_{n+1}=fx_{n}$, for each $n\in\mathbb{N}$. Applying the contractive condition for the pair $\left(x_{n-1},x_{n}\right)$, we get that

So $d\left(x_{n},x_{n+1}\right)\leq\delta d\left(x_{n-1},x_{n}\right)$, where $\delta:=\frac{k-b}{a+c}\in\left[0,\frac{1}{s}\right)$ from the theorem’s assumptions, since $0\leq k-b<\frac{a+c}{s}$.
So $d\left(x_{n},x_{n+1}\right)\leq\delta^{n}d\left(x_{0},x_{1}\right)$. Since $\delta\in\left[0,\frac{1}{s}\right)$, it follows that $\lim_{n\rightarrow\infty}d\left(x_{n},x_{n+1}\right)=0$.
Also, by a routine argument (by reductio ad absurdum), it follows easily that $x_{n}\neq x_{n+1}$, for each $n\in\mathbb{N}$ and that $x_{n}\neq x_{m}$, for each $n\neq m$.
The next step is to show that the sequence ( $x_{n}$ ) is b-rectangular Cauchy. We will use (Lemma 1.6) and we shall apply it on three different cases
(1) Case $c>0$ : Let’s suppose that the sequence ( $x_{n}$ ) is not b-rectangular Cauchy. Then, there exists $\varepsilon>0$ and two sequences of nonnegative real numbers ( $m(k)$ ) and $(n(k))$, such that the assumptions from (Lemma 1.6) are satisfied.
Now, we will apply the contraction condition for $x=x_{m(k)}$ and $y=x_{n(k)-2}$. It follows that
$ad\left(x_{m(k)},x_{m(k)+1}\right)+bd\left(x_{n(k)-2},x_{n(k)-1}\right)+cd\left(x_{m(k)+1},x_{n(k)-1}\right)\leq kd\left(x_{m(k)},x_{n(k)-2}\right)$
$cd\left(x_{m(k)+1},x_{n(k)-1}\right)\leq kd\left(x_{m(k)},x_{n(k)-2}\right)-ad\left(x_{m(k)},x_{m(k)+1}\right)-bd\left(x_{n(k)-2},x_{n(k)-1}\right)$.
Because $c>0$, we have that
$d\left(x_{m(k)+1},x_{n(k)-1}\right)\leq\frac{k}{c}d\left(x_{m(k)},x_{n(k)-2}\right)-\frac{a}{c}d\left(x_{m(k)},x_{m(k)+1}\right)-\frac{b}{c}d\left(x_{n(k)-2},x_{n(k)-1}\right)$.
Now, we want to apply the limit superior. We make the following necessary remark and consider the following cases
If $a\geq 0$, then $-\frac{a}{c}\leq 0$, so $-\frac{a}{c}d\left(x_{m(k)},x_{m(k)+1}\right)\leq 0$, so an upper bound for this element is 0 .

If $a\leq 0$, then $-\frac{a}{c}\geq 0$, so $-\frac{a}{c}d\left(x_{m(k)},x_{m(k)+1}\right)\geq 0$. Applying the limit superior, we get that

The same reasoning can be made about the sign of the coefficient $b$ and about the limit superior of the sequence $\left(d\left(x_{n(k)-2},x_{n(k)-1}\right)\right)$ as a subsequence of $\left(d\left(x_{n},x_{n-1}\right)\right)$.
Case (A): When $k\geq 0$.
Since $k\geq 0$, we have that $\frac{k}{c}\geq 0$. We know that $\limsup_{k\rightarrow\infty}d\left(x_{m(k)},x_{n(k)-2}\right)\leq\varepsilon$. Multiplying by $\left(\frac{k}{c}\right)$ and taking the limit superior, we get that

From (Lemma 1.6), it follows that $\frac{\varepsilon}{s}\leq\limsup_{k\rightarrow\infty}d\left(x_{m(k)+1},x_{m(k)-1}\right)\leq\frac{k}{c}\varepsilon$, so $\frac{1}{s}\leq\frac{k}{c}$. This is a contradiction with the assumption that in this case we have $\frac{k}{c}<\frac{1}{s}$.
Case (B): When $k\leq 0$.
In this case we have that $\frac{k}{c}\leq 0$, so $\frac{k}{c}d\left(x_{m(k)},x_{n(k)-2}\right)\leq 0$, then we can take 0 as an upper bound for it. By (Lemma 1.6), we have that $\frac{\varepsilon}{s}\leq 0$. Since $\varepsilon>0$ and $s\geq 1$, we got a contradiction.
Now, in the two cases from above, we have shown that $\left(x_{n}\right)$ is b-rectangular Cauchy. Moreover, we have said that $x_{n}\neq x_{m}$, for each $n\neq m$.
Since ( $X,d$ ) is complete, it implies that there exists $u\in X$, such that $x_{n}\rightarrow u$, i.e.

Now, we shall show that $u$ is a fixed point for $f$

Since $c>0$, then

So

Taking the limit when $n\rightarrow\infty$, we get

so $(c+sa)d(u,fu)\leq 0$. Furthermore, since $c>0$ and $0<(a+c)<(a+cs)$, then $u$ is a fixed point for $f$.
(2) Case $c<0$ : We have that

So

This is a case of expansive-type mapping. By (Lemma 1.6), there exists $\varepsilon>0$, such that for every $k\in\mathbb{N}$, there exists $(m(k)),(n(k))$ two sequences of nonnegative real numbers such that the assumptions in the already mentioned lemma are true. By b-rectangular inequality, we have that

Dividing by $s\geq 1$, we obtain the following

Case (C): When $k<0$ : Here we have that $\frac{k}{c}\geq 0$. Multiplying by $\left(\frac{k}{c}\right)$, it implies that

Now, we apply the contractive condition for $x=x_{m(k)-1}$ and $y=x_{n(k)-3}$, i.e.
$d\left(x_{m(k)},x_{n(k)-2}\right)\geq\frac{k}{c}d\left(x_{m(k)-1},x_{n(k)-3}\right)-\frac{a}{c}d\left(x_{m(k)-1},x_{m(k)}\right)-\frac{b}{c}d\left(x_{n(k)-3},x_{n(k)-2}\right)$.
So, combining the above inequalities, we get that

From the limit superior, we have get the following

We have the same reasoning for $d\left(x_{m(k)-1},x_{m(k)}\right)$, with coefficient $-\frac{k}{c}$. Also, for coefficients $a$ and $b$, we have that
If $a\geq 0$, then $-\frac{a}{c}\geq 0$, so $\left(-\frac{a}{c}\right)d\left(x_{m(k)-1},x_{m(k)}\right)\geq 0$, so we can make the lower bound 0 .
If $a\leq 0$, then $-\frac{a}{c}\leq 0$, so $\left(-\frac{a}{c}\right)d\left(x_{m(k)-1},x_{m(k)}\right)\leq 0$, so taking the limit superior, it follows that:

Same remarks can be made about the coefficient $b$ and for $d\left(x_{n(k)-3},x_{n(k)-2}\right)$.
By (Lemma 1.6), we get that