In this section, we will present some preliminary notions and fixed point results for single-valued self-mappings satisfying some altering distance type conditions in a complete metric space.

In [6], Khan, Swaleh and Sessa gave sufficient conditions such that an operator has a unique fixed point. This contractive-type operator satisfy the condition

for each elements $x,y$ of a complete metric space $(X,d)$, where $\psi:[0,\infty)\rightarrow[0,\infty)$ is endowed with the following properties

Furthermore, Alber and Guerre-Delabriere in [1] gave a different generalization, for mappings satisfying the assumption

Then in [19], Rhoades showed that the last assumption is not necessary for the existence and uniqueness of the fixed points of the above self-mappings. Generalizations for this type of mappings were done by Dutta et al. in [4] for self-mappings $f$ defined on a complete metric space ( $X,d$ ), satisfying

A very interesting approach was done by Amini-Harandi and Petruşel in [2], where the authors studied sufficient conditions for the existence and uniqueness of the fixed points for an operator $T$ satisfing the following assumption

where the self-mappings $u$ and $v$ defined on $[0,\infty)$ satisfy some relaxed conditions. The authors also gave some interesting corollaries showing that their theorem is a real generalization of the already presented type of mappings.

Moreover, regarding the weakly contractive condition for a single-valued operator, Rhoades et al. [20] presented other types of generalizations in the framework of partially ordered metric spaces. As an example for this type of mappings we suppose that

where the operators $\psi,\varphi$ have the following properties:
i) $\varphi,\psi$ are both positive on $(0,\infty)$ with $\psi(0)=\varphi(0)$,
ii) $\varphi(t)-\psi(t)<t$,
iii) $\varphi$ upper semicontinuous and nondecreasing,
iv) $\psi$ lower semicontinuous and nonincreasing.

Last, but not least, for fixed point results involving other generalizations of the weakly contractive condition for single-valued mappings we refer to [10], [17] and [23].

Concerning the case of multi-valued operators, T. Lazar et al. [8] presented an exhaustive study of some qualitative properties concerning Reich type multi-valued operators. Moreover, V. Lazăr [7] extended the results concerning the case of multi-valued $\varphi$-type contractions.
T.P. Petru and M. Boriceanu [12] gave some fixed point results for $\varphi$-contractions in a set endowed with two metrics.

In all the articles [8], [7] and [12], the comparison function $\varphi$ used for the case of $\varphi$-contractions satisfy the following properties:
i) $\varphi(0)=0$,
ii) $\varphi(t)>0$ for $t>0$,
iii) $\varphi^{k}(t)\rightarrow 0$, for each $t>0$ for $k\rightarrow\infty$.

Notice that $\varphi$ is not necessarily continuous on $[0,\infty)$, but in [12] the continuity of the mapping was additionally assumed. Furthermore, an important property of comparison functions is the fact that $\varphi(t)<t$, for each $t>0$.

As a conclusion, there are two distinct classes of mappings involved in the generalizations of contractivetype operators. There are comparison functions, on one hand, in many cases denoted by $\varphi$. On the other, there is the case of altering distance functions for which the most important conditions are continuity or semicontinuity properties and a certain monotonicity. We also notice that the weakly contractive mappings used in [20] are combination of these types of self-mappings.

Regarding the case of multi-valued operators, in 2011, Kamran and Kiran [5] presented some rezults involving altering distance type functionals. In this article, more precisely, in [Theorem 4.2] in [5], a special type of altering distance function denoted by $\theta$ was used. This mapping satisfies the following conditions on an interval $[0,A)$, where A is real number strictly greater than 0 , i.e.,
(i) $\theta$ is nondecreasing on $[0,A)$,
(ii) $\theta(t)>0$, for each $t\in(0,A)$,
(iii) $\theta$ subadditive on $(0,A)$ and
(iv) $\boldsymbol{\theta}(at)\leq\boldsymbol{a}\boldsymbol{\theta}(t)$, for each $a>0$ and $t\in[0,A)$.

Also, in 2012, Liu et al. [9] gave a similar theorem, namely [Theorem 2.3], were the functional $\varphi$ is similar to the functional $\alpha$ from [5]. From the same article we observe that the conditions put upon the altering distance mapping $\theta$ are somewhat different. For the sake of completeness, we recall them here

Finally, concerning weakly contractive ( $\varphi-\psi$ ) contractive type multivalued operators, G. Petruşel et.al. in [16] presented a fixed point result for this kind of operators in the context of complete ordered b-metric spaces with coefficient $s\geq 1$, along with some theorems involving coupled fixed points. From [Theorem 2.2] of [16], a self multivalued operator $T:X\rightarrow P_{cl}(X)$ was defined by a contractive-type inequality, i.e.

and sufficient conditions for the existence of fixed points for this kind of operators were studied. Here, the altering distance function $\varphi:[0,\infty)\rightarrow[0,\infty)$ satisfy the following

Also, the other altering distance function $\psi:[0,\infty)\rightarrow[0,\infty)$ satisfy

Moreover, we recognize that in contrast to the original usage of altering functions as in [4], the conditions from [16] on $\psi$ were relaxed and the conditions on the mapping $\varphi$ were made more restrictive, since the condition ( $ii_{\varphi}$ ) is a comparison type condition. So, in this sense, these weakly contractive-type selfmappings are a combination of altering distances and comparison functions as the operators defined in [20].

Finally, in [3] the authors presented some fixed point theorems for multivalued operators and in [18] Popescu et.al. extended these types of comparison based multivalued operators, i.e. $T:X\rightarrow P_{b,cl}(X)$, such that $H(Tx,Ty)\leq\varphi(d(x,y))$, for each $x,y$ from the complete metric space $(X,d)$. Here, the mapping $\varphi$ satisfies the following assumptions
(1) $\varphi(x)\leq x$, for each $x\in[0,\infty)$,
(2) $\varphi(x+y)\leq\varphi(x)+\varphi(y)$, for all $x,y\in[0,\infty)$,
(3) $\varphi(x)=0$ if and only if $x=0$ and
(4) for any $\varepsilon>0$, there exists $\delta>\varepsilon$ such that $\varepsilon<t<\delta$ implies $\varphi(t)\leq\varepsilon$.

Furthermore, condition (4) defined for the above mapping $\varphi$ can be considered as a local type comparison function. Also, the authors in [18] extended the results of the authors of [3] from the case of hyperconvex metric spaces to the usual metric spaces. Furthermore, since they worked in a less restrictive framework, they have put an important assumption regarding the well known diameter functional and so they used the condition that the multivalued operator has bounded values.

In the last part of this section, fundamental notions and concepts for the fixed point theory of multivalued operators are used. For a general perspective regarding these terminologies for multi-valued mappings, we refer to [7], [11], [13], [15], [21] and [22].

Let $X$ be a nonempty set. First of all, we shall use the following class of sets
$\mathscr{P}(X):=\{Y/Y\subset X\},\quad P(X):=\{Y\in\mathscr{P}(X)/Y\neq\emptyset\}$.
Now, for the case when ( $X,d$ ) is a metric space, we recall that $Y\subseteq X$ is bounded if and only if $\delta(Y):=\sup\{d(a,b)/a,b\in Y\}<+\infty$, where $\delta$ is the usual diameter functional on $[0,\infty)$.

Now, we define the following families of sets:
$P_{b}(X):=\{Y\in P(X)/Y$ bounded $\},\quad P_{cl}(X):=\{Y\in P(X)/Y$ closed $\}$,
$P_{cp}(X):=\{Y\in P(X)/Y$ compact $\},\quad P_{b,cl}(X):=P_{b}(X)\cap P_{cl}(X)$.
Moreover, if $Y\subseteq X$ and $T:Y\rightarrow P(X)$ is a multi-valued operator, then we define the following useful symbols. $F_{T}:=\{x\in Y/x\in Tx\}$ is the fixed point set of the operator $T,(SF)_{T}:=\{x\in Y/\{x\}=Tx\}$ denotes the strict fixed point set of the multi-valued operator $T$ and by $\operatorname{Graph}(T):=\{(x,y)\in Y\times X/y\in Tx\}$ we denote the graph of the multivalued operator $T$. Also, if $T:X\rightarrow P(X)$, then by $T^{0}:=1_{X}$, $T^{1}:=T,\ldots,T^{n+1}:=T\circ T^{n}$, for $n\in\mathbb{N}$ we denote the iterates of the operator $T$, where $T(A):=\bigcup_{a\in A}Ta$, for $A\subset X$.

Furthermore, we shall frequently use the following generalized functionals in the next section, so we shall recall them.
The gap functional $D:\mathscr{P}(X)\times\mathscr{P}(X)\rightarrow\mathbb{R}_{+}\cup\{+\infty\}$ is defined as

The excess generalized functional $\rho:\mathscr{P}(X)\times\mathscr{P}(X)\rightarrow\mathbb{R}_{+}\cup\{+\infty\}$ is defined as

The Hausdorff-Pompeiu generalized functional $H:\mathscr{P}(X)\times\mathscr{P}(X)\rightarrow\mathbb{R}_{+}\cup\{+\infty\}$ is defined as

For the sake of completeness, we recall now a very useful concept in fixed point theory for multi-valued operators. This is a basic concept known by the name of multivalued weakly Picard operator. It is defined as follows.

Definition 1.1. Let ( $X,d$ ) be a metric space and $T:X\rightarrow P(X)$ is called a multivalued weakly Picard operator, briefly a MWP operator, if for each $x\in X$ and for each $y\in Tx$, there exists a sequence $\left(x_{n}\right)_{n\in\mathbb{N}}\in X$, such that:
(i) $x_{0}=x$ and $x_{1}=y$,
(ii) $x_{n+1}\in Tx_{n}$, for each $n\in\mathbb{N}$ and
(iii) the sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ is convergent to a fixed point of $T$.

Finally, a sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ defined by the properties (i) and (ii) is called a sequence of successive approximations.

The first main result of this section concerns the fixed points of ( $\varphi-\psi$ ) multivalued operators. Let us mention first that, in [16], G. Petruşel et.al. presented some fixed point results for this type of multivalued operators on complete ordered b-metric spaces. In the same article, the authors constructed a sequence of successive approximations and showed that this sequence is convergent to a fixed point of the multivalued operator. The proof of [Theorem 2.2] in [16] contains a certain gap. The authors consider an $\tilde{\varepsilon}$ at each step and then, using the reductio ad absurdum argument, chose $\tilde{\varepsilon}$, such that $\tilde{\varepsilon}<\lim_{n\rightarrow\infty}\psi\left(\delta_{n}\right)$. Since $\tilde{\varepsilon}$ was already constructed at each step (as $\tilde{\varepsilon_{n}}$ ) and the sequence $\delta_{n}$ was not given there, this technique is not valid. Furthermore, trying to show that the sequence ( $x_{n}$ ) is Cauchy, the authors used the fact that $d\left(x_{m(k)+1},x_{n(k)+1}\right)\leq H\left(T\left(x_{m(k)}\right),T\left(x_{n(k)}\right)\right)+\tilde{\varepsilon}$, which, by the well known lemma of Nadler, is not necessarily true.

Our first purpose is to correct these arguments, by imposing the additional assumption that $\delta\left(T^{n}x\right)\rightarrow 0$ as $n\rightarrow\infty$. Our assumption is inspired by an idea from [3] and [18]. Based on this assumption and using the technique from [16], we give a fixed point theorem for ( $\varphi-\psi$ ) multivalued operators. At the same time, we relax some conditions on the altering distance functions (such as continuity) and we get rid off the property of comparison functions, i.e. $\varphi(t)<t$.

Theorem 2.1. Let ( $X,d$ ) be a complete metric space and $T:X\rightarrow P_{b,cl}(X)$ a multivalued operator, that satisfies the following

where the mappings $\varphi$ and $\psi$ satisfy

Also, suppose that $\delta\left(T^{n}x\right)\rightarrow 0$ as $n\rightarrow\infty$, for each $x\in X$. Then, the multi-valued operator $T$ has at least one fixed point $x^{*}\in F_{T}$. Moreover, if $(SF)_{T}\neq\emptyset$, then $F_{T}=(SF)_{T}=\left\{x^{*}\right\}$.

Proof. (i) Let $x_{0}\in X$ be arbitrary taken and let $x_{1}\in Tx_{0}$. Also, consider the sequence $\left(\boldsymbol{\delta}_{n}\right)_{n\in\mathbb{N}}$, with $\delta_{n}>0$, for each $n\in\mathbb{N}$, with $\lim_{n\rightarrow\infty}\delta_{n}=0$. Define $\varepsilon_{1}\leq\min\left\{\delta_{1},t_{1}\right\}$, where

Now, $t_{1}$ is well defined, because $x_{0}$ and $x_{1}$ were already constructed. Furthermore, we can suppose that $x_{0}\neq x_{1}$, because, if $x_{0}$ and $x_{1}$ were identical, then $x_{0}\in Tx_{0}$, so $x_{0}$ was a fixed point for the operator $T$. Now, for $x_{1}\in Tx_{0}$ and $\varepsilon_{1}$, there exists $x_{2}\in Tx_{1}$, such that

By induction, we can create the sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ as follows.
Let $x_{n-1}\in Tx_{n-2}$ and $x_{n}\in Tx_{n-1}$ already constructed. For $x_{n}\in Tx_{n-1}$ and $\varepsilon_{n}>0$, there exists $x_{n+1}\in Tx_{n}$, such that

So, we obtain that

Also, we use $\varepsilon_{n}$, such that $\varepsilon_{n}\leq\min\left\{\delta_{n},t_{n}\right\}$, with

Now, if there exists $n\in\mathbb{N}$, such that $x_{n-1}=x_{n}$, then $x_{n-1}\in Tx_{n-1}$, i.e. $x_{n-1}\in F_{T}$ and the proof is over. So, we can suppose that $x_{n-1}\neq x_{n}$, for each $n\in\mathbb{N}$, so $\rho_{n}>0$, for each $n\in\mathbb{N}$. Furthermore, at each step we can suppose that $H\left(Tx_{n-1},Tx_{n}\right)\neq 0$. Assuming the contrary, we obtain that $Tx_{n-1}=Tx_{n}$, so $x_{n}\in F_{T}$ and the proof is over. In this way, we applied in a valid way assumption (H5). We know that

It follows that

The last inequality is strict, because if it was not a strict inequality, then $\psi\left(\rho_{n-1}\right)=0$, that lead to $\rho_{n-1}=0$, which is false. So, defining $t_{n}$ as above, it follows that $t_{n}>0$, for each $n\in\mathbb{N}$. Furthermore,

which lead to the fact that $\varphi\left(\rho_{n}\right)<\varphi\left(\rho_{n-1}\right)$, which implies $\varphi\left(\rho_{n}\right)\leq\varphi\left(\rho_{n-1}\right)$, for each $n\in\mathbb{N}$. It follows that sequence $\left(\varphi\left(\rho_{n}\right)\right)_{n\in\mathbb{N}}$ is decreasing and bounded below by 0 , because $\varphi$ takes values in $[0,\infty)$. So, we get that there exists $\tau\geq 0$, such that $\tau=\lim_{n\rightarrow\infty}\varphi\left(\rho_{n}\right)$. Furthermore, since $\varphi\left(\rho_{n}\right)\leq\varphi\left(\rho_{n-1}\right)$, for each $n\in\mathbb{N}$, we obtain that $\rho_{n}\leq\rho_{n-1}$. So $\left(\rho_{n}\right)_{n\in\mathbb{N}}$ is a decreasing sequence and is also bounded below by 0 . Hence there exists $\rho^{*}\geq 0$, such that $\rho^{*}=\lim_{n\rightarrow\infty}\rho_{n}$. We have that

which implies that $\varphi\left(\rho_{n}\right)\leq\varphi\left(\rho_{n-1}\right)-\psi\left(\rho_{n-1}\right)+\delta_{n}$. Taking $\limsup_{n\rightarrow\infty}$, one sees that

In this way, $\left(x_{n}\right)_{n\in\mathbb{N}}$ is asymptotically regular, namely a pseudo-Cauchy sequence.

The next step is to show that the Picard sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ is a Cauchy sequence. Let us suppose the contrary that the sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ is not a Cauchy sequence. Then, there exists $\varepsilon>0$, there exists $n_{k},m_{k}>k$, such that $n_{k}>m_{k}>k$, where $n_{k}$ is the lowest element with property that

Define $\tilde{\varepsilon}>0$, such that $\tilde{\varepsilon}<\psi(\varepsilon)$. Also, from Nadler’s lemma, we know that, for $\tilde{\varepsilon}$ and $x_{m_{k}}\in Tx_{m_{k}-1}$, there exists $v\in Tx_{n_{k}-1}$, such that

Without loss of generality, we may assume that $v\neq x_{n_{k}}$, with both $v$ and $x_{n_{k}}\in Tx_{n_{k}-1}$. Applying functional $\varphi$, one obtains that

Also, we know that $d\left(x_{m_{k}},x_{n_{k}}\right)\geq\varepsilon$, so $\varphi\left(d\left(x_{m_{k}},x_{n_{k}}\right)\right)\geq\varphi(\varepsilon)$.
Now, since we used assumption (H5), we supposed that $H\left(Tx_{m_{k}-1},Tx_{n_{k}-1}\right)\neq 0$. Assuming the contrary, we obtain that $Tx_{m_{k}-1}=Tx_{n_{k}-1}$, which shows that $x_{m_{k}},x_{n_{k}}$ are both in $Tx_{n_{k}-1}$. Then, we have that

as $k\rightarrow\infty$ and we get a contradiction. The usage of diameter functional $\delta$ is also explained below, in a more detailed manner. Applying the triangle inequality, we have that

Now, we make the following remark : using (H5), we have assumed that $d\left(x_{m_{k}},v\right)\neq 0$. Assuming the contrary, we let $d\left(x_{m_{k}},v\right)=0$. So, $x_{m_{k}}=v\in Tx_{n_{k}-1}$. Then $x_{m_{k}}$ and $x_{n_{k}}\in Tx_{n_{k}-1}$. At the same time, it follows that

as $k\rightarrow\infty$. Hence, we get a contradiction. Since $v$ and $x_{n_{k}}$ are in $Tx_{n_{k}-1}$, one finds that $d\left(v,x_{n_{k}}\right)\leq\delta\left(Tx_{n_{k}-1}\right)$. It follows that

Since we want to use the assumption regarding the diameter functional $\delta$, we make the following remark, that for each $n\in\mathbb{N},Tx_{n}\subseteq T^{n+1}x_{0}$.
For $n=0$, we have $Tx_{0}\subseteq T^{1}x_{0}=Tx_{0}$, which is valid.
For $n=1$, it follows that $Tx_{1}\subseteq T^{2}x_{0}=(T\circ T)x_{0}=T\left(Tx_{0}\right)=T(A)=\bigcup_{x\in A}Tx=\bigcup_{x\in Tx_{0}}Tx$, where $A:=Tx_{0}$. Since $x_{1}\in Tx_{0}$ and $Tx_{1}\subseteq Tx_{1}$, the the affirmation is valid also for $n=1$. By induction, one can easily show that, for each $n\in\mathbb{N},Tx_{n}\subseteq T^{n+1}x_{0}$. Now, using the above property, it follows that $\delta\left(Tx_{n}\right)\leq$
$\delta\left(T^{n+1}x_{0}\right)\rightarrow 0$ as $n\rightarrow\infty$, i.e. $\lim_{n\rightarrow\infty}\delta\left(Tx_{n}\right)=0$. Hence, we obtain that $\lim_{k\rightarrow\infty}\delta\left(Tx_{n_{k}}\right)=0$, because $\left(x_{n_{k}}\right)_{k\in\mathbb{N}}$ is a subsequence of $\left(x_{n}\right)_{n\in\mathbb{N}}$. As in [2], by an easy argument based on triangle inequality, one can show that $\lim_{k\rightarrow\infty}d\left(x_{m_{k}},x_{n_{k}}\right)=\varepsilon$. So

since $d\left(x_{m_{k}-1},x_{n_{k}-1}\right)_{k\in\mathbb{N}}$ is a subsequence of $d\left(x_{m_{k}},x_{n_{k}}\right)_{k\in\mathbb{N}}$. By hypothesis (H2), i.e. $\varphi$ is usc and $\psi$ is lsc, it follows the following chain of inequalities

Since $\tilde{\varepsilon}<\psi(\varepsilon)$, we obtain a contradiction. $\mathrm{S}\left(x_{n}\right)_{n\in\mathbb{N}}$ is Cauchy. Because ( $X,d$ ) is complete, one sees that there exists $x^{*}\in X$, such that $\lim_{n\rightarrow\infty}x_{n}=x^{*}$.

The next step is to show that $x^{*}\in F_{T}$. Using triangle inequality and the fact that the gap functional is nonexpansive (see [14]), one sees that

where we have taken $\limsup_{n\rightarrow\infty}$. It follows that