Well-Posedness of the Fixed Point Problem of Multifunctions of Metric Spaces

Abstract

We consider a class of metrics which are equivalent to the Hausdorff metric in some sense to establish the well-posedness of fixed point problems associated with multifunctions of metric spaces, satisfying various generalized contraction conditions. Examples are provided to justify the applicability of new results.

1. Introduction and Preliminaries

Because of its various applications in mathematics and many other related fields, fixed-point theory has evolved as an important and active area of research. For instance, fixed-point theory has been used to solve various geometric problems on Riemann surfaces, see [1,2] and references therein. In metric fixed point theory, the Banach contraction principle (known as BCP) [3] is among the earliest tools that, due to its usefulness, has been developed further in many different directions. For the historical background of the topic, we refer to [4,5,6,7,8,9,10,11,12,13,14,15,16,17]).

Samet et al. [15] generalized the BCP for self mappings and Nadler [18] extended it for multivalued mappings. Gabeleh and Markin [19] obtained some very useful consequences of Nadler’s result in connection with the best proximity point theory of multivalued mappings. Hasanzade et al. [4] generalized Nadler’s theorem by extending Samet’s result for multivalued mappings. Kirk and Shahzad [12] proved fixed-point results via metrics sequentially equivalent to the Hausdorff metric. Ali et al. [20] obtained multivalued versions of results in [12] via metrics which are (strongly) semi-sequentially equivalent to the Hausdorff metric.

Suzuki [16] generalized the BCP and characterized the completeness of the underlying metric space. Note that the BCP does not have this property; that is, it does not characterize the completeness of underlying metric space ([8], Example 3). Ali and Abbas [21] introduced generalized multivalued Suzuki-type contractions and proved some fixed-point theorems in the settings of b-metric spaces.

The concept of well-posedness holds great significance as it plays a very important role in the theory of nonlinear analysis. Well-posedness of problems is important in establishing the convergence of algorithms in various problems, for instance, variational inequalities, equilibrium, and inverse problems. Due to its usefulness, this concept has been studied in various contexts, and we refer the reader to [22] and the references therein. Petrusel and Rus [23] considered the well-posedness of the FPP. Moreover, Petrusel et al. [24] (compare [25]) defined the well-posedness of the FPP in a generalized sense.

We fix certain notations and present some introductory concepts that will be utilized to prove the main conclusions presented in this article. The symbols $\mathbb{R},$ ${\mathbb{R}}^{+},$ ${\mathbb{R}}_{+},$ $\mathbb{N},{\mathbb{N}}_1,$ U denote the set of reals, nonnegative reals, positive reals, positive integers, nonnegative integers, and a nonempty set, respectively. The collections of subsets of U are defined as follows.

• $P\left(U\right)=\left\{V:V\subset U\right\}.$
• $P_1\left(U\right)=\left\{V\in P\left(U\right):V\ne \varnothing \right\}.$
• $P_c\left(U\right)=\left\{V\in P_1\left(U\right):V\mathrm{is}\mathrm{closed}\right\}.$
• $P_{cb}\left(U\right)=\left\{V\in P_1\left(U\right):V\mathrm{is}\mathrm{bounded}\mathrm{and}\mathrm{closed}\right\}.$

Let $(U,p)$ be a metric space. If

$$\delta (V,W)=\underset{v\in V}{sup}\underset{w\in W}{inf}p\left(v,w\right)$$

for all $V,W\in P_1\left(U\right),$ then the functions $\mathcal{D},H,H^{+}:P\left(U\right)\times P\left(U\right)\to [0,\infty ]$ are defined as follows:

$$\begin{array}{c}\mathcal{D}\left(V,W\right)=\left\{\begin{array}{c}\underset{v\in V,w\in W}{inf}p\left(v,w\right),\mathrm{if}V\ne \varnothing \mathrm{and}W\ne \varnothing ,\hfill \\ 0,\mathrm{if}V=W=\varnothing ,\hfill \\ \infty ,\mathrm{otherwise},\hfill \end{array}\right.\hfill \\ \\ H\left(V,W\right)=\left\{\begin{array}{c}max\left\{\delta (V,W),\delta (W,V)\right\},\mathrm{if}V\ne \varnothing ,W\ne \varnothing ,\hfill \\ 0,\mathrm{if}V=W=\varnothing ,\hfill \\ \infty ,\mathrm{otherwise},\hfill \end{array}\right.\hfill \\ \mathrm{and}\hfill \\ H^{+}\left(V,W\right)=\left\{\begin{array}{c}\frac{\delta (W,V)+\delta (V,W)}{2},\mathrm{if}V\ne \varnothing ,W\ne \varnothing ,\hfill \\ 0,\mathrm{if}W=V=\varnothing ,\hfill \\ \infty ,\mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

Note that H and $H^{+}$ are metrics on $P\left(U\right)$, and $\mathcal{D}$ and H are known as the gap function and Pompeiu Hausdorff generalized function, respectively. Suppose $S:U\to P\left(U\right)$ is the mapping, then $F\left(S\right)$ (a set of fixed points of the mapping S) and $F^{*}\left(S\right)$ (a strict fixed point set of S) are provided as follows:

$$F\left(S\right)=\left\{z\in U:z\in Sz\right\},\mathrm{and}{F}^{*}\left(S\right)=\left\{y\in U:\left\{y\right\}=Sy\right\}.$$

Definition 1

(compare [12]). Let $(U,p)$ be a metric space. A metric $H_1$ on $P_{cb}\left(U\right)$ is sequentially equivalent to the Hausdorff metric H (Shortly SE-H) if, for $V\in P_{cb}\left(U\right)$ and a sequence $\left\{V_q\right\}\subset P_{cb}\left(U\right),$ we have the following:

$$\underset{q\to \infty }{lim}{H}_1\left(V_q,V\right)=0\mathit{iff}\underset{q\to \infty }{lim}H\left(V_q,V\right)=0.$$

Kirk and Shahzad [12] generalized the Nadler result using metrics SE-H.

Theorem 1

([12]). Let $\left(U,p\right)$ be a complete metric space, and $H_1$ is SE-H on $P_{cb}\left(Y\right).$ Suppose $S:U\to P_{cb}\left(U\right)$ satisfies the following:

1.

$0<r<1$, such that:

$$H_1\left(Sx,Sy\right)\le rp\left(x,y\right)$$

for all $x,y\in U$.

2.

For all $x\in U$ and $y\in Sx,$

$$\mathcal{D}\left(y,Sy\right)\le H_1\left(Sy,Sx\right).$$

Then, $F\left(S\right)$ is nonempty.

Consider the slight modification in Definition 1.

Definition 2

([20]). A metric $H_1$ on $P_{cb}\left(U\right)$ is semi-sequentially equivalent to H (Shortly SSE-H) for $V\in P_{cb}\left(U\right)$ and sequence $\left\{V_q\right\}\subset P_{cb}\left(U\right),$ we obtain the following:

$$\underset{q\to \infty }{lim}{H}_1\left(V_q,V\right)=0\mathit{implies}\underset{q\to \infty }{lim}H\left(V_q,V\right)=0.$$

Definition 3

([20]). A metric $H_1$ on $P_{cb}\left(U\right)$ is strongly semi-sequentially equivalent to the Hausdorff metric H (known as SSSE-H) for $V\in P_{cb}\left(U\right)$ and a sequence of sets $\left\{V_q\right\}\subset P_{cb}\left(U\right),$

$$H\left(V_q,V\right)\le H_1\left(V_q,V\right).$$

Remark 1.

Let $H_1$ be a metric on $P_{cb}\left(U\right)$. Note the following:

$$\begin{array}{c}{H}_1\mathrm{is}\mathit{SE}\text{-}H⟹H_1\mathit{is}\mathit{SSE}\text{-}H\mathit{and}\hfill \\ H_1\mathit{is}\mathit{SSSE}\text{-}H⟹H_1\mathit{is}\mathit{SSE}\text{-}H.\hfill \end{array}$$

Also, note that

$$\frac{1}{2}H(V,W)\le H^{+}(V,W)\le H(V,W)$$

for all $V,W$ in $P\left(U\right)$; hence, $H^{+}$ is SE-H and SSSE-H

Example 1.

Let $U=\mathbb{R}$ and H be a Hausdorff metric on $P_{cb}\left(U\right)$ induced by the metric $d(x,y)=\left|x-y\right|.$ Further, define metric $\mathcal{H}$ on $P_{cb}\left(U\right)$ as follows:

$$\mathcal{H}(V,W)=\left\{\begin{array}{c}1,\mathit{if}V\ne W\\ 0,\mathit{if}V=W\end{array}\right.$$

for $V,W$ in $P_{cb}\left(U\right).$ Consider that $V_q=\left[0,1+\frac{1}{q}\right]$ and $V=[0,1]$ for $q\in \mathbb{N}.$ Also, note the following:

$$\underset{q\to \infty }{lim}H(V_q,V)=0\mathit{and}\underset{q\to \infty }{lim}\mathcal{H}(V_q,V)\ne 0.$$

Hence, H is not SSE-$\mathcal{H},$ and $\mathcal{H}$ is SSE-$H.$

If $\left(U,p\right)$ is a metric space, mapping $S:U\to P_1\left(U\right)$ is continuous at point $u\in U$ for any sequence $\left\{u_n\right\}$ in U, where ${lim}_{n\to \infty }p(u_n,u)=0$ implies ${lim}_{n\to \infty }H(S{u}_n,Su)=0.$ Let $S:U\to P_1\left(U\right)$ and $\alpha :U\times U\to \left[0,\infty \right)$, define the following:

$${\alpha }_{*}\left(Sx,Sy\right)=inf\left\{\alpha \left(v,q\right):v\in Sx,q\in Sy\right\}.$$

Mapping $S:U\to P_1\left(U\right)$ is ${\alpha }_{*}$-admissible if

$$\alpha \left(x,y\right)\ge 1\Rightarrow {\alpha }_{*}\left(Sx,Sy\right)\ge 1.$$

Consider the following class of functions:

$$\mathsf{\Phi }=\left\{\begin{array}{c}\varphi :\varphi :\left[0,\infty \right)\to \left[0,\infty \right),\mathrm{such}\mathrm{that}\hfill \\ {\sum }_{q=1}^{\infty }{\varphi }^q\left(s\right)<\infty ,\mathrm{for}\mathrm{each}s>0\mathrm{and},\hfill \\ \varphi \mathrm{is}\mathrm{strictly}\mathrm{increasing}.\hfill \end{array}\right\}$$

where ${\varphi }^q$ is the $q^{th}$ iterate of $\varphi .$ Further, if $\varphi \in \mathsf{\Phi },$ then $\varphi$ is continuous for all $\alpha \in {\mathbb{R}}^{+},$ ${lim}_{n\to \infty }{\varphi }^n\left(\alpha \right)=0$, and $\alpha \in {\mathbb{R}}_{+},$ $\varphi \left(\alpha \right)<\alpha$ (see the details [26]).

Samet et al. [15] extended the BCP by defining $\left(\alpha -\varphi \right)$ contraction condition. Hasanzade et al. [4] generalized the results of Samet for multivalued mappings of metric spaces.

Definition 4.

Let $\left(U,p\right)$ be a complete metric space, where $\alpha :U\times U\to {\mathbb{R}}^{+}$ is function $\varphi \in \mathsf{\Phi }$. Mapping $S:U\to P_c\left(U\right)$ is an $\left({\alpha }_{*}-\varphi \right)$ contractive multifunction if

$${\alpha }_{*}\left(Sx,Sy\right)H\left(Sx,Sy\right)\le \varphi \left(p\left(x,y\right)\right)$$

for all $x,y\in U.$

Further, they obtained the following theorem.

Theorem 2

([4]). Let $\left(U,p\right)$ be a complete metric space and mapping $S:U\to P_c\left(U\right)$ a $\left({\alpha }_{*}-\varphi \right)$ contractive multifunction. Moreover, if

(i) 

mapping S is ${\alpha }_{*}$-admissible;

(ii) 

there is $z_0\in U$ and $z_1\in S{z}_0$, such that $\alpha \left(z_0,z_1\right)\ge 1;$

(iii) 

there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for $q\in {\mathbb{N}}_1$, and $z_q\to z$ as $q\to \infty$ implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in {\mathbb{N}}_1;$

then, $F\left(S\right)$ is nonempty.

Ali and Abbas [21] introduced generalized contractive multifunctions as follows.

Definition 5

([21]). Let $\left(U,p\right)$ be a metric space. Mapping $S:U\to P_c\left(U\right)$ is generalized Suzuki-type $\left({\alpha }_{*}-\varphi \right)$-contractive multifunction if there is a $\varphi \in \mathsf{\Phi }$, such that

$$\frac{1}{2}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right)\mathit{implies}{\alpha }_{*}\left(Sx,Sy\right)H\left(Sx,Sy\right)\le \varphi \left({\mathcal{M}}_1\left(x,y\right)\right)$$

for all $x,y\in U,$ where

$${\mathcal{M}}_1\left(x,y\right)=max\left\{p\left(x,y\right),\mathcal{D}\left(x,Sx\right),\mathcal{D}\left(y,Sy\right),{\displaystyle \frac{\mathcal{D}\left(x,Sy\right)+\mathcal{D}\left(y,Sx\right)}{2}}\right\}.$$

Theorem 3.

Let $\left(U,p\right)$ be a complete metric space and $S:U\to P_c\left(U\right)$ be a generalized Suzuki-type $\left({\alpha }_{*}-\varphi \right)$-contractive multifunction. Moreover,

(i) 

S is ${\alpha }_{*}$-admissible and there exist $z_0\in U$, $z_1\in S{z}_0$, such that $\alpha \left(z_0,z_1\right)\ge 1;$

(ii) 

if there is a sequence $\left\{z_q\right\}$ in U such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in {\mathbb{N}}_1$ and $z_q\to z$ as $q\to \infty$ implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in {\mathbb{N}}_1;$

then, $F\left(S\right)$ is nonempty.

In [27], the following notion was introduced.

Definition 6

([27]). Mapping $S:U\to P_{cb}\left(U\right)$ is a $H^{+}$ contraction if there is an $0<r<1$, such that

$$H^{+}\left(Sx,Sy\right)\le rp\left(x,y\right)$$

for all $x,y\in U,$ and for every $x\in U$ and $y\in Sx$ and $\epsilon >0,$ there is $z\in Sy$ and we have

$$p\left(y,z\right)\le H^{+}\left(Sx,Sy\right)+\epsilon .$$

We present the following notions from [23] with a slight modification for metrics which are SSE-H.

Definition 7

([23]). If $\left(U,p\right)$ is a metric space, $H_1$ is an SSE-H, and $Z\in P_1\left(U\right),$ the FPP of mapping $S:Z\to P_c\left(U\right)$ is well posed w.r.t.

1.

$\mathcal{D}$ if $F\left(S\right)=\left\{y^{*}\right\}$ and if $\left\{y_q\right\}\subset Z$ and $\underset{q\to \infty }{lim}\mathcal{D}\left(y_q,S{y}_q\right)=0$ then $\underset{q\to \infty }{lim}{y}_q=y^{*}.$

2.

$H_1$ if $F^{*}\left(S\right)=\left\{y^{*}\right\}$ and if $\left\{y_q\right\}\subset Z$ and $\underset{q\to \infty }{lim}{H}_1\left(y_q,S{y}_q\right)=0$ then $\underset{q\to \infty }{lim}{y}_q=y^{*}.$

We present the following notions from [24] with a slight modification for metrics which are SSE-H.

Definition 8

([24]). If $\left(U,p\right)$ is a metric space, $H_1$ is an SSE-H, and $Z\in P_1\left(U\right),$ the FPP of mapping $S:Z\to P_c\left(U\right)$ is well-posed in the generalized sense w.r.t.

1.

$\mathcal{D}$ if $F\left(S\right)\ne \varnothing$ and if $\left\{y_q\right\}\subset Z$ and $\underset{q\to \infty }{lim}\mathcal{D}\left(y_q,S{y}_q\right)=0$, then there exists a subsequence $\left\{y_{q_i}\right\}$ of $\left\{y_q\right\}$, such that $\underset{i\to \infty }{lim}{y}_{q_i}=y^{*}\in F\left(S\right).$

2.

$H_1$ if $F^{*}\left(S\right)\ne \varnothing$ and if $\left\{y_q\right\}\subset Z$ and $\underset{q\to \infty }{lim}{H}_1\left(y_q,S{y}_q\right)=0$, then there exists a subsequence $\left\{y_{q_i}\right\}$ of $\left\{y_q\right\}$, such that $\underset{i\to \infty }{lim}{y}_{q_i}=y^{*}\in F\left(S\right).$

Dung and Le Hang proved the well-posedness of the FPP of multivalued mappings ([25], Theorem 2.2).

Consider the following notations for multifunction $S:U\to P_{cb}\left(U\right)$:

$$\begin{array}{ccc}\hfill \mathcal{M}\left(x,y\right)& =& max\left\{\begin{array}{c}p\left(x,y\right),\mathcal{D}\left(x,Sx\right),\mathcal{D}\left(y,Sy\right),\hfill \\ {\displaystyle \frac{\mathcal{D}\left(x,Sy\right)+\mathcal{D}\left(y,Sx\right)}{2}},{\displaystyle \frac{\mathcal{D}\left(x,Sx\right)\mathcal{D}\left(y,Sy\right)}{1+p\left(x,y\right)}}\hfill \end{array}\right\},\hfill \\ \hfill {\mathcal{M}}^{*}\left(x,y\right)& =& max\left\{p\left(x,y\right),\mathcal{D}\left(x,Sx\right),{\displaystyle \frac{\mathcal{D}\left(x,Sx\right)\mathcal{D}\left(y,Sy\right)}{1+\mathcal{D}\left(x,y\right)}}\right\},\mathrm{and}\hfill \\ \hfill \mathcal{N}\left(x,y\right)& =& max\left\{\begin{array}{c}p\left(x,y\right),\mathcal{D}\left(x,Sx\right),{\displaystyle \frac{\mathcal{D}\left(y,Sy\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(x,Sy\right)+\mathcal{D}\left(y,Sx\right)}{2}},{\displaystyle \frac{\mathcal{D}\left(x,Sx\right)\mathcal{D}\left(y,Sy\right)}{1+p\left(x,y\right)}}\hfill \end{array}\right\}\hfill \end{array}$$

for all $x,y\in U.$

2. Existence of Fixed Points of Multifunctions

We start with the following result, in which we establish the existence of solutions for the FPP of multifunctions.

Theorem 4.

If $\left(U,p\right)$ is a complete metric space, $H_1$ is an SSE-H on $P_c\left(U\right)$, and the mapping $S:U\to P_c\left(U\right)$ satisfies the following:

(i) 

${\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right)$ implies

$${\alpha }_{*}\left(Sx,Sy\right)H_1\left(Sx,Sy\right)\le \varphi \left({\mathcal{M}}^{*}\left(x,y\right)\right)$$

for all $x,y\in U$ and some $\varphi \in \mathsf{\Phi }.$

(ii) 

For $x\in U$ and $y\in Sx,$

$$\mathcal{D}\left(y,Sy\right)\le H_1\left(Sy,Sx\right).$$

(iii) 

Mapping S is ${\alpha }_{*}$-admissible.

(iv) 

$z_0\in U$ and $z_1\in S{z}_0$, such that $\alpha \left(z_0,z_1\right)\ge 1.$

(v) 

Either of the following are applicable: (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in \mathbb{N}$, or (b) mapping S forms a continuous hold.

Then, $F\left(S\right)$ is nonempty.

Proof. 

As in (iv), $z_0\in U$ and $z_1\in S{z}_0$, such that $\alpha \left(z_0,z_1\right)\ge 1$, which implies ${\alpha }_{*}\left(S{z}_0,S{z}_1\right)\ge 1.$ If $z_1=z_0$, then $z_0=z_1\in S{z}_0$; that is, $F\left(S\right)$ is nonempty and there is nothing further to prove. So, assume $z_0\ne z_1.$ Then, by (i) and (ii), we obtain the following:

$${\displaystyle \frac{1}{2}}\mathcal{D}\left(z_0,S{z}_0\right)\le p\left(z_0,z_1\right);$$

thus, we have

$$\begin{array}{ccc}\hfill \mathcal{D}\left(z_1,S{z}_1\right)& \le & {\alpha }_{*}\left(S{z}_0,S{z}_1\right)\mathcal{D}\left(z_1,S{z}_1\right)\hfill \\ & \le & {\alpha }_{*}\left(S{z}_0,S{z}_1\right)H_1\left(S{z}_0,S{z}_1\right)\hfill \\ & \le & \varphi \left({\mathcal{M}}^{*}\left(z_0,z_1\right)\right).\hfill \end{array}$$

If $\mathcal{D}\left(z_1,S{z}_1\right)=0$, then $z_1\in S{z}_1$ implies that $F\left(S\right)$ is nonempty. So, assume $\mathcal{D}\left(z_1,S{z}_1\right)>0.$ Further, from the given condition, we obtain the following:

$$\begin{array}{ccc}\hfill 0& <& \mathcal{D}\left(z_1,S{z}_1\right)\le \varphi \left({\mathcal{M}}^{*}\left(z_0,z_1\right)\right)\hfill \\ & =& \varphi \left(max\left\{p\left(z_0,z_1\right),\mathcal{D}\left(z_0,S{z}_0\right),{\displaystyle \frac{\mathcal{D}\left(z_0,S{z}_0\right)p\left(z_1,S{z}_1\right)}{1+p\left(z_0,z_1\right)}}\right\}\right)\hfill \\ & \le & \varphi \left(max\left\{p\left(z_0,z_1\right),\mathcal{D}\left(z_1,S{z}_1\right)\right\}\right).\hfill \end{array}$$

That is,

$$0<\mathcal{D}\left(z_1,S{z}_1\right)\le \varphi \left(max\left\{p\left(z_0,z_1\right),\mathcal{D}\left(z_1,S{z}_1\right)\right\}\right).$$

If

$$max\left\{p\left(z_0,z_1\right),\mathcal{D}\left(z_1,S{z}_1\right)\right\}=\mathcal{D}\left(z_1,S{z}_1\right),$$

then

$$0<\mathcal{D}\left(z_1,S{z}_1\right)\le \varphi \left(\mathcal{D}\left(z_1,S{z}_1\right)\right).$$

As $\mathcal{D}\left(z_1,S{z}_1\right)>0$ and $\varphi \in \mathsf{\Phi },$

$$0<\mathcal{D}\left(z_1,S{z}_1\right)<\mathcal{D}\left(z_1,S{z}_1\right)$$

is a contradiction. Hence,

$$0<\mathcal{D}\left(z_1,S{z}_1\right)\le \varphi \left(p\left(z_0,z_1\right)\right).$$

Choose $z_2\in S{z}_1$ and $v_1>1$, such that

$$0<\mathcal{D}\left(z_1,S{z}_1\right)\le p\left(z_1,z_2\right)<v_1\mathcal{D}\left(z_1,S{z}_1\right).$$

Thus,

$$0<p\left(z_1,z_2\right)<v_1\mathcal{D}\left(z_1,S{z}_1\right)\le v_1\varphi \left(p\left(z_0,z_1\right)\right)=v_1\varphi \left(b_0\right),$$

where $b_0=p\left(z_0,z_1\right)$. Note that, $z_2\ne z_1$ and

$$\alpha \left(z_1,z_2\right)\ge {\alpha }_{*}\left(S{z}_0,S{z}_1\right)\ge 1.$$

Thus, $\alpha \left(z_1,z_2\right)\ge 1$, and hence, ${\alpha }_{*}\left(S{z}_1,S{z}_2\right)\ge 1.$ As $\varphi \in \mathsf{\Phi },$

$$\varphi \left(p\left(z_1,z_2\right)\right)<\varphi \left(v_1\varphi \left(b_0\right)\right).$$

Set

$$v_2={\displaystyle \frac{\varphi \left(v_1\varphi \left(b_0\right)\right)}{\varphi \left(p\left(z_1,z_2\right)\right)}}>1.$$

Now, if $z_2\in S{z}_2,$ then the proof is finished. Let $z_2\notin S{z}_2$. Proceeding as above, we obtain the following:

$$0<\mathcal{D}\left(z_2,S{z}_2\right)\le \varphi \left(p\left(z_1,z_2\right)\right)$$

and $z_3\in S{z}_2$, such that

$$0<p\left(z_2,z_3\right)<v_2\mathcal{D}\left(z_2,S{z}_2\right)\le v_2\varphi \left(p\left(z_1,z_2\right)\right)=\varphi \left(v_1\varphi \left(b_0\right)\right).$$

(1)

If $z_3=z_2,$ then $z_2\in F\left(S\right).$ So, consider $z_3\ne z_2.$ As S is ${\alpha }_{*}$-admissible and

$$\alpha \left(z_2,z_3\right)\ge {\alpha }_{*}\left(S{z}_1,S{z}_2\right)\ge 1,$$

${\alpha }_{*}\left(S{z}_2,S{z}_3\right)\ge 1$. Using (1), we obtain the following:

$$\varphi \left(p\left(z_2,z_3\right)\right)<{\varphi }^2\left(v_1\left(\varphi \left(b_0\right)\right)\right).$$

Set

$$v_3={\displaystyle \frac{{\varphi }^2\left(v_1\varphi \left(b_0\right)\right)}{\varphi \left(p\left(z_2,z_3\right)\right)}}>1.$$

Now, if $z_3\in S{z}_3,$ then the proof is finished. Let $z_3\notin S{z}_3.$ Continuing, we obtain sequence $\left\{z_q\right\}$ in $U,$ which satisfies $z_{q+1}\in S{z}_q,$ $z_{q+1}\ne z_q$ and $\alpha \left(z_{q+1},z_{q+2}\right)\ge 1$ such that

$$0<p\left(z_{q+1},z_{q+2}\right)\le {\varphi }^{q+1}\left(v_1\varphi \left(b_0\right)\right).$$

As $\varphi \in \mathsf{\Phi },$ so for every $\epsilon >0,$ and $k>q,$ there is an $n_1\in \mathbb{N}$ such that

$$\begin{array}{ccc}\hfill p\left(z_q,z_k\right)& \le & p\left(z_q,z_{q+1}\right)+p\left(z_{q+1},z_{q+2}\right)+···+p\left(z_{k-1},z_k\right)\hfill \\ & =& \sum _{i=q}^{k-1}p\left(z_i,z_{i+1}\right)\le \sum _{i=q}^{k-1}{\varphi }^i\left(v_1\varphi \left(b_0\right)\right)\le \sum _{i=q}^{\infty }{\varphi }^i\left(v_1\varphi \left(b_0\right)\right)<\epsilon \hfill \end{array}$$

for all $k,q\ge n_1.$ This proves that $\left\{z_q\right\}$ is Cauchy, so $z_q$ converges to some z in $U.$ Assume (a) holds true in (v) above. In order to show $z\in F\left(S\right),$ assume $w\notin Sw$ (on contrary) for all $w\in U;$ that is, $\mathcal{D}\left(w,Sw\right)>0$ for all $w\in U.$ We claim that either

$$\frac{1}{2}\mathcal{D}\left(z_q,S{z}_q\right)\le p\left(z_q,z\right)$$

(2)

or

$$\frac{1}{2}\mathcal{D}\left(z_{q+1},S{z}_{q+1}\right)\le p\left(z_{q+1},z\right)$$

(3)

holds for all $q\in \mathbb{N}.$ If not, there is a $q_0\in {\mathbb{N}}_0$ so that

$$\frac{1}{2}\mathcal{D}\left(z_{q_0},S{z}_{q_0}\right)>p\left(z_{q_0},z\right)$$

(4)

and

$$\frac{1}{2}\mathcal{D}\left(z_{q_0+1},S{z}_{q_0+1}\right)>p\left(z_{q_0+1},z\right).$$

(5)

Now, (4) and (5) imply

$$\begin{array}{ccc}\hfill p\left(z_{q_0},z_{q_0+1}\right)& \le & p\left(z_{q_0},z\right)+p\left(z,z_{q_0+1}\right)\hfill \\ & <& \frac{1}{2}\mathcal{D}\left(z_{q_0},S{z}_{q_0}\right)+\frac{1}{2}\mathcal{D}\left(z_{q_0+1},S{z}_{q_0+1}\right)\hfill \\ & \le & \frac{1}{2}p\left(z_{q_0},z_{q_0+1}\right)+\frac{1}{2}\varphi \left(p\left(z_{q_0},z_{q_0+1}\right)\right)\hfill \\ & <& \frac{1}{2}p\left(z_{q_0},z_{q_0+1}\right)+\frac{1}{2}p\left(z_{q_0},z_{q_0+1}\right)\hfill \\ & =& p\left(z_{q_0},z_{q_0+1}\right),\hfill \end{array}$$

which is a contradiction. Hence, either (2) or (3) holds for ${\mathbb{N}}_1,$ where ${\mathbb{N}}_1$ is an infinite subset of $\mathbb{N}.$ Assume (2) holds for all $n\in {\mathbb{N}}_1,$ then there is a subsequence $\left\{z_{q_k}\right\}$ of $\left\{z_q\right\}$, such that

$$\frac{1}{2}\mathcal{D}\left(z_{q_k},S{z}_{q_k}\right)\le p\left(z_{q_k},z\right).$$

As $\alpha \left(z_q,z_{q+1}\right)\ge 1$ and $z_q\to z$, this provides $\alpha \left(z_{q_k},z_{q_{k+1}}\right)\ge 1$ and $z_{q_k}\to z$ when $k\to \infty ;$ hence, by given assumption, $\alpha \left(z_{q_k},z\right)\ge 1$ for all $k\in N.$ As mapping S is ${\alpha }_{*}$-admissible so ${\alpha }_{*}\left(S{z}_{q_k},Sz\right)\ge 1$,

$${\displaystyle \frac{1}{2}}p\left(z_{q_k},S{z}_{q_k}\right)\le p\left(z_{q_k},z\right)$$

implies

$$\begin{array}{ccc}\hfill H_1\left(S{z}_{q_k},Sz\right)& \le & {\alpha }_{*}\left(S{z}_{q_k},Sz\right)H_1\left(S{z}_{q_k},Sz\right)\hfill \\ & \le & \varphi \left({\mathcal{M}}^{*}\left(z_{q_k},z\right)\right)\hfill \\ & =& \varphi \left(max\left\{p\left(z_{q_k},z\right),\mathcal{D}\left(z_{q_k},S{z}_{q_k}\right),{\displaystyle \frac{\mathcal{D}\left(z_{q_k},S{z}_{q_k}\right)\mathcal{D}\left(z,Sz\right)}{1+p\left(z_{q_k},z\right)}}\right\}\right)\hfill \\ & \le & \varphi \left(max\left\{p\left(z_{q_k},z\right),p\left(z_{q_k},z_{q_{k+1}}\right),{\displaystyle \frac{\mathcal{D}\left(z_{q_k},z_{q_{k+1}}\right)\mathcal{D}\left(z,Sz\right)}{1+p\left(z_{q_k},z\right)}}\right\}\right).\hfill \end{array}$$

As q tends to ∞, we have

$$\underset{k\to \infty }{lim}{H}_1\left(S{z}_{q_k},Sz\right)\le \underset{k\to \infty }{lim}\varphi \left(max\left\{\begin{array}{c}p\left(z_{q_k},z\right),p\left(z_{q_k},z_{q_{k+1}}\right),\\ {\displaystyle \frac{p\left(z_{q_k},z_{q_{k+1}}\right)\mathcal{D}\left(z,Sz\right)}{1+p\left(z_{q_k},z\right)}}\end{array}\right\}\right).$$

That is, ${lim}_{k\to \infty }{H}_1\left(S{z}_{q_k},Sz\right)=0$ as $\varphi$ is continuous at $0.$ As $H_1$ is SSE-H,

$$\underset{k\to \infty }{lim}H\left(S{z}_{q_k},Sz\right)=0.$$

Hence,

$$\mathcal{D}\left(z,Sz\right)=\underset{k\to \infty }{lim}\mathcal{D}\left(z_{q_{k+1}},Sz\right)\le \underset{k\to \infty }{lim}H\left(S{z}_{q_k},Sz\right)=0.$$

That is, $z\in Sz.$ Assume (b) holds true in (v). That is, S is continuous. Then, ${lim}_{q\to \infty }p(z_q,z)=0$ implies ${lim}_{k\to \infty }H\left(S{z}_q,Sz\right)=0.$ Hence,

$$\mathcal{D}\left(z,Sz\right)=\underset{q\to \infty }{lim}\mathcal{D}\left(z_{q+1},Sz\right)\le \underset{k\to \infty }{lim}H\left(S{z}_{q_k},Sz\right)=0.$$

That is, $z\in Sz.$ □

From the above theorem, we obtain important corollaries as follows.

Corollary 1.

If $\left(U,p\right)$ is a complete metric space, $H_1$ is an SSE-H on $P_c\left(U\right)$, and mapping $S:U\to P_c\left(U\right)$ satisfies

$${\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right),,\mathit{which}\mathit{implies}{\alpha }_{*}\left(Sx,Sy\right)H_1\left(Sx,Sy\right)\le \varphi \left(p\left(x,y\right)\right)$$

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }.$ Further, for $x\in U$ and $y\in Sx$

$$\mathcal{D}\left(y,Sy\right)\le H_1\left(Sy,Sx\right),$$

and mapping S is ${\alpha }_{*}$-admissible; therefore, $z_0\in U$ and $z_1\in S{z}_0$, such that $\alpha \left(z_0,z_1\right)\ge 1.$ Moreover, (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in \mathbb{N}$ or (b) S is continuous. Then, $F\left(S\right)$ is nonempty.

Corollary 2.

If $\left(U,p\right)$ is a complete metric space, $H_1$ is an SE-H on $P_c\left(U\right)$, and mapping $S:U\to P_c\left(U\right)$ satisfies

$${\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right),\mathit{which}\mathit{implies}{\alpha }_{*}\left(Sx,Sy\right)H_1\left(Sx,Sy\right)\le \varphi \left(p\left(x,y\right)\right)$$

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }.$ Further, for $x\in U$ and $y\in Sx$,

$$\mathcal{D}\left(y,Sy\right)\le H_1\left(Sy,Sx\right),$$

and mapping S is ${\alpha }_{*}$-admissible; therefore, $z_0\in U$ and $z_1\in S{z}_0$, such that $\alpha \left(z_0,z_1\right)\ge 1.$ Moreover, (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in \mathbb{N}$ or (b) S is continuous. Then, $F\left(S\right)$ is nonempty.

Corollary 3.

If $\left(U,p\right)$ is a complete metric space, $H_1$ is an SSSE-H on $P_c\left(U\right)$, and mapping $S:U\to P_c\left(U\right)$ satisfies

$${\alpha }_{*}\left(Sx,Sy\right)H_1\left(Sx,Sy\right)\le \varphi \left({\mathcal{M}}^{*}\left(x,y\right)\right)$$

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }.$ Further, for $x\in U$ and $y\in Sx$,

$$\mathcal{D}\left(y,Sy\right)\le H_1\left(Sy,Sx\right),$$

and mapping S is ${\alpha }_{*}$-admissible; therefore, $z_0\in U$ and $z_1\in S{z}_0$, such that $\alpha \left(z_0,z_1\right)\ge 1.$ Moreover, (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in \mathbb{N}$ or (b) S is continuous. Then, $F\left(S\right)$ is nonempty.

Corollary 4.

If $\left(U,p\right)$ is a complete metric space, $H_1$ is an SE-H on $P_c\left(U\right)$, and mapping $S:U\to P_c\left(U\right)$ satisfies

$${\alpha }_{*}\left(Sx,Sy\right)H_1\left(Sx,Sy\right)\le \varphi \left({\mathcal{M}}^{*}\left(x,y\right)\right)$$

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }.$ Further, for $x\in U$ and $y\in Sx$,

$$\mathcal{D}\left(y,Sy\right)\le H_1\left(Sy,Sx\right),$$

and mapping S is ${\alpha }_{*}$-admissible; therefore, $z_0\in U$ and $z_1\in S{z}_0$, such that $\alpha \left(z_0,z_1\right)\ge 1.$ Moreover, (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in \mathbb{N}$ or (b) S is continuous. Then, $F\left(S\right)$ is nonempty.

Theorem 1 for single-valued mappings was originally proposed by Kirk and Shahzad [12]; after slight modification and generalizing, we obtained the following consequence.

Theorem 5.

If $\left(U,p\right)$ is a complete metric space, ρ is an SSE-p on U, and mapping $S:U\to U$ satisfies

$${\displaystyle \frac{1}{2}}p\left(x,Sx\right)\le p\left(x,y\right),\mathit{which}\mathit{implies}{\alpha }_{*}\left(Sx,Sy\right)\rho \left(Sx,Sy\right)\le \varphi \left({\mathcal{M}}^{*}\left(x,y\right)\right)$$

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }.$ For $x\in U$,

$$p\left(Sx,S^{2}y\right)\le \rho \left(Sx,S^{2}y\right).$$

Further, mapping S is α-admissible; therefore, $z_0\in U$ exist, such that $\alpha \left(z_0,S{z}_0\right)\ge 1.$ Moreover, (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in \mathbb{N}$ or (b) mapping S is continuous. Then, $F\left(S\right)$ is a singleton.

Proof. 

The existence of fixed points is guaranteed by Theorem 4. For uniqueness, suppose that there are $u_1,u_2\in F\left(S\right)$, such that $u_1\ne u_1.$ That is, $S{u}_1=u_1$ and $S{u}_2=u_2$, such that $u_1\ne u_2.$ As

$${\displaystyle \frac{1}{2}}p\left(u_1,S{u}_1\right)\le p\left(u_1,u_1\right)=0\le p\left(u_1,u_2\right),$$

$$\begin{array}{ccc}\hfill p\left(u_1,u_2\right)& =& p\left(S{u}_1,S^2{u}_2\right)\le \alpha \left(S{u}_1,S{u}_2\right)p\left(S{u}_1,S^2{u}_2\right)\hfill \\ & \le & \alpha \left(S{z}_0,S{z}_1\right)\rho \left(S{u}_1,S^2{u}_2\right)\hfill \\ & =& \alpha \left(S{z}_0,S{z}_1\right)\rho \left(S{u}_1,S{u}_2\right)\hfill \\ & \le & \varphi \left({\mathcal{M}}^{*}\left(u_1,u_2\right)\right)\hfill \\ & =& \varphi \left(p\left(u_1,u_2\right)\right)\hfill \\ & <& p\left(u_1,u_2\right)\hfill \end{array}$$

is a contradiction. Then, $F\left(S\right)$ is a singleton. □

In the next theorem, we prove a more general contraction condition compared to the one we used in Theorem 4 but with metrics which are SSSE-H on $P_c\left(U\right).$

Theorem 6.

If $\left(U,p\right)$ is a complete metric space, $H_1$ is an SSSE-H on $P_c\left(U\right)$, and mapping $S:U\to P_c\left(U\right)$ satisfies

$$\begin{array}{c}{\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right)\hfill \\ ,\mathit{which}\mathit{implies}\hfill \\ {\alpha }_{*}\left(Sx,Sy\right)H_1\left(Sx,Sy\right)\le \varphi \left(\mathcal{M}\left(x,y\right)\right)\hfill \end{array}$$

(6)

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }.$ Therefore, $z_0\in U$ and $z_1\in S{z}_0$ exist, such that $\alpha \left(z_0,z_1\right)\ge 1.$ Further, for $x\in U$ and $y\in Sx,$,

$$\mathcal{D}\left(y,Sy\right)\le H_1\left(Sy,Sx\right),$$

and mapping S is ${\alpha }_{*}$-admissible. Moreover, (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in \mathbb{N},$ or (b) mapping S is continuous. Then, $F\left(S\right)$ is nonempty.

Proof. 

Along similar lines, as in the proof of Theorem 4, we obtain a Cauchy sequence $\left\{z_q\right\}$ in U, which satisfies $z_{q+1}\in S{z}_q,$ $z_{q+1}\ne z_q$ such that $z_q$ converges to some $z\in U$. Suppose (a) holds. Further, assume, on the contrary, that there is no fixed point of $S.$ That is, $\mathcal{D}\left(z,Sz\right)>0.$ Again, along similar lines as in the proof of Theorem 4, a subsequence $\left\{z_{q_k}\right\}$ of $\left\{z_q\right\}$ exists, which satisfies

$$\underset{k\to \infty }{lim}{H}_1\left(S{z}_{q_k},Sz\right)\le \varphi \left(\mathcal{D}\left(z,Sz\right)\right).$$

$H_1$ is a SSSE-H on $P_c\left(U\right)$, so

$$\begin{array}{ccc}\hfill \mathcal{D}\left(z,Sz\right)& =& \underset{k\to \infty }{lim}\mathcal{D}\left(z_{q_k},Sz\right)\hfill \\ & \le & \underset{k\to \infty }{lim}H\left(S{z}_{q_k},Sz\right)\hfill \\ & \le & \underset{k\to \infty }{lim}{H}_1\left(S{z}_{q_k},Sz\right)\hfill \\ & \le & \varphi \left(\mathcal{D}\left(z,Sz\right)\right)<\mathcal{D}\left(z,Sz\right)\hfill \end{array}$$

is a contradiction. Consequently, $z\in Sz.$ As in the proof of Theorem 4, the conclusion holds whenever (b) holds. □

Corollary 5.

If $\left(U,p\right)$ is a complete metric space, $H_1$ is an SSSE-H on $P_c\left(U\right).$ If mapping $S:U\to P_c\left(U\right)$ satisfies

$$\begin{array}{c}{\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right),\hfill \\ \mathit{which}\mathit{implies}\hfill \\ {\alpha }_{*}\left(Sx,Sy\right)H_1\left(Sx,Sy\right)\le \varphi \left(p\left(x,y\right)\right)\hfill \end{array}$$

(7)

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }$. Therefore, $z_0\in U$ and $z_1\in S{z}_0$ exist, such that $\alpha \left(z_0,z_1\right)\ge 1.$ Further, for $x\in U$ and $y\in Sx,$

$$\mathcal{D}\left(y,Sy\right)\le H_1\left(Sy,Sx\right),$$

and mapping S is ${\alpha }_{*}$-admissible. Moreover, (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in \mathbb{N},$ or (b) mapping S is continuous. Then, $F\left(S\right)$ is nonempty.

Corollary 6.

If $\left(U,p\right)$ is a complete metric space and mapping $S:U\to P_c\left(U\right)$ satisfies

$$\begin{array}{c}{\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right),\hfill \\ \mathit{this}\mathit{implies}\mathit{that}\hfill \\ {\alpha }_{*}\left(Sx,Sy\right)H\left(Sx,Sy\right)\le \varphi \left({\mathcal{M}}^{*}\left(x,y\right)\right)\hfill \end{array}$$

(8)

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }.$ Therefore, $z_0\in U$ and $z_1\in S{z}_0$ exist, such that $\alpha \left(z_0,z_1\right)\ge 1.$ Further, mapping S is ${\alpha }_{*}$-admissible and (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in \mathbb{N},$ or (b) mapping S is continuous. Then, $F\left(S\right)$ is nonempty.

Corollary 7.

If $\left(U,p\right)$ is a complete metric space, $H_1$ is a SSSE-H on $P_c\left(U\right).$ If mapping $S:U\to P_c\left(U\right)$ satisfies

$${\alpha }_{*}\left(Sx,Sy\right)H_1\left(Sx,Sy\right)\le \varphi \left(\mathcal{M}\left(x,y\right)\right)$$

(9)

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }.$ Therefore, $z_0\in U$ and $z_1\in S{z}_0$ exist, such that $\alpha \left(z_0,z_1\right)\ge 1.$ Further, for $x\in U$ and $y\in Sx,$

$$\mathcal{D}\left(y,Sy\right)\le H_1\left(Sy,Sx\right),$$

and mapping S is ${\alpha }_{*}$-admissible. Moreover, (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q\in \mathbb{N},$ or (b) mapping S is continuous. Then, $F\left(S\right)$ is nonempty.

Now we present the following example to explain Theorem 6 and show that some results that already exist are not applicable in this situation.

Example 2.

Let $U=\left\{z_1,z_2,z_3,z_4,z_5\right\}.$ Define $p:U\times U\to {\mathbb{R}}^{+}$ as

$$\begin{array}{c}p\left(z_3,z_5\right)=p\left(z_2,z_5\right)=p\left(z_2,z_4\right)=p\left(z_3,z_4\right)=6,\hfill \\ p\left(z_2,z_3\right)=p\left(z_1,z_4\right)=p\left(z_1,z_5\right)=8,\hfill \\ p\left(z_1,z_3\right)=p\left(z_1,z_2\right)=5,p\left(z_4,z_5\right)=3,\hfill \\ p\left(z,z\right)=0\mathit{and}\hfill \\ p\left(m,n\right)=p\left(n,m\right)\mathit{for}\mathit{all}m,n\in U.\hfill \end{array}$$

Define $S:U\to P_c\left(U\right)$ by

$$Sz=\left\{\begin{array}{c}\left\{z_1\right\}\mathit{if}z\ne z_4,z_5,\hfill \\ \left\{z_2\right\}\mathit{if}z=z_4,\hfill \\ \left\{z_3\right\}\mathit{if}z=z_5,\hfill \end{array}\right.$$

and $\alpha :U\times U\to {\mathbb{R}}^{+}$ by $\alpha \left(z_q,z_k\right)=1$ for all $q,k\in \left\{1,2,3,4,5\right\}.$ Set $\varphi \left(m\right)={\displaystyle \frac{8}{9}}m$ to $m\in {\mathbb{R}}^{+},$ then $\varphi \in \mathsf{\Phi }.$ S is ${\alpha }_{*}$-admissible. Note that $H^{+}$ is SSE-H and

$${\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,z\right),\mathit{which}\mathit{implies}{\alpha }_{*}\left(Sx,Sz\right)H^{+}\left(Sx,Sz\right)\le \varphi \left(\mathcal{M}\left(x,z\right)\right)$$

holds for all $x,z\in \left\{z_1,z_2,z_3\right\}$ as

$$H^{+}\left(Sx,Sz\right)=0\le \varphi \left(\mathcal{M}\left(x,z\right)\right)=\frac{8}{9}p\left(x,z\right).$$

If $x=z_1$ and $z=z_4$, then

$$\begin{array}{ccc}\hfill H^{+}\left(S{z}_1,S{z}_4\right)& =& 5\le \varphi \left(\mathcal{M}\left(z_1,z_4\right)\right)\hfill \\ & =& \varphi \left(max\left\{\begin{array}{c}p\left(z_1,z_4\right),\mathcal{D}\left(z_1,S{z}_1\right),\mathcal{D}\left(z_4,S{z}_4\right),\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_1,S{z}_4\right)+\mathcal{D}\left(z_4,S{z}_1\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_1,S{z}_1\right)\mathcal{D}\left(z_4,S{z}_4\right)}{1+p\left(z_1,z_4\right)}}\hfill \end{array}\right\}\right)\hfill \\ & =& \varphi \left(max\left\{8,0,6,{\displaystyle \frac{5+8}{2}},0\right\}\right)=\varphi \left(8\right)=7.11.\hfill \end{array}$$

If $x=z_1$ and $z=z_5$, then

$$\begin{array}{ccc}\hfill H^{+}\left(S{z}_1,S{z}_5\right)& =& 5\le \varphi \left(\mathcal{M}\left(z_1,z_5\right)\right)\hfill \\ & =& \varphi \left(max\left\{\begin{array}{c}p\left(z_1,z_5\right),\mathcal{D}\left(z_1,S{z}_1\right),\mathcal{D}\left(z_4,S{z}_5\right),\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_1,S{z}_5\right)+\mathcal{D}\left(z_5,S{z}_1\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_1,S{z}_1\right)\mathcal{D}\left(z_5,S{z}_5\right)}{1+p\left(z_1,z_5\right)}}\hfill \end{array}\right\}\right)\hfill \\ & =& \varphi \left(max\left\{8,0,6,{\displaystyle \frac{5+8}{2}},0\right\}\right)=\varphi \left(8\right)=7.11.\hfill \end{array}$$

If $x=z_2$ and $z=z_4,$ then

$$\begin{array}{ccc}\hfill H^{+}\left(S{z}_2,S{z}_4\right)& =& p\left(z_1,z_2\right)=5\le \varphi \left(\mathcal{M}\left(z_2,z_4\right)\right)\hfill \\ & =& \varphi \left(max\left\{\begin{array}{c}p\left(z_2,z_4\right),\mathcal{D}\left(z_2,S{z}_2\right),\mathcal{D}\left(z_4,S{z}_4\right),\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_2,S{z}_4\right)+\mathcal{D}\left(z_4,S{z}_2\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_2,S{z}_2\right)\mathcal{D}\left(z_4,S{z}_4\right)}{1+p\left(z_2,z_4\right)}}\hfill \end{array}\right\}\right)\hfill \\ & =& \varphi \left(max\left\{6,5,6,{\displaystyle \frac{0+8}{2}},{\displaystyle \frac{5\times 6}{1+6}}\right\}\right)=\varphi \left(6\right)=5.333.\hfill \end{array}$$

If $x=z_3$ and $z=z_5,$ then

$$\begin{array}{ccc}\hfill H^{+}\left(S{z}_3,S{z}_5\right)& =& p\left(z_1,z_3\right)=5\le \varphi \left(\mathcal{M}\left(z_3,z_5\right)\right)\hfill \\ & =& \varphi \left(max\left\{\begin{array}{c}p\left(z_3,z_5\right),\mathcal{D}\left(z_3,S{z}_3\right),\mathcal{D}\left(z_5,S{z}_5\right),\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_3,S{z}_5\right)+\mathcal{D}\left(z_5,S{z}_3\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_3,S{z}_3\right)\mathcal{D}\left(z_5,S{z}_5\right)}{1+p\left(z_3,z_5\right)}}\hfill \end{array}\right\}\right)\hfill \\ & =& \varphi \left(max\left\{6,5,6,{\displaystyle \frac{0+8}{2}},{\displaystyle \frac{5\times 6}{1+6}}\right\}\right)=\varphi \left(6\right)=\frac{48}{9}.\hfill \end{array}$$

If $x=z_2$ and $z=z_5$, then

$$\begin{array}{ccc}\hfill H^{+}\left(S{z}_2,S{z}_5\right)& =& p\left(z_1,z_3\right)=5\le \varphi \left(\mathcal{M}\left(z_2,z_5\right)\right)\hfill \\ & =& \varphi \left(max\left\{\begin{array}{c}p\left(z_2,z_5\right),\mathcal{D}\left(z_2,S{z}_2\right),\mathcal{D}\left(z_5,S{z}_5\right),\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_2,S{z}_5\right)+\mathcal{D}\left(z_5,S{z}_2\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_2,S{z}_2\right)\mathcal{D}\left(z_5,S{z}_5\right)}{1+p\left(z_2,z_5\right)}}\hfill \end{array}\right\}\right)\hfill \\ & =& \varphi \left(max\left\{6,5,6,{\displaystyle \frac{8+8}{2}},{\displaystyle \frac{5\times 6}{1+6}}\right\}\right)=\varphi \left(8\right)=7.11.\hfill \end{array}$$

If $x=z_3$ and $z=z_4$, then

$$\begin{array}{ccc}\hfill H^{+}\left(S{z}_3,S{z}_4\right)& =& p\left(z_1,z_2\right)=5\le \varphi \left(\mathcal{M}\left(z_3,z_4\right)\right)\hfill \\ & =& \varphi \left(max\left\{\begin{array}{c}p\left(z_3,z_4\right),\mathcal{D}\left(z_3,S{z}_3\right),\mathcal{D}\left(z_4,S{z}_4\right),\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_3,S{z}_4\right)+\mathcal{D}\left(z_4,S{z}_3\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_3,S{z}_3\right)\mathcal{D}\left(z_4,S{z}_4\right)}{1+p\left(z_3,z_4\right)}}\hfill \end{array}\right\}\right)\hfill \\ & =& \varphi \left(max\left\{6,5,6,{\displaystyle \frac{8+8}{2}},{\displaystyle \frac{5\times 6}{1+6}}\right\}\right)=\varphi \left(8\right)=7.11.\hfill \end{array}$$

If $x=z_4$ and $z=z_5$, then

$$\begin{array}{ccc}\hfill H^{+}\left(S{z}_4,S{z}_5\right)& =& p\left(z_2,z_3\right)=8\le \varphi \left(\mathcal{M}\left(z_4,z_5\right)\right)\hfill \\ & =& \varphi \left(max\left\{\begin{array}{c}p\left(z_4,z_5\right),\mathcal{D}\left(z_4,S{z}_4\right),\mathcal{D}\left(z_5,S{z}_5\right),\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_4,S{z}_5\right)+\mathcal{D}\left(z_5,S{z}_4\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_4,S{z}_4\right)\mathcal{D}\left(z_5,S{z}_5\right)}{1+p\left(z_4,z_5\right)}}\hfill \end{array}\right\}\right)\hfill \\ & =& \varphi \left(max\left\{3,6,6,{\displaystyle \frac{6+6}{2}},{\displaystyle \frac{6\times 6}{1+3}}\right\}\right)=\varphi \left(9\right)=8.\hfill \end{array}$$

Hence, using Theorem 6, S is fixed.

Remark 2.

In the above example, please note that

$${\displaystyle \frac{1}{2}}\mathcal{D}\left(z_4,S{z}_4\right)=3=p\left(z_4,z_5\right),$$

but

$$\begin{array}{ccc}\hfill H\left(S{z}_4,S{z}_5\right)& =& p\left(z_2,z_3\right)=8\nleqq \varphi \left(6\right)\hfill \\ & =& \varphi \left(max\left\{3,6,6,{\displaystyle \frac{6+6}{2}}\right\}\right)\hfill \\ & =& \varphi \left(max\left\{\begin{array}{c}p\left(z_4,z_5\right),\mathcal{D}\left(z_4,S{z}_4\right),\mathcal{D}\left(z_5,S{z}_5\right),\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_4,S{z}_5\right)+\mathcal{D}\left(z_5,S{z}_4\right)}{2}}\hfill \end{array}\right\}\right)\hfill \\ & =& \varphi \left(\mathcal{M}\left(z_4,z_5\right)\right)\hfill \end{array}$$

for any $\varphi \in \mathsf{\Phi }.$ Hence, Theorems 2 and 3 are not applicable in this example.

In the following, we present the result for metric $H^{+}.$

Theorem 7.

If $\left(U,p\right)$ is a complete metric space, $S:U\to P_c\left(U\right)$ is mapping that satisfies

$${\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right),\mathit{which}\mathit{implies}{\alpha }_{*}\left(Sx,Sy\right)H^{+}\left(Sx,Sy\right)\le \varphi \left(\mathcal{N}\left(x,y\right)\right)$$

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }.$ Mapping S is ${\alpha }_{*}$-admissible; therefore, $z_0\in U$ and $z_1\in S{z}_0$ exist, such that $\alpha \left(z_0,z_1\right)\ge 1.$ Further, for $x\in U$ and $y\in Sx,$

$$\mathcal{D}\left(y,Sy\right)\le H^{+}\left(Sy,Sx\right).$$

Moreover, (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q,$ or (b) S is continuous. Then, $F\left(S\right)$ is nonempty.

Proof. 

As in the proof of Theorem 4, we obtain a Cauchy sequence $\left\{z_q\right\}$ in U, which satisfies $z_{q+1}\in S{z}_q,$ $z_{q+1}\ne z_q$ such that $z_q$ converges to some $z\in U$. Assume there is no fixed point of $S.$ That is, $\mathcal{D}\left(z,Sz\right)>0.$ Again, along similar lines to the proof of Theorem 4, a subsequence $\left\{z_{q_k}\right\}$ of $\left\{z_q\right\}$ exists, such that

$$\frac{1}{2}\mathcal{D}\left(z_{q_k},S{z}_{q_k}\right)\le p\left(z_{q_k},z\right).$$

Assume (a) holds; therefore, $\alpha \left(z_q,z_{q+1}\right)\ge 1$ and $z_q\to z$ as $q\to \infty$, which implies $\alpha \left(z_{q_k},z\right)\ge 1$ for all $k\in N.$ As mapping S is ${\alpha }_{*}$-admissible, ${\alpha }_{*}\left(S{z}_{q_k},Sz\right)\ge 1$. Consequently, we obtain

$$\begin{array}{ccc}\hfill H\left(S{z}_{q_k},Sz\right)& \le & 2H^{+}\left(S{z}_{q_k},Sz\right)\le 2{\alpha }_{*}\left(S{z}_{q_k},Sz\right)H^{+}\left(S{z}_{q_k},Sz\right)\hfill \\ & \le & 2\varphi \left(\mathcal{N}\left(z_q,z\right)\right)\hfill \\ & =& 2\varphi \left(max\left\{\begin{array}{c}p\left(z_{q_k},z\right),\mathcal{D}\left(z_{q_k},S{z}_{q_k}\right),{\displaystyle \frac{\mathcal{D}\left(z,Sz\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_{q_k},Sz\right)+\mathcal{D}\left(z,S{z}_{q_k}\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(z_{q_k},S{z}_{q_k}\right)\mathcal{D}\left(z,Sz\right)}{1+p\left(z_{q_k},z\right)}}\hfill \end{array}\right\}\right).\hfill \end{array}$$

Applying limit $q\to \infty$, we also obtain the following:

$$\begin{array}{c}\mathcal{D}\left(z,Sz\right)\le 2\varphi \left(\frac{\mathcal{D}\left(z,Sz\right)}{2}\right)\\ <\mathcal{D}\left(z,Sz\right),\end{array}$$

which provides a contradiction, so we obtain $z\in Sz.$ As in the proof of Theorem 4, the conclusion holds whenever (b) holds. □

Corollary 8.

If $\left(U,p\right)$ is a complete metric space and $S:U\to P_c\left(U\right)$ is mapping that satisfies

$${\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right),\mathit{this}\mathit{implies}{\alpha }_{*}\left(Sx,Sy\right)H\left(Sx,Sy\right)\le \varphi \left(\mathcal{N}\left(x,y\right)\right)$$

for every $x,y\in U$ and for some $\varphi \in \mathsf{\Phi }.$ Further, S is ${\alpha }_{*}$-admissible; therefore, $z_0\in U$ and $z_1\in S{z}_0$ exist, such that $\alpha \left(z_0,z_1\right)\ge 1.$ Moreover, (a) if there is a sequence $\left\{z_q\right\}$ in U, such that $\alpha \left(z_q,z_{q+1}\right)\ge 1$ for all $q\in \mathbb{N}$ and $z_q\to z$ when $q\to \infty ,$ this implies $\alpha \left(z_q,z\right)\ge 1$ for all $q,$ or (b) S is continuous. Then, $F\left(S\right)$ is nonempty.

Proof. 

As $H^{+}(V,W)\le H(V,W)$, the result follows Theorem 7. □

3. Well-Posedness of the FPP of Multifunctions

Now, we discuss some results about the well-posedness of the FPP.

Theorem 8.

Let $\left(U,p\right)$ be a complete metric space, $H_1$ an SSE-H, and $Y\in P_c\left(U\right),$ $S:Y\to P_c\left(U\right)$ satisfies,

(i) 

$F^{*}\left(S\right)\ne \varnothing$;

(ii) 

for all $x,y\in Y,$

$${\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right)\mathit{implies}{\alpha }_{*}\left(Sx,Sy\right)H_1\left(Sx,Sy\right)\le \varphi \left(\mathcal{M}\left(x,y\right)\right);$$

(iii) 

for all $x,z\in Y$ and $y\in Sx$,

$$\mathcal{D}\left(y,Sz\right)\le H_1\left(Sz,Sx\right);$$

(iv) 

$\alpha \left(x,y\right)\ge 1$ and ${\alpha }_{*}\left(Sy,S{x}_q\right)\ge 1$ for all $x,y\in F\left(S\right)$ and for any sequence $\left\{x_q\right\}$ in Y, satisfying $\underset{q\to \infty }{lim}\mathcal{D}\left(x_q,S{x}_q\right)=0$,

then

(a) 

$y^{*}\in U$, such that $F^{*}\left(S\right)=F\left(S\right)=\left\{y^{*}\right\};$

(b) 

the given FPP is well posed w.r.t. $\mathcal{D}$ and generalized well posed w.r.t. $\mathcal{D}$ as well;

(c) 

the given FPP is well posed w.r.t. $H_1$ and generalized well posed w.r.t. $H_1$ as well.

Proof. 

(a) Let $y^{*}\in F^{*}\left(S\right).$ We claim that $y^{*}=y$ for all $y\in F\left(S\right)$. On the contrary, if $z^{*}\in F\left(S\right)$, such that $y^{*}\ne z^{*},$ then

$${\displaystyle \frac{1}{2}}\mathcal{D}\left(y^{*},S{y}^{*}\right)={\displaystyle \frac{1}{2}}p\left(y^{*},y^{*}\right)=0\le p\left(y^{*},z^{*}\right)$$

and implies

$$\begin{array}{ccc}\hfill p\left(y^{*},z^{*}\right)& =& \mathcal{D}\left(S{y}^{*},z^{*}\right)=\mathcal{D}\left(z^{*},S{y}^{*}\right)\le H_1\left(S{y}^{*},S{z}^{*}\right)\hfill \\ & \le & {\alpha }_{*}\left(S{y}^{*},S{z}^{*}\right)H_1\left(S{y}^{*},S{z}^{*}\right)\hfill \\ & \le & \varphi \left({\mathcal{M}}^{*}\left(y^{*},z^{*}\right)\right)\hfill \\ & =& \varphi \left(max\left\{\begin{array}{c}p\left(y^{*},z^{*}\right),\mathcal{D}\left(y^{*},S{y}^{*}\right),\mathcal{D}\left(z^{*},S{z}^{*}\right)\hfill \\ {\displaystyle \frac{\mathcal{D}\left(y^{*},S{z}^{*}\right)+\mathcal{D}\left(z^{*},S{y}^{*}\right)}{2}},\hfill \\ {\displaystyle \frac{\mathcal{D}\left(y^{*},S{y}^{*}\right)\mathcal{D}\left(z^{*},S{z}^{*}\right)}{1+\mathcal{D}\left(y^{*},z^{*}\right)}}\hfill \end{array}\right\}\right)\hfill \\ & =& \varphi \left(p\left(y^{*},z^{*}\right)\right)<p\left(y^{*},z^{*}\right)\hfill \end{array}$$

is a contradiction. Hence, $y^{*}=y$ for all $y\in F\left(S\right).$ Consequently,

$$F\left(S\right)=F^{*}\left(S\right)=\left\{y^{*}\right\}.$$

(b) From (a), we obtain $F\left(S\right)=F^{*}\left(S\right)=\left\{y^{*}\right\}.$ Suppose $\underset{q\to \infty }{lim}\mathcal{D}\left(x_q,S{x}_q\right)=0.$ As

$$\frac{1}{2}\mathcal{D}\left(y^{*},S{y}^{*}\right)=0\le p\left(y^{*},x_q\right)$$

for all $x\in S{x}_q,$ we obtain

$$\begin{array}{ccc}\hfill p\left(x_q,y^{*}\right)& \le & p\left(x_q,x\right)+p\left(x,y^{*}\right)\hfill \\ & =& p\left(x_q,x\right)+\mathcal{D}\left(S{y}^{*},x\right)\hfill \\ & \le & p\left(x_q,x\right)+H_1\left(S{y}^{*},S{x}_q\right)\hfill \\ & \le & p\left(x_q,x\right)+{\alpha }_{*}\left(S{y}^{*},S{x}_q\right)H_1\left(S{y}^{*},S{x}_q\right),\hfill \end{array}$$

which implies

$$\begin{array}{ccc}\hfill p\left(x_q,y^{*}\right)& \le & \underset{x\in S{x}_q}{inf}p\left(x_q,x\right)+{\alpha }_{*}\left(S{y}^{*},S{x}_q\right)H_1\left(S{y}^{*},S{x}_q\right)\hfill \\ & =& \mathcal{D}\left(x_q,S{x}_q\right)+\varphi \left({\mathcal{M}}^{*}\left(y^{*},x_q\right)\right)\hfill \\ & \le & \mathcal{D}\left(x_q,S{x}_q\right)+\varphi \left(max\left\{\begin{array}{c}p\left(x_q,y^{*}\right),\mathcal{D}\left(y^{*},S{y}^{*}\right),\mathcal{D}\left(x_q,S{x}_q\right),\hfill \\ {\displaystyle \frac{\mathcal{D}\left(y^{*},S{x}_q\right)+\mathcal{D}\left(x_q,S{y}^{*}\right)}{2}}\hfill \\ {\displaystyle \frac{\mathcal{D}\left(x_q,S{x}_q\right)\mathcal{D}\left(y^{*},S{y}^{*}\right)}{1+\mathcal{D}\left(x_q,y^{*}\right)}}\hfill \end{array}\right\}\right)\hfill \\ & =& \mathcal{D}\left(x_q,S{x}_q\right)+\varphi \left(max\left\{\begin{array}{c}p\left(x_q,y^{*}\right),\mathcal{D}\left(x_q,S{x}_q\right),\hfill \\ {\displaystyle \frac{p\left(y^{*},x_q\right)+\mathcal{D}\left(x_q,S{x}_q\right)+p\left(x_q,y^{*}\right)}{2}}\hfill \end{array}\right\}\right)\hfill \\ & =& \mathcal{D}\left(x_q,S{x}_q\right)+\varphi \left(max\left\{\begin{array}{c}p\left(x_q,y^{*}\right),\mathcal{D}\left(x_q,S{x}_q\right),\hfill \\ p\left(x_q,y^{*}\right)+{\displaystyle \frac{\mathcal{D}\left(x_q,S{x}_q\right)}{2}}\hfill \end{array}\right\}\right),\hfill \end{array}$$

which further implies

$$\begin{array}{ccc}\hfill \underset{q\to \infty }{limsup}p\left(x_q,y^{*}\right)& \le & \underset{q\to \infty }{lim\; sup}\varphi \left(max\left\{\begin{array}{c}p\left(x_q,y^{*}\right),\mathcal{D}\left(x_q,S{x}_q\right),\hfill \\ p\left(x_q,y^{*}\right)+{\displaystyle \frac{\mathcal{D}\left(x_q,S{x}_q\right)}{2}}\hfill \end{array}\right\}\right)\hfill \\ & =& \varphi \left(\underset{q\to \infty }{lim\; sup}p\left(x_q,y^{*}\right)\right).\hfill \end{array}$$

Since $\varphi \in \mathsf{\Phi }$; therefore,

$$\underset{q\to \infty }{lim\; sup}p\left(x_q,y^{*}\right)=0,$$

hence $\underset{q\to \infty }{lim}p\left(x_q,y^{*}\right)=0$. Consequently,

$$\underset{q\to \infty }{lim}{x}_q=y^{*}\in F\left(S\right),$$

which implies that the FPP is well-posed w.r.t. $\mathcal{D}$, and particularly, we obtain the generalized well-posedness w.r.t. $\mathcal{D}$ of FPP as well. (c) From (a), we obtain $F\left(S\right)=F^{*}\left(S\right)=\left\{x^{*}\right\}.$ Assume that $\underset{q\to \infty }{lim}{H}_1\left(\left\{x_q\right\},S{x}_q\right)=0.$ As $H_1$ is SSE-H, we obtain

$$\underset{q\to \infty }{lim}H\left(x_q,S{x}_q\right)=0.$$

Consequently, we obtain

$$\underset{q\to \infty }{lim}\mathcal{D}\left(x_q,S{x}_q\right)=0.$$

Along similar lines as in part (b), we obtain

$$\underset{q\to \infty }{lim}{x}_q=y^{*}\in F\left(S\right).$$

Hence, the FPP is well-posed w.r.t. $H_1$, and particularly, we obtain the generalized well-posedness w.r.t. $H_1$ of the FPP as well. □

Theorem 9.

If $\left(U,p\right)$ is a complete metric space, $H_1$ is an SSE-H, and $Y\in P_c\left(U\right),$ $S:Y\to P_c\left(U\right)$ satisfies the following:

(i) 

$F^{*}\left(S\right)\ne \varnothing$;

(ii) 

for all $x,y\in Y,$

$${\displaystyle \frac{1}{2}}\mathcal{D}\left(x,Sx\right)\le p\left(x,y\right)\mathit{implies}{\alpha }_{*}\left(Sx,Sy\right)H_1\left(Sx,Sy\right)\le \varphi \left({\mathcal{M}}^{*}\left(x,y\right)\right);$$

(iii) 

for all $x,z\in Y$ and $y\in Sx$,

$$\mathcal{D}\left(y,Sz\right)\le H_1\left(Sz,Sx\right);$$

(iv) 

$\alpha \left(x,y\right)\ge 1$, ${\alpha }_{*}\left(Sy,S{x}_q\right)\ge 1$ for all $x,y\in F\left(S\right)$ and for any sequence $\left\{x_q\right\}$ in Y satisfying $\underset{q\to \infty }{lim}\mathcal{D}\left(x_q,S{x}_q\right)=0$,

then

(a) 

$y^{*}\in U$ exists, such that $F^{*}\left(S\right)=F\left(S\right)=\left\{y^{*}\right\};$

(b) 

the given FPP is well posed w.r.t. $\mathcal{D}$ and generalized well posed w.r.t. $\mathcal{D}$ as well;

(c) 

the given FPP is well posed w.r.t. $H_1$ and generalized well posed w.r.t. $H_1$ as well.

Proof. 

As

$${\mathcal{M}}^{*}\left(x,y\right)\le \mathcal{M}\left(x,y\right),$$

the result follows from Theorem 8. □

4. Conclusions

In this paper, classes of metrics which are semi-sequentially equivalent to the Hausdorff metric (SSE-H) were considered. An example of a metric was provided for which the Hausdorff metric was not SSE. Further, the problem of the existence of fixed points of multifunctions of metrics, which are SSE-H, was solved. An example has been provided to show that the obtained results properly generalize some existing results in the literature. Moreover, we established the well-posedness of fixed point problems of multifunctions of SSE-H metrics. In the future, we intend to extend the exploration in the context of quasi-metric spaces and asymmetric normed spaces. As asymmetry has become an essential part of many natural phenomena, asymmetric distance structures can play important roles in solving problems in the context of asymmetry.