The first result of this section is the following one:
Theorem 3
Let
\(\left({X, {\varvec{F}}, \varDelta_{M}} \right)\)
be a complete Menger space and let
\(\{T_{n}\}\)
be a sequence of operators such that:
-
1.
The operators
\(T_{n} :X \to X\); \(\forall n \in {\varvec{Z}}_{0+}\)
of the sequence {T
n
} are all strict
k-contractions for some real constant
\(k \in (0,1)\),
-
2.
\(\{T_{n}\} \to T\)
for some
\(T :X \to X\).
Then, \(T: X \to X\) is a strict k-contraction and \(\left\{{x_{n}^{*}} \right\} \to x^{*}\) a.s., where \(F_{{T_{n}}} = \left\{{x_{n}^{*}} \right\}\); \(\forall n \in {\varvec{Z}}_{+}\), and F
T
= {x
*}. Furthermore, \(\left\{{x_{n}} \right\} \to x^{*}\) a.s., where x
n+1 = T
n
x
n
; \(\forall n \in {\varvec{Z}}_{0+}\) for any given x
0 ∈ X.
Proof
We have that \(F_{{T_{n} x, T_{n} y}} (t) \ge F_{x,y} \left({k^{-1} t} \right)\); \(\forall n \in {\varvec{Z}}_{0+},\forall t \in {\varvec{R}}_{+}\) for any \(x, y \in X\), and
$$\begin{aligned} F_{Tx,Ty} (t) & \ge \varDelta_{M} \left({F_{{Tx, T_{n} x}} (t/2), F_{{T_{n} x, Ty}} (t/2)} \right) \hfill \\ & \ge \varDelta_{M} \left({F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({F_{{T_{n} x, T_{n} y}} (t/4), F_{{T_{n} y, Ty}} (t/4)} \right)} \right) \hfill \\ & \ge \varDelta_{M} \left({F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({F_{x,y} \left({k^{-1} t/4} \right), F_{{T_{n} y, Ty}} (t/4)} \right)} \right);\quad \forall n \in {\varvec{Z}}_{0+},\; \forall t \in {\varvec{R}}_{+} \hfill \\ \end{aligned}$$
(15)
Thus, since \(\{T_{n}\} \to T\), \(T_{n} : X \to X\) are strict k-contractions, then everywhere continuous, and \(\varDelta_{M} : [0,1] \times [0,1] \to [0,1]\) is a continuous triangular norm,
$$\begin{aligned} F_{Tx,Ty} (t) & \ge \varDelta_{M} \left({\mathop{lim \; inf}\limits_{n \to \infty} F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({F_{x,y} \left({k^{-1} t/4} \right), \mathop{lim \; inf}\limits_{n \to \infty} F_{{T_{n} y, Ty}} (t/4)} \right)} \right) \hfill \\ &\ge \varDelta_{M} \left({\mathop{lim \; inf}\limits_{n \to \infty} F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({F_{x,y} \left({k^{-1} t/4} \right), \mathop{lim \; inf}\limits_{n \to \infty} F_{{T_{n} y, Ty}} (t/4)} \right)} \right) \hfill \\ &= \varDelta_{M} \left({1, \varDelta_{M} \left({F_{x,y} \left({k^{-1} t/4} \right), 1} \right)} \right) \hfill \\ &= \varDelta_{M} \left({F_{x,y} \left({k^{-1} t/4} \right), 1} \right) \hfill \\ &= F_{x,y} \left({k^{-1} t/4} \right);\quad \forall n \in {\varvec{Z}}_{0+},\;\forall t \in {\varvec{R}}_{+},\;\forall x, y \in X \hfill \\ \end{aligned}$$
(16)
so that
$$F_{{T^{n} x, T^{n} y}} (t) \ge F_{x,y} \left({k^{- n} t/4} \right);\forall n \in {\varvec{Z}}_{0+},\quad \forall t \in {\varvec{R}}_{+},\; \forall x, y \in X$$
(17)
$$\mathop {lim}\limits_{n \to \infty} F_{{T^{n} x, T^{n} y}} (t) = 1;\quad \forall t \in {\varvec{R}}_{+},\; \forall x, y \in X$$
(18)
Eq. (17), leading to (18), establishes that \(T: X \to X\) is a strict k-contraction. It has to be proved that it has a unique fixed point x
*. Assume on the contrary that there are two \(x^{*}, \bar{x}^{*} \left({\ne x^{*}} \right) \in F(T)\) so that
$$F_{{\bar{x}^{*}, x^{*}}} (t) = F_{{T^{n} \bar{x}^{*}, T^{n} x^{*}}} (t) \ge F_{{\bar{x}^{*}, x^{*}}} \left({k^{- n} t} \right);\quad \forall n \in {\varvec{Z}}_{0+},\; \forall t \in {\varvec{R}}_{+}$$
$$F_{{\bar{x}^{*}, x^{*}}} (t) = \mathop {lim}\limits_{n \to \infty} F_{{T^{n} \bar{x}^{*}, T^{n} x^{*}}} (t) \ge \mathop {lim}\limits_{n \to \infty} F_{{\bar{x}^{*}, x^{*}}} \left({k^{- n} t} \right) = 1;\quad \forall t \in {\varvec{R}}_{+}$$
and then \(\bar{x}^{*} = x^{*}\) by the property 1 of (1) of the PM space \(\left({X,{\varvec{F}}} \right)\).On the other hand, one gets by taking y = y(x) = Tx and z
n
= z
n
(x) = T
n
x for any given x ∈ X that
$$\mathop {lim}\limits_{n \to \infty} F_{{T^{n + 1} x, T^{n} x}} (t) = \mathop {lim}\limits_{n \to \infty} F_{{Tz_{n}, z_{n}}} (t) = 1;\quad \forall t \in {\varvec{R}}_{+},\; \forall x \in X$$
and, for any given \(\lambda \in (0,1)\) and \(t \in {\varvec{R}}_{+}\), there is \(N_{0} = N_{0} \left({\varepsilon, \lambda} \right) \in {\varvec{Z}}_{0+}\) such that \(F_{{z_{n + 1}, z_{n}}} (t) > 1 - \lambda; \forall n\left({\in {\varvec{Z}}_{0+}} \right) \ge N_{0}, \forall t \in {\varvec{R}}_{+}\) so that \(\left\{{z_{n}} \right\}\) is a Cauchy sequence which converges to some limit point z
* = z
*(x) ∈ X since \(\left({X, {\varvec{F}}, \varDelta_{M}} \right)\) is complete. Since the fixed point x
*of \(T: X \to X\) is unique, any limit point z
* = z
*(x) ∈ X of any sequence \(\left\{{T^{n} x} \right\}\) for any arbitrary initial point x ∈ X is the fixed point x
* of \(T: X \to X\). It remains to prove that \(\left\{{x_{n}} \right\} \to x^{*}\) a.s., where x
n+1 = T
n
x
n
;\(\forall n \in {\varvec{Z}}_{0+}\), for any given initial point x
0 ∈ X. Note that
$$\begin{aligned} & F_{{x_{n + 1}, x^{*}}} (t) \\ & \quad \ge \varDelta_{M} \left({F_{{x_{n + 1}, T_{n + 1}^{m} x_{n + 1}}} (t/2), \varDelta_{M} \left({F_{{T_{n + 1}^{m} x_{n + 1}, T^{m} x_{n + 1}^{*}}} (t/4), F_{{T^{m} x_{n + 1}^{*}, T^{m} x^{*}}} (t/4)} \right)} \right) \\ &\quad \ge \varDelta_{M} \left({F_{{x_{n + 1}, T_{n + 1}^{m} x_{n + 1}}} (t/2), \varDelta_{M} \left({\varDelta_{M} \left({F_{{T_{n + 1}^{m} x_{n + 1}, T_{n + 1}^{m} x_{n + 1}^{*}}} (t/8), F_{{T^{m} x_{n + 1}^{*},T_{n + 1}^{m} x_{n + 1}^{*}}} (t/8)} \right), F_{{T^{m} x_{n + 1}^{*}, T^{m} x^{*}}} (t/4)} \right)} \right) \\ &\quad \ge \varDelta_{M} \left({F_{{x_{n + 1}, T_{n + 1}^{m} x_{n + 1}}} (t/2), \varDelta_{M} \left({\varDelta_{M} \left({F_{{x_{n + 1}, x_{n + 1}^{*}}} \left({k^{- m} t/8} \right), F_{{T^{m} x_{n + 1}^{*},T_{n + 1}^{m} x_{n + 1}^{*}}} (t/8)} \right), F_{{x_{n + 1}^{*}, x^{*}}} \left({k^{- m} t/4} \right)} \right)} \right); \\ & \qquad \forall n \in {\varvec{Z}}_{0+},\; \forall m \in {\varvec{Z}}_{+},\; \forall t \in {\varvec{R}}_{+} \\ \end{aligned}$$
(19)
Note also that
$$\begin{aligned}F_{{T^{m} x_{n + 1}^{*},T_{n + 1}^{m} x_{n + 1}^{*}}} (t) & = F_{{T^{p} \left({T^{m - p} x_{n + 1}^{*}} \right), T_{n}^{p} \left({T_{n + 1}^{m - p} x_{n + 1}^{*}} \right)}} (t) \\ & = F_{{T\left({T^{m - 1} x_{n + 1}^{*}} \right), T_{n} \left({T_{n + 1}^{m - 1} x_{n + 1}^{*}} \right)}} (t) \end{aligned}$$
(20)
for any \(p,n, \left({m \ge p} \right) \in {\varvec{Z}}_{0+}, \forall t \in {\varvec{R}}_{+}\) and, since \(T_{n} :X \to X; \forall n \in {\varvec{Z}}_{0+}\) and \(T :X \to X\) are all strict k-contractions, they are also continuous so that if \(\{T_{n}\} \to T\), then \(\left\{{T_{n} x} \right\} \to Tx\) a.s. and \(lim_{n \to \infty} F_{{T_{n} x, Tx}} (t) = 1; \forall t \in {\varvec{R}}_{+}\) as n → ∞ for any given x ∈ X. Then, \(\left\{{T_{n}^{p} \left({T_{n + 1}^{m - p} x_{n + 1}^{*}} \right)} \right\} \to T^{p} \left({T^{m - p} x_{n + 1}^{*}} \right)\) a.s. as m → ∞, \(\forall n \in {\varvec{Z}}_{0+}\) since \(T_{n} :X \to X; \forall n \in {\varvec{Z}}_{0+}\) and \(T :X \to X\) are strict k-contractions. This implies from (20) that
$$\mathop {lim}\limits_{m \to \infty} F_{{T^{m} x_{n + 1}^{*}, T_{n + 1}^{m} x_{n + 1}^{*}}} (t) = 1;\quad \forall t \in {\varvec{R}}_{+},\; \forall n \in {\varvec{Z}}_{0+}$$
(21)
On the other hand,
$$\left\{{T_{n}^{m} x_{n}} \right\} \to x_{n}^{*} \; {\rm a.s.\;as\;}m \to \infty;\quad \forall n \in {\varvec{Z}}_{0+}$$
(22)
since \(\{T_{n}\}\) is a sequence of strict k-contractive operators with \(F_{{T_{n}}} = \left\{{x_{n}^{*}} \right\}; \forall n \in {\varvec{Z}}_{0+}\). Furthermore, one has:
$$\mathop {lim}\limits_{m \to \infty} F_{{x_{n + 1}, x_{n + 1}^{*}}} \left({k^{- m} t} \right) = \mathop {lim}\limits_{m \to \infty} F_{{x_{n + 1}^{*}, x^{*}}} \left({k^{- m} t/4} \right) = 1;\quad \forall t \in {\varvec{R}}_{+}$$
(23)
In a similar way to (19), we get:
$$\begin{aligned} F_{{x_{n + 1}^{*}, x^{*}}} (t) & = F_{{T_{n + 1}^{m} x_{n + 1}^{*}, T^{m} x^{*}}} (t) \\ & \ge \varDelta_{M} \left({F_{{T^{m} x_{n + 1}^{*}, x^{*}}} (t/2), F_{{T^{m} x_{n + 1}^{*}, T_{n + 1}^{m} x_{n + 1}^{*}}} (t/2)} \right) \hfill \\ & \ge \varDelta_{M} \left({F_{{x_{n + 1}^{*}, x^{*}}} \left({k^{- m} t/2} \right), F_{{T^{m} x_{n + 1}^{*}, T_{n + 1}^{m} x_{n + 1}^{*}}} (t/2)} \right);\quad \forall m \in {\varvec{Z}}_{0+}, \; \forall t \in {\varvec{R}}_{+} \hfill \\ \end{aligned}$$
(24)
and taking limits as m → ∞, one gets that \(lim_{n \to \infty} F_{{T^{m} x_{n + 1}^{*}, T_{n + 1}^{m} x_{n + 1}^{*}}} (t/2) = lim_{n \to \infty} F_{{x_{n + 1}^{*}, x^{*}}} (t) = 1; \forall t \in {\varvec{R}}_{+}\) so that \(\left\{{x_{n}^{*}} \right\} \to x^{*}\) a.s.. On the other hand, the use of (21)–(23) in (19) as well as \(lim_{n \to \infty} F_{{x_{n + 1}^{*}, x^{*}}} (t) = 1; \forall t \in {\varvec{R}}_{+}\) got from (24) yields for all \(t \in {\varvec{R}}_{+}\), since \(\varDelta_{M} : [0,1] \times [0,1] \to [0,1]\) is a continuous triangular norm,
$$\begin{aligned} & \mathop{lim\;inf}\limits_{n \to \infty} F_{{x_{n + 1}, x^{*}}} (t) \hfill \\ & \quad \ge \mathop{lim\;inf}\limits_{n, m \to \infty} \varDelta_{M} \left({F_{{x_{n + 1}, T_{n + 1}^{m} x_{n + 1}}} (t/2), \varDelta_{M} \left({\varDelta_{M} \left({F_{{x_{n + 1}, x_{n + 1}^{*}}} \left({k^{- m} t/8} \right), F_{{T^{m} x_{n + 1}^{*},T_{n + 1}^{m} x_{n + 1}^{*}}} (t/8)} \right), F_{{x_{n + 1}^{*}, x^{*}}} \left({k^{- m} t/4} \right)} \right)} \right) \hfill \\ &\quad \ge \varDelta_{M} \left({\mathop {lim}\limits_{n, m \to \infty} F_{{x_{n + 1}, T_{n + 1}^{m} x_{n + 1}}} (t/2), \varDelta_{M} \left({\varDelta_{M} \left({\mathop {lim}\limits_{m \to \infty} F_{{x_{n + 1}, x_{n + 1}^{*}}} \left({k^{- m} t/8} \right), \mathop {lim}\limits_{m \to \infty} F_{{T^{m} x_{n + 1}^{*},T_{n + 1}^{m} x_{n + 1}^{*}}} (t/8)} \right), \mathop {lim}\limits_{m \to \infty} F_{{x_{n + 1}^{*}, x^{*}}} \left({k^{- m} t/4} \right)} \right)} \right) \hfill \\ &\quad \ge \varDelta_{M} \left({1, \varDelta_{M} \left({\varDelta_{M} \left({1, 1} \right), 1} \right)} \right) = 1 \hfill \\ \end{aligned}$$
(25)
so that \(\exists lim_{n \to \infty} F_{{x_{n}, x^{*}}} (t) = 1\); \(\forall t \in {\varvec{R}}_{+}\) and \(\left\{{x_{n}} \right\} \to x^{*}\) a.s..\(\square\)
As it occurs in the deterministic counterpart, (Berinde 2007), the uniform convergence of a sequence of operators \(\{T_{n}\}\) can be weakened if such operators possess certain additional contractive properties. See also Istratescu (1981). In this case, it is possible to get some close properties to those proved in “Main results concerning the uniform convergence of operators” section for the case of uniform convergence. Firstly, two definitions which are then used follow below:
Definition 2
(Berinde 2007) A non-decreasing function \(\varphi : {\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\) (i.e. \(\varphi\)(t
1) ≤ \(\varphi\)(t
2) if t
1 ≤ t
2) is said to be a comparison function if \(\left\{{\varphi^{n} (t)} \right\} \to 0, \forall t \in {\varvec{R}}_{+}\). If, furthermore, \(\left({t - \varphi (t)} \right) \to + \infty\) as t → + ∞ then it is said to be a strict comparison function.
Example 2
Note that \(\varphi (t) = \frac{\lambda (t) t}{1 + \lambda (t) t}\) for \(t \in {\varvec{R}}_{0+}\) with \(\lambda : {\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\) being such that \(\lambda (t) t\) is non-decreasing is a strict comparison function since it is non-decreasing and \(\varphi^{n} (t) = \varphi \left({\varphi^{n - 1} (t)} \right) = \frac{\lambda (t) t}{1 + n \lambda (t) t}\) for all \(n \in {\varvec{Z}}_{+}\) implying that \(\left\{{\varphi^{n} (t)} \right\} \to 0; \forall t \in {\varvec{R}}_{0+}\) as n → ∞.
Example 3
Let \(\left({X,{\varvec{F}}} \right)\) be a PM-space, let \(T: X \to X\) be a mapping on X and let \(\varphi :X \times X \times {\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\) be defined as \(\varphi_{x,y} (t) = \frac{\lambda (t) \left(F_{x,y}^{- 1} (t) - 1\right)}{1 + \lambda (t) \left(F_{x,y}^{-1} (t) - 1\right)}; \forall x, y \in X, \forall t \in {\varvec{R}}_{0+}\) leading to the n-the composite function with itself resulting to be \(\varphi_{x,y}^{n} (t) = \frac{\lambda (t) \left(F_{x,y}^{-1} (t) - 1\right)}{1 + n\lambda (t) \left(F_{x,y}^{-1} (t) -1\right)}; \forall t \in {\varvec{R}}_{0+}, \forall n \in {\varvec{Z}}\) which satisfies \(\left\{{\varphi^{n} (t)} \right\} \to 0; \forall t \in {\varvec{R}}_{0+}\) as n → ∞. Then, \(\varphi :X \times X \times {\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\) is a strict comparison function for any \(x, y \in X\) provided that λ(0) = 0 and \(\lambda (t) \left({F_{x,y}^{-1} (t) - 1} \right)\) is non-decreasing for all \(t \in {\varvec{R}}_{0+}\) for each pair \(\left({x,y} \right) \in X \times X\). Note, in particular, that \(\varphi\)
x,x
(t) = 0; \(\forall t \in {\varvec{R}}_{0+}\); ∀x ∈ X so that the null-function \(\varphi\) is both non-increasing and non-decreasing.
Definition 3
Let \(\left({X,{\varvec{F}}} \right)\) be a PM-space. Then, \(G : X \to X\) is said to be a strict \(\varphi\)-contraction if \(G_{Tx,Ty}^{-1} (t) \le 1 + \varphi \left({G_{x,y}^{-1} (t) - 1} \right)\), ∀x, y ∈ X,\(\forall t \in {\varvec{R}}_{+}\) for some strict comparison function \(\varphi :{\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\).
The next result follows:
Theorem 4
Let
\(\left({X, {\varvec{F}}, \varDelta_{M}} \right)\)
be a complete Menger space and let
\(\{T_{n}\}\)
be a sequence of operators such that:
-
1.
The operators
\(T_{n} :X \to X\)
of the sequence {T
n
} are all strict
\(\varphi\)
-contractions,
-
2.
\(\{T_{n}\} \to T\)
for some
\(T :X \to X\).
Then, \(T: X \to X\) is a strict \(\varphi\)-contraction and \(\left\{{x_{n}^{*}} \right\} \to x^{*}\) a.s., where \(F_{{T_{n}}} = \left\{{x_{n}^{*}} \right\}; \forall n \in {\varvec{Z}}_{+}\) and F
T
= {x
*}. Furthermore \(\left\{{x_{n}} \right\} \to x^{*}\) a.s., where x
n+1 = T
n
x
n
; \(\forall n \in {\varvec{Z}}_{0+}\) for any given x
0 ∈ X.
Proof
We have \(F_{{T_{n} x,T_{n} y}}^{-1} (t) - 1 \le \varphi \left({F_{x,y}^{-1} (t) - 1} \right)\); ∀x, y ∈ X,\(\forall t \in {\varvec{R}}_{+}, \forall n \in {\varvec{Z}}_{0+}\) for some strict comparison function \(\varphi :{\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\), since all the operators of the sequence {T} are strict
n
\(\varphi\)-contractions, what is equivalent to
$$F_{T_{n} x, T_{n} y} (t) \ge \frac{1}{1 + \varphi \left(F_{x,y}^{- 1} (t) - 1\right)};\quad \forall x, y \in X,\; \forall t \in {\varvec{R}}_{+}, \; \forall n \in {\varvec{Z}}_{0+}$$
and then
$$\begin{aligned} F_{Tx,Ty} (t) & \ge \varDelta_{M} \left({F_{{Tx, T_{n} x}} (t/2), F_{{T_{n} x, Ty}} (t/2)} \right) \hfill \\ &\ge \varDelta_{M} \left({F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({F_{{T_{n} x, T_{n} y}} (t/4), F_{{T_{n} y, Ty}} (t/4)} \right)} \right) \hfill \\ &\ge \varDelta_{M} \left({F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({\frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}}, F_{{T_{n} y, Ty}} (t/4)} \right)} \right);\\ & \quad \forall n \in {\varvec{Z}}_{0+},\;\forall t \in {\varvec{R}}_{+} \hfill \\ \end{aligned}$$
(26)
Thus, since \(\{T_{n}\} \to T\), \(T_{n} : X \to X\) all strict \(\varphi\)-contractions, then everywhere continuous, and \(\varDelta_{M} : [0,1] \times [0,1] \to [0,1]\) is a continuous triangular norm, one gets:
$$\begin{aligned} F_{Tx,Ty} (t) &\ge \mathop{lim\;inf}\limits_{n \to \infty} \varDelta_{M} \left({F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({\frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}}, F_{{T_{n} y, Ty}} (t/4)} \right)} \right) \hfill \\ &\ge \varDelta_{M} \left({\mathop{lim\;inf}\limits_{n \to \infty} F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({\frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}}, \mathop{lim\;inf}\limits_{n \to \infty} F_{{T_{n} y, Ty}} (t/4)} \right)} \right) \hfill \\ &= \varDelta_{M} \left({1, \varDelta_{M} \left({\frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}}, 1} \right)} \right) \hfill \\ &= \varDelta_{M} \left({\frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}}, 1} \right) \hfill \\ &= \frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}};\quad \forall t \in {\varvec{R}}_{+},\; \forall x, y \in X \hfill \\ \end{aligned}$$
(27)
so that
$$\begin{aligned} F_{Tx, Ty}^{-1} (t) & \le 1 + \varphi \left(F_{x, y}^{-1} (t/4) -1 \right) \\ F_{T^{n}x, T^{n} y}^{-1} (t) & \le 1 + \varphi \left(F_{T^{n - 1} x, T^{n - 1}y}^{- 1} (t/4) - 1 \right) \le \cdots \le 1 + \varphi^{n} \left(F_{x, y}^{- 1} \left(t/2^{n + 1} \right) - 1 \right); \\ & \qquad \forall n \in {\varvec{Z}}_{0+}, \; \forall t \in {\varvec{R}}_{+},\; \forall x, y \in X\end{aligned}$$
(28)
since \(lim_{n \to \infty} \varphi^{n} \left({F_{x,y}^{-1} \left({t/2^{n + 1}} \right) - 1} \right) = 0\) since \(\left\{{\varphi^{n} (t)} \right\} \to 0; \forall t \in {\varvec{R}}_{+}\). Then, \(lim_{n \to \infty} F_{x,y}^{-1} \left({t/2^{n + 1}} \right) = 1, \forall t \in {\varvec{R}}_{+}\), ∀x, y ∈ X. Thus, one also has that:
$$\mathop {lim}\limits_{n \to \infty} F_{{T^{n} x, T^{n} y}} (t) = 1;\quad \forall t \in {\varvec{R}}_{+},\; \forall x, y \in X$$
(29)
Eq. (28), leading to (29), establishes that \(T: X \to X\) is a strict \(\varphi\)-contraction. It has to be proved that it has a unique fixed pointx
*. Assume on the contrary that there are two points \(x^{*}, \bar{x}^{*} \left({\ne x^{*}} \right) \in F(T)\) so that
$$F_{{\bar{x}^{*}, x^{*}}} (t) = F_{{T^{n} \bar{x}^{*}, T^{n} x^{*}}} (t) = \mathop {lim}\limits_{n \to \infty} F_{{T^{n} \bar{x}^{*}, T^{n} x^{*}}} (t) \le 1 + \mathop {lim}\limits_{n \to \infty} \varphi^{n} \left({F_{x,y}^{-1} \left({t/2^{n + 1}} \right) - 1} \right) = 1;\quad \forall t \in {\varvec{R}}_{+}$$
(30)
and then \(\bar{x}^{*} = x^{*}\) by the property 1 of (1) of the PM space \(\left({X,{\varvec{F}}} \right)\). On the other hand, one gets from (29) by taking y = y(x) = Tx and z
n
= z
n
(x) = T
n
x for any given x ∈ X that
$$\mathop {lim}\limits_{n \to \infty} F_{{T^{n + 1} x, T^{n} x}} (t) = \mathop {lim}\limits_{n \to \infty} F_{{Tz_{n}, z_{n}}} (t) = 1;\quad \forall t \in {\varvec{R}}_{+},\; \forall x \in X$$
and, for any given \(\lambda \in (0,1)\) and \(t \in {\varvec{R}}_{+}\), there is \(N_{0} = N_{0} \left({\varepsilon, \lambda} \right) \in {\varvec{Z}}_{0+}\) such that \(F_{{z_{n + 1}, z_{n}}} (t) > 1 - \lambda; \forall n\left({\in {\varvec{Z}}_{0+}} \right) \ge N_{0}, \forall t \in {\varvec{R}}_{+}\) so that \(\left\{{z_{n}} \right\}\) is a Cauchy sequence which converges to some limit point z
* = z
*(x) ∈ X since \(\left({X, {\varvec{F}}, \varDelta_{M}} \right)\) is complete. Since the fixed point x
*of \(T: X \to X\) is unique, any limit point z
* = z
*(x) ∈ X of any sequence \(\left\{{T^{n} x} \right\}\) for any arbitrary initial point x ∈ X is the fixed point x
* of \(T: X \to X\). It remains to prove that \(\left\{{x_{n}} \right\} \to x^{*}\) a.s., where x
n+1 = T
n
x
n
; \(\forall n \in {\varvec{Z}}_{0+}\), for any given initial point x
0 ∈ X. Note that, since T
n
is a strict \(\varphi\)-contraction with \(F_{{T_{n}}} = \left\{{x_{n}^{*}} \right\}; \forall n \in {\varvec{Z}}_{0+}\), we can perform the two next replacements to a close set of inequalities to those got in (19) and (24) within the proof of Theorem 1
$$F_{{T_{n + 1}^{m} x_{n + 1}, T_{n + 1}^{m} x_{n + 1}^{*}}} (t) \to \frac{1}{{1 + \varphi^{m} \left({F_{{x_{n + 1}, x_{n + 1}^{*}}}^{-1} (t) - 1} \right)}}$$
$$F_{{x_{n + 1}^{*}, x^{*}}} \left({k^{- m} t} \right) \to \frac{1}{{1 + \varphi^{m} \left({F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t) - 1} \right)}}$$
for all \(t \in {\varvec{R}}_{+}\) and \(m \in {\varvec{Z}}_{0+}\). Thus, one gets instead of (24),
$$\begin{aligned} F_{{x_{n + 1}^{*}, x^{*}}} (t) & = F_{{T_{n + 1}^{m} x_{n + 1}^{*}, T^{m} x^{*}}} (t) \ge \varDelta_{M} \left({F_{{T^{m} x_{n + 1}^{*}, x^{*}}} (t/2), F_{{T^{m} x_{n + 1}^{*}, T_{n + 1}^{m} x_{n + 1}^{*}}} (t/2)} \right) \hfill \\ &\ge \varDelta_{M} \left({\frac{1}{{1 + \varphi^{m} \left({F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t/2) - 1} \right)}}, F_{{T^{m} x_{n + 1}^{*}, T_{n + 1}^{m} x_{n + 1}^{*}}} (t/2)} \right);\quad \forall n,m \in {\varvec{Z}}_{0+},\; \forall t \in {\varvec{R}}_{+} \hfill \\ \end{aligned}$$
(31)
Since \(\left\{{T_{n}^{m}} \right\} \to T^{m}\) then \(lim_{n \to \infty} F_{{T^{m} x_{n + 1}^{*}, T_{n + 1}^{m} x_{n + 1}^{*}}} (t/2) = 1; \forall t \in {\varvec{R}}_{+}; \forall m \in {\varvec{Z}}_{0+}, \forall t \in {\varvec{R}}_{+}\) and the above constraint implies that
$$\mathop{lim\;inf}\limits_{n \to \infty} \left({\frac{1}{{F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t)}} - \frac{1}{{1 + \varphi^{m} \left({F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t/2) - 1} \right)}}} \right) \ge 0;\quad \forall m \in {\varvec{Z}}_{0+},\;\forall t \in {\varvec{R}}_{+}$$
Equivalently, \({lim\;sup}_{n \to \infty} \left({F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t) - 1 - \varphi \left({F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t/2) - 1} \right)} \right) \le 0;\forall t \in {\varvec{R}}_{+}\). Since, furthermore, F
−1(t) is non-increasing and \(\varphi\)(t) is non-decreasing, one has:
$$\begin{aligned} &\mathop {lim\;sup}\limits_{n \to \infty} \left({F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t) - 1 - \varphi \left({F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t) - 1} \right)} \right) \hfill \\ &\quad \le \mathop{lim\;sup}\limits_{n \to \infty} \left({F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t) - 1 - \varphi \left({F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t/2) - 1} \right)} \right) \le 0;\quad \forall t \in {\varvec{R}}_{+}; \hfill \\ \end{aligned}$$
Define \(\sigma_{\eta} = sup \left\{{\sigma \in {\varvec{R}}_{0+} :\sigma - \varphi \left(\sigma \right) \le \eta} \right\}\) which is defined for any given \(\eta \in {\varvec{\bar{R}}}_{0+}\) (the extended nonnegative real semi-line) from the property of strict \(\varphi\)-contractions, since \(\varphi\) is a strict comparison function, \({lim\;sup}_{t \to + \infty} \left({t - \varphi^{m} (t)} \right) \le lim_{t \to + \infty} \left({t - \varphi (t)} \right) = + \infty; \forall n \in {\varvec{Z}}_{0+}\) as t → + ∞. It turns out that \(lim_{\eta \to 0} \sigma_{\eta} = 0\). Now, define \(\sigma_{n} = F_{{x_{n}^{*}, x^{*}}}^{-1} (t) - 1; \forall n \in {\varvec{Z}}_{0+}, \forall t \in {\varvec{R}}_{+}\). Taking into account that
$$\mathop {lim\;sup}\limits_{n \to \infty} \left({F_{{x_{n}^{*}, x^{*}}}^{-1} (t) - 1 - \varphi \left({F_{{x_{n}^{*}, x^{*}}}^{-1} (t) - 1} \right)} \right) \le 0;\quad \forall t \in {\varvec{R}}_{+}$$
(32)
One concludes that \(lim_{\eta \to 0} \sigma_{\eta} = lim_{\eta \to 0} \left({F_{{x_{n}^{*}, x^{*}}}^{-1} (t) - 1} \right) = 0; \forall t \in {\varvec{R}}_{+}\) so that {x
*
n
} → x
* a.s..
Also, one can get, instead of (19) in the proof of Theorem 1, that
$$\begin{aligned} F_{{x_{n + 1}, x^{*}}} (t) & \ge \varDelta_{M} \left({F_{{x_{n + 1}, T_{n + 1}^{m} x_{n + 1}}} (t/2), \varDelta_{M} \left({F_{{T_{n + 1}^{m} x_{n + 1}, T^{m} x_{n + 1}^{*}}} (t/4), F_{{T^{m} x_{n + 1}^{*}, T^{m} x^{*}}} (t/4)} \right)} \right) \\ & \ge \varDelta_{M} \left({F_{{x_{n + 1}, T_{n + 1}^{m} x_{n + 1}}} (t/2), \varDelta_{M} \left({\varDelta_{M} \left({F_{{T^{m} x_{n + 1}^{*},T_{n + 1}^{m} x_{n + 1}^{*}}} (t/8), \frac{1}{{1 + \varphi^{m} \left({F_{{x_{n + 1}, x_{n + 1}^{*}}}^{-1} (t/8) - 1} \right)}}} \right), \frac{1}{{1 + \varphi^{m} \left({F_{{x_{n + 1}^{*}, x^{*}}}^{-1} (t/4) - 1} \right)}}} \right)} \right); \forall n \in {\varvec{Z}}_{0+},\forall m \in {\varvec{Z}}_{+},\; \forall t \in {\varvec{R}}_{+} \\ \end{aligned}$$
and then \(lim_{n \to \infty} \varphi^{n} \left({F_{x,y}^{-1} \left({t/2^{n + 1}} \right) - 1} \right) = 0\) since \(\left\{{\varphi^{n} (t)} \right\} \to 0\); \(\forall t \in {\varvec{R}}_{+}\). Take limits in the above expression by using the continuity of the minimum triangular norm and the fact that \(\varphi\) is a strict \(\varphi\)-comparison function by using close arguments to those used to get (32). We then conclude in a similar way the validity of (21) to (25) by replacing the conditions of k-contractions by conditions of strict \(\varphi\)-contractions so that there exist the limits \(lim_{n \to \infty} F_{{x_{n}^{*}, x^{*}}} (t)\)
\(= lim_{n \to \infty} F_{{x_{n}, x^{*}}} (t) = 1\); \(\forall t \in {\varvec{R}}_{+}\) and \(\left\{{x_{n}} \right\} \to x^{*}\) a.s..\(\square\)
A close result to Theorem 4 is now got in the case when \(\{T_{n}\} \begin{array}{*{20}c} {_{\to}} \\ {^{\to}} \\ \end{array} T\), with \(T_{n} : X \to X\), so that \(T : X \to X\) is a strict \(\varphi\)-contraction without requesting that all the elements of the sequence {T
n
} be strict \(\varphi\)-contraction.
Theorem 5
Let
\(\left({X, {\varvec{F}}, \varDelta_{M}} \right)\)
be a complete Menger space and let
\(\{T_{n}\}\)
be a sequence of operators such that:
-
1.
\(\{T_{n}\} \begin{array}{*{20}c} {_{\to}} \\ {^{\to}} \\ \end{array} T\)
such that
\(T_{n} :X \to X\); \(\forall n \in {\varvec{Z}}_{0+}\)
for some
\(T :X \to X\)
which is a strict
\(\varphi\)
-contraction,
-
2.
\(x_{n}^{*} \in F_{{T_{n}}} \ne \varnothing; \forall n \in {\varvec{Z}}_{0+}\)
Then, {T
n
} has a subsequence of strict \(\varphi\)-contractions and \(\left\{{x_{n}^{*}} \right\} \to x^{*}\) a.s., where \(F_{{T_{n}}} = \left\{{x_{n}^{*}} \right\}; \forall n \in {\varvec{Z}}_{+}\) and F
T
= {x
*}. Furthermore \(\left\{{x_{n}} \right\} \to x^{*}\) a.s., where x
n+1 = T
n
x
n
; \(\forall n \in {\varvec{Z}}_{0+}\) for any given x
0 ∈ X.
Proof
We have \(F_{Tx,Ty}^{-1} (t) - 1 \le \varphi \left({F_{x,y}^{-1} (t) - 1} \right)\); ∀x, y ∈ X,\(\forall t \in {\varvec{R}}_{+}, \forall n \in {\varvec{Z}}_{0+}\) for some strict comparison function \(\varphi :{\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\), since \(T: X \to X\) is a strict \(\varphi\)-contraction, equivalently, \(F_{Tx,Ty} (t) \ge \frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t) - 1} \right)}}\); ∀x, y ∈ X,\(\forall t \in {\varvec{R}}_{+}, \forall n \in {\varvec{Z}}_{0+}\), and then
$$\begin{aligned} F_{{T_{n} x, T_{n} y}} (t) & \ge \varDelta_{M} \left({F_{{T_{n} x, Tx}} (t/2), F_{{Tx, T_{n} y}} (t/2)} \right) \hfill \\ &\ge \varDelta_{M} \left({F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({F_{Tx,Ty} (t/4), F_{{T_{n} y, Ty}} (t/4)} \right)} \right) \hfill \\ &\ge \varDelta_{M} \left({F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({\frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}}, F_{{T_{n} y, Ty}} (t/4)} \right)} \right);\quad \forall n \in {\varvec{Z}}_{0+},\; \forall t \in {\varvec{R}}_{+}. \hfill \\ \end{aligned}$$
Thus, since \(\{T_{n}\} \begin{array}{*{20}c} {_{\to}} \\ {^{\to}} \\ \end{array} T\) and \(T : X \to X\) is a strict \(\varphi\)-contraction, then everywhere continuous, and \(\varDelta_{M} : [0,1] \times [0,1] \to [0,1]\) is a continuous triangular norm, one gets:
$$\begin{aligned} \mathop{lim\;inf}\limits_{n \to \infty} F_{{T_{n} x, T_{n} y}} (t) & \ge \mathop{lim\;inf}\limits_{n \to \infty} \varDelta_{M} \left({F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({\frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}}, F_{{T_{n} y, Ty}} (t/4)} \right)} \right) \hfill \\ &\ge \varDelta_{M} \left({\mathop{lim\;inf}\limits_{n \to \infty} F_{{Tx, T_{n} x}} (t/2), \varDelta_{M} \left({\frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}}, \mathop{lim\;inf}\limits_{n \to \infty} F_{{T_{n} y, Ty}} (t/4)} \right)} \right) \hfill \\ &= \varDelta_{M} \left({1, \varDelta_{M} \left({\frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}}, 1} \right)} \right) = \frac{1}{{1 + \varphi \left({F_{x,y}^{-1} (t/4) - 1} \right)}};\quad \forall t \in {\varvec{R}}_{+},\; \forall x, y \in X \hfill \\ \end{aligned}$$
so that one gets the following recursion:
$$\begin{aligned} & \mathop{lim\;sup}\limits_{m \to \infty} \left(\mathop{lim\;sup}\limits_{n \to \infty} F_{T_{n}^{m}x, T_{n}^{m}y}^{- 1}(t) \right) \\ &\quad \le 1 + \mathop{lim\;sup}\limits_{m \to \infty} \left(\mathop{lim\;sup}\limits_{n \to \infty} \varphi \left(F_{{T_{n}}^{m - 1}x, T_{n}^{m - 1} y}^{- 1} (t/4) - 1 \right) \right) \\ &\quad \le \cdots \le 1 + \mathop{lim\;sup}\limits_{m \to \infty} \left(\mathop{lim\;sup}\limits_{n \to \infty} \varphi^{m} \left(F_{x, y}^{- 1} \left(t/2^{m + 1}\right) - 1 \right) \right) = 1 \\ \end{aligned}$$
since \(lim_{m \to \infty} \varphi^{m} \left({F_{x,y}^{-1} \left({t/2^{n + 1}} \right) - 1} \right) = 0\) since \(\left\{{\varphi^{m} (t)} \right\} \to 0; \forall t \in {\varvec{R}}_{+}\). Then, \(lim_{m \to \infty} F_{x,y}^{-1} \left({t/2^{m + 1}} \right) = 1, \forall t \in {\varvec{R}}_{+}\), ∀x, y ∈ X, and \(lim_{n \to \infty} F_{{T^{n} x, T^{n} y}} (t) = 1; \forall t \in {\varvec{R}}_{+}\), ∀x, y ∈ X. So, there is a subsequence \(\left\{{T_{{n_{n}}}} \right\}\) of \(\{T_{n}\}\) whose elements are strict \(\varphi\)-contractions and the elements of such a subsequence have unique fixed points \(\left\{{x_{{n_{k}}}^{*}} \right\}\). Eq. (32) can also be got under the conditions of this theorem so that one concludes that {x
*
n
} → x
* a.s.. The remaining of the proof is close to its counterpart in Theorem 4.\(\square\)
We now reformulate close results to the above ones associated with strict \(\varphi\)-contractions via a dual class of contractions referred to as dual strict \(\varphi\)-contractions which operate directly on contractive conditions on the probability density function instead on its inverse. For that purpose, we first introduce the concept of dual strict comparison function as follows:
Definition 4
A non-increasing function \(\varphi : {\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\) (i.e. \(\varphi\)(t
1) ≥ \(\varphi\)(t
2) if t
1 ≤ t
2) is said to be a dual comparison function if \(\left\{{\varphi^{n} (t)} \right\} \to 1, \forall t \in {\varvec{R}}_{+}\). If, furthermore, \(\left({t - \varphi (t)} \right) \to + \infty\) as t → + ∞ then it is said to be a dual strict comparison function.
Definition 5
Let (X, F) be a PM-space. Then, \(G : X \to X\) is said to be a dual strict \(\varphi\)-contraction if \(G_{Tx,Ty} (t) \ge \varphi \left({G_{x,y} (t)} \right)\), ∀x, y ∈ X,\(\forall t \in {\varvec{R}}_{+}\) for some dual strict comparison function \(\varphi :{\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\).
Note that if T is any strict k-contraction for any given \(k \in (0,1)\) then
$$F_{Tx,Ty} (t) \ge F_{x,y} \left({k^{-1} t} \right) \ge F_{x,y} (t) = \varphi \left({F_{x,y} (t)} \right);\forall x,y \in X;\quad \forall t \in {\varvec{R}}_{+}$$
if \(\varphi\) ≡ 1 which is a dual strict comparison function since F is non-decreasing and left-continuous.
If \(\left\{{\varphi^{n} (t)} \right\} \to 1\), although \(\varphi :{\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\) be non-necessarily unity but a dual strict comparison function, then
$$F_{{T^{n + 1} x,T^{n + 1} y}} (t) \ge F_{Tx,Ty} \left({k^{- n} t} \right) \ge \varphi \left({F_{x,y} \left({k^{- n} t} \right)} \right) = \varphi^{n} \left({F_{x,y} (t)} \right);\quad\forall x,y \in X,\;\forall t \in {\varvec{R}}_{+}.$$
Since \(\left\{{\varphi^{n} (t)} \right\} \to 1, \forall t \in {\varvec{R}}_{+}\) because it is a dual strict comparison function, all the limits of the above chain equalize unity. So, if \(T:X \to X\) is any strict k-contraction then it is also a dual strict \(\varphi\)-contraction. The converse is not true in general. Assume now that \(T:X \to X\) is a dual strict \(\varphi\)-contraction for some dual strict comparison function \(\varphi\) so that \(F_{Tx,Ty} (t) \ge \varphi \left({F_{x,y} (t)} \right); \forall x,y \in X, \forall t \in {\varvec{R}}_{+}\). Since F is non-decreasing and \(\varphi\) is non-increasing, we have for \(k \in (0,1)\):
$$F_{x,y} \left({k^{-1} t} \right) \ge F_{x,y} (t);\varphi \left({F_{x,y} \left({k^{-1} t} \right)} \right) \le \varphi \left({F_{x,y} (t)} \right);\quad \forall x,y \in X,\;\forall t \in {\varvec{R}}_{+}$$
so that \(F_{Tx,Ty} (t) \ge \varphi \left({F_{x,y} (t)} \right) \ge \varphi \left({F_{x,y} \left({k^{-1} t} \right)} \right); \forall x,y \in X, \forall t \in {\varvec{R}}_{+}\) and any given k ∈ (0, 1). One then gets that \(F_{{T^{n} x,T^{n} y}} (t) \ge \varphi^{n} \left({F_{x,y} \left({k^{- n} t} \right)} \right)\) so that \(F_{{T^{n} x,T^{n} y}} (t) \to 1\) as n → ∞; ∀x, y ∈ X, \(\forall t \in {\varvec{R}}_{+}\). Then, if \(T:X \to X\) is a \(\varphi\)-contraction, it is not a strict k-contraction, in general.
The next result follows:
Theorem 6
Let
\(\left({X, {\varvec{F}}, \varDelta_{M}} \right)\)
be a complete Menger space and let
\(\{T_{n}\}\)
be a sequence of operators such that:
-
1.
The operators
\(T_{n} :X \to X; \forall n \in {\varvec{Z}}_{0+}\)
of the sequence {T
n
} are all dual strict
\(\varphi\)
-contractions,
-
2.
\(\{T_{n}\} \to T\)
for some
\(T :X \to X\).
Then, \(T: X \to X\) is a dual strict \(\varphi\)-contraction and \(\left\{{x_{n}^{*}} \right\} \to x^{*}\) a.s., where \(F_{{T_{n}}} = \left\{{x_{n}^{*}} \right\}\); \(\forall n \in {\varvec{Z}}_{+}\) and F
T
= {x
*}. Furthermore \(\left\{{x_{n}} \right\} \to x^{*}\) a.s., where x
n+1 = T
n
x
n
; \(\forall n \in {\varvec{Z}}_{0+}\) for any given x
0 ∈ X.
Proof
We have \(F_{{T_{n} x,T_{n} y}} (t) \ge \varphi \left({F_{x,y} (t)} \right)\); ∀x, y ∈ X,\(\forall t \in {\varvec{R}}_{+}, \forall n \in {\varvec{Z}}_{0+}\) for some strict comparison function \(\varphi :{\varvec{R}}_{0+} \to {\varvec{R}}_{0+}\), since all the operators of the sequence {T
n
} are dual strict \(\varphi\)-contractions; ∀x, y ∈ X,\(\forall t \in {\varvec{R}}_{+}\). Under close steps to those used in the proof of Theorem 4, we get instead of (28):
$$F_{T^{n}x, T^{n}y} (t) \ge \varphi \left(F_{T^{n - 1} x, T^{n - 1}y} (t/4)\right) \ge \cdots \ge \varphi^{n} \left(F_{x, y} \left(t/2^{n + 1}\right)\right);\quad \forall n \in {\varvec{Z}}_{0+}, \; \forall t \in {\varvec{R}}_{+},\forall x, y \in X$$
(33)
and \(lim_{n \to \infty} \varphi^{n} \left({F_{x,y} \left({t/2^{n + 1}} \right)} \right) = 1\), since \(\left\{{\varphi^{n} (t)} \right\} \to 1; \forall t \in {\varvec{R}}_{+}\). Then \(lim_{n \to \infty} F_{{T^{n} x, T^{n} y}} (t) = 1; \forall t \in {\varvec{R}}_{+}\), ∀x, y ∈ X and one can conclude in a similar way to Theorem 4 that T, which is the point-wise limit of the sequence \(\{T_{n}\}\) of dual strict \(\varphi\)-contractions, is also a dual strict \(\varphi\)-contraction. We also can prove the remaining properties of the statement in a close way to the proof of Theorem 4.\(\square\)
It turns out that a similar result to Theorem 6, under the guidelines of Theorem 5, can be directly formulated for the case when \(\{T_{n}\} \begin{array}{*{20}c} {_{\to}} \\ {^{\to}} \\ \end{array} T\) with \(T_{n} :X \to X\) where the limit operator \(T :X \to X\) is a dual strict \(\varphi\)-contraction.