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Periodic and fixed points of the Leader-type contractions in quasi-triangular spaces
Fixed Point Theory and Applications volume 2016, Article number: 85 (2016)
Abstract
Let \(C=\{C_{\alpha}\}_{\alpha\in\mathcal{A}}\in[1;\infty)^{\mathcal{A}}\) with index set \(\mathcal{A}\). A quasi-triangular space \((X,\mathcal{P}_{C;\mathcal{A}})\) is a set X with family \(\mathcal{P}_{C;\mathcal{A}}=\{p_{\alpha}:X^{2}\rightarrow[0,\infty),\alpha \in \mathcal{A}\}\) satisfying \(\forall_{\alpha\in\mathcal{A}}\forall _{u,v,w\in X}\{p_{\alpha}(u,w)\leq C_{\alpha}[p_{\alpha }(u,v)+p_{\alpha }(v,w)]\}\). In \((X,\mathcal{P}_{C;\mathcal{A}})\), using the left (right) families \(\mathcal{J}_{C;\mathcal{A}}\) generated by \(\mathcal {P}_{C;\mathcal{A}}\) (\(\mathcal{P}_{C;\mathcal{A}}\) is a particular case of \(\mathcal {J}_{C;\mathcal{A}}\)), we establish theorems concerning left (right) \(\mathcal {P}_{C;\mathcal{A}}\)-convergence, existence, periodic point, fixed point, and (when \((X,\mathcal{P}_{C;\mathcal{A}})\) is separable) uniqueness for \(\mathcal{J}_{C;\mathcal{A}}\)-contractions and weak \(\mathcal {J}_{C;\mathcal{A}}\)-contractions \(T:X\rightarrow X\) satisfying \(\forall_{x,y\in X}\forall _{\alpha\in\mathcal{A}}\forall_{\varepsilon>0}\exists_{\eta >0}\exists _{r\in\mathbb{N}}\forall_{s,l\in\mathbb{N}}\{J_{\alpha }(T^{[s]}(x),T^{[l]}(y)) < \eta+\varepsilon \Rightarrow C_{\alpha }J_{\alpha}(T^{[s+r]}(x),T^{[l+r]}(y))<\varepsilon\}\) and \(\exists _{w^{0}\in X}\forall_{\alpha\in\mathcal{A}}\forall_{\varepsilon >0}\exists_{\eta>0}\exists_{r\in\mathbb{N}} \forall_{s,l\in \mathbb{N}}\{J_{\alpha}(T^{[s]}(w^{0}), T^{[l]}(w^{0}))<\eta+\varepsilon \Rightarrow C_{\alpha}J_{\alpha }(T^{[s+r]}(w^{0}),T^{[l+r]}(w^{0}))<\varepsilon\}\), respectively. The spaces \((X,\mathcal{P}_{C;\mathcal{A}})\), in particular, generalize metric, ultrametric, quasi-metric, ultra-quasi-metric, b-metric, partial metric, partial b-metric, pseudometric, quasi-pseudometric, ultra-quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra-quasi-gauge, and partial quasi-gauge spaces. Results are new in all these spaces. Examples are provided.
1 Introduction
Contractions are among the most important objects studied in fixed point theory. The study concerning convergence and fixed points of several types contractions in metric spaces and also in various more general distance spaces has undergone remarkable developments in the last years. The effect has been a still-ongoing series of results that are far stronger and more general and optimal than those known before.
The following theorem is one of the first, important, central, simpler, and very inspired result of this theory.
Theorem 1.1
Let \((X,d)\) be a complete metric space. If \(T:X\rightarrow X\) satisfies
then: (i) T has a unique fixed point w in X; and (ii) for each \(w^{0}\in X\), \(\lim_{m\rightarrow\infty}d(T^{[m]}(w^{0}), w)=0\).
The maps satisfying condition (1.1) are called in the literature Banach contractions.
In metric spaces, not necessarily complete, by changing condition (1.1), there are many different interesting generalizations of Theorem 1.1 in the literature. Significant papers here are Burton [3], Rakotch [4], Geraghty [5, 6], Matkowski [7–9], Walter [10], Dugundji [11], Tasković [12], Dugundji and Granas [13], Browder [14], Krasnosel’skiĭ et al. [15], Boyd and Wong [16], Mukherjea [17], Meir and Keeler [18], Leader [19], Jachymski [20, 21], Jachymski and Jóźwik [22], and many others.
Among the papers mentioned, the following is especially remarkable.
Theorem 1.2
(Leader [19], Theorem 3)
Let \((X,d)\) be a metric space, and let \(T:X\rightarrow X\) be a map with a complete graph (i.e., closed in \(Y^{2} \), where Y is the completion of X). The following hold:
-
(A)
T has a contractive fixed point in X if and only if
$$\begin{aligned} &\forall_{x,y\in X}\forall_{\varepsilon>0}\exists_{\eta\in(0,\infty ]} \exists_{r\in\mathbb{N}}\forall_{i,j\in\mathbb{N}}\bigl\{ d\bigl(T^{[i]}(x),T^{[j]}(y) \bigr)< \varepsilon+\eta \\ &\quad\Rightarrow d\bigl(T^{[i+r]}(x),T^{[j+r]}(y) \bigr)< \varepsilon\bigr\} . \end{aligned}$$(1.2) -
(B)
T has a fixed point in X if and only if
$$\begin{aligned} &\exists_{x\in X}\forall_{\varepsilon>0}\exists_{\eta\in(0,\infty ]} \exists_{r\in\mathbb{N}}\forall_{i,j\in\mathbb{N}}\bigl\{ d\bigl(T^{[i]}(x),T^{[j]}(x) \bigr)< \varepsilon+\eta \\ &\quad\Rightarrow d\bigl(T^{[i+r]}(x),T^{[j+r]}(x) \bigr)< \varepsilon\bigr\} . \end{aligned}$$(1.3)Moreover, if \(x,\varepsilon,\eta\), and r are as in (1.3) and if \(\exists _{w\in X}\{\lim_{m\rightarrow\infty}d(T^{[m]}(x),w)=0\}\), then \(\forall _{i\in\mathbb{N}}\{d(T^{[i]}(x),T^{[i+r]}(x))\leq\eta\Rightarrow d(T^{[i+r]}(x),w)\leq\varepsilon\}\).
By a contractive fixed point of \(T:X\rightarrow X\) we mean a fixed point w of T in X such that, for each \(w^{0}\in X\), \(\lim_{m\rightarrow\infty}d(T^{[m]}(w^{0}),w)=0\). The maps satisfying conditions (1.2) and (1.3) are called in the literature Leader contractions and weak Leader contractions, respectively.
Remark 1.1
Let \((X,d)\) be a metric space.
-
(a)
We refer to Jachymski [20, 21] and Jachymski and Jóźwik [22] for a few theorems in \((X,d)\) showing how natural Leader contractions are and how results of Leader [19] generalize the papers of Banach [1], Caccioppoli [2], Burton [3], Rakotch [4], Geraghty [5, 6], Matkowski [7–9], Walter [10], Dugundji [11], Tasković [12], Dugundji and Granas [13], Browder [14], Krasnosel’skiĭ et al. [15], Boyd and Wong [16], Mukherjea [17], Meir and Keeler [18], and many others.
-
(b)
It is worth noticing that Leader’s method does not require the complete assumption of \((X,d)\) and that the statements become more elegant and the most general.
Remark 1.2
Notice that:
-
(a)
Leader’s proof was based on the observation that \((X,d)\) are topological and Hausdorff, d vanishes on the diagonal, is symmetric, and satisfies triangle inequality, and the map d is continuous.
-
(b)
If we remove some of these conditions, then the situation is much more complicated.
Definition 1.1
Let X be a nonempty set.
-
(A)
A quasi-pseudometric on X is a map \(p:X^{2}\rightarrow [0,\infty)\) such that: \(\forall_{u\in X}\{p(u,u)=0\}\); and \(\forall _{u,v,w\in X}\{p(u,w)\leq p(u,v)+p(v,w)\}\). For given quasi-pseudometric p on X, a pair \((X,p)\) is called quasi-pseudometric space, and \((X,p)\) is called Hausdorff if \(\forall_{u,v\in X}\{u\neq v\Rightarrow p(u,v)>0\vee p(v,u)>0\}\).
-
(B)
Each family \(\mathcal{P}=\{p_{\alpha}:\alpha\in\mathcal {A}\}\) of quasi-pseudometrics \(p_{\alpha}:X^{2}\rightarrow[ 0,\infty)\), \(\alpha\in\mathcal{A}\), is called a quasi-gauge on X.
-
(C)
Let the family \(\mathcal{P}=\{p_{\alpha}:\alpha\in\mathcal {A}\} \) be a quasi-gauge on X. The topology \(\mathcal{T(P)}\) having as a subbase the family \(\mathcal{B}(\mathcal{P})=\{B(u,\varepsilon_{\alpha }):u\in X, \varepsilon_{\alpha}>0, \alpha\in\mathcal{A}\}\) of all balls \(B(u,\varepsilon_{\alpha})=\{v\in X:p_{\alpha}(u,v)<\varepsilon _{\alpha}\}\), \(u\in X\), \(\varepsilon_{\alpha}>0\), \(\alpha\in \mathcal{A}\), is called the topology induced by \(\mathcal{P}\) on X.
-
(D)
(Dugundji [23], Reilly [24]) A topological space \((X,\mathcal{T})\) such that there is a quasi-gauge \(\mathcal{P}\) on X with \(\mathcal{T}=\mathcal{T(P)}\) is called a quasi-gauge space and is denoted by \((X,\mathcal{P})\).
-
(E)
A quasi-gauge space \((X,\mathcal{P})\) is called Hausdorff if the quasi-gauge \(\mathcal{P}\) has the property \(\forall_{u,v\in X}\{ u\neq v\Rightarrow\exists_{\alpha\in\mathcal{A}}\{p_{\alpha}(u,v)>0\vee p_{\alpha}(v,u)>0\}\}\).
Remark 1.3
Each quasi-uniform space and each topological space is a quasi-gauge space (Reilly [24], Theorems 4.2 and 2.6). The quasi-gauge spaces are spaces with asymmetric structures.
Let X be a (nonempty) set. A distance on X is a map \(p:X^{2}\rightarrow[0;\infty)\). A set X, together with a distance on X, is called a distance space (see [25, 26]).
Before proceeding further, let us recall the following:
Definition 1.2
([27])
Let X be a (nonempty) set, let \(\mathcal{A}\) be an index set, and let \(C=\{C_{\alpha}\}_{\alpha\in\mathcal{A}}\in [1;\infty)^{\mathcal{A}}\).
-
(A)
We say that a family \(P_{C;\mathcal{A}}=\{p_{\alpha},\alpha \in A\}\) of distances \(p_{\alpha}:X^{2}\rightarrow[0,\infty)\), \(\alpha \in A\), is a quasi-triangular family on X if
$$ \forall_{\alpha\in\mathcal{A}}\forall_{u,v,w\in X}\bigl\{ p_{\alpha }(u,w)\leq C_{\alpha}\bigl[p_{\alpha}(u,v)+p_{\alpha}(v,w)\bigr]\bigr\} . $$A quasi-triangular space \((X,P_{C;\mathcal{A}})\) is a set X together with a quasi-triangular family \(P_{C;\mathcal{A}}\) on X.
-
(B)
Let \((X,P_{C;\mathcal{A}})\) be a quasi-triangular space. We say that \(P_{C;\mathcal{A}}\) is separating if
$$ \forall_{u,w\in X}\bigl\{ u\neq w\Rightarrow\exists_{\alpha\in\mathcal{A} }\bigl\{ p_{\alpha}(u,w)>0\vee p_{\alpha}(w,u)>0\bigr\} \bigr\} . $$(1.4) -
(C)
We say that a family \(\mathcal{L}_{C;\mathcal{A}}=\{l_{\alpha },\alpha\in\mathcal{A}\}\) of distances \(l_{\alpha}:X^{2}\rightarrow [0,\infty)\), \(\alpha\in\mathcal{A}\), is an ultra-quasi-triangular family on X if
$$ \forall_{\alpha\in\mathcal{A}}\forall_{u,v,w\in X}\bigl\{ l_{\alpha }(u,w)\leq C_{\alpha}\max\bigl\{ l_{\alpha}(u,v),l_{\alpha}(v,w)\bigr\} \bigr\} . $$An ultra-quasi-triangular space \((X,\mathcal {L}_{C;\mathcal{A}})\) is a set X together with the ultra-quasi-triangular family \(\mathcal{L}_{C;\mathcal{A}}\) on X.
-
(D)
We say that a family \(\mathcal{S}_{C;\mathcal{A}}=\{s_{\alpha },\alpha\in\mathcal{A}\}\) of distances \(s_{\alpha}:X^{2}\rightarrow [0,\infty)\), \(\alpha\in\mathcal{A}\), is a partial quasi-triangular family on X if
$$ \forall_{\alpha\in\mathcal{A}}\forall_{u,v,w\in X}\bigl\{ s_{\alpha }(u,w)\leq C_{\alpha}\bigl[s_{\alpha}(u,v)+s_{\alpha}(v,w) \bigr]-s_{\alpha}(v,v)\bigr\} . $$A partial quasi-triangular space \((X,\mathcal {S}_{C;\mathcal{A}})\) is a set X together with a partial quasi-triangular family \(\mathcal{S}_{C;\mathcal{A}}\) on X.
-
(E)
We say that a family \(P_{\mathcal{A}}=\{p_{\alpha},\alpha \in A\} \) of distances \(p_{\alpha}:X^{2}\rightarrow[0,\infty)\), \(\alpha\in A\), is a triangular family on X if
$$ \forall_{\alpha\in\mathcal{A}}\forall_{u,v,w\in X}\bigl\{ p_{\alpha }(u,w)\leq p_{\alpha}(u,v)+p_{\alpha}(v,w)\bigr\} . $$A triangular space \((X,P_{\mathcal{A}})\) is a set X together with a triangular family \(P_{\mathcal{A}}\) on X.
Remark 1.4
There are several reasons for studying the quasi-triangular spaces.
-
(a)
First, in the spaces \((X,\mathcal{P}_{C;\mathcal{A}})\), in general, the distances \(p_{\alpha}:X^{2}\rightarrow[0,\infty )\), \(\alpha\in\mathcal{A}\), do not vanish on the diagonal, are asymmetric, and do not satisfy the triangle inequality (i.e., the properties \(\forall _{\alpha\in\mathcal{A}}\forall_{u\in X}\{p_{\alpha}(u,u)=0\}\), or \(\forall_{\alpha\in\mathcal{A}}\forall_{u,w\in X}\{p_{\alpha }(u,w)=p_{\alpha}(w,u)\}\), or \(\forall_{\alpha\in\mathcal{A}}\forall _{u,v,w\in X}\{p_{\alpha}(u,w)\leq p_{\alpha}(u,v)+p_{\alpha}(v,w)\} \) do not necessarily hold).
-
(b)
Second, these spaces are not necessarily topological, or Hausdorff, or sequentially complete.
-
(c)
Finally, these spaces generalize ultra-quasi-triangular and partial quasi-triangular spaces (in particular, generalize metric, ultrametric (Roovij [28]), quasi-metric (Wilson [29]), ultra-quasi-metric, b-metric (Bakhtin [30], Czerwik [31]), partial metric (Matthews [32]), partial b-metric (Shukla [33]), pseudometric, quasi-pseudometric, ultra-quasi-pseudometric (Künzi and Otafudu [34]), partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge (Dugundji [11]), ultra-gauge, partial gauge, quasi-gauge (Reilly [24]), ultra-quasi-gauge, and partial quasi-gauge spaces).
Remarks 1.1-1.4 are a well-motivated argument to study several fundamental problems concerning the fixed point theory in these very general spaces \((X,\mathcal{P}_{C;\mathcal{A}})\). Some results in this direction are given in Włodarczyk [27, 35]. Based on some ideas presented in [27, 35–38], we can ask the following question.
Question 1.1
Is there a theorem of Leader type in \((X,\mathcal{P}_{C;\mathcal{A}})\)?
The main results of this paper (see Theorems 3.1 and 3.2) provide conditions under which the answer to this question is ‘Yes.’
More precisely, the subject of this paper is the constructions of contractions and weak contractions of Leader type and the study of convergence, existence, approximation, periodic point, fixed point, and uniqueness properties of these contractions and weak contractions in quasi-triangular spaces \((X,\mathcal{P}_{C;\mathcal{A}})\). In these very general spaces, by elementary tools and by a natural argument, using radically new and original technique, we derive unexpectedly richer conclusions from very weak hypotheses (see Theorems 3.1 and 3.2). This paper shows that the answer indeed is affirmative and inspires also new ways of looking at old problems in quasi-triangular spaces.
2 Left (right) families \(\mathcal{J}_{C;\mathcal{A}}\) generated by \(\mathcal{P}_{C;\mathcal{A}}\), left (right) \(\mathcal {J}_{C;\mathcal{A}}\)-convergences, and left (right) \(\mathcal {J}_{C;\mathcal{A}}\)-sequential completeness in quasi-triangular spaces \((X,\mathcal{P}_{C;\mathcal{A}})\)
Let \(\mathcal{P}_{C;\mathcal{A}}\) be a quasi-triangular family on X. It is natural to define the notions of left (right) families \(\mathcal{J}_{C;\mathcal{A}}\) generated by \(\mathcal {P}_{C;\mathcal{A}}\), which provide new structures on X.
Definition 2.1
Let \((X,\mathcal{P}_{C;\mathcal{A}})\) be a quasi-triangular space.
-
(A)
The family \(\mathcal{J}_{C;\mathcal{A}}=\{J_{\alpha}: \alpha \in\mathcal{A}\}\) of distances \(J_{\alpha}:X^{2}\rightarrow[ 0,\infty)\), \(\alpha\in\mathcal{A}\), is said to be a left (right) family generated by \(\mathcal{P}_{C;\mathcal{A}}\) if:
- (\(\mathcal{J}\)1):
-
\(\forall_{\alpha\in\mathcal {A}}\forall _{u,v,w\in X}\{J_{\alpha}(u,w)\leq C_{\alpha}[J_{\alpha }(u,v)+J_{\alpha }(v,w)]\}\); and furthermore
- (\(\mathcal{J}\)2):
-
for any sequences \((u_{m}:m\in \mathbb{N}) \) and \((v_{m}:m\in\mathbb{N})\) in X satisfying
$$ \forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty }\sup _{n>m}J_{\alpha}(u_{m},u_{n})=0 \Bigr\} \quad\Bigl(\forall_{\alpha \in \mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty}\sup _{n>m}J_{\alpha }(u_{n},u_{m})=0 \Bigr\} \Bigr) $$and
$$ \forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty}J_{\alpha }(v_{m},u_{m})=0 \Bigr\} \quad\Bigl(\forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty}J_{\alpha}(u_{m},v_{m})=0 \Bigr\} \Bigr), $$we have
$$ \forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty}p_{\alpha }(v_{m},u_{m})=0 \Bigr\} \quad\Bigl(\forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty}p_{\alpha}(u_{m},v_{m})=0 \Bigr\} \Bigr). $$
-
(B)
\(\mathbb{J}_{(X,\mathcal{P}_{C;\mathcal{A}})}^{L}\) \((\mathbb {J}_{(X,\mathcal{P}_{C;\mathcal{A}})}^{R})\) is the set of all left (right) families \(\mathcal{J}_{C;\mathcal{A}}\) on X generated by \(\mathcal{P}_{C;\mathcal{A}}\).
Remark 2.1
Let \((X,\mathcal{P}_{C;\mathcal{A}})\) be a quasi-triangular space. Then we have:
-
(a)
\(\mathcal{P}_{C;\mathcal{A}}\in\mathbb{J}_{(X,\mathcal {P}_{C;\mathcal{A}})}^{L}\cap\mathbb{J}_{(X,\mathcal{P}_{C;\mathcal{A}})}^{R}\).
-
(b)
The structures on X determined by left (right) families \(\mathcal{J}_{C;\mathcal{A}}\) generated by \(\mathcal{P}_{C;\mathcal{A}}\) are more general than the structure on X determined by \(\mathcal{P}_{C;\mathcal{A}}\).
-
(c)
If \(\mathcal{J}_{C;\mathcal{A}}\in\mathbb{J}_{(X,\mathcal {P}_{C;\mathcal{A}})}^{L}\cup\mathbb{J}_{(X,\mathcal{P}_{C;\mathcal{A}})}^{R}\), then \((X,\mathcal{J}_{C;\mathcal{A}})\) is a quasi-triangular space.
Definition 2.2
Let \((X,\mathcal{P}_{C;\mathcal{A}})\) be a quasi-triangular space, and let \(\mathcal{J}_{C;\mathcal{A}}\) be a left (right) family generated by \(\mathcal{P}_{C;\mathcal{A}}\).
-
(A)
We say that a sequence \((u_{m}:m\in\mathbb{N})\subset X\) is left (right) \(\mathcal{J}_{C;\mathcal {A}}\) -Cauchy sequence if
$$ \forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty }\sup _{n>m}J_{\alpha}(u_{m},u_{n})=0 \Bigr\} \quad \Bigl(\forall_{\alpha \in \mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty}\sup _{n>m}J_{\alpha }(u_{n},u_{m})=0 \Bigr\} \Bigr). $$ -
(B)
Let \(u\in X\) and \((u_{m}:m\in\mathbb{N})\subset X\). We say that a sequence \((u_{m}:m\in\mathbb{N})\) is left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-convergent to u if
$$ u\in LIM_{(u_{m}:m\in\mathbb{N})}^{L-\mathcal{J}_{C;\mathcal{A}}}\neq \varnothing \quad\bigl(u\in LIM_{(u_{m}:m\in\mathbb{N})}^{R-\mathcal {J}_{C;\mathcal{A}}}\neq\varnothing\bigr), $$where
$$\begin{aligned}& LIM_{(u_{m}:m\in\mathbb{N})}^{L-\mathcal{J}_{C;\mathcal{A}}} =\Bigl\{ x\in X:\forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty }J_{\alpha }(x,u_{m})=0\Bigr\} \Bigr\} \\& \Bigl(LIM_{(u_{m}:m\in\mathbb{N})}^{R-\mathcal{J}_{C;\mathcal{A}}} =\Bigl\{ x\in X:\forall_{\alpha\in\mathcal{A}} \Bigl\{ \lim_{m\rightarrow\infty }J_{\alpha }(u_{m},x)=0\Bigr\} \Bigr\} \Bigr). \end{aligned}$$ -
(C)
We say that a sequence \((u_{m}:m\in\mathbb{N})\subset X\) is left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-convergent in X if
$$ LIM_{(u_{m}:m\in\mathbb{N})}^{L-\mathcal{J}_{C;\mathcal{A}}}\neq \varnothing \quad\bigl(LIM_{(u_{m}:m\in\mathbb{N})}^{R-\mathcal{J}_{C;\mathcal{A}}} \neq\varnothing\bigr). $$ -
(D)
If every left (right) \(\mathcal{J}_{C;\mathcal {A}}\)-Cauchy sequence \((u_{m}:m\in\mathbb{N})\subset X\) is left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-convergent in X (i.e., \(LIM_{(u_{m}:m\in\mathbb{N})}^{L-\mathcal{J}_{C;\mathcal{A}}}\neq \varnothing\) (\(LIM_{(u_{m}:m\in\mathbb{N})}^{R-\mathcal{J}_{C;\mathcal {A}}}\neq\varnothing\))), then \((X,\mathcal{P}_{C;\mathcal{A}})\) is called left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-sequential complete.
-
(E)
We say that \((X,\mathcal{P}_{C;\mathcal{A}})\) is left (right) Hausdorff if for each left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-convergent in X sequence \((u_{m}:m\in \mathbb{N})\), the set
$$ LIM_{(u_{m}:m\in\mathbb{N})}^{L-\mathcal{P}_{C;\mathcal{A}}} \quad \bigl(LIM_{(u_{m}:m\in\mathbb{N})}^{R-\mathcal{P}_{C;\mathcal{A}}} \bigr) $$is a singleton.
Remark 2.2
Let \((X,\mathcal{P}_{C;\mathcal{A}})\) be a quasi-triangular space. It is clear that if \((u_{m}:m\in\mathbb{N})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-convergent in X, then
for each subsequence \((v_{m}:m\in\mathbb{N})\) of \((u_{m}:m\in\mathbb{N})\).
The following relations between \(\mathcal{J}_{C;\mathcal{A}}\) and \(\mathcal{P}_{C;\mathcal{A}}\) are interesting.
Theorem 2.1
([27])
Let \((X,\mathcal{P}_{C;\mathcal{A}})\) be a quasi-triangular space. Let \(E\subset X\) be a set containing at least two different points, and let \(\{\mu_{\alpha}\}_{\alpha\in\mathcal {A}}\in (0;\infty)^{\mathcal{A}}\) where \(\forall_{\alpha\in\mathcal{A}}\{\mu _{\alpha}\geq\delta_{\alpha}(E)/(2C_{\alpha})\}\) and \(\forall _{\alpha \in\mathcal{A}}\{\delta_{\alpha}(E)=\sup\{p_{\alpha}(u,w):u,w\in E\}\}\). If \(\mathcal{J}_{C;\mathcal{A}}=\{J_{\alpha}:\alpha\in\mathcal{A}\}\) where, for each \(\alpha\in\mathcal{A}\), the distance \(J_{\alpha }:X^{2}\rightarrow[0,\infty)\) is defined by
then \(\mathcal{J}_{C;\mathcal{A}}\) is the left and right family generated by \(\mathcal{P}_{C;\mathcal{A}}\).
Remark 2.3
This result shows that Definition 2.1 is correct and that \(\mathbb{J}_{(X,\mathcal{P}_{C;\mathcal{A}})}^{L}\backslash\{\mathcal {P}_{C;\mathcal{A}}\}\neq\varnothing\) and \(\mathbb{J}_{(X,\mathcal{P}_{C;\mathcal{A}})}^{R}\backslash\{\mathcal{P}_{C;\mathcal{A}}\}\neq\varnothing\).
Theorem 2.2
([27])
Let \((X,\mathcal{P}_{C;\mathcal {A}})\) be a quasi-triangular space, and let \(\mathcal{J}_{C;\mathcal{A}}\) be the left (right) family generated by \(\mathcal{P}_{C;\mathcal{A}}\). If \(\mathcal {P}_{C;\mathcal{A}}\) is separating on X (i.e., (1.4) holds), then \(\mathcal{J}_{C;\mathcal{A}}\) is separating on X, that is,
3 Statement of results
Recall that a single-valued dynamic system is defined as a pair \((X,T)\), where X is a certain space, and T is a single-valued map \(T:X\rightarrow X\), that is, \(\forall_{x\in X}\{T(x)\in X\}\). By \(Fix(T)\) and \(Per(T)\) we denote the sets of all fixed points and periodic points of T, respectively, that is, \(Fix(T)=\{ w\in X:w=T(w)\}\) and \(Per(T)=\{w\in X:w=T^{[q]}(w)\mbox{ for some }q\in\mathbb {N}\}\). For each \(w^{0}\in X\), a sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup \mathbb{N})\) is called a Picard iteration starting at \(w^{0}\) of the system \((X,T)\).
In this section, in the quasi-triangular spaces \((X,\mathcal {P}_{C;\mathcal{A}})\), using left (right) families \(\mathcal{J}_{C;\mathcal{A}}\) generated by \(\mathcal{P}_{C;\mathcal{A}}\), we construct the \(\mathcal{J}_{C;\mathcal {A}}\)-contractions \((X,T)\) and weak \(\mathcal {J}_{C;\mathcal{A}}\)-contractions \((X,T)\) of Leader type, and we formulate the left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-convergence, existence, approximation, uniqueness, periodic point, and fixed point theorems for such contractions.
The following terminology will be much used in the sequel.
Definition 3.1
Let \((X,\mathcal{P}_{C;\mathcal{A}})\) be a quasi-triangular space. Suppose that \(\mathcal{J}_{C;\mathcal{A}}\) is the left (right) family generated by \(\mathcal{P}_{C;\mathcal{A}}\).
-
(A)
A single-valued dynamic system \((X,T)\) is said to be left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-admissible on a set \(W^{L-\mathcal{J}_{C;\mathcal{A}}}\) (\(W^{R-\mathcal{J}_{C;\mathcal{A}}}\)) if the following two conditions hold:
-
(A1)
For each \(w^{0}\in X\) satisfying
$$\begin{aligned} &\forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty }\sup _{n>m}J_{\alpha}\bigl(w^{m},w^{n} \bigr) =0\Bigr\} \end{aligned}$$(3.1)$$\begin{aligned} &\Bigl(\forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim_{m\rightarrow\infty }\sup _{n>m}J_{\alpha}\bigl(w^{n},w^{m} \bigr) =0\Bigr\} \Bigr), \end{aligned}$$(3.2)we have
$$\begin{aligned}& \exists_{w\in X}\forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim_{m\rightarrow \infty}J_{\alpha} \bigl(w,w^{m}\bigr)=0\Bigr\} \quad\bigl(\mbox{i.e. }LIM_{(w^{m}:m\in \{0\}\cup\mathbb{N})}^{L-\mathcal{J}_{C;\mathcal{A}}}\neq\varnothing\bigr) \end{aligned}$$(3.3)$$\begin{aligned}& \Bigl(\exists_{w\in X}\forall_{\alpha\in\mathcal{A}}\Bigl\{ \lim _{m\rightarrow \infty}J_{\alpha}\bigl(w^{m},w\bigr)=0\Bigr\} \quad\bigl(\mbox{i.e. }LIM_{(w^{m}:m\in \{0\}\cup\mathbb{N})}^{R-\mathcal{J}_{C;\mathcal{A}}}\neq\varnothing\bigr) \Bigr), \end{aligned}$$(3.4)where \(w^{m}=T^{[m]}(w^{0})\), \(m\in\{0\}\cup\mathbb{N}\); and
-
(A2)
There exists \(W^{L-\mathcal{J}_{C;\mathcal{A}}}\in 2^{X}\) (\(W^{R-\mathcal{J}_{C;\mathcal{A}}}\in2^{X}\)) satisfying
$$\begin{aligned}& W^{L-\mathcal{J}_{C;\mathcal{A}}} =\bigl\{ w^{0}\in X:w^{0}\mbox{ satisfies (3.1) and (3.3)}\bigr\} \\& \bigl(W^{R-\mathcal{J}_{C;\mathcal{A}}} =\bigl\{ w^{0}\in X:w^{0} \mbox{ satisfies (3.2) and (3.4)}\bigr\} \bigr). \end{aligned}$$
-
(A1)
-
(B)
If \((X,T)\) is left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-admissible on \(W^{L-\mathcal{J}_{C;\mathcal{A}}}\in2^{X}\) \((W^{R-\mathcal{J}_{C;\mathcal{A}}}\in2^{X})\) and \(w^{0}\in W^{L-\mathcal{J}_{C;\mathcal{A}}}\) \((w^{0}\in W^{R-\mathcal{J}_{C;\mathcal{A}}})\), then we say that \((X,T)\) is left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-admissible in a point \(w^{0}\).
Here \(2^{X}\) denotes the family of all nonempty subsets of a space X.
Remark 3.1
Let \((X,\mathcal{P}_{C;\mathcal{A}})\) be a quasi-triangular space, and let \(\mathcal{J}_{C;\mathcal{A}}\) be the left (right) family generated by \(\mathcal{P}_{C;\mathcal{A}}\). Let \((X,T)\) be a single-valued dynamic system. If \((X,\mathcal{P}_{C;\mathcal{A}})\) is left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-sequentially complete, then \((X,T)\) is left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-admissible on some set \(W^{L-\mathcal{J}_{C;\mathcal{A}}}\in2^{X}\) \((W^{R-\mathcal{J}_{C;\mathcal{A}}}\in2^{X})\).
Next, we introduce the following concept of left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-closed single-valued dynamic systems in \((X,\mathcal{P}_{C;\mathcal{A}})\).
Definition 3.2
Let \((X,\mathcal{P}_{C;\mathcal{A}})\) be a quasi-triangular space. Let \((X,T)\) be a single-valued dynamic system, \(T:X\rightarrow X\), and let \(q\in\mathbb{N}\). The single-valued dynamic system \((X,T^{[q]})\) is said to be left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-closed on \(U\in2^{X}\) \((V\in2^{X})\) if for each \(w^{0}\in U\) \((w^{0}\in V)\) such that the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-converging in X (thus, \(LIM_{(w^{m}:m\in \{0\}\cup\mathbb{N})}^{L-\mathcal{P}_{C;\mathcal{A}}}\neq\varnothing \) \((LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{R-\mathcal{P}_{C;\mathcal {A}}}\neq \varnothing)\)) and having subsequences \((v_{m}:m\in\mathbb{N})\) and \((u_{m}:m\in\mathbb{N})\) satisfying \(\forall_{m\in\mathbb{N}}\{v_{m}=T^{[q]}(u_{m})\}\), the following property holds: there exists \(x\in LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal{P}_{C;\mathcal {A}}}\) \((y\in LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{R-\mathcal{P}_{C;\mathcal {A}}})\) such that \(x=T^{[q]}(x)\) \((y=T^{[q]}(y))\).
For the definition and properties of closed maps in topological spaces, see [39, 40].
The main results of this paper are the following theorems.
Theorem 3.1
Let \((X,\mathcal{P}_{C;\mathcal{A}})\) be a quasi-triangular space, and let \((X,T)\) be a single-valued dynamic system with \(T:X\rightarrow X\). Assume that the following three conditions hold:
-
(H1)
\(\mathcal{J}_{C;\mathcal{A}}\) is the left (right) family generated by \(\mathcal{P}_{C;\mathcal{A}}\).
-
(H2)
\((X,T)\) is a \(\mathcal{J}_{C;\mathcal{A}}\)-contraction on X, that is,
$$\begin{aligned} &\forall_{x,y\in X}\forall_{\alpha\in\mathcal{A}}\forall _{\varepsilon >0} \exists_{\eta>0}\exists_{r\in\mathbb{N}}\forall_{s,l\in\mathbb {N}}\bigl\{ J_{\alpha}\bigl(T^{[s]}(x),T^{[l]}(y)\bigr)< \varepsilon+ \eta \\ &\quad\Rightarrow C_{\alpha}J_{\alpha }\bigl(T^{[s+r]}(x),T^{[l+r]}(y) \bigr)< \varepsilon\bigr\} . \end{aligned}$$(3.5) -
(H3)
There exists a set \(W^{L-\mathcal{J}_{C;\mathcal{A}}}\in 2^{X}\) \((W^{R-\mathcal{J}_{C;\mathcal{A}}}\in2^{X})\) such that \((X,T)\) is left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-admissible on \(W^{L-\mathcal {J}_{C;\mathcal{A}}}\) \((W^{R-\mathcal{J}_{C;\mathcal{A}}})\).
Then the following statements hold:
-
(A)
For each \(w^{0}\in W^{L-\mathcal{J}_{C;\mathcal{A}}}\) \((w^{0}\in W^{R-\mathcal{J}_{C;\mathcal{A}}})\), there exists a point \(w\in X\) such that the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-convergent to w.
-
(B)
If the single-valued dynamic system \((X,T^{[q]})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-closed on \(W^{L-\mathcal {J}_{C;\mathcal{A}}}\) \((W^{R-\mathcal{J}_{C;\mathcal{A}}})\) for some \(q\in\mathbb{N}\), then:
-
(B1)
\(Fix(T^{[q]})\neq\varnothing\);
-
(B2)
For each \(w^{0}\in W^{L-\mathcal{J}_{C;\mathcal{A}}}\) \((w^{0}\in W^{R-\mathcal{J}_{C;\mathcal{A}}})\), there exists a point \(w\in Fix(T^{[q]})\) such that the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-convergent to w; and
-
(B3)
\(\forall_{\alpha\in\mathcal{A}}\forall_{v\in Fix(T^{[q]})}\{J_{\alpha}(v,T(v))=J_{\alpha}(T(v),v)=0\}\).
-
(B1)
-
(C)
If the family \(\mathcal{P}_{C;\mathcal{A}}=\{p_{\alpha },\alpha \in\mathcal{A}\}\) is separating on X and if the single-valued dynamic system \((X,T^{[q]})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-closed on \(W^{L-\mathcal{J}_{C;\mathcal{A}}}\) \((W^{R-\mathcal{J}_{C;\mathcal{A}}})\) for some \(q\in\mathbb{N}\), then:
-
(C1)
There exists a point \(w\in X\) such that \(Fix(T^{[q]})=Fix(T)=\{w\}\);
-
(C2)
For each \(w^{0}\in W^{L-\mathcal{J}_{C;\mathcal{A}}}\) \((w^{0}\in W^{R-\mathcal{J}_{C;\mathcal{A}}})\), the sequence \((w^{m}=T^{[m]}(w^{0}):m\in \{0\}\cup\mathbb{N})\) is left (right) \(\mathcal {P}_{C;\mathcal{A}}\)-convergent to w; and
-
(C3)
\(\forall_{\alpha\in\mathcal{A}}\{J_{\alpha}(w,w)=0\}\).
-
(C1)
Theorem 3.2
Let \((X,\mathcal{P}_{C;\mathcal{A}})\) be a quasi-triangular space, and let \((X,T)\) be a single-valued dynamic system with \(T:X\rightarrow X\). Assume that the following three conditions hold:
-
(H1)
\(\mathcal{J}_{C;\mathcal{A}}\) is the left (right) family generated by \(\mathcal{P}_{C;\mathcal{A}}\).
-
(H2)
\((X,T)\) is a weak \(\mathcal{J}_{C;\mathcal{A}}\)-contraction on X, that is, there exists \(w^{0}\in X\) such that
$$\begin{aligned} &\forall_{\alpha\in\mathcal{A}}\forall_{\varepsilon>0}\exists_{\eta >0} \exists_{r\in\mathbb{N}}\forall_{s,l\in\mathbb{N}}\bigl\{ J_{\alpha } \bigl(T^{[s]}\bigl(w^{0}\bigr),T^{[l]} \bigl(w^{0}\bigr)\bigr)< \varepsilon+\eta \\ &\quad \Rightarrow C_{\alpha}J_{\alpha }\bigl(T^{[s+r]}\bigl(w^{0} \bigr),T^{[l+r]}\bigl(w^{0}\bigr)\bigr)< \varepsilon\bigr\} . \end{aligned}$$(3.6) -
(H3)
There exists a set \(W^{L-\mathcal{J}_{C;\mathcal{A}}}\in 2^{X}\) \((W^{R-\mathcal{J}_{C;\mathcal{A}}}\in2^{X})\) such that \((X,T)\) is left (right) \(\mathcal{J}_{C;\mathcal{A}}\)-admissible on \(W^{L-\mathcal {J}_{C;\mathcal{A}}}\) \((W^{R-\mathcal{J}_{C;\mathcal{A}}})\) and \(w^{0}\in W^{L- \mathcal{J}_{C;\mathcal{A}}}\) \((w^{0}\in W^{R-\mathcal{J}_{C;\mathcal{A}}})\).
Then the following statements hold:
-
(A)
There exists a point \(w\in X\) such that the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-convergent to w.
-
(B)
If the single-valued dynamic system \((X,T^{[q]})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-closed on \(W^{L-\mathcal {J}_{C;\mathcal{A}}}\) \((W^{R-\mathcal{J}_{C;\mathcal{A}}})\) for some \(q\in\mathbb{N}\), then:
-
(B1)
\(Fix(T^{[q]})\neq\varnothing\);
-
(B2)
There exists a point \(w\in Fix(T^{[q]})\) such that the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-convergent to w; and
-
(B3)
If \(w^{0}\in Fix(T^{[q]})\), then \(\forall_{\alpha\in\mathcal{A}}\{J_{\alpha}(w^{0},T(w^{0}))=J_{\alpha}(T(w^{0}),w^{0})=0\}\).
-
(B1)
-
(C)
If the family \(\mathcal{P}_{C;\mathcal{A}}=\{p_{\alpha },\alpha \in\mathcal{A}\}\) is separating on X and if the single-valued dynamic system \((X,T^{[q]})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-closed on \(W^{L-\mathcal{J}_{C;\mathcal{A}}}\) \((W^{R-\mathcal{J}_{C;\mathcal{A}}})\) for some \(q\in\mathbb{N}\), then:
-
(C1)
There exists a point \(w\in X\) such that \(Fix(T^{[q]})=Fix(T)=\{w\}\);
-
(C2)
The sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left (right) \(\mathcal{P}_{C;\mathcal{A}}\)-convergent to w; and
-
(C3)
If \(w^{0}=w\), then \(\forall_{\alpha\in\mathcal{A}}\{J_{\alpha }(w^{0},w^{0})=0\}\).
-
(C1)
4 Proofs of Theorems 3.1 and 3.2
Proof of Theorem 3.1
The proof is divided into eleven steps and is only in the case of ‘left’; we omit the proof in the case of ‘right’ since it is based on an analogous technique.
For all \(u^{0},v^{0}\in X\), \(\alpha\in\mathcal{A}\), and \(k\in\mathbb{N}\), we define
where \(u^{m}=T^{[m]}(u^{0})\) and \(v^{m}=T^{[m]}(v^{0})\), \(m\in\{0\}\cup \mathbb{N}\).
Step 1. We have the following property:
Indeed, let \(u^{0},v^{0}\in X\) be arbitrary and fixed. If we assume that \(\alpha\in\mathcal{A}\) and \(\varepsilon>0\) are arbitrary and fixed, then, using (3.5) for \(x=u^{0}\) and \(y=v^{0}\), we obtain \(\exists_{\eta _{1}>0}\exists_{r_{1}\in\mathbb{N}}\forall_{s,l\in\mathbb{N}}\{ J_{\alpha }(u^{s},v^{l})<\varepsilon+\eta_{1}\Rightarrow C_{\alpha}J_{\alpha }(u^{s+r_{1}},v^{l+r_{1}})<\varepsilon\}\), and, using (3.5) for \(x=v^{0}\) and \(y=u^{0}\), we obtain \(\exists_{\eta_{2}>0}\exists_{r_{2}\in \mathbb{N}}\forall_{s,l\in\mathbb{N}}\{J_{\alpha}(v^{s},u^{l})<\varepsilon +\eta _{2}\Rightarrow C_{\alpha}J_{\alpha }(v^{s+r_{2}},u^{l+r_{2}})<\varepsilon \}\). Hence, putting \(\eta=\min\{\eta_{1},\eta_{2}\}\), we have
and
This gives (4.5).
Step 2. We show that
and
Indeed, if (4.6) is false, then
With this choice of \(u^{0},v^{0},\alpha_{0}\), and \(\varepsilon_{0}\) we can use hypothesis (3.5); then there exist \(\eta_{0}>0\) and \(r_{0}\in \mathbb{N}\) such that
Further, by (4.1), \(\delta_{\mathcal{J}_{C;\mathcal{A}};\alpha _{0},k_{0}}(u^{0},v^{0})=\inf\{\Delta_{\mathcal{J}_{C;\mathcal {A}};\alpha _{0},k_{0}}(u^{0},v^{0},n):n\in\mathbb{N}\}\). This implies, using (4.8), that
Next, in view of (4.3), we have that \(\Delta_{\mathcal{J}_{C;\mathcal {A}};\alpha_{0},k_{0}}(u^{0},v^{0},n_{0})=\max\{J_{\alpha _{0}}(u^{s},v^{l}):n_{0}\leq s,l\leq n_{0}+k_{0}\}\). Thus,
and, using (4.9), we get \(\forall_{n_{0}\leq s,l\leq n_{0}+k_{0}}\{C_{\alpha_{0}}J_{\alpha _{0}}(u^{s+r_{0}},v^{l+r_{0}})<\varepsilon_{0}\}\), which we can write as
Now, note that
in view of (4.3). Consequently,
Finally, from (4.8), (4.1), and (4.10) it follows that
which is impossible. Therefore, (4.6) holds.
The proof of (4.7) is identical to the proof of (4.6) and is omitted.
Step 3. Let \(u^{0}\in X\), \(\alpha\in\mathcal{A}\), and \(\varepsilon>0\) be arbitrary and fixed. Let \(u^{m}=T^{[m]}(u^{0})\) for \(m\in\{0\}\cup\mathbb{N}\), and let
We show that
and
By (4.1), (4.2), (4.6), and (4.7), \(\delta_{\mathcal{J}_{C;\mathcal{A}};\alpha,r}(u^{0},u^{0})=\inf\{\Delta_{\mathcal{J}_{C;\mathcal {A}};\alpha ,r}(u^{0},u^{0},n):n\in\mathbb{N}\}=0\), \(\gamma_{\mathcal {J}_{C;\mathcal{A}};\alpha,r}(u^{0},u^{0})=\inf\{\Gamma_{\mathcal{J}_{C;\mathcal {A}};\alpha ,r}(u^{0},u^{0},n):n\in\mathbb{N}\}=0\), and \(\forall_{n\in\mathbb{N}}\{\Delta_{\mathcal{J}_{C;\mathcal{A}};\alpha ,r}(u^{0},u^{0},n)=\Gamma_{\mathcal{J}_{C;\mathcal{A}};\alpha,r}(u^{0},u^{0},n)\}\). Then there exists \(n_{0}\in\mathbb{N}\) such that (4.12) holds.
By (4.3) and (4.4) we see that (4.12) implies
First, we establish that
If (4.15) is false, then \(\exists_{l\geqslant n_{0}}\{C_{\alpha }J_{\alpha }(u^{n_{0}+r},u^{l})\geq\varepsilon\}\), that is,
Denote
Of course, in view of (4.14), this gives
Note that
Next, note that also
Otherwise, \(l_{0}\leq n_{0}+r\), and by (4.13) we get \(C_{\alpha }J_{\alpha}(u^{n_{0}+r},u^{l_{0}})\leq\max\{C_{\alpha}J_{\alpha }(u^{i},u^{j}): n_{0}\leq i,j\leq n_{0}+r\} =C_{\alpha}\Delta_{\mathcal{J}_{C;\mathcal{A}};\alpha,r}(u^{0},u^{0},n_{0})<\min\{ \varepsilon ,\eta\}\leq\varepsilon\), which, in view of (4.16), (4.17), and (4.19), is impossible. Thus, (4.20) holds.
In view of (4.18) and (4.20), we have that \(n_{0}< l_{0}-r< l_{0}\), and, consequently, using (4.16) and (4.17), we conclude that
Next, using (\(\mathcal{J}\)1) of Definition 2.1, (4.13), (4.14), and (4.21), we obtain
Hence, since r satisfies (4.11), we get \(C_{\alpha}J_{\alpha }(u^{n_{0}+r},u^{l_{0}})<\varepsilon\). In view of (4.16) and (4.17), this is impossible.
Consequently, (4.15) holds.
We can show in a similar way that
the proof of (4.22) is identical to the proof of (4.15) and is omitted.
To establish (4.13), we see that by (\(\mathcal{J}\)1) of Definition 2.1, (4.22), and (4.15) we obtain
Step 4. Let \(w^{0}\in W^{L-\mathcal{J}_{C;\mathcal{A}}}\) be arbitrary and fixed. Define the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N)}\). We show that
Indeed, let \(\alpha_{0}\) and \(\varepsilon_{0}\) be arbitrary and fixed. By (3.5) we get
Next, by (4.3) and (4.4) we have \(\forall_{\alpha\in\mathcal {A}}\forall _{k\in\mathbb{N}}\forall_{n\in\mathbb{N}}\{\Delta_{\mathcal{J}_{C;\mathcal{A}};\alpha,k}(w^{0},w^{0},n)=\Gamma_{\mathcal{J}_{C;\mathcal {A}};\alpha,k}(w^{0}, w^{0},n)\}\). Moreover, by Step 2 we have \(\forall _{\alpha\in\mathcal{A}}\forall_{k\in\mathbb{N}}\{\delta_{\mathcal {J}_{C;\mathcal{A}};\alpha,k}(w^{0},w^{0})=\gamma_{\mathcal{J}_{C;\mathcal{A} };\alpha,k}(w^{0},w^{0})=0\}\). Hence, it follows that there exists \(n_{0}\in \mathbb{N}\) such that
Now, from (4.24), using Step 3, we get \(\forall_{s,l\geq n_{0}}\{ J_{\alpha _{0}}(w^{s},w^{l})<2(\varepsilon_{0}/4)=\varepsilon_{0}/2\}\). This proves that (4.23) holds.
Step 5. Let \(w^{0}\in W^{L-\mathcal{J}_{C;\mathcal{A}}}\) be arbitrary and fixed. Define the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N)}\). We show that
and
Indeed, in view of (4.23), we obtain, in particular, that
and
From this it follows that
and
and hence (4.25) and (4.26) hold.
Step 6. Statement (A) holds.
Indeed, let \(w^{0}\in W^{L-\mathcal{J}_{C;\mathcal{A}}}\) be arbitrary and fixed. Define the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb {N)}\). By (4.25), Definition 2.2, and hypothesis (H3) we get that this sequence is left \(\mathcal{J}_{C;\mathcal{A}}\)-convergent in X, that is, there exists a nonempty set \(LIM_{(w^{m}:m\in\{0\}\cup\mathbb {N})}^{L-\mathcal{J}_{C;\mathcal{A}}}\subset X\) such that
However, by hypothesis (H1), \(\mathcal{J}_{C;\mathcal{A}}\) is left family generated by \(\mathcal{P}_{C;\mathcal{A}}\). Therefore, fixing \(w\in LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal{J}_{C;\mathcal{A}}}\), defining \((u_{m}=w^{m}:m\in\{0\}\cup\mathbb{N})\) and \((v_{m}=w:m\in \{0\}\cup\mathbb{N})\), and using (4.25) and (4.27) we obtain
By Definition 2.1 this gives \(\forall_{\alpha\in\mathcal{A}}\{\lim_{m\rightarrow\infty}p_{\alpha}(v_{m},u_{m})=0\}\), which means that
Therefore, \(LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L- \mathcal{P}_{C;\mathcal{A}}}\neq\varnothing\).
Step 7. Conclusions (B1) and (B2) hold.
Let \(w^{0}\in W^{L-\mathcal{J}_{C;\mathcal{A}}}\) be arbitrary and fixed. Define the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\).
First, we prove that \(Fix(T^{[q]})\neq\varnothing\) and that there exists a point \(w\in Fix(T^{[p]})\) such that this sequence is left \(\mathcal{P}_{C;\mathcal{A}}\)-convergent to w. Indeed, by statement (A), \(LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal {P}_{C;\mathcal{A}}}\neq\varnothing\). Next, for this sequence, we have \(w^{mq+k}=T^{[q]}(w^{(m-1)q+k})\) for \(k=1,2,\ldots,q\) and \(m\in\mathbb{N}\). Let, in the sequel, \(k=1,2,\ldots,q\) be arbitrary and fixed. Defining \((z_{m}=w^{m-1+q}:m\in\mathbb{N})\), we see that \(\varnothing\neq LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal{P}_{C;\mathcal{A}}}=LIM_{(z_{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal{P}_{C;\mathcal {A}}}\) and that the sequences \((v_{m}=w^{mq+k}:m\in\mathbb{N})\) and \((u_{m}=w^{(m-1)q+k}:m\in\mathbb{N})\) satisfy \(\forall_{m\in\mathbb{N}}\{ v_{m}=T^{[q]}(u_{m})\}\) and, as subsequences of \((w^{m}:m\in\{0\}\cup\mathbb{N})\), are left \(\mathcal{P}_{C;\mathcal{A}}\)-convergent to each point of \(w\in LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal{P}_{C;\mathcal{A}}}\). Clearly, by Remark 2.2, \(LIM_{(z_{m}:m\in\mathbb{N})}^{L-\mathcal{P}_{C;\mathcal{A}}}\subset LIM_{(v_{m}:m\in\mathbb{N})}^{L-\mathcal {P}_{C;\mathcal{A}}}\) and \(LIM_{(z_{m}:m\in\mathbb{N})}^{L-\mathcal{P}_{C;\mathcal {A}}}\subset LIM_{(u_{m}:m\in\mathbb{N})}^{L-\mathcal{P}_{C;\mathcal{A}}}\). Therefore, since \(T^{[q]}\) is left \(\mathcal{P}_{C;\mathcal{A}}\)-closed on \(W^{L-\mathcal{J}_{C;\mathcal{A}}}\), in virtue of Definition 3.2, we get \(\exists _{w\in LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal {P}_{C;\mathcal{A}}}=LIM_{(z_{m}:m\in\mathbb{N})}^{L-\mathcal{P}_{C;\mathcal{A}}}}\{w=T^{[q]}(w)\}\). Consequently, \(Fix(T^{[q]})\neq\varnothing\), and there exists a point \(w\in Fix(T^{[q]})\) such that the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left \(\mathcal{P}_{C;\mathcal{A}}\)-convergent to w, so (B1) and (B2) hold.
Step 8. Conclusion (B3) holds.
Suppose that
If \(J_{\alpha_{0}}(w,T(w))>0\), then, putting
by (3.5) we get
Since
we have
Hence, using (4.29) for \(s=q\) and \(l=q+1\), we get
We note that then, in particular, since \(C_{\alpha_{0}}\geq1\),
Next, by (4.31), using (4.29) for \(s=q+r_{0}\) and \(l=q+1+r_{0}\), we have
Hence, since \(C_{\alpha_{0}}\geq1\),
Using (4.31) and (4.32), by induction we have that (4.28) gives
Putting \(m=q\) in (4.33), we find
Finally, by (4.28), (4.31), and (4.34) we obtain that
a contradiction. Therefore, \(J_{\alpha_{0}}(w,T(w))=0\).
Similarly, we prove that \(J_{\alpha_{0}}(T(w),w)=0\). We proved that (B3) holds.
Step 9. Conclusion (C1) holds.
First, we show that \(Fix(T^{[q]})=Fix(T)\neq\varnothing\). Indeed, let \(w\in Fix(T^{[q]})\). Then, by (B3), \(\forall _{\alpha \in\mathcal{A}}\{J_{\alpha}(w,T(w))=J_{\alpha}(T(w),w)=0\}\). By Theorem 2.2 this gives \(w=T(w)\), i.e. \(w\in Fix(T)\). Consequently, \(Fix(T^{[q]})=Fix(T)\neq\varnothing\).
Next, we show that \(Fix(T)=\{w\}\) for some \(w\in X\). Otherwise, \(u,v\in Fix(T)\) and \(u\neq v\) for some \(u,v\in X\); recall that, by the preceding, \(Fix(T)\neq\varnothing\). Then, by Theorem 2.2 there exists \(\alpha _{0}\in \mathcal{A}\) such that \(J_{\alpha_{0}}(u,v)>0\) or \(J_{\alpha_{0}}(v,u)>0\). Suppose \(J_{\alpha_{0}}(u,v)>0\). Then, for \(\varepsilon_{0}=J_{\alpha _{0}}(u,v)>0\), by (3.5) there exist \(\eta_{0}>0\) and \(r_{0}\in\mathbb {N}\) such that
However, for all \(s,l\in\mathbb{N}\), we have \(J_{\alpha _{0}}(T^{[s]}(u),T^{[l]}(v))=J_{\alpha_{0}}(u,v)=\varepsilon _{0}<\varepsilon_{0}+\eta_{0}\), and thus by (4.35) we get \(0<\varepsilon _{0}= J_{\alpha_{0}}(u,v)=J_{\alpha _{0}}(T^{[s+r_{0}]}(u),T^{[l+r_{0}]}(v))<C_{\alpha_{0}}J_{\alpha _{0}}(T^{[s+r_{0}]}(u),T^{[l+r_{0}]}(v))<\varepsilon_{0}\), which is impossible. We obtain a similar implication in the case where \(J_{\alpha _{0}}(v,u)>0\). Therefore, \(Fix(T)=\{w\}\) for some \(w\in X\), so (C1) holds.
Step 10. Conclusion (C2) holds.
Step 9 with (B) means that, for each \(w^{0}\in W^{L-\mathcal{J}_{C;\mathcal{A}}}\), the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup \mathbb{N})\) is left \(\mathcal{P}_{C;\mathcal{A}}\)-convergent to w. Thus, (C2) holds.
Step 11. Conclusion (C3) holds.
Finally, we show that \(\forall_{\alpha\in\mathcal {A}}\{J_{\alpha}(w,w)=0\}\), where \(Fix(T)=\{w\}\). Indeed, if we assume that there exists \(\alpha_{0}\in\mathcal{A}\) such that \(J_{\alpha _{0}}(w,w)>0\), then, denoting \(\varepsilon_{0}=J_{\alpha_{0}}(w,w)>0\), by (3.5) there exist \(\eta_{0}>0\) and \(r_{0}\in\mathbb{N}\) such that
However, for all \(s,l\in\mathbb{N}\), we have \(J_{\alpha _{0}}(T^{[s]}(w),T^{[l]}(w))=J_{\alpha_{0}}(w,w)=\varepsilon _{0}<\varepsilon_{0}+\eta_{0}\). Thus, using (4.36), we obtain that
which is impossible. Therefore, (C3) holds.
The proof of Theorem 3.1 is complete. □
Proof of Theorem 3.2
Assume that condition (3.6) holds. Then, defining \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N),}\) where \(w^{0}\in X\) is as in (3.6), and next, using a similar argument as in the proof of Theorem 3.1 for this sequence, we have the assertions. □
5 Examples
Example 5.1
Let \(X=(0;3)\). For \(A=(1/2;2]\) and \(\gamma>0 \), we let \(p:X^{2}\rightarrow[0;\infty)\) be a distance of the form
Define \(T:X\rightarrow X\) by
(1) Notice that \((X,\mathcal{P}_{\{1\};\{1\}})\), \(\mathcal{P}_{\{1\};\{1\}}=\{p\}\), is a triangular space. See Definition 1.2(E) and [27], Example 2, p.11; p does not vanish on the diagonal, is symmetric, and is triangular.
(2) We show that \((X,T)\) is \(\mathcal{P}_{\{1\};\{1\}}\)-contraction on X, that is,
First, we claim that
Indeed, observing that
and
we see that (5.5) and (5.6) imply (5.4). Now, using (5.4) and (5.1), we obtain
In view of (5.7), we conclude that (5.3) holds.
(3) We show that \((X,T)\) is left and right \(\mathcal{P}_{\{1\};\{1\}}\)-admissible in each point \(w^{0}\in X\); thus, by Definition 3.1 , \(W^{L-\mathcal{P}_{\{1\};\{1\}}}=W^{R-\mathcal{P}_{\{1\};\{1\}}}=X\). Indeed, it is clear that, for arbitrary and fixed \(w^{0}\in X\), the sequence \((w^{m}:m\in\{0\}\cup\mathbb{N})\), where \(\forall_{m\in\{0\}\cup \mathbb{N}}\{w^{m+1}=T(w^{m})\}\) satisfies
Then, in view of (5.8) and (5.1),
and also
Thus, \(LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal{P}_{\{1\};\{1\}}}=LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{R-\mathcal{P}_{\{1\};\{1\}}}=A\neq\varnothing\).
(4) We show that the single-valued dynamic system \((X,T^{[2]})\) is left and right \(\mathcal{P}_{\{1\};\{1\}}\)-closed on \(W^{L-\mathcal{P}_{\{1\};\{1\}}}=W^{R-\mathcal{P}_{\{1\};\{1\}}}=X\). Indeed, if \(w^{0}\in X\) is arbitrary and fixed and if \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is a left or right \(\mathcal{P}_{\{1\};\{1\}}\)-converging sequence in X having subsequences \((v_{m}:m\in\mathbb{N})\) and \((u_{m}:m\in\mathbb{N})\) satisfying \(\forall_{m\in\mathbb{N}}\{v_{m}=T^{[2]}(u_{m})\}\), then by (5.4), (5.1), and (5.6) we have that \(A=LIM_{(w^{m}:m\in\{0\}\cup \mathbb{N})}^{L-\mathcal{P}_{\{1\};\{1\}}}=LIM_{(w^{m}:m\in\{0\}\cup\mathbb {N})}^{R-\mathcal{P}_{\{1\};\{1\}}}\) and \(\{1=T^{[2]}(1),2=T^{[2]}(2)\}\subset A\). Hence, in virtue of Definition 3.2, \(U=V=W^{L-\mathcal{P}_{\{1\};\{1\}}}=W^{R-\mathcal{P}_{\{1\};\{1\}}}=X\).
(5) \(\mathcal{P}_{\{1\};\{1\}}=\{p\}\) is not separating on X. This follows from Definition 1.2(B) since, for each \(x,y\in X\) such that \(A\cap\{x,y\}\neq\{x,y\}\), we have \(p(x,y)=p(y,x)=\gamma>0\).
Claim
It follows from (1)-(5) that, for \((X,\mathcal{P}_{\{1\};\{1\}})\), \(\mathcal{P}_{\{1\};\{1\}}=\{p\}\), \((X,T)\), and \(\mathcal{J}_{\{1\};\{1\}}=\mathcal{P}_{\{1\};\{1\}}\) defined by (5.1) and (5.2) and for \(q=2\), statements (A) and (B) of Theorem 3.1 hold: (a) Statement (A) holds since, for each \(w^{0}\in X\), the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left and right \(\mathcal{P}_{\{1\};\{1\}}\)-convergent to each point \(w\in A\); \(W^{L-\mathcal{P}_{\{1\};\{1\}}}=W^{R-\mathcal{P}_{\{1\};\{1\}}}=X\) by (3). (b) We have that \(Fix(T^{[2]})=\{1,2\}\); thus, conclusion (B1) holds. (c) Conclusion (B2) follows from statement (A) and conclusion (B1) since \(Fix(T^{[2]})\subset A\). (d) Conclusion (B3) holds; by (5.1), we have \(\forall _{v\in Fix(T^{[2]})=\{1,2\}}\{p(v,T(v))=p(T(v),v)=0\}\) since \(T(1)=2\), \(T(2)=1\), and thus \(T(\{1,2\})=\{1,2\}=Fix(T^{[2]})\subset A\). (e) By (5), \((X,\mathcal{P}_{\{1\};\{1\}})\) is not separable. (f) We see that \(Fix(T)=\varnothing\) and that statement (C) does not hold.
Example 5.2
Let \(X=(0;1)\cup(1;2]\), and let \(p:X^{2}\rightarrow[0;\infty)\) be of the form
Define \(T:X\rightarrow X\) by
(1) Notice that \((X,\mathcal{P}_{\{8\};\{1\}})\), \(\mathcal{P}_{\{8\};\{1\}}=\{p\}\), is a quasi-triangular space. See Definition 1.2(A) and [27], Example 1, p.10; p vanishes on the diagonal, is asymmetric, and is quasi-triangular since \(\forall _{u,v,w\in X}\{p(u,w)\leq8[p(u,v)+p(v,w)]\}\).
(2) We show that \((X,T)\) is \(\mathcal{P}_{\{8\};\{1\}}\)-contraction, that is,
Indeed, by (5.10),
Next, in the sequel, let \(\varepsilon>0\) be arbitrary and fixed, and let \(\eta=\varepsilon\). Without loss of generality, to prove (5.11), it suffices to prove that if \(x,y\in X\) and \(s,l\in\mathbb{N}\) are such that \(W_{x,y}^{s,l}<2\varepsilon\), then there exists \(r\in\mathbb{N}\) such that \(8\cdot W_{x,y}^{s+r,l+r}<\varepsilon\), where
With this aim, we consider the following cases:
Case A. Let \(x,y\in(0;1)\).
(A1) If \(s,l,r\in\mathbb{N}\) and \(T^{[s]}(x)=2-(1-x^{2})/2^{s-1}\geq T^{[l]}(y)=2-(1-y^{2})/2^{l-1}\), then \(T^{[s+r]}(x)=2-(1-x^{2})/2^{s+r-1}\geq T^{[l+r]}(y)=2-(1-y^{2})/2^{l+r-1}\), and by (5.9) and (5.13) we get that \(W_{x,y}^{s,l}=W_{x,y}^{s+r,l+r}=0\), that is, (5.11) holds.
(A2) If \(s,l,r\in\mathbb{N}\) and \(T^{[s]}(x)=2-(1-x^{2})/2^{s-1}< T^{[l]}(y)=2-(1-y^{2})/2^{l-1}\), then \(T^{[s+r]}(x)=2-(1-x^{2})/2^{s+r-1}< T^{[l+r]}(y)=2-(1-y^{2})/2^{l+r-1}\), and by (5.9) and (5.13) we get that
whenever s and l are sufficiently great, and, for such s and l,
whenever also r is sufficiently great. Thus, (5.11) holds.
Case B. Let \(x,y\in(1;2]\).
(B1) If \(s,l,r\in\mathbb{N}\) and \(T^{[s]}(x)=2-(2-x)/2^{s}\geq T^{[l]}(y)=2-(2-y)/2^{l}\), then \(T^{[s+r]}(x)=2-(2-x)/2^{s+r}\geq T^{[l+r]}(y)=2-(2-y)/2^{l+r}\), and by (5.9) and (5.13) we get that \(W_{x,y}^{s,l}=W_{x,y}^{s+r,l+r}=0\), that is, (5.11) holds.
(B2) If \(s,l,r\in\mathbb{N}\) and \(T^{[s]}(x)=2-(2-x)/2^{s}< T^{[l]}(y)=2-(2-y)/2^{l}\), then \(T^{[s+r]}(x)=2-(2-x)/2^{s+r}< T^{[l+r]}(y)=2-(2-y)/2^{l+r}\), and by (5.9) and (5.13) we get that
whenever s and l are sufficiently great, and, for such s and l,
whenever also r is sufficiently great. Thus, (5.11) holds.
Case C. Let \(x\in(0;1)\) and \(y\in(1;2]\).
(C1) If \(s,l,r\in\mathbb{N}\) and \(T^{[s]}(x)=2-(1-x^{2})/2^{s-1}\geq T^{[l]}(y)=2-(2-y)/2^{l}\), then \(T^{[s+r]}(x)=2-(1-x^{2})/2^{s+r-1}\geq T^{[l+r]}(y)=2-(2-y)/2^{l+r}\), and by (5.9) and (5.13) we get that \(W_{x,y}^{s,l}=W_{x,y}^{s+r,l+r}=0\), that is, (5.11) holds.
(C2) If \(s,l,r\in\mathbb{N}\) and \(T^{[s]}(x)=2-(1-x^{2})/2^{s-1}< T^{[l]}(y)=2-(2-y)/2^{l}\), then \(T^{[s+r]}(x)=2-(1-x^{2})/2^{s+r-1}< T^{[l+r]}(y)=2-(2-y)/2^{l+r}\), and by (5.9) and (5.13) we get that
whenever s and l are sufficiently great, and, for such s and l,
whenever also r is sufficiently great. Thus, (5.11) holds.
Case D. Let \(x\in(1;2]\) and \(y\in(0;1)\). Then, with analogous consideration as in Case C, we obtain that (5.11) holds.
(3) We show that \((X,T)\) is left and right \(\mathcal{P}_{\{8\};\{1\}}\)-admissible in each point \(w^{0}\in X\); thus, by Definition 3.1 , \(W^{L-\mathcal{P}_{\{8\};\{1\}}}=W^{R- \mathcal{P}_{\{8\};\{1\}}}=X\). Indeed, let \(w^{0}\in X\) be arbitrary and fixed, and let the sequence \((w^{m}:m\in \{0\}\cup\mathbb{N})\) be defined by \(\forall_{m\in\{0\}\cup\mathbb {N}}\{w^{m+1}=T(w^{m})\}\). We consider the cases:
Case A. Let \(w^{0}\in(0;1)\). Then \(w^{m}=2-[1-(w^{0})^{2}]/2^{m-1}< w^{n}=2-[1-(w^{0})^{2}]/2^{n-1}\) for \(n>m\). Using (5.9), we then have \(p(w^{m},w^{n})=[1-(w^{0})^{2}]^{4}(1/2^{4(m-1)})(1-1/2^{n-m})^{4}\) and \(p(w^{n},w^{m})=0\). Hence, we conclude that
and \(\lim_{m\rightarrow\infty}\sup_{n>m}p(w^{n},w^{m})=0\), respectively.
Next, observe that
thus, \(LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal{P}_{\{8\};\{1\}}}\neq\varnothing\) and \(LIM_{(w^{m}:m\in\{0\}\cup \mathbb{N})}^{R-\mathcal{P}_{\{8\};\{1\}}}\neq\varnothing\). In fact, by (5.9) we have
Moreover, by (5.9), \(\forall _{w\in (0;1)\cup(1;2)}\{\lim_{m\rightarrow\infty}p(w^{m},w)=0\}\).
Case B. Let \(w^{0}\in(1;2]\). Then \(w^{m}=2-(2-w^{0})/2^{m}< w^{n}=2-[2-w^{0}]/2^{n}\) for \(n>m\), and thus, by (5.9), \(p(w^{m},w^{n})=[2-w^{0}]^{4}(1/2^{4m})(1-1/2^{n-m})^{4}\), \(p(w^{n},w^{m})=0\). Consequently, we get
In addition, we see that \(\lim_{m\rightarrow\infty }p(2,w^{m})=\lim_{m\rightarrow\infty}[2-w^{0}]^{4}/2^{4m}=0\) and
Thus, \(\{2\}\subset LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal {P}_{\{8\};\{1\}}}\neq\varnothing\) and \((0;1)\cup(1;2)\subset LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{R-\mathcal{P}_{\{8\};\{1\} }}\neq \varnothing\).
(4) We show that \((X,T)\) is left \(\mathcal{P}_{\{8\};\{1\}}\)-closed on \(X=W^{L-\mathcal{P}_{\{8\};\{1\}}}\). Indeed, by Definition 3.2, if \(w^{0}\in X\) is arbitrary and fixed and if \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup \mathbb{N)}\) is a left \(\mathcal{P}_{\{8\};\{1\}}\)-converging sequence in X having subsequences \((v_{m}:m\in\mathbb{N})\) and \((u_{m}:m\in\mathbb{N})\) satisfying \(\forall_{m\in\mathbb{N}}\{v_{m}=T(u_{m})\}\), then by (5.9) (5.10), and (5.12) we have that \(2\in LIM_{(w^{m}:m\in\{0\}\cup\mathbb {N})}^{L-\mathcal{P}_{\{8\};\{1\}}}\) and \(2=T(2)\). Therefore, \(U=W^{L-\mathcal{P}_{\{8\};\{1\}}}=X\). Clearly, \(V=\varnothing\).
(5) \(\mathcal{P}_{\{8\};\{1\}}=\{p\}\) is separating on X. Indeed, for each \(x,y\in X\) such that \(x\neq y\), we have \(p(x,y)>0\) or \(p(y,x)>0\). Thus, (1.4) holds.
Claim
It follows from (1)-(5) that, for \((X,\mathcal{P}_{\{8\};\{1\}})\), \(\mathcal{P}_{\{8\};\{1\}}=\{p\}\), \((X,T)\), and \(\mathcal{J}_{\{8\};\{1\}}=\mathcal{P}_{\{8\};\{1\}}\) defined by (5.9) and (5.10) and for \(q=1\), statements (A), (B), and (C) of Theorem 3.1 in the left case hold: (a) We have \(Fix(T)=\{2\}\). (b) Conclusions (B3) and (C3) hold since by (5.9) we have \(p(2,T(2))=p(T(2),2)=p(2,2)=0\). (c) For each \(w^{0}\in X=W^{L-\mathcal{P}_{\{8\};\{1\}}}\), the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left \(\mathcal{P}_{\{8\};\{1\}}\)-convergent to 2.
Example 5.3
Let \(X=(0;3]\), and let \(p:X^{2}\rightarrow [0;\infty)\) be of the form
Define \(T:X\rightarrow X\) by
(1) Observe that \((X,\mathcal{P}_{\{8\};\{1\}})\), \(\mathcal{P}_{\{8\};\{1\}}=\{p\}\), is a separable quasi-triangular space.
(2) When \(\mathcal{J}_{\{8\};\{1\}}=\mathcal{P}_{\{8\};\{1\}}\), we will show that: (a) \(Fix(T)=\{1,2,3\}\), (b) for each \(w^{0}\in X\), there exists \(w\in Fix(T)\) such that the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left \(\mathcal{P}_{\{8\};\{1\}}\)-convergent to w, and (c) \(\forall_{w\in Fix(T)}\{p(w,w)=0\}\).
To prove this, we see that (5.15) implies
This means that, for each \(k=\{1,2,3\}\), \(T_{k}:X_{k}\rightarrow X_{k}\) where \(X_{k}=(k-1;k]\), and \((X_{k},\mathcal{P}_{\{8\};\{1\}})\) is a separable quasi-triangular space.
It is not hard to show that, for each \(k=\{1,2,3\}\), we have: (a) \((X_{k},T_{k})\) is a \(\mathcal{P}_{\{8\};\{1\}}\)-contraction on \(X^{k}\). (b) \((X_{k},T_{k})\) is left and right \(\mathcal{P}_{\{8\};\{1\}}\)-admissible on \(X_{k}=W^{L-\mathcal{P}_{\{8\};\{1\}}}=W^{R-\mathcal{P}_{\{8\};\{1\}}}\); if \(w^{0}\in X_{k}\) is arbitrary and fixed, then the sequence \((w^{m}=T_{k}^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N)}\) satisfies \(k\in LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal{P}_{\{8\};\{1\}}}\) and \((k-1;k)\subset LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{R-\mathcal{P}_{\{8\};\{1\}}}\). (c) \((X_{k},T_{k})\) is left \(\mathcal{P}_{\{8\};\{1\}}\)-closed on \(X_{k}=W^{L-\mathcal{P}_{\{8\};\{1\}}}\); indeed, if \(w^{0}\in X_{k}\) is arbitrary and fixed and if \((w^{m}=T_{k}^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N)}\) is a left \(\mathcal{P}_{\{8\};\{1\}}\)-converging sequence in X and having subsequences \((v_{m}:m\in\mathbb{N})\) and \((u_{m}:m\in\mathbb{N})\) satisfying \(\forall_{m\in\mathbb{N}}\{v_{m}=T_{k}(u_{m})\}\), then by (5.14)-(5.16) we have that \(k\in LIM_{(w^{m}:m\in\{0\}\cup\mathbb{N})}^{L-\mathcal{P}_{\{8\};\{1\}}}\) and \(k=T_{k}(k)\).
Claim
For each \(k=\{1,2,3\}\), all assumptions of Theorem 3.1 in the left case hold, and we see that: (a) \(Fix(T_{k})=\{ k\}\). (b) For each \(w^{0}\in X_{k}\), the sequence \((w^{m}=T_{k}^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left \(\mathcal{P}_{\{8\};\{1\}}\)-convergent to k. (c) \(p(k,k)=0\).
Example 5.4
Let \(X=(0;1)\cup(1;2)\). For \(\gamma >0\) and \(A=\{1/2,3/2\}\), set
Define \(T:X\rightarrow X\) by
(1) Observe that \((X,\mathcal{P}_{\{1\};\{1\}})\), \(\mathcal{P}_{\{1\};\{1\}}=\{p\}\), is a triangular space.
(2) \((X,T)\) is a weak \(\mathcal{P}_{\{1\};\{1\} }\)-contraction on X. More precisely, for each \(w^{0}\in A\),
In fact, by (5.18) we calculate that, for each \(m\in\{0\}\cup\mathbb {N}\),
Putting \(m\in\{0\}\cup\mathbb{N}\), this becomes
Also, from (5.17) and (5.20)-(5.23) we conclude that
Hence, \(\forall_{w^{0}\in A}\forall_{s,l\in\mathbb{N}}\{p(T^{[s]}(w^{0}),T^{[l]}(w^{0}))=0\}\). Consequently, (5.19) holds.
(3) \((X,T)\) is left and right \(\mathcal{P}_{\{1\};\{ 1\}}\)-admissible in each point \(w^{0}\in A\); thus, \(W^{L-\mathcal{P}_{\{1\};\{1\}}}=W^{R-\mathcal{P}_{\{1\};\{1\}}}=A\). To verify this, we take any \(w^{0}\in A\) and define \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup \mathbb{N)}\). Then, by (5.17) and (5.25),
(4) We show that \((X,T^{[2]})\) is left and right \(\mathcal{P}_{\{1\};\{1\}}\)-closed on \(A=W^{L-\mathcal{P}_{\{1\};\{1\}}}=W^{R-\mathcal{P}_{\{1\};\{1\}}}\). Indeed, if \(w^{0}\in A\), then by (5.25) and (5.17) the sequence \((w^{m}=T^{[m]}(w^{0}):m\in \{0\}\cup\mathbb{N})\) is contained in A, is left and right \(\mathcal {P}_{\{1\};\{1\}}\)-convergent in X to each point of A, has subsequences \((v_{m}:m\in\mathbb{N})\) and \((u_{m}:m\in\mathbb{N})\) satisfying \(\forall _{m\in\mathbb{N}}\{v_{m}=T^{[2]}(u_{m})\}\), and \(LIM_{(w^{m}:m\in\{0\} \cup \mathbb{N})}^{L-\mathcal{P}_{\{1\};\{1\}}}=LIM_{(w^{m}:m\in\{0\}\cup \mathbb{N})}^{R-\mathcal{P}_{\{1\};\{1\}}}=A=Fix(T^{[2]})\) holds.
(5) We show that the single-valued dynamic system \((X,T^{[4]})\) is also left and right \(\mathcal{P}_{\{1\};\{1\} }\)-closed on \(W^{L-\mathcal{P}_{\{1\};\{1\}}}=W^{R-\mathcal{P}_{\{1\};\{1\}}}=A\). Indeed, if \(w^{0}\in A\) is arbitrary and fixed and if \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is a left or right \(\mathcal{P}_{\{1\};\{1\}}\)-converging sequence in X having subsequences \((v_{m}:m\in\mathbb{N})\) and \((u_{m}:m\in\mathbb{N})\) satisfying \(\forall_{m\in\mathbb{N}}\{v_{m}=T^{[4]}(u_{m})\}\), then by (5.4), (5.1), (5.6), and (5.24) we have that \(A=LIM_{(w^{m}:m\in\{0\} \cup \mathbb{N})}^{L-\mathcal{P}_{\{1\};\{1\}}}=LIM_{(w^{m}:m\in\{0\}\cup \mathbb{N})}^{R-\mathcal{P}_{\{1\};\{1\}}}\) and \(A\subset X=Fix(T^{[4]})\). Hence, by virtue of Definition 3.2, \(U=V=W^{L-\mathcal{P}_{\{1\};\{1\}}}=W^{R-\mathcal{P}_{\{1\};\{1\}}}=A\).
Claim I
By (1)-(4) it follows that we may use Theorem 3.2 in the left and right cases (when \(\mathcal{J}_{\{1\};\{1\}}=\mathcal{P}_{\{1\};\{1\}}\)), and we see that statements (A) and (B) of this theorem hold. We have: (a) For each \(w^{0}\in A\) and for each \(w\in A\), the sequence \((w^{m}=T^{[m]}(w^{0}):m\in \{0\}\cup\mathbb{N})\) is left and right \(\mathcal{P}_{\{1\};\{1\}}\)-convergent to w, and thus statement (A) holds. (b) \(Fix(T^{[2]})=\{1/2,3/2\}\neq\varnothing\); thus, conclusion (B1) holds. (c) For each \(w^{0}\in A\), the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left and right \(\mathcal{P}_{\{1\};\{1\}}\)-convergent to each point \(w\in Fix(T^{[2]})\), and thus conclusion (B2) holds. (d) \(\forall_{w^{0}\in Fix(T^{[2]})}\{ p(w^{0},T(w^{0}))=p(T(w^{0}),w^{0})=0\}\), and thus conclusion (B3) holds. (e) \((X,\mathcal{P}_{\{1\};\{ 1\}}) \) is not separable, \(Fix(T)=\varnothing\), and statement (C) does not hold.
Claim II
It follows from (1)-(3) and (5) that, for \((X,\mathcal{P}_{\{1\};\{1\}})\), \(\mathcal{P}_{\{1\};\{1\} }=\{p\}\), \((X,T)\) and \(\mathcal{J}_{\{1\};\{1\}}=\mathcal{P}_{\{1\};\{ 1\}}\) defined by (5.1) and (5.2) and for \(q=4\), statements (A) and (B) of Theorem 3.1 hold: (a) Statement (A) holds since, for each \(w^{0}\in X\), the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left and right \(\mathcal{P}_{\{1\};\{1\}}\)-convergent to each point \(w\in A\); \(W^{L-\mathcal{P}_{\{1\};\{1\}}}=W^{R-\mathcal{P}_{\{1\};\{1\}}}=A\) by Theorem 3.1. (b) We have that \(Fix(T^{[4]})=X\); thus, conclusion (B1) holds. (c) Conclusion (B2) follows from statement (A) and conclusion (B1) since \(A\subset Fix(T^{[4]})\). (d) Conclusion (B3) holds; by (5.1) we have \(\forall_{w^{0}\in A\subset Fix(T^{[4]})}\{p(w^{0},T(w^{0}))=p(T(w^{0}),w^{0})=0\}\). (e) \((X,\mathcal{P}_{\{1\};\{1\}})\) is not separable, \(Fix(T)=\varnothing\), and statement (C) does not hold.
Example 5.5
Let \(X=(0;3)\), \(p:X^{2}\rightarrow [ 0;\infty)\) be of the form
and \(\mathcal{J}_{\{2\};\{1\}}=\{J\}\), \(J:X^{2}\rightarrow[ 0;\infty) \), be given by the formula
where \(\mu=9/64\) and
Let \(T:X\rightarrow X\) be given by
where
(1) \((X,\mathcal{P}_{\{2\};\{1\}})\), \(\mathcal{P}_{\{2\};\{1\}}=\{p\}\), is a separable quasi-triangular space. See [27], Example 4, p.11; p vanishes on the diagonal, is asymmetric, and is quasi-triangular since \(\forall_{u,v,w\in X}\{p(u,w)\leq 2[p(u,v)+p(v,w)]\}\).
(2) \(\mathcal{J}_{\{2\};\{1\}}\) is the left and right family generated by \(\mathcal{P}_{\{2\};\{1\}}=\{p\}\), and \(\mathcal{J}_{\{2\} ;\{1\}} \) is separating on X (see Theorems 2.1 and 2.2).
(3) \((X,T)\) is a weak \(\mathcal{J}_{\{2\};\{1\}}\)-contraction on X, that is, there exists \(w^{0}\in X \) such that
More precisely, we show that (5.35) holds for each \(w^{0}\in E\).
The proof proceeds in four steps.
Step 1. We construct \(T^{[m]}\) in \((0;1]\cup(2;3)\), \(m\in\mathbb{N}\). Using (5.28)-(5.34), for \(m\in \mathbb{N}\), we have:
Case A. If \(x\in(0;1/2)\), then \(T^{[m]}=f_{1}^{[m]}:(0;1/2)\rightarrow(0;1/2)\).
Case B. If \(x=1/2\), then \(T^{[m]}(1/2)=(f_{4}^{[m-1]}\circ f_{2})(1/2)=f_{4}^{[m-1]}(2)=2\).
Case C. If \(x\in(1/2;1)\), then \(T^{[m]}=f_{5}^{[m-1]} \circ f_{3}:(1/2;1)\rightarrow(2;3)\).
Case D. If \(x=e_{1}=1\), then \(T^{[m]}(1)=(f_{4}^{[m-1]}\circ f_{3})(1)=f_{4}^{[m-1]}(2)=2\).
Case E. If \(x\in(2;3)\), then \(T^{[m]}=f_{5}^{[m]}:(2;3)\rightarrow(2;3)\).
Step 2. We construct \(T^{[m]}\) in \((1;2] \), \(m\in\mathbb{N}\). Using (5.33), we compute that if \(m\in\mathbb {N}\), then \(f_{4}^{[m]}(x)=(3/2)^{m}(x-2)+2\), \(f_{4}^{[m]}:\mathbb{R\rightarrow R}\), \(f_{4}^{[m]}(e_{m})=1/2\), \(f_{4}^{[m-1]}(e_{m})=1\), \(e_{m}=2-(2/3)^{m-1}\), and \(f_{4}^{[m]}(e_{\infty}=2)=2\). Using (5.28)-(5.34), we therefore have:
Case A. Let \(m=1\). Since \(f_{4}(x)=(3/2)x-1=(3/2)(x-2)+2\), we see that \(f_{4}:\mathbb{R\rightarrow R}\), \(f_{4}:(1;2]\rightarrow(1/2;2]\), \(f_{4}(e_{1})=1/2\), \(e_{1}=1\), and \(f_{4}(e_{\infty}=2)=2\). Consequently,
Case B. Let \(m=2\). Then \(f_{4}^{[2]}(x)=(3/2)^{2}(x-2)+2\), \(f_{4}^{[2]}:\mathbb{R\rightarrow R}\), \(f_{4}^{[2]}(e_{2})=1/2\), \(e_{2}=2-(2/3)=4/3\), \(f_{4}(e_{2})=1\), \(f_{4}(e_{1}=1)=1/2\), \(f_{4}^{[2]}(e_{\infty}=2)=2\), and in view of (5.29), this implies
Case C. Let \(m=3\). We see that \(f_{4}^{[3]}(x)=(3/2)^{3}(x-2)+2\), \(f_{4}^{[3]}:\mathbb{R\rightarrow R}\), \(f_{4}^{[3]}(e_{3})=1/2\), \(e_{3}=2-(2/3)^{2}=14/9\), \(f_{4}^{[2]}(e_{3})=1\), \(f_{4}^{[2]}(e_{2})=1/2\), and \(f_{4}^{[3]}(e_{\infty}=2)=2\). Now (5.31) implies that
Case D. Let \(m>3\). Since \(f_{4}^{[m]}(x)=(3/2)^{m}(x-2)+2\), \(f_{4}^{[m]}:\mathbb{R\rightarrow R}\), \(f_{4}^{[m]}(e_{m})=1/2\), \(e_{m}=2-(2/3)^{m-1}\), \(f_{4}^{[m-1]}(e_{m})=1\), \(f_{4}^{[m-1]}(e_{m-1})=1/2\), and \(f_{4}^{[m]}(e_{\infty}=2)=2\). Hence,
Step 3. We describe the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N)}\), \(w^{0}\in E\). Let \(w^{0}\in E\) be arbitrary and fixed. We consider the following cases:
Case A. If \(k=0\) and \(w^{0}=e_{1}=1\), then
This is a consequence of Case B of Step 1.
Case B. If \(k\in\mathbb{N}\backslash\{1\}\) and \(w^{0}=e_{k}\), then
In fact, by Case D of Step 2 we have
Case C. If \(w^{0}=e_{\infty}=2\) and \(m\in\mathbb{N}\), then
Indeed, by Case D of Step 2 we obtain \(\forall_{m\in\mathbb{N}}\{w^{m}=T^{[m]}(2)=f_{4}^{[m]}(2)=2\}\).
Step 4. We will show that ( 5.35 ) holds on E. Let \(w^{0}\in E\) be arbitrary and fixed. Using (5.36)-(5.38) in Cases A-C of Step 3 and (5.26)-(5.28), we observe that
In view of this, we conclude that (5.35) holds.
(4) \((X,T)\) is left and right \(\mathcal{J}_{\{2\};\{1\}}\)-admissible on
Indeed, by Step 3 it is clear that, for arbitrary and fixed \(w^{0}=e_{k}\in E\), \(k\in\mathbb{N\cup\{\infty\}}\), the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N)}\) satisfies
Then, in view of (5.39), (5.26), and (5.27), we have
and
Also, we have
and
that is,
Therefore, (3.1)-(3.4) of Definition 3.1 hold.
(5) We show that the single-valued dynamic system \((X,T)\) is left and right \(\mathcal{P}_{\{2\};\{1\}}\)-closed on \(U=V=W^{L-\mathcal{J}_{\{2\};\{1\}}}=W^{R-\mathcal{J}_{\{2\};\{1\}}}=E\). Indeed, if \(w^{0}\in E\) then, by Step 3 and (5.26) we conclude that the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) satisfies
Clearly, the subsequences \((v_{m}=w^{m+1}:m\in\mathbb{N})\) and \((u_{m}=w^{m}:m\in\mathbb{N})\) of \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup \mathbb{N})\) satisfy \(\forall_{m\in\mathbb{N}}\{v_{m}=T(u_{m})\}\). We have \(2\in Fix(T)\).
Claim
By (1)-(5) it follows that, for each \(w^{0}\in E\), we may use Theorem 3.2 in the cases of left and right (when \(\mathcal{J}_{\{2\};\{1\}}\neq\mathcal{P}_{\{2\};\{1\}}\)), and we see that statements (A)-(C) of this theorem hold. We have: (a) \((X,\mathcal{P}_{\{2\};\{1\}})\) is separable. (b) For each \(w^{0}\in E\), the sequence \((w^{m}=T^{[m]}(w^{0}):m\in\{0\}\cup\mathbb{N})\) is left and right \(\mathcal{P}_{\{2\};\{1\}}\)-convergent to \(w=2\in Fix(T)=\{ 2\}\). (c) For \(w^{0}=w=2\), we have \(J(w^{0},w^{0})=0\).
Example 5.6
Let X, p, E, T, and \(\mathcal{P}_{\{2\};\{1\}}\) be such as in Example 5.5.
(1) We show that, for each \(w^{0}\in X\),
that is, \((X,T)\) is a weak \(\mathcal{P}_{\{2\};\{1\} }\)-contraction on X.
Indeed, let \(w^{0}\in(0;1/2)\cup[(1/2;3)\backslash E]\) be arbitrary and fixed. Then, for all \(s,l\in\mathbb{N}\),
Using Steps 1 and 2 of Example 5.5, we therefore have
which means that (5.40) holds.
If \(w^{0}\in\{1/2\}\cup E\), then by Steps 1 and 3 of Example 5.5 we see that
and therefore (5.40) also holds.
(2) We may not use Theorems 3.1 and 3.2 for \(\mathcal{J}_{\{2\};\{1\}}=\mathcal{P}_{\{2\};\{1\}}\) in the left and right cases since condition (A1) of Definition 3.1 in these cases does not hold. Indeed, we consider two cases:
Case A. Let \(w^{0}\in(1/2;3)\backslash E\) be arbitrary and fixed. It is clear that
and that the sequence \((w^{m}=T^{[m]}(w^{0}):m\geq m_{0}\mathbb{)}\) is increasing. Hence,
that is, property (3.1) holds, whereas \(LIM_{(w^{m}:m\in\{0\}\cup \mathbb{N})}^{L-\mathcal{P}_{\{2\};\{1\}}}=\varnothing\). In fact, since
we have
that is, property (3.3) does not hold.
Case B. Let \(w^{0}\in(0;1/2)\) be arbitrary and fixed. It is clear that
and that the sequence \((w^{m}=T^{[m]}(w^{0}):m\geq m_{0}\mathbb{)}\) is decreasing. Hence,
that is, property (3.2) holds. Note, however, that \(LIM_{(w^{m}:m\in\{ 0\}\cup \mathbb{N})}^{R-\mathcal{P}_{\{2\};\{1\}}}=\varnothing\). In fact, since \(\forall_{w\in(0;3)}\exists_{m_{0}\in\mathbb{N}}\forall_{m\geq m_{0}}\{p(w^{m},w)=(w-w^{m})^{2}\}\), we have
that is, property (3.4) does not hold.
Remark 5.1
We make the following remarks about Examples 5.5 and 5.6: (a) By Example 5.5 we observe that, for \((X,T)\), we may apply Theorem 3.2 in \((X,\mathcal{P}_{C;\mathcal{A}})\) with the left and right family \(\mathcal{J}_{C;\mathcal{A}}\) generated by \(\mathcal{P}_{C;\mathcal{A}}\) where \(\mathcal{J}_{C;\mathcal{A}}\neq\mathcal{P}_{C;\mathcal{A}}\); (b) By Example 5.6 we note, however, that, for \((X,T)\) in the left and right cases, we do not apply Theorems 3.1 and 3.2 in \((X,\mathcal{P}_{C;\mathcal{A}})\) when \(\mathcal{J}_{C;\mathcal{A}}=\mathcal{P}_{C;\mathcal{A}}\); (c) From (a) and (b) it follows that, in Theorems 3.1 and 3.2, the existence of the family \(\mathcal{J}_{C;\mathcal{A}}\) generated by \(\mathcal {P}_{C;\mathcal{A}}\) and such that \(\mathcal{J}_{C;\mathcal{A}}\neq\mathcal {P}_{C;\mathcal{A}} \) is essential.
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Włodarczyk, K. Periodic and fixed points of the Leader-type contractions in quasi-triangular spaces. Fixed Point Theory Appl 2016, 85 (2016). https://doi.org/10.1186/s13663-016-0575-7
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DOI: https://doi.org/10.1186/s13663-016-0575-7