The main purpose of this research article is to generalize Kannan-type fixed-point (FP) theorems for single-valued mappings and Chatterjea-type FP result for fuzzy mappings (FMs) in the context of complete strong b-metric spaces (MSs). Moreover, fuzzy FPs are established for Suzuki-type fuzzy contraction in the setting of complete strong b-MSs. The conclusions are supported by nontrivial examples to enhance the validity of the results obtained in this study. In addition, previous findings have been made as corollaries from the relevant literature. The numerous implications that this technique has across the literature improve and integrate our findings. Applications of some of the results obtained are also incorporated.
Hinweise
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1 Introduction
In the past few decades, a noteworthy interest in FP theory has been directed to interchanging recent metric FP results from the usual MSs to some generalized MSs, like quasi-MSs usually called b-MSs introduced by Bakhtin [3] and Czerwik [8]. The class of strong b-MSs lying between the class of b-MSs and the class of MSs was introduced by Kirk and Shahzad [16]. As compared with b-MSs, strong b-MSs have the advantage that open balls are open in the induced topology and, hence, they have given many properties that are similar to the properties of classic MSs. In 1965, Zadeh [32] introduced the notion of fuzzy logic. In the theory of traditional logic, some element does or does not belong to the set, but in fuzzy logic a number from the interval \([0, 1]\) expresses the affiliation of the element to the set. Zadeh started to research the theory of fuzzy sets (FSs) in order to deal with the issue of indeterminacy, which is a real problem that is fundamentally characterized by uncertainty. The concept of the FM was given by Heilpern [13] and for fuzzy contraction mapping in a metric linear space, a theorem was proved by him that is a fuzzy generalization of Banach’s contraction principle. Many authors such as Banach [4], Benavides et al. [5], Ciric [7], Kirk [17], Meir and Keeler [18], Nadler [23], Subrahmanyam [26], and Suzuki [27, 28] proved theorems in which every contraction mapping was a continuous function. Then, in 1968, Kannan [15] was the first who introduced the contraction mapping that was not necessarily continuous.
Fuzzy common FPs for generalized mappings were obtained by Abbas et al. [1], fuzzy FPs and common FPs were established by Azam et al. [2] and fuzzy FPs for FMs were constructed by Estruch and Vidal [10] and Frigon and O’Regan [11]. Işık et al. [14] and Mohammadi et al. [19‐22] have established valuable fixed-point and common fixed-point results using various contractive conditions for fuzzy and nonfuzzy mappings in the generalizations of metric spaces.
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Also, some other authors [24, 25, 30, 31] worked on the existence of FPs and common FPs of FMs satisfying a contractive-type condition. Fuzzy theory has been applied in several fields, for example quantum physics, nonlinear dynamical systems, population dynamics, computer programming, fuzzy stability problems, statistical convergence, functional equation, approximation theory, nonlinear equations, and many others.
Theorem 1.1
[15] Suppose\((S,d)\)is a complete MS, and\(\theta :S\rightarrow S\)is a mapping. If there exists\(x \epsilon [0,\frac{1}{2})\), satisfying
for alls, \(u \epsilon S\), thenθhas a unique FP\(r \epsilon S\)and for any\(s \epsilon S\)the sequence of iterates\(\{\theta ^{n}s\}\)converges tor.
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Further, Gornicki [12] introduced various extensions of the Kannan FP theorem. He proved the following results:
Assume ζ denotes the class of functions that satisfy the condition \(\zeta =\{\phi :(0,\infty )\rightarrow [0,\frac{1}{2}):\phi (t_{n}) \rightarrow \frac{1}{2} \text{ implies } t_{n}\rightarrow 0 \text{ as } n\rightarrow \infty \}\).
Theorem 1.3
[12] Suppose\((S,d)\)is a complete MS and\(\theta :S\rightarrow S\)is a mapping. Also, assume there exists\(\phi \epsilon \zeta \)such that for eachs, \(u \epsilon S\)with\(s\neq u\),
Then, θhas a unique FP\(r \epsilon S\)and for any\(s \epsilon S\)the sequence of iterates\(\{\theta ^{n}s\}\)converges tor.
In 2021, Doan [9] extended the results in [12] for a class of contractive mappings in strong b-MSs. He proved a new version of FP theorems for single-valued and multivalued mappings by combining the results in [15] and [29].
Theorem 1.4
[9] Suppose\((S,\varpi ,\sigma )\)is a complete strong b-MS with\(\sigma \geq 1\)and\(T:S\rightarrow S\)is a mapping. Assume there exists\(\phi \epsilon \zeta \)such that for eachs, \(u \epsilon S\)with\(s\neq u\),
Then, θhas a unique FP\(r \epsilon S\)and for any\(s \epsilon S\)the sequence of iterates\(\{\theta ^{n}s\}\)converges to r.
In this article, we obtained the idea from [9] and extended it to [6, 29]. We prove FP theorems for single-valued FMs in strong b-MS by combining the results in [6] and [9].
2 Basic concepts
We recall some results and concepts, which are necessary to understand our results.
Definition 2.1
[16] Suppose S is a nonempty set and \(\sigma \geq 1\). A mapping \(\varpi :S\times S\rightarrow [0,+\infty )\) is called a strong b-metric on S if
\(sb_{1})\)
\(\varpi (s,u)\geq 0\), \(\forall s, u\in S\);
\(sb_{2})\)
\(\varpi (s,u)= 0\) iff \(s=u\);
\(sb_{3})\)
\(\varpi (s,u)= \varpi (u,s)\)\(\forall s, u \in S \);
Assume that there exists\(x \in [0,1)\)such that\(\psi (x)d(s,\theta s)\leq d(s,u)\)implies\(d(\theta s,\theta u)\leq xd(s,u)\)for all\(s,u \in S\). Then, θhas a unique FP\(r \in S\)and for any\(s \in S\), the sequence of iterates\(\{\theta ^{n}s\}\)converges tor.
Definition 2.3
[16] Suppose \((S,\varpi ,\sigma )\) is a strong b-MS, \(\{s_{n}\}\) is a sequence in S, and \(s \in S\). Then,
(i)
If \(\lim_{n\rightarrow \infty}\varpi (s_{n},s)=0\), then \(\{s_{n}\}\) is called convergent to s. This means \(\lim_{n\rightarrow \infty}s_{n}=s\) or \(s_{n}\rightarrow s\) as \(n\rightarrow \infty \).
(ii)
If \(\lim_{n,m\rightarrow \infty}\varpi (s_{n},s_{m})=0\), then \(\{s_{n}\}\) is called a Cauchy sequence (CS) in S.
(iii)
If every CS in S converges in S then \((S,\varpi ,\sigma )\) is complete.
Proposition 2.4
[16] Suppose\((S,\varpi ,\sigma )\)is a strong b-MS and\(\{s_{n}\}\)is a sequence inS. Then,
[32] Suppose S is any arbitrary set and a function \(A:S \rightarrow [0,1]\) is a FS. The functional value \(A(s)\) is called the grade of membership of s in A. The collection of all FSs in S is denoted by \(F(S)\).
The α-cut of A is denoted by \(A_{\alpha}\) and is defined as follows:
[13] Suppose \((S,d)\) is any MS and P is an arbitrary set. θ is called FM if \(\theta : W \rightarrow F(S)\) is a function, i.e., \(\theta (p) \in F(S)\) for each \(p \in P\).
Example 2.9
Let \(P= [-9,9]\) and \(S=[-4, 4]\). Define \(T_{1}:P \longrightarrow F(S)\) by
$$ T_{1}(x) (y)= \frac{x^{2} + y^{2} }{100}. $$
Then, \(T_{1}\) is a FM. Note that \(T_{1}(x)(y)\in [0,1]\), for all \(x \in P\) and \(y\in S\). The graphical representation \(T_{1}(x)(y)\) showing the possible membership values of y in \(T_{1}(x)\) is given in Fig. 2.
Figure 2
Graph of fuzzy mapping \(T_{1} \)
×
Example 2.10
Let \(S= [-3,3]\). Define \(T_{2}:S \longrightarrow F(S)\) by
Then, \(T_{2}\) is a fuzzy mapping. Note that \(T_{2}(x)(y)\in [0,1]\), for all \(x, y\in S\). The graphical representation \(v=T_{2}(x)(y)\) showing the possible membership values of y in \(T_{2}(x)\) is shown in Figure 3.
Figure 3
Graph of fuzzy mapping \(T_{2} \)
×
Definition 2.11
Suppose \((S,d)\) is a MS and \(CB(S)\) denotes the collection of all nonempty, closed, and bounded subsets of S. Consider a map \(H:CB(S)\times CB(S) \rightarrow \mathbb{R}\). For \(C,E \in CB(S)\) define
where \(d(c,E)=\{\inf d(c,e): e \in E\}\) is the distance of c to meet E. This H is a metric on \(CB(S)\) is called the Hausdorff metric induced by the metric d.
Definition 2.12
Let \((S,\varpi ,K)\) be a strong b-MS. Let \(\theta :S\rightarrow F(S)\) be a FM on S:
where H is the Hausdroff metric on \(F(S)\) induced by ϖ, \([\theta s]_{\alpha _{\theta s}},[\theta u]_{\alpha _{\theta u}} \in F(S)\) and \(d(s,[Lu]_{\alpha _{Lu}})=\inf_{u \in [Lu]_{\alpha _{Lu}}}\varpi (s,u)\).
Lemma 2.13
[2] Suppose\((S,d,b)\)is a b-MS. Then, for\(C,E \in CB(S)\),
(i)
\(d(c,E)\leq H(C,E)\), \(c\in C\);
(ii)
For\(\varepsilon > 0\)and\(c\in C\), \(\exists e\in E\)such that
$$ d(c,e) \leq H(C,E)+ \varepsilon . $$
Theorem 2.14
Suppose\((S,d)\)is a complete MS. If\(\theta :S\rightarrow F(S)\)is a continuous FM such that\([\theta s]_{\alpha _{\theta s}}\)and\([\theta u]_{\alpha _{\theta u}}\)are closed and bounded subsets ofSsatisfying
\(\forall s,u \in S\), where\(0\leq x <\frac{1}{2}\). Then, θhas at least one FP.
3 Main results
In this section, we establish our main results.
Theorem 3.1
Suppose\((S,\varpi ,\sigma )\)is a complete strong b-MS and\(\theta :S\rightarrow S\)is a mapping. Suppose there exists\(\phi \in \zeta \)such that for each\(s,u \in S\)with\(s\neq u\),
Then, θhas a unique FP\(r \in S\)and for any\(s \in S\)the sequence of iterates\(\{\theta ^{n}s\}\)converges to r.
Proof
Fix \(s_{0} \in S\) and define a sequence \(\{s_{n}\}\) in S by \(s_{n+1}=\theta s_{n}\) for all integers \(n\geq 0\). Assume that there exists n such that \(s_{n+1}=s_{n}\), then \(s_{n}\) is a FP of θ. Therefore, suppose that \(s_{n+1} \neq s_{n}\) for all \(n\geq 0\). Set \(\varpi _{n}=\varpi (s_{n},s_{n+1})\) for all \(n\geq 0\). By hypothesis, we have
Hence, \(\varpi _{n+1}<\varpi _{n}\) for all \(n\geq 0\) and so \(\{\varpi _{n}\}\) is monotonic decreasing and bounded below, so there exists \(\eta \geq 0\) such that
as \(n,m \rightarrow \infty \), so \(\{s_{n}\}\) is a CS in S. By the completeness of S, there is \(r \in S\) such that \(\lim_{n\rightarrow \infty}s_{n}=r\), since
Then, \((S, \varpi ,\sigma =2)\) is a strong b-MS, but it is not MS, because \(7= \varpi (3,1)> \varpi (3,2)+ \varpi (2,1)=\frac{11}{5}\). Let \(\theta :S\rightarrow S\) by \(\theta 1=1\), \(\theta 2=1\), \(\theta 3=2\), and the function \(\phi \in \zeta \) given by \(\phi (t)=t\sin (t)\), \(t>0\) and \(\phi (0) \in [0,\frac{1}{2})\). Then,
Therefore, θ satisfies all the conditions of Theorem 3.1. Hence, 0 is a fixed point of θ.
If we take \(\sigma =1\) in Theorem 3.1, the strong b-MS is a usual MS, then we obtain the following corollary.
Corollary 3.3
Suppose\((S,d)\)is a complete MS and\(\theta :S\rightarrow S\)is a mapping. Assume there exists\(\phi \in \zeta \)such that for each\(s,u \in S\)with\(s\neq u\),
for all\(s,u \in S\)and\(\beta \in [0,1)\). Then, there existsrsuch that\(r\in [\theta r]_{\alpha _{\theta r}}\).
Proof
Let \(s_{1}\in [\theta s_{0}]_{\alpha _{\theta s_{0}}}\), with \([\theta s_{1}]_{\alpha _{\theta s_{1}}}\neq \phi \), where \(s_{0}\in S, [\theta s_{0}]_{\alpha _{\theta s_{0}}}\) are closed and bounded subsets of S. By using Lemma 2.13, \(\exists s_{2}\in [\theta s_{1}]_{\alpha _{\theta s_{1}}}\) such that
Now, ∃ \(s_{3}\in [\theta s_{2}]_{\alpha _{\theta s_{2}}}\) for \([\theta s_{2}]_{\alpha _{\theta s_{2}}} \neq \phi \) are closed and bounded subsets of S such that
In the limiting case, \(m,n \rightarrow \infty \),
$$ \varpi (s_{n},s_{m})=0. $$
This implies that \(\{s_{n}\}\) is a CS in S. The completeness of S implies that there exists \(r\in S\) such that \(s_{n} \rightarrow r\). We will now demonstrate that r is a FP of θ. By utilizing Lemma 2.13,
Thus, all hypotheses of Theorem 3.4 are satisfied and \(r=5\) is a unique FP of θ.
Corollary 3.6
Suppose\((S, \varpi )\)is a complete MS with and\(\theta :S \rightarrow F(S)\)is a fuzzy map. Suppose\([\theta s]_{\alpha _{\theta s}}\)and\([\theta u]_{\alpha _{\theta u}}\)are closed and bounded subsets ofSdefined as
for all\(s,u \in S\)and\(\beta \in [0,1)\). Then, there existrinSsuch that\(r\in [\theta r]_{\alpha _{\theta r}}\).
Proof
Suppose \(\{s_{n}: n\in \mathbb{N}\}\) is a sequence such that \(s_{n+1} \in [\theta s_{n}]_{\alpha _{\theta s_{n}}}\). By using Lemma 2.13, for each \(s_{1}\in [\theta s_{0}]_{\alpha _{\theta s_{0}}}\), \(\exists s_{2}\in [\theta s_{1}]_{\alpha _{\theta s_{1}}}\) such that
Here, \(\gamma =\frac{\beta}{(1-\beta \sigma )}\), where \(\beta \in (0,\frac{1}{2\sigma })\), then \(\gamma \in (0,\frac{1}{\sigma })\). By using Lemma 2.13 again,
Taking \(m,n \rightarrow \infty \), \(\Rightarrow \varpi (s_{n},s_{m})=0\). Hence, \(\{s_{n}\}\) is a CS in S. Since S is complete, so \(\exists r\in S\) such that \(s_{n} \rightarrow r\).
Using the characteristics of infimum, with \(\varepsilon =(g-1)H([C]_{\alpha _{C}},[E]_{\alpha _{E}})>0\) there exists \(e \in [E]_{\alpha _{E}}\) such that
Step 1. If \(H([\theta s_{0}]_{\alpha _{\theta s_{0}}},[\theta s_{1}]_{\alpha _{ \theta s_{1}}})=0\) then \([\theta s_{0}]_{\alpha _{\theta s_{0}}}=[\theta s_{1}]_{\alpha _{ \theta s_{1}}}\). θ. Thus, \(s_{1}\) is a FP of θ. If \(H([\theta s_{0}]_{\alpha _{\theta s_{0}}},[\theta s_{1}]_{\alpha _{ \theta s_{1}}})>0\), by Lemma 3.9 then for each \(g_{1}>1\), there exists \(s_{2} \in [\theta s_{1}]_{\alpha _{\theta s_{1}}}\) such that
Step 2. Similarly, if \(H([\theta s_{1}]_{\alpha _{\theta s_{1}}},[\theta s_{2}]_{\alpha _{ \theta s_{2}}})=0\) then \([\theta s_{1}]_{\alpha _{\theta s_{1}}}=[\theta s_{2}]_{\alpha _{ \theta s_{2}}}\). Thus, \(s_{2}\) is a FP of θ. If \(H([\theta s_{1}]_{\alpha _{\theta s_{1}}},[\theta s_{2}]_{\alpha _{ \theta s_{2}}})>0\), by Lemma 3.9 then for each \(g_{2}>1\), there exists \(s_{3} \in [\theta s_{2}]_{\alpha _{\theta s_{2}}}\) such that
Stepn. Continuing in this manner, if \(H([\theta s_{n-1}]_{\alpha _{\theta s_{n-1}}},[\theta s_{n}]_{ \alpha _{\theta s_{n}}})=0\). Thus, \(s_{n}\) is a FP of θ. If \(H([\theta s_{n-1}]_{\alpha _{\theta s_{n-1}}},[\theta s_{n}]_{ \alpha _{\theta s_{n}}})>0\), by Lemma 3.9 then for each \(g_{n}>1\), there exists \(s_{n+1} \in [\theta s_{n}]_{\alpha _{\theta s_{n}}}\) such that
The above process continues, if at step k satisfying \(H([\theta s_{k-1}]_{\alpha _{\theta s_{k-1}}},[\theta s_{k}]_{ \alpha _{\theta s_{k}}})=0\), then \(s_{k}\) is a FP of θ. If not, we obtain two sequences \(\{s_{n}\}\) and \(\{g_{n}\}\) such that \(s_{n} \in [\theta s_{n-1}]_{\alpha _{\theta s_{n-1}}}\), \(g_{n}>1\) and
Since \(\frac{1}{\sigma +1}d(s_{n-1},[\theta s_{n-1}]_{\alpha _{\theta s_{n-1}}}) \leq \frac{1}{\sigma +1}d(s_{n-1},s_{n})\leq \varpi (s_{n-1},s_{n})\) and by hypothesis, we obtain
We can choose \(g_{n}=\frac{k}{x}>1\) with \(x \in (0,k)\) and \(0< k<\frac{1}{2}\). Then, we obtain \(\varpi _{n}<\frac{k}{1-k} \varpi _{n-1}\), where \(\frac{k}{1-k}<1\) and \(\varpi _{n}= \varpi (s_{n},s_{n+1})\). Thus, \(\varpi _{n}<(\frac{k}{1-k})^{n} \varpi _{0}\) for all \(n\geq 1\). Hence,
By Proposition 2.5, \(\{s_{n}\}\) is a CS in S. Since S is complete, ∃ \(r \in S\) such that \(\lim_{n\rightarrow \infty}s_{n}=r\). Now, we show that for any \(n\geq 0\), either
Then, either (3.10) holds for infinity natural numbers n or (3.11) holds for infinity natural numbers n. Suppose (3.10) holds for infinity natural numbers n. We can choose that in that infinity set the sequence \(\{n_{k}\}\) is a monotone strictly increasing sequence of natural numbers. Therefore, sequence \(\{s_{n_{k}}\}\) is a subsequence of \(\{s_{n}\}\) and
By taking limits on both sides of the above inequality, we obtain \(d(r,[\theta r]_{\alpha _{\theta r}})=0\). This means that \(r \in [\theta r]_{\alpha _{\theta r}}\). If (3.11) holds for infinity natural numbers n, by using an argument similar to that of above we have r is a FP of θ. Suppose r̄ is another FP of θ, then \(0=\frac{1}{\sigma +1}d(r,[\theta r]_{\alpha _{\theta r}})\leq \varpi (r,\bar{r})\) and by hypothesis,
Now, since \(H([\theta (s)]_{\alpha _{\theta (s)}}, [\theta (u)]_{\alpha _{ \theta (u)}})= H(A(s), A(u))\), Theorem 3.4 can be applied to obtain required FP of A in S. □
Theorem 4.2
Suppose\((S, \varpi ,\sigma )\)is a complete strong b-MS with\(\sigma \geq 1\)and\(P :S \rightarrow CB(S)\)is a multivalued mapping such that
$$ H\bigl(P( s), P (u)\bigr) \leq \beta \varpi \bigl(s, P (u)\bigr)+ \varpi \bigl(u, P(s)\bigr), $$
(4.1)
for all\(s,u \in S\)and\(\beta \in [0,1)\). Then, there existrinSsuch that\(r\in P(r)\).
Proof
Consider an arbitrary mapping \(Q: S\rightarrow (0, 1]\). Define a FM \(\theta : S \rightarrow F(S)\) as follows:
$$ \theta (s) (g)= \textstyle\begin{cases} Q(s), & \text{if } g \in P(s) \\ 0, & \text{if } g \notin P(s). \end{cases} $$
Now, since \(H([\theta (s)]_{\alpha _{\theta (s)}}, [\theta (u)]_{\alpha _{ \theta (u)}})= H(P(s), P(u))\), Theorem 3.7 can be applied to obtain the required FP of P in S. □
Theorem 4.3
Suppose\((S, \varpi ,\sigma )\)is a complete strong b-MS and\(A :S\rightarrow CB(S)\)is a multivalued mapping. Suppose\(x \in (0,k)\)with\(0< k<\frac{1}{2}\)satisfying\(\frac{1}{\sigma +1} \varpi (s,As)\leq \varpi (s,u)\)implies\(H(A (s),A (u))\leq x\{ \varpi (s,A (s))+ \varpi (u,A (u))\}\), for all\(s,u \in S\). Then, Ahas a unique FP\(r \in S\). Moreover, for each\(s \in S\)the sequence of iterates\(\{A ^{n}s\}\)converges tor.
Proof
Consider an arbitrary mapping \(P: S\rightarrow (0, 1]\). Define a FM \(\theta : S \rightarrow F(S)\) as follows:
$$ \theta (s) (g)= \textstyle\begin{cases} P(s), & \text{if } g \in A(s) \\ 0, & \text{if } g \notin A(s). \end{cases} $$
Now, since \(H([\theta (s)]_{\alpha _{\theta (s)}}, [\theta (u)]_{\alpha _{ \theta (u)}})= H(A(s), A(u))\), Theorem 3.10 can be applied to obtain the required FP of A in S. □
5 Conclusion
FP theory is a useful theoretical tool in diverse fields, such as logic programming, functional analysis, artificial intelligence, and many others. In 2021, Doan [9] extended the results in [12] for a class of contractive mappings in strong b-MSs. He proved new versions of FP theorems for single-valued and multivalued mappings by combining the results in [15] and [29]. In this article, we obtained the idea from [9] and extended it to [6] and [29]. We have established FP theorems for fuzzy and nonfuzzy mappings in complete strong b-MS by combining results [6] and [9] and the obtained results are furnished with interesting and nontrivial examples. Moreover, some other contractions are also applied to find fuzzy and nonfuzzy fixed points. Some results for FMs and multivalued mappings are incorporated as corollaries and as applications. Moreover, other direct consequences are obtained as well. We hope these existence results will provide an appropriate environment to approximate further operator equations in applied science.
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The authors declare no competing interests.
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