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New Lfuzzy fixed point techniques for studying integral inclusions
Journal of Inequalities and Applications volume 2024, Article number: 83 (2024)
Abstract
The survey of the available literature shows that a lot of important invariant point problems of Banach and Heilpern types have been examined in both metric and quasimetric spaces. However, a handful of the existing results employed the recent approaches of interpolative contractions. Therefore, based on the new idea of interpolation techniques in fixed point theory, this article studies new notions of Lfuzzy contractions and investigates conditions for the existence of Lfuzzy fixed points for such mappings. On the fact that fixed points of pointtopoint mappings satisfying interpolativetype contraction are not always unique, whence making the concepts more fitted for invariant point results of crisp setvalued maps, new multivalued analogues of the key findings put forward in this work are derived. Comparative illustrations, which indicate the preeminence of the results presented herein, are constructed. From application viewpoint, one of the theorems so obtained is employed to introduce new solvability conditions of Fredholmtype integral inclusions.
1 Introduction and preliminaries
Throughout this paper, a metric space, a complete metric space, and a fixed point will hereon be written as MS, CMS, and FP, respectively.
Banach [2] introduced one of the most known metric FP concepts, commonly named the Banach contraction principle. Due to the simplicity and usefulness of this principle, it has enjoyed multiple improvements in several directions. In some refinements of Banach invariant point idea, the original inequality is reformulated (see, e.g., [3]), and in others, the topological notions of the space are modified (see [7] and the citations therein). Along the lane, a chief refinement of the contraction technique was brought up by Hardy and Rogers [8]. The paradigm of this result (in [8]) is given hereunder.
Theorem 1.1
[8] Let \(({W}, {d})\) be a CMS and S be a singlevalued mapping on W satisfying
where \({a_{1}}\), \({b_{1}}\), \({c_{1}}\), \({e_{1}}\), \({l_{1}}\) are nonnegative reals with \({a_{1}}+{b_{1}}+{c_{1}}+{e_{1}}+{l_{1}}<1\), then S has a unique FP in W.
For not too long, Roldán et al. [24] brought up new FP ideas for a host of contractions depending on two functions and a few constants, named multiparametric contractions, and noted out a good number of Hardy–Rogerstype contractions in the frame of metric and quasiMSs. Of recent as well, Karapinar et al. [14] established some common FP results for interpolative mappings availing Perovtype operators, which fulfil Suzukitype inequalities. The announcement in Theorem 1.1 has also been moved further by more than a handful of examiners. Related copies of the contraction tools were on several occasions provided by Ćirić [3], Reich [22], and Rus [26].
Definition 1.2

(i)
Rus contraction if we can find \({a_{1}}, {b_{1}}\in \mathbb{R}_{+}\) with \({a_{1}}+{b_{1}}<1\) such that for all \({h}, \mu \in {W}\),
$$ {d}({S} {h}, {S} \mu )\leq {a_{1}} {d}({h}, \mu )+{b_{1}} {d}(\mu , {S} \mu ). $$ 
(ii)
Ćirić–Reich contraction if we can find \({a_{1}}, {b_{1}},c\in \mathbb{R}_{+}\) with \({a_{1}}+{b_{1}}+c<1\) such that for all \({h}, \mu \in {W}\),
$$ {d}({S} {h}, {S} \mu )\leq {a_{1}} {d}({h}, \mu )+ {b_{1}} {d}({h}, {S} {h})+ c{d}(\mu , {S} \mu ). $$
A refined copy of the above findings is given hereunder.
Theorem 1.3
[3, 22, 26] Let \(({W}, {d})\) be a CMS and the selfmapping \({S}:{W}\longrightarrow {W}\) be a Ćirić–Reich–Rus contraction, that is,
for all \({h}, \mu \in {W}\), \({c}\in [0, \frac{1}{3} )\). Then S enjoys an FP in W.
Supported by the interpolation theory, Karapinar et al. [13] initiated the idea of interpolativetype notions as follows.
Definition 1.4
[13] Let \(({W}, {d})\) be an MS. \({S}:{W}\longrightarrow {W}\) is named an interpolative Hardy–Rogerstype contraction if we can find \({c}\in [0, 1[\) and \({a_{1}}, {b_{1}}, {c_{1}}\in ]0, 1[\) with \({a_{1}}+{b_{1}}+{c_{1}}< 1\):
for all \({h}, \mu \in {W}\setminus \mathcal{F}_{ix}({S})\), where \(\mathcal{F}_{ix}({S})\) is the collection of invariant points of S.
For related FP results employing the interpolation approach, the researcher can look up [10–12, 18]. An intersecting behavior of the known FP results of interpolative type mappings is that their FP (if it exists) is not usually one and only one (for reference, see [11, Example 1]). This makes it seemingly clear that FP theorems via the concept of interpolation are more compatible for FP theory of pointtosetvalued maps.
On the flip side, a difficulty in modeling practical problems is connected to the inconclusiveness caused by our incapability to sort events with adequacy. It is a common knowledge that earlier sciences in the bodywork of crisp sets cannot withstand imprecisions efficiently. Therefore, a struggle to ameliorate the aforementioned hurdles led to the launching of fuzzy set by Zadeh [29]. At present, the fundamental ideas of fuzzy sets have been developed and applied in various domains. In 1981, Heilpern [9] employed the idea of fuzzy sets to launch the notion of fuzzy setvalued maps and setup an FP result for fuzzy mappings, which is a fuzzy improvement of the FP theorems due to Nadler [19] and Banach [2]. Recently, by using the family of Γ functions initiated by Patel and Radenovic [28], Sessa et al. [27] introduced the idea of \(\alpha _{\Gamma}\)fuzzy contraction mappings and discussed the existence of FP for such mappings. Meanwhile, variants of Heilperntype fuzzy invariant point results have been developed (e.g., see [1, 5, 16, 17] and the references therein). A very interesting improvement of fuzzy sets by replacing the interval \([0, 1]\) of range set with a complete distributive lattice was brought up by Goguen [6] and is termed Lfuzzy set. Along the line, Rashid et al. [20] came up with the concept of Lfuzzy mappings and obtained common FP theorems through \(\beta _{FL}\)admissible pair of Lfuzzy mappings. As an improvement of the idea of Hausdorff distance and \({d}_{\infty}\)metric for fuzzy sets, Rashid et al. [21] proposed the ideas of \(D_{\alpha L}\) and \({d}^{\infty}_{L}\) distances for Lfuzzy sets and deduced some existing FP results for fuzzy setvalued and crisp setvalued maps.
Following the new interpolation approach in the study of FP results launched in [10–13], we noticed that the equivalent concepts with respect to Lfuzzy sets have not yet been studied or, at least, their analogues in fuzzy mathematics are very limited. Hence, this manuscript proposes the idea of interpolative Hardy–Rogerstype and interpolative Reich–Rus–Ćirićtype Lfuzzy contractions in MS and examines new ways for analyzing the Lfuzzy FPs of such contractions. It is worthy to indicate that FP of a singlevalued mapping enjoying the interpolativetype expression is not necessarily unique. Therefore, the interpolative techniques are more appropriate for FP theorems of multivalued maps. On this observation, some new setvalued copies of the Lfuzzy FP theorems studied herewith are discussed.
We now list a few preliminaries that are specific to our main results. Let \(({W}, {d})\) be an MS and \(\mathcal{V}({W})\) be the collation of compact subsets of W. Take \(M, U\in \mathcal{V}({W})\) and \({{r}}> 0\) be arbitrary. Then the sets \(N_{d}({{r}}, M)\) and \(E^{d}_{(M, U)}\) and the distance function \({d}(M, U)\) are respectively defined as follows:
Then the Hausdorff metric Ĥ on \(\mathcal{V}({W})\) generated by the metric d is defined as \({\widehat{H}}(M, U)=\inf E^{d}_{(M, U)} \) (see [19, P. 3]).
Recall that an ordinary subset M of W is determined by its characteristic function \(\chi _{M}\), defined by \(\chi _{M}: M\longrightarrow \{0, 1\}\):
The value \(\chi _{M}({h})\) specifies whether an element belongs to M or not. This idea was employed to define fuzzy sets by permitting an element \({h}\in A\) to take any value within \([0, 1] = I\).
Definition 1.5
[4] A relation ⪯ on a nonempty set L is termed a partial order if it is

(i)
reflexive;

(ii)
antisymmetric;

(iii)
transitive.
A set L together with a partial ordering ⪯ is named a partially ordered set (poset, for short) and is denoted by \((L, \preceq _{L})\). Recall that partial orderings are used to give an order to sets that may not have a natural one. Let L be a nonempty set and \((L, \preceq )\) be a partially ordered set. Then any two elements \(\beta , \varrho \in L\) are said to be comparable if either \(\beta \preceq \varrho \) or \(\varrho \preceq \beta \).
Definition 1.6
[4] A partially ordered set \((L, \preceq _{L})\) is named:

(i)
a lattice if \(\beta \vee \varrho \in L\), \(\beta \wedge \varrho \in L\) for any \(\beta , \varrho \in L\);

(ii)
a complete lattice if \(\bigvee \nabla \in L\), \(\bigwedge \nabla \in L\) for any \(\nabla \subseteq L\);

(iii)
distributive lattice if \(\beta \vee (\varrho \wedge \xi ) = (\beta \vee \varrho )\wedge ( \beta \vee \xi )\), \(\beta \wedge (\varrho \vee \xi )= (\beta \wedge \varrho )\vee ( \beta \wedge \xi )\), for any \(\beta , \varrho , \xi \in L\).
A partially ordered set L is named a complete lattice if for every doubleton \(\{\beta , \varrho \}\) in L, either \(\sup \{\beta , \varrho \}= \beta \bigvee \varrho \) or \(\inf \{\beta , \varrho \}= \beta \bigwedge \varrho \) exists.
Definition 1.7
[6] An Lfuzzy set ∇ on a nonempty set W is a function with domain W whose range lies in a complete distributive lattice L with top and bottom elements \(1_{L}\) and \(0_{L}\), respectively.
Remark 1.8
[6] The class of Lfuzzy sets is larger than the class of fuzzy sets as an Lfuzzy set reduces to a fuzzy set if \(L = I = [0, 1]\).
We denote the class of all Lfuzzy sets on a nonempty set W by \(L^{W}\).
Definition 1.9
[6]The \({\widehat{\tau}}_{L}\)level set of an Lfuzzy set ∇ is denoted by \([\nabla ]_{{\widehat{\tau}} L}\) and is defined as follows:
Definition 1.10
[20] Let W be an arbitrary nonempty set and Y be an MS. A mapping \({\widehat{\Psi}}: {W}\longrightarrow L^{Y}\) is named an Lfuzzy mapping. The function value \({\widehat{\Psi}}({d})(\varrho )\) is named the degree of membership of ϱ in \({\widehat{\Psi}}({d})\). For any two Lfuzzy mappings \({S}, {\widehat{\Psi}}:{W}\longrightarrow L^{Y}\), a point \(\mathfrak{o}\in {W}\) is named an Lfuzzy FP of S if we can find \({\widehat{\tau}}_{L}\in L\setminus \{0_{L}\}\) such that \(\mathfrak{o}\in [ {S} \mathfrak{o}]_{{\widehat{\tau}}_{L}}\). A point \(\mathfrak{o}\) is known as a common Lfuzzy FP of S and Ψ̂ if \(\mathfrak{o}\in [{S} \mathfrak{o}]_{{\widehat{\tau}}_{L}}\cap [{ \widehat{\Psi}} \mathfrak{o}]_{{\widehat{\tau}}_{L}}\).
If we can find \({\widehat{\tau}}_{L}\in L\setminus \{0_{L}\}\) such that \([S]_{{\widehat{\tau}}_{L}}, [U]_{{\widehat{\tau}}_{L}}\in \mathcal{V}({W})\), then we define
Observe that \(p_{{\widehat{\tau}}_{L}}\) is a nondecreasing function of \({{\widehat{\tau}}_{L}}\) (see [9]), \({d}^{\infty}_{L}\) is a metric on \(\mathcal{V}({W})\), and since \(({W}, {d})\) is complete, then \((\mathcal{V_{F}}({W}), {d}^{\infty}_{L})\) (see [9]) is also. Adding with, \(({W}, {d})\longmapsto (\mathcal{V}({W}), {\widehat{H}})\longmapsto ( \mathcal{V_{F}}({W}), {d}^{\infty}_{L})\) are isometric embeddings under \({h}\longrightarrow \{{h}\}\) and \(M\longrightarrow \chi _{M}\), respectively, where
The following observation made in [19] is useful in discussing our main idea.
Lemma 1.11
[19] Let S and U be nonempty closed and bounded subsets of an MS W. If \(a\in S\), then \({d}(a, U)\leq{\widehat{H}} (S, U)\).
2 Main results
The idea of Lfuzzy contraction of Hardy–Rogerstype is launched in this section, and the corresponding FP results are studied.
Definition 2.1
Given an MS \(({W}, {d} )\), the Lfuzzy setvalued map \({S} :{W}\longrightarrow L^{W} \) is named an interpolative Hardy–Rogerstype (IHRT) Lfuzzy contraction if we can find a mapping \({\widehat{\tau}}_{L}:{W} \longrightarrow L\setminus \{0_{L}\}\) and constants \({c} , {a_{1}} ,{b_{1}}, {c_{1}} \in ( 0, 1 ) \) with \({a_{1}}+ {b_{1}} + {c_{1}} < 1 \) such that for all \({h}, \mu \in {W} \setminus \mathcal{F}_{ix}({S})\),
where
Theorem 2.2
Let \(({W}, {d} )\) be a CMS and \({S}: {W} \rightarrow L^{{W}}\) be an IHRT Lfuzzy contraction. Suppose that \([{S} {h} ]_{{\widehat{\tau}}_{L}({h})}\) is a nonempty compact subset of W for each \({h} \in {W} \). Then S has an Lfuzzy FP in W.
Proof
Let \({h} _{0} \in {W} \) be arbitrary. Then, by hypothesis, \([ {S} {h}_{0} ]_{{\widehat{\tau}}_{L}({h}_{0})} \in \mathcal{V}({W})\). Choose \({h}_{{1}} \in [{S} {h}_{0}]_{{\widehat{\tau}}_{L}({h}_{0})}\), then for this \({h}_{1}\in {W}\), \([{S} {h}_{1}]_{{\widehat{\tau}}_{L}({h}_{1})}\) is a nonempty compact subset of W. Therefore, we can find \({h} _{2} \in [ {S} {h}_{1}] _{{\widehat{\tau}}_{L}({h}_{1})}\) such that
Setting \({h}= {h}_{0} \) and \(\mu = {h}_{1}\) in (2.1) and using the fact that the function \(\varpi ({h}) = {h}^{1{a_{1}}{b_{1}}{c_{1}}}\) is nondecreasing yields
Suppose that \({d}( {h}_{0}, {h}_{1}) \leq {d}( {h}_{1}, {h}_{2})\), then (2.3) produces
a contradiction. Therefore, \({d}({h}_{1}, {h}_{2} ) < {d}({h}_{0}, {h}_{1})\). Therefore, for \(\zeta = \sqrt{{c}}\) and \(\varrho = \zeta {d}({h}_{0}, {h}_{1})\), (2.3) yields
It follows that \({d}({h}_{1}, {h}_{2})<\varrho \) for some \({h}_{2} \in [ {S} {h}_{1}]_{{\widehat{\tau}}_{L}({h}_{1})}\). Thus, \(\varrho \in E^{{d}}_{([{S} {h}_{0}]_{{\widehat{\tau}}_{L}({h}_{0})}, [{S} {h}_{1}]_{{\widehat{\tau}}_{L}({h}_{1})} )}\). This implies that \([ {S} {h}_{0}] _{{\widehat{\tau}}_{L}} \subseteq N_{{d}}( \varrho , [{S} {h}_{0}]_{{\widehat{\tau}}_{L}({h}_{0})})\) and \({h}_{1} \in N_{{d}}( \varrho , [ {S} {h}_{1}]_{{\widehat{\tau}}_{L}({h}_{1})})\). On similar steps, we can find \({h}_{2} \in N_{{d}} ( \zeta d({h}_{0}, {h}_{1}), [ {S} {h}_{2}]_{{ \widehat{\tau}}_{L}({h}_{2})})\) and \({h}_{3} \in [ {S} {h}_{2}]_{{\widehat{\tau}}_{L}({h}_{2})}\) such that for \(\varrho ^{2} = \zeta ^{2}{d}({h}_{0} , {h}_{1}) \) we have
Therefore, \(\varrho ^{2} \in E^{{d}}_{( [{S} {h}_{1}]_{{\widehat{\tau}}_{L}({h}_{1})} , [{S} {h}_{2}]_{{\widehat{\tau}}_{L}({h}_{2})})}\). By induction, we come up with a sequence \(\{{h}_{\mathfrak{x}} \}_{\mathfrak{x}\geq 1}\) in W such that \({h}_{\mathfrak{x} +1} \in [{S} {h}_{\mathfrak{x}}] _{{ \widehat{\tau}}_{L}({h}_{\mathfrak{x}})}\) and
We now demonstrate that \(\{{h}_{\mathfrak{x}}\}_{\mathfrak{x}\geq 1}\) is Cauchy in W. So, for all \(k \geq 1\),
Passing to limit in (2.5) as \(\mathfrak{x}\longrightarrow \infty \), \(\lim_{ \mathfrak{x}\rightarrow \infty }{d}( {h}_{ \mathfrak{x}}, {h}_{\mathfrak{x}+k}) = 0\). Therefore, \(\{{h}_{\mathfrak{x}}\}_{\mathfrak{x}\geq 1}\) is a Cauchy sequence in W. By completeness of W, we can find \(\mathfrak{o} \in {W}\) such that \({h}_{\mathfrak{x}} \longrightarrow \mathfrak{o} \) as \(\mathfrak{x} \longrightarrow \infty \). Now, to show that \(\mathfrak{o} \) is an Lfuzzy FP of S, assume that \(\mathfrak{o} \notin [{S} \mathfrak{o}]_{{\widehat{\tau}}_{L}( \mathfrak{o})}\) for all \({\widehat{\tau}}_{L}\in L\). Then, replacing h, μ with \({h}_{\mathfrak{x}} \) and \(\mathfrak{o} \), respectively, in (2.1) leads to
Letting \(\mathfrak{x}\longrightarrow \infty \) in (2.6) and using the continuity of d, we obtain \({d}(\mathfrak{o}, [{S} \mathfrak{o}]_{{\widehat{\tau}}_{L}( \mathfrak{o})})= 0 \). This proves that \(\mathfrak{o} \in [ {S} \mathfrak{o} ]_{{\widehat{\tau}}_{L}( \mathfrak{o})}\) for some \({\widehat{\tau}}_{L}(\mathfrak{o})\in L\setminus \{0_{L}\}\). □
Example 2.3
Let \(L=\{a,b,c, g, s, m, n, v\}\) be such that \(a \preceq _{L} s\preceq _{L} c\preceq _{L} v\), \(a\preceq _{L} g\preceq _{L} b\preceq _{L} v\), \(s\preceq _{L} m\preceq _{L} v\), \(g\preceq _{L} m\preceq _{L}v\), \(n\preceq _{L} b\preceq _{L}v\); and each element of the doubletons \(\{c, m\}\), \(\{m,b\}\), \(\{s, n\}\), \(\{n, g\}\) is not comparable. Whence, \((L, \preceq _{L})\) is a complete distributive lattice. Let \({W}= \{2, 6, 7, 12, 20\}\) and define \({d}:{W}\times {W}\longrightarrow \mathbb{R}\) by \({d}({h},\mu )= {h}\mu \) for all \({h},\mu \in {W}\). Clearly, \(({W}, {d})\) is a CMS. Define the mapping \({S}:{W} \longrightarrow L\setminus \{a\}\) as follows:
Suppose that \({\widehat{\tau}}_{L}({h}) = v\setminus \{a\}\) for all \({h}\in {W}\). Then
To show that S is a Hardy–Rogerstype Lfuzzy contraction, take \({h}, \mu \in {W}\setminus \mathcal{F}_{ix}({S})\). Obviously, \({h}, \mu \in \{12, 20\}\). Thus,
Therefore, all the conditions of Theorem 2.2 are verified. We can see that the set of all Lfuzzy FP of S is given by \(\mathcal{F}_{ix}({S}) = \{2, 6\}\).
Turned on by Theorem 1.3 and the idea of [12, Theorem 4], we investigate the next notion of interpolative Reich–Rus–Ćirićtype (IRRCT) Lfuzzy contraction and investigate the condition for the existence of Lfuzzy FP for such contraction.
Definition 2.4
Let \(({W}, {d})\) be an MS. An Lfuzzy setvalued map \({S}:{W}\longrightarrow L^{W}\) is named IRRCT Lfuzzy contraction if we can find a mapping \({\widehat{\tau}}_{L}:{W}\longrightarrow L\setminus \{0_{L}\}\) and constants \(\widehat{\eta}\in [0, 1)\), \({a_{1}},{b_{1}}\in (0,1)\) with \({a_{1}}+{b_{1}}< 1\) such that
for all \({h},\mu \in {W}\setminus \mathcal{F}_{ix}({S})\).
Theorem 2.5
Let \(({W}, {d})\) be a CMS and \({S}:{W}\longrightarrow L^{W}\) be an IRRCT Lfuzzy contraction. Suppose further that \([{S} {h}]_{{\widehat{\tau}}_{L}({h})}\) is a nonempty compact subset of W for each \({h}\in {W}\). Then S has a fuzzy FP in W.
Proof
Let \({h}_{0}\in {W}\) be given. Then, by hypothesis, we can find \({\widehat{\tau}}_{L}({h}_{0})\in L\setminus \{0_{L}\}\) such that \([{S} {h}_{0}]_{{\widehat{\tau}}_{L}({h}_{0})}\in \mathcal{V}({W})\). By compactness of \([{S} {h}_{0}]_{{\widehat{\tau}}_{L}({h}_{0})}\), we can find \({h}_{1}\in [{S} {h}_{0}]_{{\widehat{\tau}}_{L}({h}_{0})}\) with \({d}({h}_{0}, {h}_{1})>0\) such that \({d}({h}_{0}, {h}_{1}) = {d}({h}_{0}, [{S} {h}_{0}]_{{\widehat{\tau}}_{L}({h}_{0})})\). In the same way, by assumption, we can find \({\widehat{\tau}}_{L}({h}_{1})\in L\setminus \{0_{L}\}\) such that \([{S} {h}_{1}]_{{\widehat{\tau}}_{L}({h}_{1})}\) is a nonempty compact subset of W. Thus, we can find \({h}_{2}\in [{S} {h}_{1}]_{{\widehat{\tau}}_{L}({h}_{1})}\) with \({d}({h}_{1}, {h}_{2})>0\) such that \({d}({h}_{1}, {h}_{2}) = {d}({h}_{1}, [{S} {h}_{1}]_{{\widehat{\tau}}_{L}({h}_{1})})\). In this fashion, we come up with a sequence \(\{{h}_{\mathfrak{x}}\}_{\mathfrak{x} \geq 1}\) of points of W with \({h}_{\mathfrak{x} +1}\in [{S} {h}_{\mathfrak{x}}]_{{\widehat{\tau}}_{L}({h}_{ \mathfrak{x}})}\), \({d}({h}_{\mathfrak{x}}, {h}_{\mathfrak{x} +1})>0\) such that \({d}({h}_{\mathfrak{x}}, {h}_{\mathfrak{x} +1}) = {d}({h}_{ \mathfrak{x}}, [{S} {h}_{\mathfrak{x}}]_{{\widehat{\tau}}_{L}({h}_{ \mathfrak{x}})})\). By Lemma 1.11, we have
Now, we show that \(\{{h}_{\mathfrak{x}}\}_{\mathfrak{x}\geq 1}\) is a Cauchy sequence in W. Setting \({h} = {h}_{\mathfrak{x}}\) and \(\mu = {h}_{\mathfrak{x} 1}\) in (2.7), we get
From (2.9), we have
We deduce from (2.10) that for all \(\mathfrak{x}\in \mathbb{N}\),
From (2.11), following the proof of Theorem 2.2, we infer that \(\{{h}_{\mathfrak{x}}\}_{\mathfrak{x}\geq 1}\) is a Cauchy sequence in W. The completeness of this space implies that we can find \(\mathfrak{o}\in {W}\) such that \({h}_{\mathfrak{x}}\longrightarrow \mathfrak{o}\) as \(\mathfrak{x}\longrightarrow \infty \). Now, we show that \(\mathfrak{o}\) is an Lfuzzy FP of W. For this, replacing h and μ with \({h}_{\mathfrak{x}}\) and \(\mathfrak{o}\), respectively, in (2.7) and using Lemma 1.11 gives
Letting \(\mathfrak{x}\longrightarrow \infty \) in (2.12) and using the continuity of the metric d yields \({d}(\mathfrak{o}, [ {S} \mathfrak{o}]_{{\widehat{\tau}}_{L}( \mathfrak{o})})= 0\). Therefore, \(\mathfrak{o}\in [{S} \mathfrak{o}]_{{\widehat{\tau}}_{L}( \mathfrak{o})}\). □
As an extension of the result of Heilpern [9, Theorem 3.1], next we study FP theorems of Hardy–Rogerstype Lfuzzy contraction and Reich–Rus–Ćirićtype Lfuzzy contraction, availing the interpolative technique in association with \({d}^{\infty}_{L}\)metric for Lfuzzy sets. Worthy of note is the fact that Lfuzzy FP results in the setting of \({d}^{\infty}_{L}\)metric are very paramount in computing Hausdorff dimensions. These dimensions aid us to analyze the concept of \(\varepsilon ^{ \infty}\)space, which is of enormous gain in higher energy physics.
Theorem 2.6
Let \(( {W}, {d} )\) be a CMS and \({S}: {W} \longrightarrow \mathcal{V_{F}}({W})\) be an Lfuzzy setvalued map. Suppose that the following conditions are satisfied: we can find \({c} , {a_{1}} ,{b_{1}}, {c_{1}} \in ( 0, 1 ) \) with \({a_{1}}+ {b_{1}} + {c_{1}} < 1 \) such that, for all \({h},\mu \in {W}\setminus \mathcal{F}_{ix}({S})\),
Then S has an Lfuzzy FP in W.
Proof
Let \({h} \in {W}\) be arbitrary, and define the mapping \({\widehat{\tau}}_{L}:{W} \longrightarrow L\setminus \{0_{L}\}\) by \({\widehat{\tau}}_{L}({h}) = 1_{L}\), where \(1_{L}\) is the top element of L. Then, by hypothesis, \([{S} {h}]_{1_{L}} \in \mathcal{V}({W})\). Now, for every \({h},\mu \in {W}\setminus \mathcal{F}_{ix}({S})\),
Since \([{S} {h}]_{1_{L}}\subseteq [{S} {h}]_{{\widehat{\tau}}_{L}({h})}\in \mathcal{V}({W})\), therefore \({d}({h}, [{S} {h}]_{{\widehat{\tau}}_{L}({h})} ) \leq {d}({h}, [{S} {h}]_{1_{L}}) \) for each \({\widehat{\tau}}_{L}({h}) \in L\setminus \{0_{L}\}\). It follows that \(p({h}, {S} {h} )\leq {d}({h}, [{S} {h}]_{1_{L}}) \). Thus,
Therefore, Theorem 2.2 can be invited to find \(\mathfrak{o}\in {W} \) such that \(\mathfrak{o} \in [{S} \mathfrak{o} ]_{1_{L}}\). □
By ignoring some terms in Theorem 2.6, we can obtain the next result using similar arguments.
Theorem 2.7
Let \(({W}, {d} )\) be a CMS and \({S}: {W} \longrightarrow \mathcal{V_{F}}({W})\) be an Lfuzzy setvalued map. Suppose that the following conditions are satisfied: we can find \(\widehat{\eta}\in [0, 1)\) and \({a_{1}}, {b_{1}}\in ( 0, 1 ) \) with \({a_{1}}+ {b_{1}} < 1 \) such that, for all \({h},\mu \in {W}\setminus \mathcal{F}_{ix}({S})\),
Then S has an Lfuzzy FP in W.
Example 2.8
Let \({W} = \{\wp _{\mathfrak{x}} = \frac{\mathfrak{x}(\mathfrak{x} +1)}{5}: \mathfrak{x} = 1, 2, \ldots \}\cup \{0\}\) and \({d}({h}, \mu ) = {h}\mu \) for all \({h}, \mu \in {W}\). Then \(({W}, {d})\) is a CMS. Let \(L=\{a,b,c, g, s, m, n, v\}\) be such that \(a \preceq _{L} s\preceq _{L} c\preceq _{L} v\), \(a\preceq _{L} g\preceq _{L} b\preceq _{L} v\), \(s\preceq _{L} m\preceq _{L} v\), \(g\preceq _{L} m\preceq _{L}v\), \(n\preceq _{L} b\preceq _{L}v\); and each element of the doubletons \(\{c, m\}\), \(\{m,b\}\), \(\{s, n\}\), \(\{n, g\}\) is not comparable. Then \((L, \preceq _{L})\) is a complete distributive lattice. Define an Lfuzzy setvalued map \({S}: {W}\longrightarrow \mathcal{V_{F}}({W})\) as follows:
For \({h} = 0\),
and for \({h}\in {W}\setminus \{0\}\),
Define the mapping \({\widehat{\tau}}_{L}:{W}\longrightarrow L\setminus \{a\}\) by \({\widehat{\tau}}_{L}({h}) = v\) for all \({h}\in {W}\). Then
Now, to see that the contractive condition (2.15) holds, let \({h}, \mu \in {W}\setminus \mathcal{F}_{ix}({S})\). Clearly, \({h} = \mu = \wp _{1}\). Therefore,
for all \(\widehat{\eta}\in (0, 1)\). This shows that (2.15) holds for all \({h}, \mu \in {W}\). Therefore, all the assumptions of Theorem 2.7 are satisfied. We see that S has many Lfuzzy FP in W.
In contrast, S is not a fuzzy setvalued contraction in the sense of Heilpern [9]. To show this, take \({h} = 0\) and \(\mu = \wp _{\mathfrak{x} 1}\), \(\mathfrak{x}\geq 3\), we have
Therefore, the main result of Heilpern [9] is inapplicable to this example.
3 Applications to crisp setvalued and singlevalued mappings
Let \(({W}, {d})\) be an MS, \(CB({W})\) and \(\mathcal{N}({W})\) be the classes of nonempty closed and bounded and nonempty subsets of W, respectively. A mapping \(\mho :{W}\longrightarrow \mathcal{N}({W})\) is named a multivalued contraction (see [19]) if we can find a constant \({c}\in (0, 1)\) such that \({\widehat{H}}(F {h}, F \mu )\leq {c}{d}({h}, \mu )\) for all \({h}, \mu \in {W}\). A point \(\mathfrak{o}\in {W}\) is termed an FP of ℧ if \(\mathfrak{o}\in F\mathfrak{o}\). Nadler [19, Theorem 5] noted that each multivalued contraction on a CMS enjoys an FP. Among the extensions of multivalued contractions in the sense of Nadler that we are concerned with here are the ones studied by Reich [23] and Rus [25].
Theorem 3.1
(see Rus [25]) Let \(({W}, {d})\) be a CMS and \(\mho :{W}\longrightarrow CB({W})\) be a multivalued mapping. Suppose that we can find \({a_{1}}, {b_{1}}\in \mathbb{R}_{+}\) with \({a_{1}}+{b_{1}}<1\) such that, for all \({h}, \mu \in {W}\),
Then we can find \(\mathfrak{o}\in {W}\) such that \(\mathfrak{o}\in F\mathfrak{o}\).
Theorem 3.2
(see Reich [23]) Let \(({W}, {d})\) be a CMS and \(\mho :{W}\longrightarrow CB({W})\) be a multivalued mapping. Suppose that we can find \({a_{1}}, {b_{1}}\in \mathbb{R}_{+}\) with \({a_{1}}+{b_{1}}+c<1\) such that, for all \({h}, \mu \in {W}\),
Then we can find \(\mathfrak{o}\in {W}\) such that \(\mathfrak{o}\in F\mathfrak{o}\).
Herewith, we come up with some consequences and equivalent results of our main theorems in the framework of both singlevalued and multivalued mappings. First, we present multivalued analogues of Theorems 2.2 and 2.5. They are also crisp setvalued refinements of the recently established FP theorems due to Karapinar et al. [13, Theorem 4] and Karapinar et al. [11, Corollary 1], respectively.
Corollary 3.3
Let \(({W}, {d}) \) be a CMS and \(F:{W} \longrightarrow \mathcal{V}({W}) \) be a multivalued mapping. Suppose that we can find \({c}, {a_{1}}, b,c, \in (0,1] \) with \({a_{1}}+b+c < 1 \) such that, for all \({h}, \mu \in {W} \setminus \mathcal{F}_{ix}(F) \),
Then we can find \(\mathfrak{o} \in {W} \) such that \(\mathfrak{o}\in F\mathfrak{o} \).
Proof
Consider a mapping \(\vartheta : {W} \longrightarrow L\setminus \{0_{L}\}\) and an Lfuzzy setvalued map \({S}: {W} \longrightarrow L ^{{W}}\) defined by
Taking \({\widehat{\tau}}_{L}({h}) = \vartheta ({h}) \) for all \({h}\in {W}\) leads to
Therefore, Theorem 2.2 can be used to find \(\mathfrak{o} \in {W} \) such that \(\mathfrak{o} \in F\mathfrak{o} = [{S} \mathfrak{o}]_{{ \widehat{\tau}}_{L}}\). □
Example 3.4
Let \({W} = [1, 5]\) and \({d}({h}, \mu ) = {h}\mu \) for all \({h}, \mu \in {W}\). Then \(({W}, {d})\) is a CMS. Define \(\mho :{W}\longrightarrow \mathcal{V}({W})\) by
Let \({h}, \mu \in {W}\setminus \mathcal{F}_{ix}(F)\). Clearly, \({h}, \mu \in (1, 2)\) and
Therefore, all the assumptions of Corollary 3.3 are satisfied. We see that ℧ has many FP in W.
In contrast, ℧ is not a multivalued contraction since for \({h} = 1\) and \(\mu = 2\) we have
for all \({c}\in (0, 1)\). Therefore, the result of Nadler [19, Theorem 5] is not applicable in this example to obtain an FP of ℧. In the same way, since \(\mho 1 = [1, 2] \) and \(\mho 2 = [3, 5]\), we have
Therefore,
for all \({a_{1}}, {b_{1}}\in \mathbb{R}_{+}\) satisfying \({a_{1}}+{b_{1}}<1\). That is to say, Theorem 3.1 due to Rus [25] is inapplicable to this example to find an FP of ℧.
In like manner,
for all \({a_{1}}, {b_{1}}, {c_{1}}\in \mathbb{R}_{+}\) with \({a_{1}}+{b_{1}}+{c_{1}}<1\). Therefore, Theorem 3.2 due to Reich [23] is not applicable in this case to locate any FP of ℧.
Corollary 3.5
(see Karapinar et al. [13, Theorem 4]) Let \(({W}, {d})\) be a CMS and \(f:{W} \rightarrow {W} \) be a singlevalued mapping. Suppose that we can find \({c} , {a_{1}}, {b_{1}}, {c_{1}} \in (0, 1) \) with \({a_{1}}+{b_{1}}+{c_{1}} < 1\) such that, for all \({h}, \mu \in {W} \setminus \mathcal{F}_{ix}(f)\), we have
Then we can find \(\mathfrak{o}\in {W}\) such that \(f\mathfrak{o}=\mathfrak{o}\).
Proof
Let \({\widehat{\tau}}_{L}:{W}\longrightarrow L\setminus \{0_{L}\}\) be a mapping, and define an Lfuzzy setvalued map \({S}:{W}\longrightarrow L^{W}\) as follows:
Then
Clearly, \(\{f{h}\}\in \mathcal{V}({W})\) for all \({h}\in {W}\). Note that in this case \({\widehat{H}}([{S} {h}]_{{\widehat{\tau}}_{L}({h})}, [{S} {h}]_{{ \widehat{\tau}}_{L}(\mu )}) = {d}(f{h}, f \mu )\) for all \({h}, \mu \in {W}\). Therefore, Theorem 2.2 can be invited to find \(\mathfrak{o}\in {W}\) such that \(\mathfrak{o}\in [{S} \mathfrak{o}]_{{\widehat{\tau}}_{L}( \mathfrak{o})} = \{f\mathfrak{o}\}\), which signifies further that \(\mathfrak{o}=f\mathfrak{o}\). □
By maintaining the procedure for determining Corollary 3.5, additionally, we can arrive at the following.
Corollary 3.6
(see Karapinar et al. [11, Corollary 1]) Let \(({W}, {d})\) be a CMS and \(f:{W} \rightarrow {W} \) be a singlevalued mapping. Suppose that we can find \({c} , {a_{1}}, {b_{1}}\in (0, 1) \) with \({a_{1}}+{b_{1}} < 1\) such that, for all \({h}, \mu \in {W} \setminus \mathcal{F}_{ix}(f)\), we have
Then we can find \(\mathfrak{o}\in {W} \) such that \(f\mathfrak{o}=\mathfrak{o}\)
4 Applications to Fredholmtype integral inclusions
Herewith, we apply Theorem 2.2 to investigate new conditions for the existence of solutions to a Fredholmtype integral inclusion, given as
where \({h}\in C ( [d, j], \mathbb{R} )\) is an unknown realvalued continuous function defined on \([d, j]\), τ̂ is a given realvalued continuous function, and L is a given setvalued map. The family of nonempty compact and convex subsets of \(\mathbb{R}\) is denoted by \(\digamma _{cv}(\mathbb{R})\).
Now, we study the existence of solutions of (4.1) under the following assumptions.
Theorem 4.1
Let \({W} = C([d, j],\mathbb{R})\) and suppose that:
 \((C_{1})\):

the setvalued map \(L:[d, j]\times [d, j]\times \mathbb{R}\longrightarrow \digamma _{cv}( \mathbb{R})\) is such that, for every \({h}\in {W}\), the map \(L_{h}(\intercal , t) := \mathcal{L}(\intercal ,t, {h}(t))\) is lower semicontinuous;
 \((C_{2})\):

\(\widehat{\tau}\in C ([d, j], \mathbb{R} )\);
 \((C_{3})\):

we can find a function \(\xi : (0, \infty )\longrightarrow \mathbb{R}\) such that, for all \({h}, \mu \in {W}\),
$$ \begin{aligned} {\widehat{H}} \bigl(L_{h}(\intercal , t), L_{\mu}( \intercal ,t) \bigr) \leq \pi (\intercal ,t)\xi (\intercal ) \bigl( \bigl\vert {h}(t)\mu (t) \bigr\vert \bigr)^{r} \end{aligned} $$for each \(\intercal ,t\in [d, j]\), where \(\sup_{d\leq t \leq j} (\int _{d}^{j}\pi ( \intercal , t)\,{d} s )\leq 1\), \(\pi (\intercal ,.)\in L^{1}[d, j]\) and \(r\in (0, 1)\).
Then the integral inclusion (4.1) has at least one solution in W.
Proof
Define \({d}:{W}\times {W}\longrightarrow \mathbb{R}\) by
then \(({W}, {d})\) is a complete MS. Let \({S}:{W}\longrightarrow L^{W}\) be an Lfuzzy setvalued map. Consider the \({\widehat{\tau}}_{L}\)level set of S defined as
Obviously, the set of solutions of equation (4.1) coincides with the set of Lfuzzy FP of the setvalued map S. Therefore, we need to show that under the given assumptions, S has at least one Lfuzzy FP in W. To do this, we will verify that all the assumptions of Theorem 2.2 are satisfied.
Let \({h}\in {W}\) be arbitrary. Since the setvalued map \(L_{h} : [d, j]^{2}\longrightarrow \digamma _{cv}(\mathbb{R})\) is lower semicontinuous, it follows from Michael’s selection theorem ([15, Theorem 1]) that we can find a continuous map \(\rho _{h} :[d, j]^{2}\longrightarrow \mathbb{R}\) such that \(\rho _{h}(\intercal , \rho _{h}(\intercal , t)\in L_{h}(\intercal , t)\) for each \((\intercal , t)\in [d, j]^{2}\). Thus, \(\widehat{\tau}(\intercal )+\int _{d}^{j}\rho _{h}(\intercal , t)\,{d} s \in [{S} {h}]_{{\widehat{\tau}}_{L}({h})}\). So \([{S} {h}]_{{\widehat{\tau}}_{L}({h})}\) is nonempty. One can easily see that \([{S} {h}]_{{\widehat{\tau}}_{L}({h})}\) is a compact subset of W. In other words, given that \(\widehat{\tau}\in C ([d, j] )\) and \(L_{h}(\intercal , t)\) is continuous on \([d, j]^{2}\), their range sets are also continuous for each \({h}\in {W}\).
Take \({h}_{1}, {h}_{2}\in {W}\). Then we can find \({\widehat{\tau}}_{L}({h}_{1}),{\widehat{\tau}}_{L}({h}_{2})\in L \setminus \{0_{L}\}\) such that \([{S} {h}_{1}]_{{\widehat{\tau}}_{L}({h}_{1})}\) and \([{S} {h}_{2}]_{{\widehat{\tau}}_{L}({h}_{2})}\) are nonempty compact subsets of W. Then we can find an arbitrary point \(\mu _{1}\in [{S} {h}_{1}]_{{\widehat{\tau}}_{L}({h}_{1})}\) with
This means that for each \((\intercal , t)\in [d, j]^{2}\) there exists \(\rho _{{h}_{1}}\in L_{{h}_{1}}(\intercal , t)\) such that
Since from \((C_{2})\)
for each \(\intercal ,t\in [d,j]\) and \(r\in (0, 1)\), we can find \(\rho _{{h}_{2}}\in L_{{h}_{2}}(\intercal , t)\) such that
for all \((\intercal , t)\in [d, j]^{2}\).
Now, consider the setvalued map \(\mathfrak{M}\) defined by
Taking into account the fact that from \((C_{1})\), \(\mathfrak{M}\) is lower semicontinuous, we can find a continuous map \(\rho _{{h}_{2}}:[d, j]^{2}\longrightarrow \mathbb{R}\) such that \(\rho _{{h}_{2}}(\intercal , t)\in \mathfrak{M}(\intercal , t)\) for all \((\intercal , t)\in [d, j]^{2}\). Then
Thus, \(\mu _{2}\in [{S} {h}_{2}]_{{\widehat{\tau}}_{L}({h}_{2})}\) and
Therefore,
Therefore, taking \({h} ={h}_{1}\) and \(\mu = {h}_{2}\) in (4.2) yields
Therefore, all the conditions of Theorem 2.2 are fulfilled with \(\xi (\intercal )={c} \intercal \) for all \(\intercal > 0\) and \({c}\in (0, 1)\). As a result, the conclusion of Theorem 4.1 holds good. □
5 Conclusion
New invariant point results of Lfuzzy maps were introduced and conditions under which such mappings possess FPs were studied (see Theorems 2.2, 2.5, 2.6, 2.7) in this paper. The presented results are refinements of some already announced ideas in [9–13, 19, 23, 25]. Comparative examples (Examples 2.8 and 3.4) were constructed to support the theoretical assumptions of the proposed concepts. From the usability point of consideration, Theorem 2.2 was applied to set up new condition for analyzing the existence of solutions to a Fredholmtype integral inclusion.
The findings of this work widened up the coverage of nonclassical mathematics by incorporating interpolation approaches in Lfuzzy FP theory. However, the ideas presented herein, being set up in the framework of MS, are rudimentary. Whence, they can be finetuned when studied in the setting of some generalized MS or quasiMS.
Data Availability
No datasets were generated or analysed during the current study.
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Acknowledgements
The authors, A. Aloqaily and N. Mlaiki would like to thank Prince Sultan University for paying the APC and for the support through the TAS research lab.
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Conceptualization, Mohammed Shehu Shagari and Trad Alotaibi,; Formal analysis,Hassen Aydi and Nail Mlaiki; Investigation, Ahmad Aloqialy, Hassen Aydi and Mohammed Shehu Shagari; Methodology, Trad Alotaibi and Nabil Mlaiki; Writing – original draft, Mohammed Shehu Shagari, Ahmad Aloqialy; Writing – review and editing, Hassen Aydi and Mohammed Shehu Shagari.
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Shagari, M.S., Alotaibi, T., Aydi, H. et al. New Lfuzzy fixed point techniques for studying integral inclusions. J Inequal Appl 2024, 83 (2024). https://doi.org/10.1186/s13660024031577
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DOI: https://doi.org/10.1186/s13660024031577