1. Introduction
A fixed-point theorem is an outcome indicating that a mapping
on a nonempty set has at least one fixed point (
) under some circumstances on a nonempty set
and
that can be stated in general terms. Results of this kind are most generally useful in almost all types of applied sciences. The Banach contraction principle (
) [
1] is one of the prototypical, simple, and multiuse results in
theory in a metric space (
) structure. Many studies embraced the applications and refinements of this principle in various arms by, for example, relaxing the axioms, and employing different mappings and several kinds of metric spaces (
). In this context, the work of Rhoades [
2] is useful in assembling multiple modifications of Banach-type contractive ideas.
Finding new spaces and their properties has been an inviting topic among mathematical investigators, and
b-
is presently a focus in the literature. The idea began with the work of Bakhtin [
3] and Bourbaki [
4]. Later on, Czerwik [
5] provided an assumption that was more relaxed than the known triangle inequality, and officially brought up
b-
to improve the Banach
theorem. Meanwhile, the notion of
b-
is gaining enormous refinements; see, for example, [
6,
7]. For some new surveys on the concepts of
theory in the framework of
b-
, we direct the interested reader to Karapinar [
8]. An active field of
theory is the investigation of hybrid contractions. The notion was considered in two ways: first, hybrid contraction concerns contractions involving single-valued and set-valued maps; second, harmonizing linear and nonlinear contractions. Hybrid
theory plays enormous roles in functional inclusions, optimization theory, fractal graphics, discrete dynamics for set-valued operators, and other ambits of functional analysis. For a few related works, refer to [
9,
10,
11].
As a refinement of the theory of crisp sets, the fuzzy set (
) was presented by Zadeh [
12]. Since then, to avail this concept, many researchers have advanced the theory and its roles to other arms of sciences, social sciences, and engineering. In 1981, Heilpern [
13] utilized the idea of
to initiate a class of
-valued maps, and coined the
theorem for fuzzy contraction mapping, which is a fuzzy analog of the
theorem of Nadler [
14]. Later, several researchers examined the existence of
of
-valued maps and related developments in
theory; for example, the work of Al-Mazrooei et al. [
15], Azam et al. [
16], Alansari et al. [
17], Martino and Sessa [
18], Qiu and Shu [
19], and Shehu and Akbar [
20].
Integral inclusions emerge in diverse domains in mathematical physics, control theory, critical point theory for nonsmooth energy functionals, differential variational inequalities,
arithmetic, and traffic theory (see, for instance, [
21,
22,
23]). Commonly, the first most examined aspect in the study of integral inclusions is the criteria for the occurrence of their solutions. Seeveral authors applied different
approaches and topological tools to deduce the existence results of integral inclusions in various spaces; see, for example, Appele et al. [
21], Cardinali and Papageorgiou [
23], Kannan and O’Regan [
24], Pathak et al. [
25], Sintamarian [
26], and the references therein. Almost all of the results proposed in the above papers depend on the multivalued versions of the Banach, Leray–Schauder, Matelli, Schauder, and Sadovskii-type
theorems. Moreover, the ambient space of the existence results is either a Banach space or a classical
.
With the above discussion in mind, we propose the idea of
b-hybrid fuzzy contraction (
-contraction) in a
b-
, and prove a fuzzy
theorem via this contraction. Thereafter, a few corollaries are deduced that include
theorems due to Heilpern [
13], Karapinar and Fulga [
11], Nadler [
14], and allied ones. Moving further, we employ one of our results to present a sufficient yardstick for the existence of solutions to an integral inclusion of the Fredholm type. The latter inference was coined from Sintamarian [
26]. However, our result, which was achieved through a
b-hybrid
-contraction in the body of
b-
, leads to a new existence principle that improves and complements existing ideas.
The rest of the paper is organized as follows:
Section 2 shows basic definitions and results needed in the sequel. In
Section 3, the notion of a
b-hybrid
-contraction and corresponding
ideas are discussed. Some consequences of our proposed concepts in the theory of multivalued and single-valued mappings are highlighted in
Section 4.
Section 5 establishes the new yardstick for obtaining the solutions of Fredholm-type integral inclusions. Concluding statements of the principal ideas presented here are stated in
Section 6.
2. Preliminaries
Here, we collate coherent notations, specific definitions and basic results needed hereafter. Throughout this paper,
,
and
signify sets of positive integers, non-negative real numbers, and real numbers, respectively. Most of these preliminaries are from [
5,
11,
14].
In 1993, Czerwik [
5] launched the notion of a
b-
as follows:
Definition 1 ([
5]).
Let Ξ
be a nonempty set, and be a real number. Suppose that mapping satisfies the following yardstick for all :- (i)
if and only if (self-distancy).
- (ii)
(symmetry).
- (iii)
(weighted triangle inequality).
Afterwards, is termed a b-.
Example 1 ([
27]).
Let and be defined byAfter that, is a b- with the parameter , but σ is not a continuous functional.
Definition 2 ([
4]).
Let be a b-. A sequence is termed as follows:- (i)
Convergent if and only if we can find for which as , and we write this as .
- (ii)
Cauchy if and only if as .
- (iii)
Complete if every Cauchy sequence in Ξ is convergent.
Definition 3 ([
4]).
Let be a b-. After that, a subset P of Ξ
is termed:- (i)
Compact if and only if, for every sequence of elements of P, we can find a subsequence that converges to an element of P.
- (ii)
Closed if and only if for every sequence of elements of P that converges to an element ℏ, we have .
Definition 4 ([
28]).
A nonempty subset P of Ξ
is termed proximal if, for each , we can find for which . Throughput this paper, , and depict the set of all nonempty closed and bounded subsets of , the class of all nonempty bounded proximal subsets of , and the class of nonempty compact subsets of , respectively.
Let
be a
b-
. For
, function
, defined by
is termed the Hausdorff–Pompeiu
b-metric on
generated
, for which
see [
29].
Definition 5 ([
11]).
A nondecreasing function is termed:- (i)
a c-comparison function if as for every ;
- (ii)
a b-comparison function if we can find , and a convergent non-negative series for which , for and any , for which denotes the iterate of φ.
denotes the class of functions obeying the following yardstick:
- (i)
is a b-comparison function.
- (ii)
if and only if .
- (iii)
is continuous.
Lemma 1 ([
30]).
For a comparison function , the following properties hold:- (i)
Each iterated is also a comparison function.
- (ii)
for all .
Lemma 2 ([
30]).
Let be a b-comparison function. After that, series converges for every . Remark 1 ([
11]).
From Lemma 2, every b-comparison function is a comparison function; thus, from Lemma 1, every b-comparison function satisfies . Lemma 3 ([
29]).
Let be a b-. For and , the following yardstick holds:- (i)
, .
- (ii)
.
- (iii)
.
- (iv)
.
- (v)
.
- (vi)
.
Let
depict a reference set.
in
is a function with domain
and values in
. Designed with
, the class of all
is in
. If
P is a
in
, then
is the grade of the membership of
ℏ in
P. The
-level set of a
P is depicted with
and defined as follows:
A
P, in a metric linear space
V, is an approximate quantity if and only if
is compact and convex in
V and
.
We depict the collection of all approximate quantities in
V with
. If we can find an
for which
, we define
Definition 6 ([
13]).
Let Ξ be an arbitrary set, and Y a . Mapping is termed -valued map. An -valued map P is a fuzzy subset of with membership function . Value is the grade of membership of j in . Definition 7 ([
13]).
Let be -valued maps. Point is a fuzzy of P if we can find an for which . Point ♭ is a common fuzzy of P and L if we can find an for which . The set of all of P is depicted with , and the of P and L with .
Example 2. Let and . is defined withAfter that, P is a -valued map. for all and . The graphical representation of P showing all possible grades of membership of j in the is depicted in Figure 1. Figure 1 shows that, for instance, the grade of membership of in the was . Example 3. Let . Define byAfter that, P is a -valued map. for all and . Figure 2 depicts the graphical representation of the membership values of j in . Figure 2 shows that element had a full grade of membership (i.e., 1) in the . 3. Main Results
First, we present the notion of b-hybrid -contraction in the following manner.
Definition 8. Let be a b- and be -valued maps. Pair forms a b-hybrid -contraction if, for each , we can find for which ,for which , , , with andfor which Theorem 1. Let be a complete b- and be -valued maps. Suppose that, for each , we can find for which and are bounded proximal subsets of Ξ. If pair forms a b-hybrid -contraction, then P and L enjoy a common fuzzy in Ξ.
Proof. Let
; then, by supposition, we can find
for which
.
is chosen for which
. Similarly, we can find
for which
. So, we can find
for which
. We can then find a sequence
of elements of
for which
and
By Lemma 3 and the above results,
Suppose that
for some
and
. After that,
Hence, by availing the continuity of
and Lemma 1, we have
a contradiction. It follows that
. Whence,
So,
happens to emerge the common fuzzy
of
P and
L.
Again, for
and
,
. Hence, with property
of
, we obtain
from which, on a related reasoning, the same deduction follows that
. For this purpose, we hypothesize that, for all
,
Now, regarding (
1), we set
and
,
That is,
Consider the two following cases:
Case 1:
. Suppose that
. Then, from (
4), we have
From (
1) and (
5), we have
Employing the given property of
, (
6) provides that
which is a contradiction. Hence, it follows that
. Thus, from (
6), we obtain
Setting
in (
7), we have
From (
8), utilizing Lemma 2 and the triangle inequality with respect to
, for every
,
Hence,
is a Cauchy sequence, and we can find
for which
Now, we demonstrate that ♭ is the expected common fuzzy
of
P and
L. First, assume that
. After that, with Lemma 3 and considering case
in Contractive Inequality (
1), we have
Letting
in (
10) and availing the fact that
give
and
,
yields a contradiction. Hence,
. On a related reasoning, by assuming that ♭ is not a fuzzy
of
L, and noting
we can demonstrate that
. Hence, for
, ♭ is a common fuzzy
of
P and
L.
Case 2:
. employing Inequality (
1) on the recognition of
b-comparison of
,
Assuming that
, (
11) gives
a contradiction. Hence,
by employing (
11) and (
12),
The (
13) is equivalent to (
8). So, we infer that iterative sequence
is Cauchy in
. Thus, the completeness of this space guarantees that
as
, for some
.
To realize that ♭ is a common fuzzy
of
L and
P, we apply Lemma 3 and Inequality (
1) as follows:
for which
. Hence, under this limiting case, (
14) becomes
Via criterion
of
, (
15) implies that
. Hence,
. On a related reasoning, one can demonstrate that
. Hence, ♭ is the common fuzzy
of
P and
L. □
Corollary 1. Let be a b- and be a -valued map. Assume that, for each , we can find for which is a bounded proximal subset of Ξ, andfor which , , , with andfor whichAfter that, . Proof. Put in Theorem 1. □
Example 4. Set and are defined by . After that, is a complete b- with parameter . is not an ; for example, taking and ,For , consider two -valued maps , defined as follows: If we take and for all , thenClearly, . is defined with . Then, . For , and via elementary calculation, we haveIn this case, and . Thus, all the suppositions of Corollary 1 are obeyed. Corollary 2. Let be a complete b-, and be a -valued map. Suppose that, for , we can find for which is a bounded proximal subset of Ξ. Iffor all , for which andAfter that, we can find for which . Proof. , and are taken in Theorem 1. □
Corollary 3. Let be a complete b- and be -valued maps. Assume that, for each , we can find and for whichfor which . After that, P and L have a common fuzzy in Ξ. Proof. , and are taken in Theorem 1. □
Corollary 4. Let be a complete b- and be a -valued map. Further, assume that, for all , we can find for which is a bounded proximal subset of Ξ, andfor all . After that, we can find for which . Proof. , , and are put in Theorem 1. □
Corollary 5. Let be a complete b- and be -valued maps for whichfor which , , , with andfor whichAfter that, we can find for which and . Proof. Take
and
. Then, by supposition,
and
are nonempty compact subsets of
. Now, via the definition of
and
-metric for
, we have
Since
for each
, then
for each
. It follows that
. Similarly,
. Furthermore, this implies that, for all
,
Theorem 1 can, thus, be applied to obtain
for which
, that is,
and
. □
Corollary 6. Let be a complete b- and be a -valued map. Assume that we can find for whichfor each . After that, we can find for which . Proof. Place , and in Corollary 5. □
Example 5. is endowed with metric for all . After that, Ξ is a complete b- with parameter . is not a metric; to see this, take and . After that,Now, define by Let . Then,Suppose that, without loss of generality, for all . If , then , so for all . Otherwise, for all , we haveThus, all the yardsticks of Corollary 6 hold. In this case, we can find for which . Corollary 7. Let be a complete , and be -valued maps. Assume that, for each , we can find for which and are nonempty compact subsets of Ξ, andfor which , , , with andfor whichAfter that, P and L have a common fuzzy in Ξ. Proof. Take in Theorem 1. □
Corollary 8. (Heilpern (Theorem 3.1 [13])) Let be a complete and be a -valued map. In addition, suppose that we can find for whichfor each . After that, we can find for which . Proof. Take in Corollary 6. □
4. Applications and Significance in the Theory of Multivalued and Single-Valued Mappings
Let be an and the class of nonempty subsets of . A set-valued mapping is a multivalued map. Point is a of T if . For a single-valued mapping , if , then ℏ is a of T.
In 1969, Nadler [
14] first gave a refinement of the
for multivalued map by availing the Hausdorff metric. Since then, a number of refinements in various frames of Nadler’s
theorem have been observed by several authors; see, for example, [
6,
25,
31]. Following this advancement, we obtain some consequences of the corresponding results of the previous section in the setting of multivalued and single-valued mappings.
Corollary 9. Let be a complete b- and be multivalued maps. Assume that, for each ,we can find for whichfor which , , , with andfor whichAfter that, we can find for which . Proof. Consider a mapping
and a pair of
-valued maps
defined by
and
After that, for
,
and
Hence, Theorem 1 can be applied to obtain
for which
□
Corollary 10. Let be a complete and be multivalued maps. Assume that, for each , we can find for whichfor which , , , with andfor whichAfter that, we can find for which . Proof. Take in Corollary 9. □
Corollary 11. Let be a complete b- and be a multivalued map. Assume that we can find for whichfor each . After that, we can find for which . Proof. Set , and in Corollary 9. □
The following example supports the suppositions of Corollary 11.
Example 6. Let be equipped with the metric for all . After that, Ξ is a complete b- with parameter . However, is not a metric; for instance, take and . After that,Now, define by Without loss of generality, let for all . If , then , Hence, for all . Otherwise, for all (that is, and ), consider the following cases:
Case 1: If and , then and Case 2: If and , then andHence, we infer that all the suppositions of Corollary 11 are obeyed for all . Here, . Corollary 12. (Nadler [14] Theorem 5) Let be a complete and be a multivalued map. Assume that we can find for whichfor each . After that, we can find for which . Proof. Take in Corollary 11. □
Corollary 13 ([
11] Theorem 1).
Let be a complete b-, and be single-valued mapping. Assume that, for each , we can find for whichfor which , , , with andfor whichAfter that we can find for which . Proof. for every . Consider a mapping defined as . After that, all the yardsticks of Corollary 9 are reduced to the yardstick of Corollary 13 with and , for all . Thus, by applying Corollary 9, we can find for which . The definition of implies that . Hence, . □
6. Conclusions
The basic notion of Banach’s theorem is understood as a modification of the successive approximation method that was initially used by Cauchy, Liouville, Lipschitz, Picard, and Poincar in the context of classical . However, in certain spaces, the triangle inequality cannot be obeyed. However, by taking the product of parameter with the right-hand side of the inequality, we can derive a more versatile abstract frame, namely, the b-. Following this advancement, in this work, the idea of a b-hybrid -contraction was proposed in the setting of b-. The results suggest merging several ideas in a theorem. A few of these particular cases are mentioned. We then established an existence theorem for the solutions of an integral inclusion of the Fredholm type by utilizing one of the presented results. The main ideas of this paper, discussed in a fuzzy setting, are fundamental. As possible future work, the paper can be examined in the setting of refined such as soft and rough sets, and related domains. In addition, the b- component of this work can be extended to other dislocated or quasimetric spaces.