1. Introduction and Preliminaries
In 1922, Stefan Banach [
1] established a prominent fixed point result for contractive mappings in complete metric space
.
Definition 1. ([1]) A mapping is called a contraction if ∃ such that Jleli and Samet [
2] introduced a new type of contraction and established some new fixed point theorems for such contraction in the context of generalized metric spaces.
Definition 2. Let be a function satisfying:
- ()
is nondecreasing.
- ()
For each sequence , if and only if
- ()
There exist and such that
A mapping
is said to be
-contraction if there exist the function
satisfying (
)–(
) and a constant
such that,
. We denote by
the family of functions
satisfying (
–(
) and which are continuous from the right. For more details in this direction, we refer the following [
3,
4,
5,
6,
7,
8] to the reader.
In 1969, Nadler [
9] initiated the notion of multi-valued contraction and extended Banach contraction principle from single-valued mapping to multivalued mapping.
Definition 3. ([9]) A point φ∈ is called a fixed point of the multi-valued mapping → if φ. For
let
be defined by
where
Such
H is called the generalized Hausdorff–Pompieu metric induced by the metric
and
and
indicate the classes of all nonempty, closed and closed and bounded subsets of
, respectively.
Definition 4. ([9]) A mapping Q: → is said to be a multi-valued contraction if ∃ such that Theorem 1. ([9]) Let (Q, ϱ) be a complete metric space and Q: → a multi-valued contraction; then, Q has a fixed point. In 1994, Constantin [
10] introduced a new family
of continuous functions
satisfying the following assertions:
- ()
- ()
is sub-homogeneous, that is, for all and we have
- ()
is a non-decreasing function, that is, for
we have
and if
then
and
Constantin also obtained a random fixed point theorem for multivalued mappings. Isik [
11] utilized the above-mentioned family
of functions and established fixed point theorem for multivalued mappings in complete metric space.
On the other hand, Heilpern [
12] employed the conception of fuzzy set to introduce a family of fuzzy mappings to generalize the multivalued mapping and obtained a fixed point result for fuzzy contractions in the setting of metric linear space. It is worth noting that the theorem given by Heilpern [
12] is a real generalization of Nadler’s result in the sense of fuzzy mappings. Moreover, we use the following notations that have been recorded from [
13,
14,
15,
16,
17,
18,
19,
20]:
A fuzzy set in
is a mapping with domain
and values in
. If
is a fuzzy set and
, then
is called the grade of membership of
in
. The
-level set of
is denoted by
and is defined in this way:
Here, denotes the closure of the set . Let () be the collection of all fuzzy sets in a metric space
Definition 5. [15] Let : ( A point is called a common α- fuzzy fixed point of and if ∃ so that If then it reduces to common fixed point of : (). Remark 1. Let be a metric space and let : and for each ∃ such that . Assume that there exist some and such thatfor all with Then, from Inequation (
4), we get
Since Θ is non-decreasing, we obtain
for all
with
We need the following lemmas of [
11] in the proof of our main result.
Lemma 1. [11] If and are such thatthen Lemma 2. [11] Let be a metric space and with Then, for each and for each there exists such that In this paper, we obtain certain new fixed point theorems for α-fuzzy mappings satisfying a new class of contractive conditions in the context of complete metric space. We also investigate the existence of solutions for Fredholm integral inclusion. We also provide a non-trivial example to support our new results.
3. Consequences for Fuzzy Fixed Points
Corollary 2. [7] Let be a complete metric space and let → and for each ∃ such that . Assume that there exist some and such that
∀
with Then, there exists such that Proof. Consider given by Then, the result follows from Theorem 2. □
Corollary 3. [7] Let be a complete metric space and let and for each ∃ such that . Assume that there exist some and such that
∀
with Then, there exists such that Corollary 4. Let be a complete metric space and let : → and for each ∃ such that . Assume that there exist some and such thatfor all with Then, there exists such that Proof. Consider given by Then, the result follows from Theorem 2. □
Corollary 5. Let be a complete metric space and let : → and for each ∃ such that . Assume that there exist some and such thatfor all with Then, there exists such that Proof. Consider given by Then, the result follows from Theorem 2. □
Corollary 6. Let be a complete metric space and let : → and for each ∃ such that . Assume that there exist some and and non-negative real numbers with such thatfor all with Then, there exists such that Proof. Consider given by Then, the result follows from Theorem 2. □
Corollary 7. Let be a complete metric space and let : → and for each ∃ such that . Assume that there exist some and and non-negative real number and such thatfor all with Then, there exists such that Proof. Consider given by Then, the result follows from Theorem 2. □
Corollary 8. Let be a complete metric space and let : → and for each ∃ such that . Assume that there exist some and and non-negative real numbers with such thatfor all with Then, there exists such that Proof. Consider given by Then, the result follows from Theorem 2. □
Corollary 9. [3] Let be a complete metric space and let : → and for each ∃ such that . Assume that there exist some and such thatfor all with Then, there exists such that Proof. Consider given by Then, the result follows from Theorem 2. □
Corollary 10. Let be a complete metric space and let : → and for each ∃ such that . Assume that there exist some and such thatfor all with Then, there exists such that Proof. Consider given by Then, the result follows from Theorem 2. □
Corollary 11. Let be a complete metric space and let : → and for each ∃ such that . Assume that there exist some and such thatfor all with Then, there exists such that Corollary 12. Let be a complete metric space and let : → and for each ∃ such that . Assume that there exist some and such thatfor all with Then, there exists such that Proof. Consider given by . Then, the result follows from Theorem 2. □
4. Consequences for Multivalued Mappings
Fixed point theorems for multivalued mappings can be derived from fuzzy fixed point theorems in this way.
Theorem 3. Let be a complete metric space and let . Assume that there exist some and such that
∀
. Then, F and G a have common fixed point. Proof. Define
and
by
and
Thus, Theorem 2 can be applied to obtain such that □
The main result of Isik [
11] can be deduced by taking one mapping in Theorem 3.
Corollary 13. [11] Let be a complete metric space and let . Assume that there exist some and such thatfor all . Then, there exists such that We derive the main result of Vetro [
6], which is itself a Θ-generalization of Nadler’s [
9] fixed point theorem from Theorem 3.
Corollary 14. [6] Let be a complete metric space and let . Assume that there exist some and such thatfor all Then, there exists such that Proof. Consider given by and Then, the result follows from Theorem 3. □
The following result is a Θ-generalization of Kannan [
21] fixed point theorem for multivalued mapping.
Corollary 15. Let be a complete metric space and let . Assume that there exist some and such thatfor all . Then, there exists such that Proof. Consider given by and Then, the result follows from Theorem 3. □
The following result is a Θ-generalization of Chatterjea [
22] fixed point theorem for multivalued mapping.
Corollary 16. Let be a complete metric space and let . Assume that there exist some and such thatfor all Then, there exists such that Proof. Consider given by and Then, the result follows from Theorem 3. □
The following result is a Θ-generalization of Reich [
23,
24] fixed point theorem for multivalued mapping.
Corollary 17. Let be a complete metric space and let . Assume that there exist some and and non-negative real numbers with such thatfor all Then, there exists such that Proof. Consider given by and Then, the result follows from Theorem 3. □
The following result is a Θ-generalization of Berinde’s [
25] fixed point theorem for multivalued mapping.
Corollary 18. Let be a complete metric space and let . Assume that there exist some and and non-negative real number and such thatfor all Then, there exists such that Proof. Consider given by and Then, the result follows from Theorem 3. □
The following result is a Θ-generalization of Hardy and Rogers’ [
26] fixed point theorem for multivalued mapping.
Corollary 19. Let be a complete metric space and let . Assume that there exist some and and non-negative real numbers and L with such thatfor all Then, there exists such that Proof. Consider given by and Then, the result follows from Theorem 3. □
The following result is a Θ-generalization of Ćirić [
27] fixed point theorem for multivalued mapping.
Corollary 20. Let be a complete metric space and let . Assume that there exist some and such thatfor all Then, F and G have common fixed point. Proof. Consider given by Then, the result follows from Theorem 3. □
Corollary 21. Let be a complete metric space and let . Assume that there exist some and such thatfor all Then, there exists such that The following result is a Θ-generalization of Ćirić [
28] fixed point theorem for multivalued mapping.
Corollary 22. Let be a complete metric space and let . Assume that there exist some and such thatfor all Then, there exists such that Proof. Consider given by Then, the result follows from Theorem 3. □
The following result is a Θ-generalization of Ćirić’s [
27,
28] fixed point theorem for multivalued mapping.
Corollary 23. Let be a complete metric space and let . Assume that there exist some and such that
∀
Then, F and G have common fixed point. Corollary 24. Let be a complete metric space and let . Assume that there exist some and such thatfor all with Then, there exists such that Proof. Consider given by Then, the result follows from Theorem 3. □
5. Applications
The aim of this section is to apply the established results to obtain the existence of solutions for a certain Fredholm integral inclusion.
Consider the following integral inclusion of Fredholm type,
for
where
is a given real-valued function and
a given multivalued operator, where
denotes the family of non-empty compact and convex subsets of
and
is the unknown function.
Now, consider the metric ϱ on
defined by
for all
Then,
is a complete metric space.
We assume the following:
() For each the operator is such that is lower semicontinuous in .
(
) There exists a continuous function
such that
for all
(
) There exists some
such that
Theorem 4. Under Conditions ()–(), the integral inclusion in (22) has a solution in . Proof. Let
. Define the fuzzy mapping
by
It is clear that the set of solutions of the integral inclusion in (
22) coincides with the set of fixed points of the operator
(see [
29,
30]). Hence, we have to prove that under the given conditions,
has at least one fixed point in
. For this, we check that the conditions of Corollary 11 hold true. □
Let
be arbitrary; then, there exists
. For the multivalued operator
it follows from the Michael’s selection theorem that there exists a continuous operator
such that
for each
It follows that
Hence,
It is an easy matter to show that
is closed, and thus the details are omitted (see also [
31,
32]). Moreover, since f is continuous on
and
is continuous on
, their ranges are bounded. It follows that
is also bounded. Hence,
Let
then, there exists
such that
are nonempty, closed and bounded subsets of
Let
be arbitrary such that
for
holds. This means that for all
there exists
such that
for
For all
, it follows from (
) that
This means that there exists
such that
for all
Now, we can consider the multivalued operator U defined by
Hence, by (
), U is lower semicontinuous; it follows that there exists a continuous operator
such that
for
Then,
satisfies that
That is,
and
for all
Thus, we obtain that
Interchanging the roles of
and
, we obtain that
Taking exponential on both side, we have
Taking the function
given by
and
defined by
for
, we obtain that the condition in (
21) is fulfilled. Using Corollary 11, we conclude that the given integral inclusion has a solution.