Abstract
The aim of this study is to investigate the existence of solutions for the following Fredholm integral inclusion for where is a given real-valued function and a given multivalued operator, where represents the family of non-empty compact and convex subsets of , is the unknown function and is a metric defined on . To attain this target, we take advantage of fixed point theorems for -fuzzy mappings satisfying a new class of contractive conditions in the context of complete metric spaces. We derive new fixed point results which extend and improve the well-known results of Banach, Kannan, Chatterjea, Reich, Hardy-Rogers, Berinde and Ćirić by means of this new class of contractions. We also give a significantly non-trivial example to support our new results.
Keywords:
fredholm integral inclusion; α-fuzzy mappings; Θ-contractions; fixed point; multivalued mappings MSC:
46S40; 47H10; 54H25
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 haveand 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 that
for 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 that
then
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.
2. Main Results
Theorem 2.
Let be a metric space and let : and for each ∃ such that . Assume that there exist some and such that
for all with Then, there exists such that
Proof.
Let be an arbitrary point in By hypothesis there exists such that is a nonempty, closed and bounded subset of . Let For this there exists such that is a nonempty, closed and bounded subset of
and so
Now, Lemma 1 gives that Thus, we obtain
and hence
Since is continuous function from the right, there exists a real number such that
As
by Lemma 2, there exists (obviously, such that
Thus, we have
For such that Thus, we have
and so
Now, Lemma 1 gives that Thus, we obtain
and hence
Since is continuous function from the right, there exists a real number such that
As
by Lemma 2, there exists (obviously, such that
Thus, we have
Thus, pursuing repeatedly, we get in so that , ,
and
∀ By Inequalities (14) and (15), we get
∀ Let for all From Inequality (16), we get
which implies that By (), we have Now, we claim that is a Cauchy sequence: for this, consider Condition (). From (), there exist and such that
Take . By the definition of limit, there exists such that
for all Using Inequality (17) and the above inequality, we deduce
This implies that
Then, there exists such that
for . Now, we prove that is a Cauchy sequence. For we have
Since, , converges. Therefore, as Thus, we proved that is a Cauchy sequence in . The completeness of ensures that there exists such that, that is
Now, we prove that Assuming on the contrary that , then ∃ and a subsequence of so that
for all Now, using Inequality (5) with and . Taking Remark 1 into account, we have
Passing the limit as in the above inequality, we obtain
which implies by Lemma 1 that
which is a contradiction. Hence, . Since is closed, we deduce that . Similarly, one can easily show that Thus as required. □
Example 1.
Let and define by
Clearly, satisfies all axioms of complete metric space. Define by
and
such that
for and . Let be defined by and All the hypotheses of our main Theorem 2 are satisfied to obtain for .
The following result for one fuzzy mapping is a direct consequence of our main theorem.
Corollary 1.
Let be a complete metric space and let : → and for each ∃ such that . Assume that there exist some and so that
∀ with Then, there exists such that
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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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
Then,
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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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 that
for 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.
6. Conclusions
In this paper, we obtain certain common fixed point theorems for α-fuzzy mappings satisfying a new class of contractive conditions in the setting of complete metric spaces. We derive various well known results of the literature as corollaries. The existence of solutions for the Fredholm integral inclusion is also investigated as application of our main result. We also establish a significant example to support our results. We hope that the results contained in this paper will raise new associations for those who are working in Θ-contraction, fuzzy mapping and its applications to integral inclusions.
Author Contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.
Acknowledgments
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-23-130-38). The authors, therefore, acknowledge with thanks DSR technical and financial support.
Conflicts of Interest
The authors declare that they have no competing interests.
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