Abstract
A picture fuzzy -normed linear space (), a mixture of a picture fuzzy set and an -normed linear space, is a proficient concept to cope with uncertain and unpredictable real-life problems. The purpose of this manuscript is to present some novel contractive conditions based on . By using these contractive conditions, we explore some fixed point theorems in a picture fuzzy -Banach space (). The discussed modified results are more general than those in the existing literature which are based on an intuitionistic fuzzy -Banach space () and a fuzzy -Banach space. To express the reliability and effectiveness of the main results, we present several examples to support our main theorems.
1. Introduction
In various real-life problems, for a suitable mapping, the existence of a solution and existence of a fixed point (FP) are equivalent. Thus, the existence of a FP is a proficient technique to cope with awkward and difficult problems in real-life issues. Various scholars have utilized such results in the environment of many fields [1, 2]. The extensive useful techniques capable with both algebraic and topological properties are those of a normed linear space (NLS), but the continuous maps are more proficient in the sense of NLS. Moreover, in a metric space, every contractive map is uniformly continuous. One of the fundamental applications of Banach’s contraction principle is the “Picard’s theorem,” which is the basic theorem for the existence and uniqueness of solution to the ordinary differential equations. Various scholars have utilized this application in the environment of a partial differential equation [3], in the Gauss-Seidel method for evaluating systems of linear equations [4], in the proof of the inverse function theorem [5], etc.
The theory of a fuzzy set (FS) was investigated by Zadeh [6], characterized by only positive grades restricted to . FS has achieved more success due to its ability to cope with complications and troubles. However, in some practice cases, the concept of FS cannot cope with complications and uncertainty because of lack of knowledge of the problem. Therefore, Atanassov [7] investigated the intuitionistic FS (IFS) containing both positive and negative grades, whose sum is bounded to . IFS is regarded as a more improved way to cope with complex and awkward information. Further, Cường [8] investigated the picture FS (PFS) including positive, abstinence, and negative grades, whose sum is bounded to . PFS is regarded as a more improved way to deal with even more complex information. For more related works, we may refer to References [9–16].
Keeping the advantages of the PFS, the objective of this manuscript is summarized in the following ways: (1)To present some novel contractive conditions, we used as a basis(2)By using these contractive conditions, some fixed point theorems are explored for a picture fuzzy -Banach space (). These results are more modified and more general than the existing results which are based on an intuitionistic fuzzy -Banach space () and a fuzzy -Banach space(3)To express the reliability and effectiveness of the explored approaches, we explain examples in support of the main results
The rest of this manuscript is summarized in the following ways: In Section 2, we review some basic notions like and their related properties used in the presented work. In Section 3, we describe the notion of and their fundamental properties. In Section 4, we present some novel contractive conditions based on . By using these contractive conditions, we instigate some fixed point theorems for a picture fuzzy -Banach space (). Finally, the conclusion of this manuscript is discussed in Section 5.
2. Preliminaries
The purpose of this section is to review some existing notions, like and their related properties. Throughout this section, the symbols , , , , , , , and represent the positive real numbers, real numbers, natural numbers, universal set, supporting grade, supporting against, continuous -norm, and continuous -conorm, respectively.
Definition 1. [9]. A is stated by , where is defined on , where the following conditions hold: (i)(ii)(iii) iff are linearly dependent(iv) is invariant under any permutation of (v) if (vi)(vii) is a nondecreasing function of and (viii)(ix) iff are linearly dependent(x) is invariant under any permutation of (xi) if (xii)(xiii) is a nonincreasing function of and (xiv)Further, and imply (xv)For , and are continuous functions of and are strictly increasing and strictly decreasing, respectively, on the subset of
Moreover, we explain some important theories based on convergent and Cauchy convergent sequences.
Definition 2. [9]. Consider ; then, the sequence in is convergent to based on the intuitionistic fuzzy -norm if for every and , there exists such that for all and it is represented by .
Definition 3. [9]. Let ; then, the sequence in is Cauchy convergent based on the intuitionistic fuzzy -norm if for every and , there exists such that for all and it is represented by .
3. Picture Fuzzy -Normed Linear Space
The purpose of this section is to explore some new approaches like and their related properties, which are extensively efficient for the proof of our main work in the next section. Throughout this section, the symbols , , , , , and represented the universal set, supporting grade, abstinence grade, supporting against, continuous -norm, and continuous -conorm, respectively.
Definition 4. A is stated as , where is defined on , where the following conditions hold: (i)(ii)(iii) iff are linearly dependent(iv) is invariant under any permutation of (v) if (vi)(vii) is a nondecreasing function of and (viii)(ix) iff are linearly dependent(x) is invariant under any permutation of (xi) if (xii)(xiii) is a nonincreasing function of and (xiv)(xv) iff are linearly dependent(xvi) is invariant under any permutation of (xvii) if (xviii)(xix) is a nonincreasing function of and (xx)Further, and ; then, (xxi)For , , , and are continuous functions of and also strictly increasing and strictly decreasing, respectively, on the subset of
Moreover, we explain some important theories based on convergent and Cauchy convergent sequences.
Definition 5. For a , the sequence in is convergent to based on the picture fuzzy -norm if for every and , there exists such that for all and it is represented by .
Definition 6. For a , the sequence in is Cauchy convergent based on the picture fuzzy -norm if for every and , there exists such that for all and it is represented by .
Remark 7. The following assumptions are important for our main results. (1)Suppose is the set of functions such that (i) is continuous and nondecreasing(ii)(2)Suppose is the set of functions , such that (i) is continuous and nonincreasing(ii)(3)Suppose is the set of functions such that (i) is continuous and strictly increasing(ii)(4)Suppose is the set of functions with such that (i) is continuous and strictly decreasing(ii)
4. Contractive Mappings Based on the Picture Fuzzy -Banach Space
Based on the definitions introduced in Section 3, we describe some contractive mappings using the named as picture fuzzy -normed contractive mapping () and verify it with the help of numerical examples.
Definition 8. For a , the mapping is called , if for all .
Further, based on equation (5) and using Remark 7, we explore the following results, which are very helpful for future work.
Theorem 9. For a , we define such that where , , and , for all with . Then, possesses a unique fixed point in .
Proof. Let with . By using Remark 7 and inequality (6), we get
Further, we write the above equations as
It is clear from the above analysis that is a bounded nondecreasing sequence while and are bounded nonincreasing sequences. Then, the limit of these equations exists. Hence,
By using the induction on , we have
As , we have
Supposing , we have
Similarly, from abstinence and falsity grades, we have
By using the above analysis, we write, if and , then
Similarly, solving the grades of abstinence and falsity, we have
Therefore,
Similarly, dealing with the grades of abstinence and falsity, we have
Then, for all ,
Also, we find
Since is arbitrary and the sequence is Cauchy, hence they are convergent. Therefore, .
Suppose ; then, there exists such that
Moreover, doing the same process to abstinence and falsity grades, we obtain
for all . Hence,
for all . Therefore,
for all . Hence, ; that is, has a fixed point in . Next, we prove its uniqueness. For this, we suppose is another fixed point of in ; then,
for all and ; then,
for all . Hence, . Thus, has a unique fixed point in .
Example 10. For a Banach space , we define a mapping such that for all ,
We know that , , and . We consider that , , and , where and . Now, we describe the picture fuzzy -norm , , and :
We consider that
The first three parts are discussed for the truth grade. We have the following cases:
Case 1. Suppose ; then,
Further, we write
Therefore, we get
Case 2. Suppose ; then,
Therefore, we get
Case 3. Suppose and ; then,
Similarly, we can prove these conditions for abstinence and falsity grades. Hence, the solution is completed. Further, we instigate more results based on to show the proficiency of the discussed results.
Theorem 11. For a , the grade of truth, abstinence, and falsity satisfies the conditions of Definition 4. Now, we define the decreasing mapping and increasing mappings and , such that and with , such that where , , and , for all with . Then, has a unique fixed point in .
Proof. Let with . By using Remark 7 and inequality (35), we get
Further, we write the above equations as
It is clear from the above analysis that is a bounded nondecreasing sequence and and are the bounded nonincreasing sequences. Then, the limit of these equations exists. We suppose that
Therefore, we have
Then, the limit of these equations also exists. We have
If , , and , then
Therefore,
Similarly, we can find that
And it is clear that ; then,
Again,
Thus, we get
which is a contradiction; hence,
Suppose . We have
Similarly, for abstinence and falsity grades, we have
By using the above analysis, we get, if and , then
Similarly, resolving the grades of abstinence and falsity, we have
Therefore,
Also, we note
Then, for all ,
Further, we find
Since is arbitrary and the sequence is Cauchy, hence they are convergent. Therefore, .
Suppose ; then, there exists such that
Similarly, observing for abstinence and falsity grades, we have
for all . Hence,
for all . Therefore,
for all . Hence, ; that is, has a fixed point in . Next, we prove the uniqueness of the fixed point. For this, we suppose is another fixed point in ; then,
Hence, ,
Therefore, , , and . It is a contradiction; thus, for all and , we obtain
for all . Hence, . Hence, has a unique fixed point in .
Example 12. For a Banach space , we define the decreasing mapping and increasing mappings and , such that and , and are such that for all , where , , and . Suppose that is nondecreasing and are nonincreasing functions with for all . Further, define picture fuzzy -norm as in Example 10. Consider that
By using the three cases of Example 10 and using Theorem 11, we explore that the function has a unique fixed point in . Hence, the solution is completed. Further, we have utilized more results based on to show the proficiency of the proven approaches.
Theorem 13. Let . Let the grade of truth, abstinence, and falsity satisfy the conditions of Definition 4. Now, we define the mapping , such that where , , and , for all with and . Then, has a unique fixed point in .
Proof. Let with . By using Remark 7 and equation (66), we get Further, from the above equations, we obtain It is clear from the above analysis that is a bounded nondecreasing sequence and the sequences and are bounded and nonincreasing. Then, the limit of these equations exists. We suppose that Therefore, we have Then, the limit of these equations also exists. We have If , , and , then Therefore, Similarly, we can find Clearly, ; hence, Also, we write Thus, we get Similarly, It is a contradiction; hence, The rest of the proof to express for all the equation can be obtained using the similar technique of Theorem 9 and Theorem 11; that is, has a fixed point in . Next, we prove the uniqueness of the fixed point. For this, we suppose is another fixed point in ; then, Hence, , Therefore, , , and . It is a contradiction; thus, as , we get for all . Hence, . Thus, has a unique fixed point in .
Example 14. For a Banach space , we define the mapping such that for all , where , , and . Suppose is nondecreasing and are nonincreasing functions with for all and . Further, define picture fuzzy -norm as in Example 10. Consider that
We explore that the function has a unique fixed point in . Hence, the solution is completed.
5. Conclusion
A picture fuzzy set is more proficient and more capable than an intuitionistic fuzzy set and fuzzy to cope with uncertain and unpredictable information in realistic issues. Keeping the advantages of the picture fuzzy set and a -norm linear space, the manuscript made the following advancements in the existing literature: (1)The novel picture fuzzy -norm linear space and its basic properties are explored(2)Some novel contractive conditions based on are presented. By using these contractive conditions, we have explored some fixed point theorems for a picture fuzzy -Banach space (). It was observed that these results are more modified and more general than the existing ones in the literature, which are based on intuitionistic fuzzy -Banach spaces () and fuzzy -Banach spaces(3)The reliability and effectiveness of the obtained main theorems are expressed, and several examples are presented afterwards
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
We declare that we do not have any commercial or associative interests that represent conflicts of interest in connection with this manuscript. There are no professional or other personal interests that can inappropriately influence our submitted work.
Authors’ Contributions
All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.