Research Article | Open Access

Awais Asif, Hassen Aydi, Muhammad Arshad, Zeeshan Ali, "A Novel Picture Fuzzy -Banach Space with Some New Contractive Conditions and Their Fixed Point Results", *Journal of Function Spaces*, vol. 2020, Article ID 6305856, 12 pages, 2020. https://doi.org/10.1155/2020/6305856

# A Novel Picture Fuzzy -Banach Space with Some New Contractive Conditions and Their Fixed Point Results

**Academic Editor:**Serkan Araci

#### 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