In this paper we provide three new type of contractive conditions and establish fixed point theorems related to them in the setting of an intuitionistic fuzzy n-Banach space. Our results improve and generalize several classical results existing in literature. Several examples have been provided in support of non-triviality of our results.
In numerous mathematical problems, the existence of a solution is equivalent to the existence of a fixed point for a suitable map. Thus, the existence of a fixed point is of paramount importance in several areas of mathematics and many other sciences. Fixed point theorems provide conditions under which maps have solutions. The most significant structures endowed with both algebraic and topological properties are those of normed linear spaces (NLSs) while the most important maps between two normed linear spaces are the continuous maps. Further, it is well known that in a metric space, every contraction map is uniformly continuous. One of the major application of Banach’s contraction principle is “Picard’s theorem”, which is the basic existence and uniqueness theorem for ordinary differential equations. There are more applications of this principle to partial differential equations [31], Gauss-Seidel method of solving system of linear equations [30], proof of inverse function theorem [10], Google’s Page Rank algorithm [19] etc.
Because of its topological structure, a NLS enables us to study contraction maps as well as fixed point results in it. A natural question arises whether we can provide contractive conditions which imply existence and uniqueness in the most general setting of an intuitionistic fuzzy n-normed linear space (IFnNLS). A complete IFnNLS is called an intuitionistic fuzzy n-Banach space (IFnBS).
The fuzzy set theory was introduced by Zadeh [32] in 1965, in order to explain the situations where data are imprecise or vague. On the other hand in 1986, a generalized form of fuzzy set namely, intuitionistic fuzzy set was introduced by Atanassaov [1] which deals with both the degree of membership (belongingness) and non-membership (non-belongingness) of an elements within a set.
Situations where the crisp norm is unable to measure the length of a vector accurately, the notion of fuzzy norm happens to be useful. The idea of a fuzzy norm on a linear space was introduced by Katsaras [15] in 1984. In 1992, Felbin [9] introduced an alternative idea of a fuzzy norm whose associated metric is of Kaleva and Seikkala [14] type. In 1994, another notion of fuzzy norm on a linear space was given by Cheng and Moderson [3] whose associated metric is that of Kramosil and Michalek [18] type. Again in 2003, following Cheng and Mordeson, Bag and Samanta [2] introduced another concept of fuzzy normed linear space. In this way, there has been a systematic development of fuzzy normed linear spaces (FNLSs) and one of the important development over FNLS is the notion of intuitionistic fuzzy normed linear space (IFNLS) [24]. Vijayabalaji and Narayanan [28] extended n-normed linear space to fuzzy n-normed linear space while the concept of IFnNLS was introduced by Vijayabalaji et al. [29]. Some more recent work in similar context may be found in [4–8, 26]. Recently, continuity and Banach contraction principle have been studied in an IFnNLS by the present authors [17].
The study of fixed point theory in fuzzy metric spaces was initiated by Heilpern [12]. He established a fuzzy extension of the Banach’s contraction principle. Contractive mappings in fuzzy metric spaces have been recently studied by Saheli [25].
In the present paper we provide some new contractive conditions in an IFnBS and establish existence and uniqueness theorems for fixed points. Our results extend and generalize several results from classical contractive mappings [13, 23].
Preliminaries
First we recall some basic definitions and examples which are useful for the current work.
The notion of n-norm was defined by Gunawan and Mashadi [11], whereas an IFnNLS was defined by Vijayabalaji et al. [29].
Definition 1.1. [29] An IFnNLS is the five-tuple (X, μ, ν, ∗ , ∘), where X is a linear space over a field , ∗ is a continuous t-norm, ∘ is a continuous t-conorm, μ, ν are fuzzy sets on Xn × (0, ∞), μ denotes the degree of membership and ν denotes the degree of non-membership of (x1, x2, …, xn, t) ∈ Xn × (0, ∞) satisfying the following conditions for every (x1, x2, …, xn) ∈ Xn and s, t > 0:
μ (x1, x2, …, xn, t) + ν (x1, x2, …, xn, t) ≤ 1,
μ (x1, x2, …, xn, t) >0,
μ (x1, x2, …, xn, t) =1 if and only if x1, x2, …, xn are linearly dependent,
μ (x1, x2, …, xn, t) is invariant under any permutation of x1, x2, …, xn,
if ,
,
μ (x1, x2, …, xn, ·) is non-decreasing function of and ,
ν (x1, x2, …, xn, t) <1,
ν (x1, x2, …, xn, t) =0 if and only if x1, x2, …, xn are linearly dependent,
ν (x1, x2, …, xn, t) is invariant under any permutation of x1, x2, …, xn,
if ,
,
ν (x1, x2, …, xn, ·) is non-increasing function of and .
Also assume that
μ (x1, x2, … , xn, t) >0 and ν (x1, x2, … , xn, t) <1, for all t > 0 implies x = 0.
For x ≠ 0, μ (x1, x2, …, xn, ·) and ν (x1, x2, …, xn, ·) are continuous functions of and μ (x1, x2, …, xn, ·) and ν (x1, x2, …, xn, ·) are respectively strictly increasing and strictly decreasing on the subset {t : 0 < μ (x1, x2, …, xn, t) , ν (x1, x2, …, xn, t) <1} of .
Sen and Debnath [27] gave a new definition of convergence of sequences in an IFnNLS as follows.
Definition 1.2. [27] Let (X, μ, ν, ∗ , ∘) be an IFnNLS. We say that a sequence x = {xk} in X is convergent to l ∈ X with respect to the intuitionistic fuzzy n-norm (μ, ν) n if, for every ɛ > 0, t > 0 and y1, y2, …, yn-1 ∈ X, there exists such that μ (y1, y2, …, yn-1, xk - l, t) >1 - ɛ and ν (y1, y2, …, yn-1, xk - l, t) < ɛ for all k ≥ k0. It is denoted by (μ, ν) n - lim xk = l or as k→ ∞.
Definition 1.3. [27] Let (X, μ, ν, ∗ , ∘) be an IFnNLS. Then the sequence x = {xk} in X is called a Cauchy sequence with respect to the intuitionistic fuzzy n-norm (μ, ν) n if, for every ɛ > 0, t > 0 and y1, y2, …, yn-1 ∈ X, there exists such that μ (y1, y2, …, yn-1, xk - xm, t) >1 - ɛ and ν (y1, y2, …, yn-1, xk - xm, t) < ɛ for all k, m ≥ k0.
Some notations
Here we introduce some notations which are essential for our current work.
Notation 2.1. Let S1 be the set of all functions ψ : [0, + ∞) ⟶ [0, + ∞) satisfying the following conditions:
ψ is continuous and non-decreasing.
ψ (t) =0 if and only if t = 0.
Notation 2.2. Let S2 be the set of all functions ψ : [0, + ∞) ⟶ [0, + ∞) satisfying the following conditions:
ψ is continuous and non-increasing.
ψ (t) =0 if and only if t = 0.
Notation 2.3. Let T1 be the set of all functions φ : [0, + ∞) ⟶ [0, + ∞) satisfying the following conditions:
φ is continuous and strictly increasing.
φ (t) =0 if and only if t = 0.
Notation 2.4. Let T2 be the set of all functions φ : [0, + ∞) ⟶ [0, + ∞) satisfying the following conditions:
φ is continuous and strictly decreasing.
φ (t) =0 if and only if t = 0.
Main results
Now we discuss our main results.
First we define contraction mapping in IFnBS.
Definition 3.1. Let (X, μ, ν, ∗ , ∘) is an IFnBS. A selfmap f : X ⟶ X is called an intuitionistic fuzzy n-normed contraction mapping, if
for all x, y ∈ X, t > 0 and x1, x2, …, xn-1 ∈ X.
We now study fixed point theorems on three different types of self-maps along with the functions considered in the sets S1 and S2 and also establish the uniqueness of fixed points for such mappings.
Theorem 3.2.Let (X, μ, ν, ∗ , ∘) be an IFnBS and f : X ⟶ X an intuitionistic fuzzy n-normed contraction mapping such that for all x, y ∈ X; , x1, x2, …, xn-1 ∈ X and α ∈ (0, 1],
μ (x1, x2, …, xn-1, x - y, t) ≥ α impliesthat μ (x1, x2, …, xn-1, f (x) - f (y) , t - ψ1 (t)) ≥ α
and
ν (x1, x2, …, xn-1, x - y, t) <1 - α impliesthat ν (x1, x2, …, xn-1, f (x) - f (y) , t - ψ2 (t)) <1 - α,
where ψ1 ∈ S1 and ψ2 ∈ S2.
Then f has a unique fixed point in X.
Proof. Suppose x0 ∈ X and xn+1 = f (xn), for all .
Suppose that t > 0. Then using the properties of S1 and S2 and condition 1, we have
and
for all x, y ∈ X.
Therefore,
and
for all .
Hence {μ (x1, x2, …, xn-1, xn+1 - xn, t)} is a bounded non-decreasing sequence and {ν (x1, x2, …, xn-1, xn+1 - xn, t)} is a bounded non-increasing sequence.
Thus, and exist.
Now, we have
and
By induction on n, we have,
for all .
As n⟶ ∞, we have,
and , for all t > 0.
Let t > 0, ɛ > 0 and such that
and
and
and
If and then
and similarly,
Therefore
for all .
So,
and
for all n, m ≥ N.
Since ɛ is arbitrary, {xn} is Cauchy and hence it is convergent.
Therefore,
Let ɛ > 0 and t > 0. Then there exists such that
and
for all n ≥ n0.
Hence
and
for all n ≥ n0.
Therefore,
for all t > 0.
Hence f (x) = x.
i.e. f has a fixed point in X.
Next, to prove the uniqueness of the fixed point, let y be another fixed point of f in X. Suppose t > 0, we have,
and
for all .
As n⟶ ∞, we have,
for all t > 0.
Hence, x = y.
Thus f has a unique fixed point in X. □
Example 3.3. Let (X, ∥ · ∥) be a Banach space and f : X ⟶ X be a function such that for all x, y ∈ X,
where ψ1 ∈ S1 and ψ2 ∈ S2.
Assume that ψ1 (βt) ≤ βψ1 (t) and ψ2 (βt) ≥ βψ2 (t), for all t ∈ [0, + ∞), β ∈ [0, 1].
Now, define an intuitionistic fuzzy n-norm μ, ν as follows:
and
Suppose that x, y ∈ X, t > 0, α ∈ (0, 1], x1, x2, …, xn-1 ∈ X and
First we consider three cases for μ:
Case 1: Let 0< t ≤ ∥ x1, x2, …, xn-1, x - y ∥.
So, .
Hence, α ∥ x1, x2, …, xn-1, x - y ∥ ≤ t.
Therefore,
Thus,
Case 2: Let ∥x1, x2, …, xn ∥ < t.
Then,
So,
Case 3: Let t ≤ 0 and μ (x1, x2, …, xn-1, xn, t) =0, then
Next, we consider three similar cases for ν and finally, by the Theorem 3.2, we conclude that f has a fixed point in X.
Theorem 3.4.Suppose (X, μ, ν, ∗ , ∘) is an IFnBS such that μ, ν satisfies condition (xv) of the Definition 1.1 and γ1 : (0, + ∞) ⟶ [0, 1) is a decreasing, γ2 : (0, + ∞) ⟶ [0, 1) is an increasing function and f : X ⟶ X is a selfmap such that for all x, y ∈ X, t > 0, x1, x2, …, xn-1 ∈ X and α ∈ (0, 1],
andwhereφ1 ∈ T1 and φ2 ∈ T2.
Then f has a unique fixed point in X.
Proof. Suppose x0 ∈ X and xn+1 = f (xn), for all .
Suppose that t > 0, using the condition of T1 and T2 and contraction mapping, we have
and
for all x, y ∈ X.
Therefore,
and
for all .
Hence {μ (x1, x2, …, xn-1, xn+1 - xn, t)} is a bounded non-decreasing sequence and {ν (x1, x2, …, xn-1, xn+1 - xn, t)} is a bounded non-increasing sequence.
Thus, and exist.
Assume that there exists t > 0, such that
and
Since
and
for all s > 0, it follows that-
and
for all .
As s ⟶ ∥ x1, x2, …, xn-1, xn+1 - xn ∥ β1 we have,
and as s ⟶ ∥ x1, x2, …, xn-1, xn+1 - xn ∥ β2 we have,
As n⟶ ∞, we have
and
which is a contradiction.
Hence,
for all t > 0.
Let t > 0, ɛ > 0 and such that
and
If
then
and similarly,
Therefore,
for all n ≥ N.
So,
and similarly,
for all n, m ≥ N.
Since ɛ is arbitrary, {xn} is Cauchy and hence it is convergent.
Assume that
Let ɛ > 0 and t > 0. Then there exists such that
and
for all n ≥ n0.
Hence
and similarly,
for all n ≥ n0.
Therefore,
for all t > 0.
Hence f (x) = x.
Therefore f has a fixed point in X.
To prove the uniqueness of the fixed point, let y be another fixed point of f in X.
If there exists t > 0 such that 0 < μ (x1, x2, …, xn-1, x - y, t) , ν (x1, x2, …, xn-1, x - y, t) <1, then
and
Therefore,
and
Hence
and
Therefore,
which is a contradiction.
Thus, μ (x1, x2, …, xn-1, x - y, t) =1 and ν (x1, x2, …, xn-1, x - y, t) =0, for all t > 0.
This implies, x = y.
Hence, f has a unique fixed point in X. □
Example 3.5. Let (X, ∥ · ∥) be a Banach space and γ1 : (0, + ∞) ⟶ [0, 1) be a decreasing function, γ2 : (0, + ∞) ⟶ [0, 1) be an increasing function and f : X ⟶ X be a selfmap such that for all x, y ∈ X,
where φ1 ∈ T1 and φ2 ∈ T2.
Assume that γ1φ1 is a non-decreasing and γ2φ2 is a non-increasing function and
and
for all t ∈ [0, + ∞), β1, β2 ∈ [0, 1].
Define an intuitionistic fuzzy n-norm μ, ν as in Example 3.3.
Suppose that x, y ∈ X, t > 0, α ∈ (0, 1], x1, x2, …, xn-1 ∈ X and
Finally, considering 3 cases for both μ and ν similar to Example 3.3 and using Theorem 3.4, we can conclude that f has a unique fixed point in X.
Theorem 3.6.Suppose (X, μ, ν, ∗ , ∘) is an IFnBS such that μ, ν satisfies condition (xv) of the Definition 1.1 and f : X ⟶ X is a self-map such that for all x, y ∈ X, t > 0, x1, x2, …, xn-1 ∈ X and α ∈ (0, 1],
and
whereφ1, φ1 ∈ T1, φ2, φ2 ∈ T2 and φ1 (t) ≥ φ1 (t)), φ2 (t) ≥ φ2 (t)), for all t > 0.
Then f has a unique fixed point in X.
Proof. Suppose x0 ∈ X and xn+1 = f (xn), for all .
Suppose that t > 0. Then using the condition of T1 and T2 and contraction mapping, we have,
and
for all x, y ∈ X.
Therefore,
and similarly,
for all .
Hence {μ (x1, x2, …, xn-1, xn+1 - xn, t)} is a bounded non-decreasing sequence and {ν (x1, x2, …, xn-1, xn+1 - xn, t)} is a bounded non-increasing sequence.
As s ⟶ ∥ x1, x2, …, xn-1, xn+1 - xn ∥ β1 and s ⟶ ∥ x1, x2, …, xn-1, xn+1 - xn ∥ β2, we have
and
As n⟶ ∞, we have
and
which is a contradiction.
Hence,
and
for all t > 0.
The rest of the proof to show that f (x) = x can be obtained using similar techniques as in Theorems 3.2 and 3.4.
Therefore f has a fixed point in X.
To prove the uniqueness of the fixed point, let y be another fixed point of f in X.
If there exists t > 0 such that 0 < μ (x1, x2, …, xn-1, x - y, t) , ν (x1, x2, …, xn-1, x - y, t) <1, then
and
Therefore,
and
Hence and implies φ1t = φ1 (t) - φ1 (t) and φ1t = φ2 (t) - φ2 (t) .
Therefore,
which is a contradiction.
Thus, μ (x1, x2, …, xn-1, x - y, t) =1 and ν (x1, x2, …, xn-1, x - y, t) =0, for all t > 0, implies, x = y.
Hence, f has a unique fixed point in X. □
Example 3.7. Let (X, ∥ · ∥) be a Banach space and f : X ⟶ X be a function such that for all x, y ∈ X,
and
where φ1, φ1 ∈ T1 and φ2, φ2 ∈ T2.
Assume that φ1 - φ1 is a non-decreasing and φ2 - φ2 is a non-increasing function and
and
for all t ∈ [0, + ∞), β1, β2 ∈ [0, 1].
Now, defining an intuitionistic fuzzy n-norm μ, ν as in Example 3.3, and assuming that x, y ∈ X, t > 0, α ∈ (0, 1], x1, x2, …, xn-1 ∈ X and
we can conclude that f has a unique fixed point in X.
Conclusion
In this paper we have established three new contractive conditions and introduced fixed point results related to them in an IFnBS. Our results are the extended intuitionistic fuzzy versions of some classical contractive conditions. These results will be helpful in finding new fuzzy iteration schemes for solving fuzzy differential and integral equations.
Footnotes
Acknowledgments
The authors are immensely thankful to the reviewers and the Associate Editor for their constructive feedback towards the improvement of the paper.
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