Fixed Point Theorem for Neutrosophic Triplet Partial Metric Space
Abstract
:1. Introduction
2. Preliminaries
- (i)
- (a, a) =(b, b) =(a, b) =(b, a)a = b;
- (ii)
- (a, a)(a, b);
- (iii)
- (a, b) =(b, a);
- (iv)
- (a, c)(a, b) +(b, c) − (b, b);
- (i)
- There is neutral element (neut(x)) for x ∈ N such thatx*neut(x) = neut(x)* x = x.
- (ii)
- There is anti element (anti(x)) for x ∈ N such thatx*anti(x) = anti(x)* x = neut(x).
- (a)
- (a, b) ≥ 0
- (b)
- If a = b, then(a, b) = 0
- (c)
- (a, b) =(a, b)
- (d)
- If there exists any element c ∊ M such that(a, c) ≤(a, c*neut(b)), then(a, c*neut(b)) ≤(a, b) +(b, c).
3. Neutrosophic Triplet Partial Metric Space
- (i)
- 0(a, a)(a, b)
- (ii)
- If(a, a) =(a, b) =(b, b) = 0, then there exits any a, b such that a = b.
- (iii)
- (a, b) =(a, b)
- (iv)
- If there exists any element b ∊ A such that(a, c) ≤(a, c#neut(b)), then(a, c#neut(b)) ≤(a, b) +(b, c) −(b, b)
- (i), (ii) and (iii) are apparent.
- (iv) Let ∅ be empty element of P(X). Then,(X, Y) =(X, Y ∪ ∅) since for(X, Y ∪ ∅) =(X, Y) = max{m(X), m(Y)}. Also, it is clear that
- max{m(X), m(Y)} ≤ max{m(X), m(Z)}+ max{m(Z), m(Y)} – max {m(∅), m(∅)}.
- (i)
- (X, X) = = m(X) = (X, Y), since for d(X,X) = 0. Thus; 0 (X, X) (X, Y) for X, Y ∈ P(A).
- (ii)
- If (X, X) = (X, Y) = (Y, Y) = 0, then
- (iii)
- = = = 0 and ) = 0. Where, m(X) = 0, m(Y) = 0 and = 0. Thus, X = Y = (empty set).
- (iv)
- (X, Y) = = = (Y, X), since for (X, Y)= (Y, X).
- (v)
- We suppose that there exists any Z ∈ P(A) such that m(Y) and ≤ . Thus,From (1), ≤ . Since (P(A), #), d) is a NTMS,From (1), (2)
- (i)
- Since for (a, a) = 0, 0 (a, a) = (a, a) + k = k (a, b) = (a, b) + k. Thus;
- (ii)
- 0 (a, a) (a, b).
- (iii)
- There do not exists a, b ∈ A such that (a, a) = (a, b) = (b, b) = 0 since for k ∈ and (a, a) = 0.
- (iv)
- (a, b) = (a, b) + k = (b, a) + k, since for (a, b) = (b, a).
- (v)
- Suppose that there exists any element c ∊ A such that (a, b) ≤ (a, b#neut(c)). Then (a, b) + k ≤ (a, b#neut(c)) + k. Thus,
- (i)
- There exists any element c ∊ A such that(a, b) ≤(a, b*neut(c)); ⩝a, b ∊ A.
- (ii)
- There exists k in [0, 1) such that(m(a), m(b))k.(a, b); ⩝a, b ∊ A.
- (m(∅), m(∅)) =({x}, {x}) = 10, 2.(∅, ∅) = 1, 5
- (m(∅), m({x})) =({x}, {x, y}) = 10, 2.(∅, {x}) = 1, 5
- (m(∅), m({x, y})) =({x}, {x, y}) = 10, 2.(∅, {x, y}) = 1, 5
- (m({x}), m({x})) =({x, y}, {x, y}) = 00, 2.({x}, {x}) = 0, 5
- (m({x}), m({x, y})) =({x, y}, {x, y}) = 00, 2.({x}, {x,y}) = 0, 5
- (m({x, y}), m({x, y})) =({x, y}, {x, y}) = 00, 2.({x, y}, {x, y}) = 0, 5
- Thus, m is a contraction for ((A,),)
- (, ) = (m(), m()) c. (, ) and
- (, ) = (m(), m()) c. (, ) . (, ). From mathematical induction, n m;
- (, ) = (m(), m()) c.(, ) . (, ). Thus; from (8) and definition of NTPMS,
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Şahin, M.; Kargın, A.; Çoban, M.A. Fixed Point Theorem for Neutrosophic Triplet Partial Metric Space. Symmetry 2018, 10, 240. https://doi.org/10.3390/sym10070240
Şahin M, Kargın A, Çoban MA. Fixed Point Theorem for Neutrosophic Triplet Partial Metric Space. Symmetry. 2018; 10(7):240. https://doi.org/10.3390/sym10070240
Chicago/Turabian StyleŞahin, Memet, Abdullah Kargın, and Mehmet Ali Çoban. 2018. "Fixed Point Theorem for Neutrosophic Triplet Partial Metric Space" Symmetry 10, no. 7: 240. https://doi.org/10.3390/sym10070240