ABSTRACT
In this paper, we extend the concept of compatible maps of type J, (J-1) and (J-2) and prove fixed point theorems for compatible mappings in Non-Archimedian Intuitionistic Menger Probabilistic Metric Space. An example and an application to functional equation is provided to support the theoretical results.
Keywords:
common fixed point; non-Archimedean intuitionistic Menger probabilistic metric spaces; compatible maps
1 INTRODUCTION
The notion of probabilistic metric spaces (PMS) was introduced by Menger (1942MENGER K. 1942. Statistical metrics. Proceedings of the National Academy of Sciences of the United States of America, 28(12): 535.) as a generalization of a metric space together with the concept of compatible maps of type (J-1) and type (J-2), which are equivalent to compatible maps under certain conditions, and illustrated some common fixed point theorems for such maps in this Space. It is also of fundamental importance in the probabilistic functional analysis. Cho et al. (1997CHO CYJ, SIK HK & SHIH-SEN C. 1997. Common fixed point theorems for compatiable mappings of type (a) in non-Archimedean Menger PMˆ spaces. Mathematica japonicae, 46(1): 169-179.) introduced the concepts of compatible maps in Non-Archimedian Menger Probabilistic Metric Space N-AIMPS proved some fixed point theorems for these maps. Many authors like Sehgal & Bharucha-Reid (1972SEHGAL VM & BHARUCHA-REID A. 1972. Fixed points of contraction mappings on probabilistic metric spaces. Mathematical Systems Theory, 6(1): 97-102.), Hadzic (1980HADZIC O. 1980. A note on I. Istratescu’s fixed point theorem in non-Archimedean Menger spaces. Bull. Math. Soc. Sci. Math. Rep. Soc. Roum., 24(72): 277-280.), Chauhan & Sharma (2021CHAUHAN SS & SHARMA R. 2021. Six maps with a common fixed point in b-metric space.Journal of Physics: Conference Series, 2089(1): 012044.), Sharma & Garg (2020SHARMA R & GARG AK. 2020. A common fixed point theorem in non-archimedean menger probabilitistic metric space. Advances inMathematics: Scientific Journal, 9(8): 5593-5600.), Jafari & Shams (2015JAFARI S & SHAMS M. 2015. Some fixed point results in non-Archimedean probabilistic Menger space, p. 579.), Krishnakumar & Sanatammappa (2016KRISHNAKUMAR R & SANATAMMAPPA NP. 2016. Study on two Non-Archimedian Menger probability metric space. International Journal of Statistics and Applied Mathematics, 1(3): 1-4.), Khan (2011KHAN MA. 2011. Common fixed point theorems in non-Archimedean Menger PM-spaces.International Mathematical Forum, 6(40): 1993-2000.), Krishnakumar & Sanatammappa (2018KRISHNAKUMAR R & SANATAMMAPPA NP. 2018. Common fixed point theorems in NonArchimedean Menger PM space. International Journal of Mathematical Trends and Technology, 57(6): 377-381.), Roldán López de Hierro et al. (2021)ROLDÁN LÓPEZ DE HIERRO AF, FULGA A, KARAPINAR E & SHAHZAD N. 2021. Proinov-type fixed-point results in non-Archimedean fuzzy metric spaces. Mathematics, 9(14): 1594., Gupta et al. (2022GUPTA V, CHAUHAN SS & SANDHU IK. 2022. Banach contraction theorem on extended fuzzy cone b-metric space. Thai Journal of Mathematics, 20(1): 177-194.) have proved fixed point theorems in various metric spaces. We further expanded and generalized the results of (Devi et al., 2018DEVI VM, JEYARAMAN M & MUTHULAKSHMI L. 2018. Fixed point theorems in NonArchimedean intuitionistic Menger PM-spaces. International Journal of Pure and Applied Mathematics, 119(12): 14687-14704.) by using different conditions for three compatible maps in Non-Archimedian Intuitionistic Menger Probabilistic Metric Space and obtained common fixed point theorems for nine maps along with an example and application to enhance the results in this space.
2 PRELIMINARIES
Definition 1.(Devi et al., 2018DEVI VM, JEYARAMAN M & MUTHULAKSHMI L. 2018. Fixed point theorems in NonArchimedean intuitionistic Menger PM-spaces. International Journal of Pure and Applied Mathematics, 119(12): 14687-14704.) A triple (X, F, G) is said to be Non-Archimedian Intuitionistic Probabilistic Metric Space (shortly NAIPM-space) if X is a non empty set and F is a probabilistic distance and G is a probabilistic non-distance on X satisfying the following conditions: for all x, y, z ∈ X and t, s ≥ 0,
A 5-tuple (X, F, G, ∗, ◊) is said to be Non-Archimedian Intuitionistic Menger Probabilistic Metric Space if (X, F, G) is a NAIPM-space and in addition the following inequalities hold for all x, y, z ∈ X and t, s > 0,
where * is a continuous t-norm and ◊ is a continuous t-conorm.
Lemma 1.(Devi et al., 2018DEVI VM, JEYARAMAN M & MUTHULAKSHMI L. 2018. Fixed point theorems in NonArchimedean intuitionistic Menger PM-spaces. International Journal of Pure and Applied Mathematics, 119(12): 14687-14704.) If a function Φ : [0, ∞) → [0, ∞) satisfies the condition (Φ), then we have
(a) For all t ≥ 0, lim n→∞ Φn (t) = 0, where Φn (t) is the n-th iteration of Φ(t),
(b) If {t n } is a non-decreasing sequence of real numbers and t n+1 ≤ Φ(t n ), n = 1, 2, 3, ... then lim n→∞ t n = 0. In particular, if t ≤ Φ(t) for all t ≥ 0 then t = 0.
3 MAIN RESULT
In this section first we modify the definition of Non-Archimedian Intuitionistic Probabilistic Metric Space for three variables as follows:
Definition 2.A triple (X, F, G) is said to be modified Non-Archimedian Intuitionistic Probabilistic Metric Space (shortly N-AIPMS) if X is a non empty set, F is a probabilistic distance and G is a probabilistic non-distance on X satisfying the following conditions for all x, y, z ∈ X and t, s ≥ 0,
A 5-tuple (X, F, G, ∗, ◊) is said to be modified Non-Archimedian Intuitionistic Menger Probabilistic Metric Space (shortly N-AIMPMS) if (X, F, G) is a N-AIPMS and in addition the following inequalities hold for all x, y, z ∈ X and t, s > 0,
where * is a continuous t-norm and ◊ is a continuous t-conorm.
Lemma 2.Let {y n } be a sequence in X, such that lim n→∞ F y n,yn+1,yn+2 (t) = 1 and lim n→∞ G y n,yn+1,yn+2 (t) = 0 for all t > 0. If {y n } is not a Cauchy sequence in X, then there exists ε 0 > 0, t 0 > 0 and three sequence {m i }, {n i }, {p i } of positive integers such that
-
mi> ni + 1 and n i → ∞ as i → ∞, n i > p i + 1 and p i → ∞ as i → ∞;
-
Fymi ,yni ,ypi (t 0) < 1 − ε 0 and F y mi−2 ,yni−1 ,ypi (t 0) ≥ 1 − ε 0 , i = 1, 2, 3...
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Gymi ,yni ,ypi (t 0) > ε 0 and G y mi−2 ,yni−1 ,ypi (t 0) ≤ ε 0 , i = 1, 2, 3...
Definition 3.Self maps A,B and C on a N-AIMPMS (X, F, G, ∗, ◊) are said to be compatible if g(F AB xn ,BCxn ,CAxn (t)) → 0 and h(G AB xn ,BCxn ,CAxn (t)) → 1 for all t > 0, whenever {x n } is a sequence in X such that Ax n , Bx n ,Cx n → q for some q in X, as n → ∞.
Definition 4.Self maps A,B and C on a N-AIMPMS (X, F, G, ∗, ◊) are said to be compatible of type (J) if g(F AB xn ,BCxn ,CCxn (t)) → 0 and g(F CB xn ,BAxn ,AAxn (t)) → 0 and h(G AB xn ,BCxn ,CCxn (t)) → 1 and h(G CB xn ,BAxn ,AAxn (t)) → 1 for all t > 0, whenever {x n } is a sequence in X such that Ax n , Bx n ,Cx n → q for some q in X, as n → ∞.
Definition 5.Self maps A,B and C on a N-AIMPMS (X, F, G, ∗, ◊) are said to be compatible of type (J-1) if g(F AB xn ,BCxn ,CCxn (t)) → 0 and h(G AB xn ,BCxn,CCxn (t)) → 1 for all t > 0, whenever {x n } is a sequence in X such that Ax n , Bx n ,Cx n → q for some q in X, as n → ∞.
Definition 6.Self maps A,B and C on a N-AIMPMS (X, F, G, ∗, ◊) are said to be compatible of type (J-2) if g(F CB xn ,BAxn ,AAxn (t)) → 0 and h(G CB xn ,BAxn ,AAxn (t)) → 1 for all t > 0, whenever {x n } is a sequence in X such that Ax n , Bx n ,Cx n → q for some q in X, as n → ∞.
Proposition 1.Let A,B and C are Self maps on a N-AIMPMS (X, F, G, ∗, ◊),
(a) If C is continuous, then the pair (A,C) or (B,C) are compatible of type (J-1).
Proof. Let {x n } be a sequence in X such that Ax n , Bx n ,Cx n → q for some q in X as n → ∞ and let the pair (A,C) or (B,C) be compatible of type (J-1) and (J-2). Since C is continuous, we have A,B,C are pair wise compatible and C is continuous,
and so,
Then A,B,C are compatible of type J-I.
If A,B,C are pairwise compatible of type J-I and A,B,C are continuous.
To prove: A,B,C are pairwise compatible.
Hence, the mappings A,B,C are compatible of type (J-1).
Note: Similarly, proof can be done for the other possible cases for mappings A,B,C.
Theorem 1.Let A,B,C,D,P,Q,R,U and V be Self maps on a complete N-AIMPMS (X, F, G, ∗, ◊) satisfying:
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(a)
-
(b)
-
(c)
-
(d)
-
for all x, y ∈ X and t > 0 , where a function ϕ, φ: [0, ∞) → [0, ∞) satisfies the condition (ϕ ) and (φ);
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(e) AB = BA,CD = DC,UV = VU, PB = BP, QD = DQ, RV = VR;
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(f) Either P or AB is continuous;
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(g) The pairs (P, AB), (Q,CD), (R,UV ) are mutually compatible of type (J). Then, A,B,C,D,P,Q,R,U,V have a unique common fixed point.
Proof. Let x 0 be an arbitrary point in X. By (a) there exists x 1 , x 2 , x 3 ∈ X such that
Px0 = CDx 1 = UVx 2 = y 0 ,
Qx1 = UVx 1 = ABx 2 = y 1 ,
Rx2 = ABx 2 = CDx 3 = y 2 .
Inductively, we can construct sequences {x n }, {y n }, {z n } ∈ X such that,
Px2n = CDx 2n+1 = UVx 2n+2 = y 2n ,
Qx2n+1 = UVx2n+2 = ABx2n+3 = y2n+1,
Rx2n+2 = ABx 2n+3 = CDx 2n+4 = y 2n+2 , for n = 0, 1, 2, ....
Step-1
We shall show that sequence {y n } is a Cauchy sequence.
Since, Px 2n = CDx 2n+1 = UVx 2n+2 , using (b), (c), (d), we have
and
Since, Qx 2n+1 = UVx 2n+2 = ABx 2n+3 , we also have
and
Thus,
and
Hence,
and
Therefore, from Lemma 2.2;
Suppose {y n } is not a Cauchy sequence. Since g is strictly decreasing from lemma 2.3, so there exists ε 0 > 0, t 0 > 0 and three sequences {m k }, {n k }, {p k } of positive integers such that
-
and >,
-
and and
Therefore,
and,
Letting k → ∞ we have,
On the other hand, we have
Without loss of generality, assume that all the sequences {m k }, {n k }, {p k } are even, using (c), (d) we have,
and
Substituting this in (3.1.3), letting k → ∞ and using (3.1.1), (3.1.2), we have
This is a contradiction. Hence, {y n } is a Cauchy sequence. Since, (X, F, G, ∗, ◊) is complete, it converges to a point q in X. Also its subsequences converges as follows: Px 2n → q, ABx 2n → q, Qy 2n+1 → q,CDy 2n+1 → q, Rz 2n+2 → q,UVz 2n+2 → q.
Case-1 AB is continuous and (P, AB),(Q, CD), (R, UV) are compatible of type (J-1).
Since AB is continuous AB(AB)x 2n → ABq and (AB)Px 2n → ABq and (P,AB) are compatible of type (J-1), PPx 2n → ABq.
Step-2
By taking x = Px 2n , y = x 2n+1 , z = x 2n+2 , we have,
Similarly,
This implies that, as n → ∞
and
Thus by lemma-2.2,
g(FABq,q,q (t)) = 0 and h(G ABq,q,q (t)) = 1 for all t > 0 and it follows that q = ABq.
Step-3
By taking x = q, y = x 2n+1 , z = x 2n+2 , we have,
Similarly,
This implies that, as n → ∞
and
this gives that q = Pq and therefore, q = ABq = Pq.
Step-4
By taking x = Bq, y = x 2n+1 , z = x 2n+2 ,in (c) and (d) using (e) we have,
Similarly,
This implies that, as n → ∞
and
gives, q = Bq. Since, q = ABq, we have q = Aq and therefore, q = Aq = Bq = Pq.
Step-5
Since, such that q = Qq = Cdq.
By taking x = x 2n , y = j, z = x 2n+1 in (c) and (d), we get
Similarly,
This implies that, as n → ∞
Similarly,
This implies that q = Q j. Hence, CD j = q = Q j. Since, (Q,CD) is compatible of type (J-1), we have Q(CD) j = CD(CD) j. Thus, CDq = Qq
Step-6
By taking x = x 2n , y = q, z = x 2n+1 in (c) and (d), we have
Similarly,
This implies that, as n → ∞
Similarly,
This means that q = Qq. Hence, CD j = q = Qq. Since, CDq = Qq, we have q = CDq. Therefore,
q = Aq = Bq = Pq = Qq = CDq.
Step-7
By taking x = x 2n , y = Dq, z = x 2n+1 in (c) and (d), we have
Similarly,
This implies that, as n → ∞
Similarly,
This gives, q = Dq. Since, q = CDq, we have q = Dq. Therefore, q = Aq = Bq = Cq = Dq = Pq = Qq.
Step-8
Since, , there exists w ∈ X such that q = Pq = UVw.
By taking x = x 2n , y = x 2n+1 , z = w in (c) and (d), we have
Similarly,
This implies that, as n → ∞
Similarly,
This gives q = Rw. Hence, Uvw = q = Rw. Since, (R, ST) is compatible of type (J-1), we have R(UV)w = UV (UV)w. Thus, UVq = Rq.
Step-9
By taking x = x 2n , y = x 2n+1 , z = q in (c), (d) and using (e), we have
Similarly,
This implies that, as n → ∞
Similarly,
This gives q = Rq. We have, q = UVq. Therefore, q = Aq = Bq = Cq = Dq = Pq = Qq = Rq = UVq
Step-10
By taking x = x 2n , y = x 2n+1 , z = Vq in (c), (d) and using (e), we have
Similarly,
This implies that, as n → ∞
Similarly,
This implies q = Vq. Since q = UVq, we get q = Vq. Therefore, q = Aq = Bq = Cq = Dq = Pq = Qq = Rq = Uq = Vq , that is common fixed point of A,B,C,D,P,Q,R,U,V. Similarly, it is clear that q is also the common fixed point of A,B,C,D,P,Q,R,U,V in the case AB is continuous and (P,AB), (Q,CD), (R,UV) are compatible of type (J-2).
Case-2 AB is continuous and (P, AB),(Q, CD), (R, UV) are compatible of type (J-1). Since P is continuous and (P, AB), (Q,CD), (R,UV ) are compatible of type (J-1).
As P is continuous, PPx 2n → Pq and P(AB)x 2n → Pq . Since, (P, AB) is compatible of type (J-1), AB(AB)x 2n → Pq.
Step-11
By taking x = ABx 2n , y = x 2n+1 , z = x 2n+2 in (c) and (d), we have
Similarly,
This implies that, as n → ∞
Similarly,
This means that q = Pq. Now using the step (4-9), we have, q = Qq = CDq = Cq = Dq = Rq = UVq = Uq = Vq.
Step-12
Since there exists l ∈ X such that q = Rq = ABl.
By taking x = l, y = x 2n+1 , z = x 2n+2 in (c) and (d), we have
Similarly,
This implies that, as n → ∞
and
This gives q = Rl. Since, q = Rq, we have q = Aq = ABl = Pl = ABl. Since P is continuous and (P, AB), (Q,CD), (R,UV) are compatible of type (J-1), we have, Pq = ABq . Also,q = Bq follows from step-3. Thus, q = Aq = Bq = Pq. Hence, q is common fixed point of the nine maps in this case also. Similarly, it is clear that q is also the common fixed point of A,B,C,D,P,Q,R,U,V in the case P is continuous and (P,AB), (Q,CD), (R,UV) are compatible of type (J-2).
Step-13
For uniqueness, let u, v(u, v ≠ q) be another common fixed point of A,B,C,D,P,Q,R,U,V. Taking x = q, y = u, z = v in (c) and (d), we have
Similarly,
This implies that, as n → ∞
Similarly,
So, we have, q = u = v. This completes the proof of the theorem.
If we take A = B = C = D = U = V = I X (the identity map on X) in Theorem-3.1, we have the following results:
Corollary-3.2
Let P, Q, R are self maps on complete N-AIMPMS (X, F, G, ∗, ◊). If and
Therefore,
and
for all x, y, z ∈ X and t > 0, where functions ϕ, φ: [0, ∞) → [0, ∞) satisfies the condition (ϕ) and (φ). Then, P, Q, R have a unique common fixed point. □
Example-3.3
Suppose . Define F: X × X → N by
for x, y, z ∈ X . It is easy to verify that (X, F, G, ∗, ◊) is a N-AIMPMS. Now assume t, q, j, x, y, z, w, l ∈ X . Then we have,
4 APPLICATION TO FUNCTIONAL EQUATIONS
There are many types of nonlinear functional equations for which fixed point theorems have been used to demonstrate their existence.
Let J and K be Banach spaces, be a state space and be a decision space. Now, by using the fixed point results obtained in previous section, we have;
where .
Let H(L) denote the space of all bounded real valued function on L. Clearly, this space endowed with the metric given by
for all t, q, j ∈ H(L), is a complete metric space.
Now, define
where t, q, j ∈ H(L), then (H(L), F, G, ∗, ◊) is a complete N-AIMPMS with
where .
Functional equations is an essential tool for describing the nature of physical universe. It has multiple real-world applications from sports to engineering to astronomy and space travel, which can be solved by reducing them to equivalent fixed-point problems. Newton’s Laws of motion and gravitational, astronomical science, Investment plans, global mappings, constructing tracks, Relation of income and market prediction are based on Functional equations. One example of a functional equation that may be used in the real world is the Euler-Lagrange equation. The shortest path between two points on a manifold is one way to approach this problem. The equation is a line when the two points are on a Cartesian plane. A great circle is obtained if the two points lie on a sphere. Airlines don’t typically travel on straight paths due to this, so the journey is in fact longer. Geographically, the shortest distance/path is called a geodesic in both cases.
5 CONCLUSIONS
We obtained common fixed point theorems for nine maps by introducing three types of compatible maps in N-AIMPMS and expanded the results of Devi et al. (2018DEVI VM, JEYARAMAN M & MUTHULAKSHMI L. 2018. Fixed point theorems in NonArchimedean intuitionistic Menger PM-spaces. International Journal of Pure and Applied Mathematics, 119(12): 14687-14704.) under certain conditions using the concept of compatible maps of type (J-1) and (J-2). An example and application is provided to stake the applicability of our results.
References
- CHAUHAN SS & SHARMA R. 2021. Six maps with a common fixed point in b-metric space.Journal of Physics: Conference Series, 2089(1): 012044.
- CHO CYJ, SIK HK & SHIH-SEN C. 1997. Common fixed point theorems for compatiable mappings of type (a) in non-Archimedean Menger PMˆ spaces. Mathematica japonicae, 46(1): 169-179.
- DEVI VM, JEYARAMAN M & MUTHULAKSHMI L. 2018. Fixed point theorems in NonArchimedean intuitionistic Menger PM-spaces. International Journal of Pure and Applied Mathematics, 119(12): 14687-14704.
- GUPTA V, CHAUHAN SS & SANDHU IK. 2022. Banach contraction theorem on extended fuzzy cone b-metric space. Thai Journal of Mathematics, 20(1): 177-194.
- HADZIC O. 1980. A note on I. Istratescu’s fixed point theorem in non-Archimedean Menger spaces. Bull. Math. Soc. Sci. Math. Rep. Soc. Roum., 24(72): 277-280.
- JAFARI S & SHAMS M. 2015. Some fixed point results in non-Archimedean probabilistic Menger space, p. 579.
- KHAN MA. 2011. Common fixed point theorems in non-Archimedean Menger PM-spaces.International Mathematical Forum, 6(40): 1993-2000.
- KRISHNAKUMAR R & SANATAMMAPPA NP. 2016. Study on two Non-Archimedian Menger probability metric space. International Journal of Statistics and Applied Mathematics, 1(3): 1-4.
- KRISHNAKUMAR R & SANATAMMAPPA NP. 2018. Common fixed point theorems in NonArchimedean Menger PM space. International Journal of Mathematical Trends and Technology, 57(6): 377-381.
- MENGER K. 1942. Statistical metrics. Proceedings of the National Academy of Sciences of the United States of America, 28(12): 535.
- ROLDÁN LÓPEZ DE HIERRO AF, FULGA A, KARAPINAR E & SHAHZAD N. 2021. Proinov-type fixed-point results in non-Archimedean fuzzy metric spaces. Mathematics, 9(14): 1594.
- SEHGAL VM & BHARUCHA-REID A. 1972. Fixed points of contraction mappings on probabilistic metric spaces. Mathematical Systems Theory, 6(1): 97-102.
- SHARMA R & GARG AK. 2020. A common fixed point theorem in non-archimedean menger probabilitistic metric space. Advances inMathematics: Scientific Journal, 9(8): 5593-5600.
Publication Dates
-
Publication in this collection
11 Dec 2023 -
Date of issue
2023
History
-
Received
19 June 2023 -
Accepted
21 Sept 2023