A Research on Common Fixed Point Theorems with Generalized Spaces

After the observed Banach contraction principle (BCP) in 1922, there have been various results in the writing managing mappings fulfilling the contraction conditions of different kinds including even nonlinear articulations. One of its vital generalizations is given by Jungck (1976) in which he built up regular fixed point theorems for driving pair of maps. This idea of commutativity of maps is casual to weakly commutativity, similarity, weakly similarity, and so forth.. Branciari (2002) acquired a fixed point hypothesis for map fulfilling a simple of Banach contraction principle for integral sort imbalance. This result was additionally generalized by numerous creators. The motivation behind this section is to enhance and generalize some current normal fixed point results in generalized metric spaces and- fuzzy metric spaces for the maps fulfilling integral write contractive conditions. To begin with, we acquire some fixed point results for half and half match of single and multivalued maps in the settings of b - metric spaces. From there on, we characterize- fuzzy metric space and acquire some fixed point theorems in it. Some current results are determined as special cases. We initially demonstrate some basic fixed point and occurrence point theorems for half breed combine of maps in b - metric spaces fulfilling basic property (E. An.) and an integral disparity.

Theorem 2.1. Letbe a complete b - metric space and and such that (ii) (iii) The pairs and satisfy the common property, (iv) for all

(2.1)

where is Lebesgue integrable mapping which is summable. non-negative and such that

(2.2)

and

with where If and are closed subspace of then (1) and have a coincidence point.

(3) and F have a common fixed point provided that is F – weakly commuting at and for where, = {.r : ,r is a coincidence point of and F}. (4) g and G have a ordinary fixed point provide that g is G- weakly commute at and for (5) F and G have a ordinary fixed point provided (3) and (4) are true. Proof. Let From (i) we can construct a series in Xsuch that for all It follows from equation (2.1) that where, Thus, Similarly, where, Thus, we have proved that for all Hence for all and noting, a constant, we have

C- 7

Then is a Cauchy sequence. Since is a Cauchy sequence, there exist asatisfying Since and are closed, there exist such that A similar argument proves that If and then Thus and satisfy, common property We claim that To prove it, we take in (2.1), where, Taking the limit as we obtain where, Since it follows from the definitions of Hausdorff metric that which implies that On the other hand, by condition (iii) again we have,

where, and similarly, we obtain Hence Thus and F have a coincidence point and and G have a coincidence point This completes the proofs of part (1) and (2). Furthermore, by virtue of condition (3), we obtain and Thus This proves (3). A similar argument proves (4). Then (5) holds immediately. If we put and in Theorem 2.1. the following result of Liu et. al. (2005) is obtained. Corollary 2.1. Let be a complete metric space and and such that (i) (ii) The pairs and satisfy the common property Let be a constant, such that for all in If fX and gX are closed subspace of X, then (1) f and F have a coincidence point. (2) g and G have a coincidence point, (3) and F have a common fixed point provided that is F weakly commuting at and for (4) and G have a common fixed point provided that is G- weakly commuting at and for (5) and G have a common fixed point provided (3) and (4) are true. As an application of Theorem 2.1 we obtain the following result. space and and such that (i) (ii) The pairs and satisfy the common property (iii) For all whereis a Lebesgue integrable mapping which is suimnable, non-negative and such that for each

(2.4)

with where If and are closed subspace of then (1) and have a coincidence point, (2) and have a coincidence point, (3) and F have a common fixed point provided that is F - weakly commuting at and for (4) and G have a common fixed point provided that is G- weakly commuting at and for (5) and G have a common fixed point provided (3) and (4) are true. Proof. From equation (2.4). we have But, so,

Let Following (2.4), it is easy to see that Thus by Theorem 2.1, we arrive at the conclusions of Theorem 2.2. If we put and in Theorem 2.2, we get the following result of Liu et al(2005), Corollary 2.2. Let be a complete metric space and and such that (i) (ii) The pairs and satisfy the common property Let be a constant, such that for all and If and are closed subspace of then (i) and F have a coincidence point. (ii) and G have a coincidence point. (iii) and F have a common fixed point provided that is F - weakly commuting at and for (iv) and G have a common fixed point provided that is G- weakly commuting at and for (v) and G have a common fixed point provided (3) and (4) are true. Now we give a common fixed point result for four self mappings in a b- metric space. Theorem 2.3. Let be a complete 6-metric space and be such that (i) (iii) for all where is a Lebesgue integrable mapping which is suimnable, non-negative and such that and

(2.5)

with where If or is closed, then A. B, 5 and T have a common fixed point. Proof. Let From (i) we can construct a sequence in X such that and As in Theorem 2.1. we can prove that is a Cauchy sequence. Since X is complete, the sequence converges to a point in X. Consequently, the subsequences and also converge to the same limit Now suppose that is closed. Then since there exists a point such that Then by using (2.5), with and we get where, Letting we get

which is a contradiction. This implies that Therefore Hence by the weak compatibility of the pair it in unediately follows that Next, we shall show thatis a common fixed point of B and T. By setting and hi (2.5) we have where, Letting and noting that and we get where, Thus, which is a contradiction, so Thus we have = i.e., is a conunon fixed point of B and T. Further, implies that Therefore there exists a point such that We now show that Indeed, by setting and in (2.5) and taking we get Hence Then by the weak compatibility of the pair we immediately have Hence Now. by setting and in (2.5) and following the earlier arguments, it can easily be verified that is a common fixed point of A and S as well. Hence is a common fixed pomt of A.B.S and T.

B. S and T can easily be verified. In fact, if is another common fixed point of the given mappings, then by setting and in (2.5) we get where, Thus we get,

z=z‘ and is a unique common fixed point

In 1965, Zacleh (1965) presented fuzzy set, which is additionally generalized to intuitionistic fuzzy set by Atauassov (1986) and - fuzzy set by Goguen (1967), Thereafter a few creators characterized fuzzy metric spaces in various ways. Subsequently a few fuzzy fixed point theorems are likewise settled in their new settings. We first review the preliminaries required for ensuing results. Definition 2.1. Let be a continuous map with respect to each variable. Then is called a 5-action if and only if it satisfies the following conditions (i) and for all (ii) if and or and (iii) for each and for each, there exists such that where (iv) for all Definition 2.2 . Let X be a nonempty set. A mapping is called a - metric on Ar with respect to B - action, if satisfies the following (i) if

(iii) for all Then is called a - metric space. Remark 2.1. If then the - metric space becomes a metric space. Remark 2.2. If then the- metric space is the 6-metric space. Now we define - fuzzy metric space following the definition given by George and Veeramani, Definition 2.3. The 3-triplet is said to be an -fuzzy metric space, if A' is an arbitrary (non- empty) set. is a continuous - norm 011 and is an -fuzzy set 011 with respect to B~ action. satisfying the following conditions for eveiy in X and in (i)

(iii) (iv) (v) is continuous and In this case is called a -fuzzy metric space. If is an intuitionistic fuzzy set, then the 3- tuple is said to be a - intuitionistic fuzzy metric space. Example 2.1. Let endowed with the usual metric and Let for all where is a complete lattice and let be the intuitionistic

Then is an intuitionistic fuzzy metric space. condition: for all implies Lemma 2.1. Let satisfies the following condition It is nondecreasing andfor all where denotes the iterate ofthenfor all Lemma 2.2. Let be a - fuzzy metric space. Then is no decreasing with respect to , for all in Proof. Suppose for some Then Now from the definition, we have, for each and for each there exists such that where So, since Then. since Also, we have for all Thus we have a contradiction. Lemma 2.3. Let be a - fuzzy metric space. Define by for each and Then we have (i) For any there exists such that for any (ii) The sequence is convergent to, r with respect to - fuzzy metric M if and only if it is Cauchy with Proof. For the first part, we can find for every such that Using the definition of - fuzzy metric space, we have

for eveiy which implies that Since is arbitrary, we have

This proves (i). For (ii), note that since M is continuous in its third place, is not an element of the set as soon as Hence we have, for eveiy This completes the proof. Lemma 2.4. Let A, B. S. T. I and J be mappings from- fuzzymetric space into itself satisfying (i) (ii) where satisfies condition and is a continuous function such that for each and for all Then the sequence defined by is a Cauchy sequence in X Proof. For we have, for which implies that, for eveiy we have From Lemma 2.3, for eveiy there exists such that Thus is a Cauchy sequence in X. Now we prove the following common fixed point theorem. Theorem 2.4. Let A, B. S. T. I and be mappings from a complete-fuzzy metric space into itself satisfying (i), (ii) of Lemma 2.4. and property Suppose that one of A. B. S, T. I and is complete and pairs and are weakly compatible, then A, B. S. T. I and have a unique common fixed point. Proof. By Lemma 2.4. is a Cauchy sequence and since A'is complete, therefore, converges to some point Consequently, the subsequences and of also converges to Assume that is complete, so there exists a pointin X such that If from (ii), we have On the other hand, by Lemma

2.2,

Taking the limit, we get, which is a contradiction. Thus we have.

If we have where, But by Lemma 2.2, Thus we have, Hence. Since the pairs and are weakly compatible, we have and i.e., If and or. We get. By Lemma 2.2, so we have which is a contradiction. Hence Similarly, we can prove that Therefore, is a common fixed point of A. B. S, T. I and J. For uniqueness, let if possible be another common fixed point of A. B. .S', T. I and J. Then there exists such that 1 and But, Thus, By Lemma 2.2, Hence, for all Since, has the property it follows that therefore i.e., is a unique common fixed point of A, B. S. T. I and J. This completes the proof If we put i.e., the identity map and and 111 Theorem 2.4. then the following result of Mamo et al (2012) is deduced. Corollary 2.3.. Let A and T be mappings from a complete fuzzy metric space into itself satisfying and Suppose that either or is complete and A and T are weakly compatible on X. Then A and T have a unique common fixed point. In the following Theorem we relax the condition of completeness on the space. Theorem 2.5. Let A. B. S, T. I and J be mappings from a - fuzzy metric space into itself satisfying (i), (ii) of Lemma 2.4 and property (C). Suppose that one of A, B. S. T. I and J is complete and pairs and are weakly compatible, then A. B. S. T. I and J have a unique common fixed point. Proof. From the proof of Theorem 2.4, we conclude that is a Cauchy sequence in A'. Assume that is complete subspace of X. Then the subsequence of must get a limit in Let it be and As is a Cauchy sequence containing a convergent subsequence, therefore the sequence also converges implying thereby the convergence of subsequence of the convergent sequence. Now we have,

where, Now by Lemma 2.2, Taking the limit we get, which is a contradiction. Thus we have, which shows that pair has a point of coincidence. Since there exists such that If we have We have, By Lemma 2.2, Thus we have, a contradiction. Hence, Since the pairs and are weakly compatible, we have and and If and We set By Lemma 2.2, so we have which is a contradiction. Hence

Then there exists such that 1 and But, Thus, By Lemma 2.2, Hence, for all Since. M has the property it follows that therefore is a unique common fixed point of A. B. S, T, I and J. This completes the proof. If we put identity map and and in Theorem 2.5, then the following result of Manro et al is obtained. Corollary 2.3. Let A and T be mappings from a fuzzy metric space into itself satisfying andSupp ose that either or is complete and A and T are wejakly compatible on X. Then A and T have a unique common fixed point.

(i) 5-weakly commuting property, (ii) 5-weakly commuting property of type (iii) 5-weakly coimnuting property of type (iv) weakly coimnuting property. Proof, (i) Since all the conditions of Theorem 2.4 are satisfied, the existence of coincidence points for both the pairs and are ensured. Let be an arbitrary point of coincidence for the pair and . Then using 5-weakly commutativity, one gets and From Lemma 2.2, and So we have and Thus the pairsand are weakly compatible. Now applying Theorem 2.4, we conclude that A, B, S. T. I and J have a unique conunon fixed point. (ii) In case pair satisfying property 5- weakly commutativity of type we have and we have From Lemma 2.2, So, This implies that Similarly, if pair satisfies property 5-weakly commutativity of type we get (iii) In case pair satisfying property 5- weakly commutativity of type we have and we have From Lemma 2.2, This implies that Similarly, if pair satisfies property 5-weak commutativity of type we get Similarly, we can show that, if pairs and are weakly commuting then and also commutes at their point of coincidence. Now, in view of Theorem 2.4. in all five cases, A. B, S. T. I and J have a unique common fixed point. This completes the proof.

In this examination, we have displayed the most critical fixed point theorems in the expository investigation of problems in the connected science: Banach's and Schauder's fixed point theorem. In our unique situation, Brouwer's fixed point might be considered as a result which empowers us to proof the Schauder fixed point theorem. The fixed point theorem of Darbo is the focal result of a bound together approach for Banach's and Schauder's theorem.

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Research Scholar, OPJS University, Churu, Rajasthan