Common Fixed Point Theorems for Six Mappings Satisfying Almost Generalized (S, T)-Contractive Condition in Partially Ordered Metric Spaces
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1 Annals of Pure Applied Mathematics Vol. 1, No., 016, ISSN: X (P), (online) Published on 7 November Annals of Common Fixed Point Theorems for Six Mappings Satisfying Almost Generalized (S, T)-Contractive Condition in Partially Ordered Metric Spaces R.K.Sharma 1, V. Raich C.S.Chauhan 3 1, Department of Mathematics, Govt. Holkar Science College (DAVV) Indore-45001, Madhya Pradesh, India raj_rma@yahoo.co.in, drvivekraich@gmail.com 3 Department of Applied Mathematics, Institute of Engineering & Tech. (DAVV) Indore-45001, Madhya Pradesh, India cschauhan0iet@gmail.com Received 14 October 016; accepted 31 October 016 Abstract. The existence of coincidence point common fixed point theorems for six self-mappings satisfying almost generalized (S, T)-contractive condition in the setup of partially ordered complete metric spaces have been proved. Our result unify generalize the earlier results of Aghajani et al. others. A related example is also furnished. Keywords: Common fixed points, partially ordered metric spaces, almost contraction, weakly annihilator, dominating map AMS Mathematics Subject Classification (010): 54H5, 47H10 1. Introduction In 1968, Kannan [10] was the first who proved the existence of a fixed point for a map that can have a discontinuity in a domain, however the maps involved in every case were continuous at the fixed point. On the other h after the classical result of Jungck [7] of common fixed point of two commuting maps, Sessa [11] initiated the weaker condition than that of commutativity namely weak commutativity of maps proved the result regarding common fixed point consideration of such maps. Of course two commuting mappings are weakly commuting but the converse is not true always. Further a weaker condition of these notions namely, compatibility of maps has been introduced by Jungck [8] proved result regarding common fixed points of such maps. Jungck [8] also demonstrated that commuting mappings are weakly commuting weakly commuting are compatible but neither implication is reversible. Recently Jungck & Rhoades [9] has introduced a weaker class among all commutative conditions namely weakly compatibility maps or coincidently commutativity of maps gave results regarding common fixed points of such maps. 143
2 R.K.Sharma, V.Raich C.S.Chauhan Abbas et al. [] introduced the notions namely weak annihilator dominating maps proved some results of common fixed points by using these notions in the framework of partially ordered metric space. Berinde in [4, 5] initiated the concept of almost contractions. Further by introducing the concept of almost generalized contractive condition Ćirić et al. [6] extend the concept of almost contractions to a pair of self-maps. Also Aghajani et al. [3] generalized the notion of almost generalized contraction by introducing the notion of almost generalized,-contraction. The aim of this paper is to establish coincidence point common fixed point results for six mappings which satisfy almost generalized,-contractive condition in the setting of partially ordered metric spaces. Infact, we have generalized the results of Aghajani et al. [3] many others.. Preliminaries We start this section by some basic definitions results which are used in sequel. Definition.1. Let, be a partially ordered set,: then (.1.1) an ordered pair (f, g) are called partially weakly increasing if for all. (cf. []) (.1.) a map is called weak annihilator of if for all. (cf. []) (.1.3) a map is called dominating if for all. (cf. []) Definition.. [9] Let, be a metric space then mappings, : are said to be weakly compatible if they commute at their coincidence points, that is, if = for some then =. Berinde [5] introduced the notion of weak contraction which further renamed as almost contraction by Berinde [4] defined as: Definition.3 [4, 5] A self-map on a metric space is said to an almost contraction or, contraction if there exist a constant 0,1 some 0 such that (.3.1),,+,,,. Remark.4. [4, 5] Due to the symmetry of the distance, the almost contraction condition (.3.1) implicitly includes the following dual one,.,+,,,, obtained from (.3.1) by formally replacing,, by,,, respectively then interchanging. Berinde in [5] established some fixed point theorems for almost contractions in complete metric spaces shown that any strict contraction such as Kannan [10] mapping as well as a large class of quasi-contractions are all almost contractions. By generalizing the notion of almost contraction Ćirić et al. [6] gave: Definition.5. [6] Let, be a metric space : is said to be an almost contraction with respect to : if there exists a constant 0,1 some 0 such that,,+,,,. 144
3 Common Fixed Point Theorems for Six Mappings Satisfying Almost Generalized (S, T)-Contractive Condition in Partially Ordered Metric Spaces Remark.6. [6] If one take = Identity map on, then the above definition reduced to the notion almost contraction. Further, Ćirić et al. [6] introduced the notion namely almost generalized contraction defined as: Definition.7. [6] Let, be a metric space,: are said to satisfy almost generalized contractive condition if there exists a constant [0,1 some 0 such that (.7.1),,,,,,,,, +,,,,,,,,,. Theorem.8. [6] Let, be a partially ordered set there exists a metric d on such that, is a complete metric space. Let, be strictly weakly increasing mappings with respect to satisfying (.7.1) for every comparable,. If either or is continuous then have a common fixed point in. Aghajani et al. [3] generalized the notion of almost generalized contraction by introducing the notion namely almost generalized,-contraction as: Definition.9. [3] Let,, be self-maps on a metric space,, then are said to satisfy almost generalized,-contractive condition if there exist a constant [0,1 some 0 such that (.9.1),,+,,,=,,,,,,,,,,=,,,,,,,,,. Remark.10. [3] If one take == Identity map on, then above definition reduced to almost generalized contractive condition. Theorem.11. [3] Let, be a partially ordered set there exists a metric on such that, is a complete metric space. Let,,,: satisfying the condition (.9.1) for each pair of comparable elements, (.11.1). (.11.) are dominating, weak annihilators of, respectively. (.11.3) there exists a non-decreasing sequence with for all implies that. (.11.4) pairs (, (, are weakly compatible. (.11.5) one of,, is a closed subspace of, then,, have a unique common fixed point. 3. Main result Our main result is generalization of result of [3] for six self-maps as opposed to four maps satisfying almost generalized,-contractive condition in partially ordered complete metric space. 145
4 R.K.Sharma, V.Raich C.S.Chauhan Theorem 3.1. Let,, be an ordered complete metric space. Let,,,,,: satisfying (.11.3) (3.1.1). (3.1.),,+,,,=,,,,,,,,,,=,,,,,,,, for each pair of comparable elements,, [0,1 some 0. (3.1.3) (i) the pairs (, (, are partially weakly increasing. (ii) are dominating, weak annihilators of, respectively. (3.1.4) one of,, is a closed subspace of, then (i) have a coincidence point in, (ii) have a coincidence point in. (3.1.5) pairs (, (, are weakly compatible then (iii),, have a unique common fixed point in. Furthermore if (3.1.6) the pairs,,,,,,,,,, commute at the common fixed point of,, then,,,, have a unique common fixed point in. Proof. Let be an arbitrary point in, since then there exists such that =. Also since then there exists such that =. Inductively we can construct the sequences in such that = = = = for all =0,1,,3.... From (3.1.3), we have = =. Thus 0, we obtain.... Now we claim that is a Cauchy sequence in. If =, for some, then from (3.1.), we have, =,, +,,, =max,,,,,,,, =,,,,,,, +, =0,,,0,,,,, =min,,,,,,, =min,,,,,,, =0. Hence,,, since [0,1 yields that =. Further by using the similar arguments, we have = so on. Thus 146
5 Common Fixed Point Theorems for Six Mappings Satisfying Almost Generalized (S, T)-Contractive Condition in Partially Ordered Metric Spaces turns out to be a constant sequence is the common fixed point of,,. If we suppose, >0, for every since = = are comparable, then from (3.1.), we have, =,, +, (3.1), =,,,,,,, +, =,,,,,,, +,,,,., =min,,,,,,, =min,,,,,,0=0. Therefore from (3.1), we have, =,,,, (3.) Now,,, = either, or, If,,, =, then from (3.), we have,,, which is a contradiction, since [0,1. Therefore,,. Similarly it can be proved that,,. Therefore for all n 1, we have,,. Inductively for all n 1, we have,,,, By triangle inequality for >, we have,, +, +, + +,, 0 as ( since [0,1, yields that is a Cauchy sequence in. By the completeness of, the Cauchy sequence its subsequences are also converges to some in, i.e., lim =lim =lim = lim = lim = lim =. Suppose is closed then there exists such that =. From (3.1.3), since as =. Using (3.1.), we have,,+, (3.3),=,,,,,,,+, 147
6 R.K.Sharma, V.Raich C.S.Chauhan =,,,,,,,,,=min,,,,,,, =min,,,,,,,. Letting, we have lim,=,, lim, =0. Therefore from (3.3) as, we have,, yields that =. So ==. Now by weakly compatibility of pair,, ===. Using (3.1.), we have,=,,+, (3.4),=,,,,,,,+, =,,,,,,,+,,=min,,,,,,, =min,,,,,,,. Letting, we have lim,=,, lim,=0. Therefore from (3.4) as, we have,,, yields that =. Hence =. (3.5) Since then there exists a point such that ==. From (3.1.3) since = implies that. Using (3.1.), we have,=,,+, (3.6),=,,,,,,,, =,,,,,,,, =0,,,0,, =,,,=min,,,,,,, =min,,0,,,,=0. Therefore from (3.6), we have,, yields that =. Now by weakly compatibility of pair,, ===. Using (3.1.), we have,=,,+, (3.7) 148
7 Common Fixed Point Theorems for Six Mappings Satisfying Almost Generalized (S, T)-Contractive Condition in Partially Ordered Metric Spaces,=,,,,,,,, =,,,,,,,, =,,0,0,,=,,,=min,,,,,,, =min 0,0,,,,=0. Therefore from (3.7), we have,,, yields that == (3.8) Hence from (3.5) (3.8), we have ====, i.e. is the common fixed point of,,. For the uniqueness of suppose be another common fixed point of,, then from (3.1.), we have,=,,+, (3.9),=,,,,,,,, =,,0,0,,=,,=min,,,,,,, =min 0,0,,,,=0. Therefore from (3.9), we have,,, yields that =, i.e. is the unique common fixed point of,,. The proof is similar for the cases in which one of, is a closed subspace of. From (3.1.6) by commutativity of,,,,, we have ===, ==, ====, ==, which shows that are the common fixed points of (,. But have a unique fixed point, then ==== (3.10) Again from (3.1.6) by commutativity of,,,,, we have ===, ==, ===, ==, which shows that are the common fixed points of (,. But have unique fixed point, then ==== (3.11) Using (3.1.), (3.10) (3.11), we have,=,,+, (3.1),=,,,,,,,, =,,,,,,,, =max0,0,0,0=0 149
8 R.K.Sharma, V.Raich C.S.Chauhan,=min,,,,,,, =min,,,,,,,=0. Therefore from (3.1), we have, 0=0, yields that = (3.13) Hence from (3.10), (3.11) (3.13), we have ======, i.e. is the unique common fixed point of self-mappings,,,,. If we take = in Theorem 3.1, we have the following corollary: Corollary 3.. Let,, be an ordered complete metric space. Let,,,,: satisfying (.11.3) (3..1). (3..),,+,,,=,,,,,,,,,,=,,,,,,, for every comparable elements,, [0,1 some 0. (3..3) (i) the pairs (, (, are partially weakly increasing. (ii) are dominating, weak annihilators of. (3..4) one of, is a closed subspace of, then (i) have a coincidence point in, (ii) have a coincidence point in. (3..5) pairs (, (, are weakly compatible then (iii), have a unique common fixed point in. Furthermore if (3..6) the pairs,,,,,,,,,, commute at the common fixed point of,, then,,, have a unique common fixed point in. Corollary 3.3. (3.3.1) If we take == Identity mappings in Theorem 3.1, we can obtain Theorem.10 of Aghajani et al. [3]. (3.3.) If we take == = in Theorem 3.1, we can obtain Corollary.3 of Aghajani et al. [3]. (3.3.3) If we take == == Identity mappings in Theorem 3.1, we can obtain Theorem.1 of Ćirić et al. [6]. Remark 3.4. Here note that for every self-map, (,) is weakly compatible, is identity dominating map so by taking == in Theorem 3.1, we have the following result. Corollary 3.5. Let,, be an ordered complete metric space. Let,: surjective maps such that for all satisfying (.10.3),,+,,,=,,,,,,,,,,=,,,,,,, 150
9 Common Fixed Point Theorems for Six Mappings Satisfying Almost Generalized (S, T)-Contractive Condition in Partially Ordered Metric Spaces for every comparable elements,, [0,1 some 0, then have a unique common fixed point in. Now we illustrate the following example in support of Theorem 3.1. Example.6. Let =[0, with relation given by,=, we define a new ordering on such that,,. Then,, is an ordered complete metric space. Define,,,,,: such that =ln1+,=,=ln1+, =4, = 1 = 1. Then by routine calculation we can see that all the conditions of Theorem 3.1 are satisfied 0 is the unique common fixed point of,,,,. Acknowledgement: Authors are thankful to the referee editor of the journal for their positive comments valuable suggestions to improve the manuscript. REFERENCES 1. M.Abbas D.Ilic, Common fixed points of generalized almost nonexpansive mappings, Filomat, 4(3) (010) M. Abbas, T.Nazir, S.Radenović, Common fixed points of four maps in partially ordered metric spaces, App. Math. Let., 4(9) (011) A.Aghajani, S. Radenović J.R. Roshan, Common fixed point results for four maps satisfying almost generalized (S,T)-contractive condition in partially ordered metric spaces, App. Math. Comp., 18(9) (01) V.Berinde M.Pacurar, Fixed points continuity of almost contractions, Fixed Point Theory, 9(1) (008) V.Berinde, Approximating fixed points of weak contractions using the Picard iteration, In Non. Anal. Forum, 9 (004) L.Ćirić, M.Abbas, R.Saadati, N.Hussain, Common fixed points of almost generalized contractive mappings in ordered metric spaces, App. Math. Comp., 17(1) (011) G.Jungck, Commuting mappings fixed points, The American Mathematical Monthly, 83(4) (1976) G.Jungck, Compatible mappings common fixed points (), International Journal of Mathematics Mathematical Sciences, 11() (1988) G.Jungck B.E.Rhoades, Fixed point for set valued functions without continuity. Indian Journal of Pure Appl. Math., 9(3) (1998) R.Kannan, Some results on fixed points, Bulletin of the Calcutta Mathematical Society, 10 (1968) S.Sessa, On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math., 3(46) (198)
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