On a Fixed Point Theorem for a Cyclical Kannan-type Mapping
aa r X i v : . [ m a t h . GN ] J a n ON A FIXED POINT THEOREM FOR A CYCLICALKANNAN-TYPE MAPPING
MITROPAM CHAKRABORTY AND S. K. SAMANTA
Abstract.
This paper deals with an extension of a recent result by the au-thors generalizing Kannan’s fixed point theorem based on a theorem of Vit-torino Pata. The generalization takes place via a cyclical condition. Introduction
Somewhat in parallel with the renowned Banach contraction principle (see, forinstance, [3]), Kannan’s fixed point theorem has carved out a niche for itself in fixedpoint theory since its inception in 1969 [4]. Let ( X, d ) be a metric space. If wedefine T : X → X to be a Kannan mapping provided there exists some λ ∈ (0 , such that(1.1) d ( T x, T y ) ≤ λ d ( x, T x ) + d ( y, T y )] for each x, y ∈ X , then Kannan’s theorem essentially states that:Every Kannan mapping in a complete metric space has a unique fixed point.To see that the two results are independent of each other, one can turn to [10],e.g., and Subrahmanyam has shown in [13] that Kannan’s theorem characterizesmetric completeness, i.e.: if every Kannan mapping on a metric has a fixed point,then that space must necessarily be complete.Kirk et al. [6] introduced the so-called cyclical contractive conditions to general-ize Banach’s fixed point theorem and some other fundamental results in fixed pointtheory. Further works in this aspect, viz. the cyclic representation of a completemetric space with respect to a map, have been carried out in [12, 7, 5]. Pata in[8], however, extended Banach’s result in a totally different direction and ended upproving that if ( X, d ) is a complete metric space and T : X → X a map such thatthere exist fixed constants Λ ≥ , α ≥ , and β ∈ [0 , α ] with(1.2) d ( T x, T y ) ≤ (1 − ε ) d ( x, y ) + Λ ε α ψ ( ε )[1 + k x k + k y k ] β for every ε ∈ [0 , and every x, y ∈ X (where ψ : [0 , → [0 , ∞ ] is an increasingfunction that vanishes with continuity at zero and k x k : = d ( x, x ) , ∀ x ∈ X , withan arbitrarily chosen x ∈ X ), then T has a unique fixed point in X . CombiningPata’s theorem and the cyclical framework, Alghamdi et al. have next come upwith a theorem of their own [1].On the one hand, proofs of cyclic versions of Kannan’s theorem were given in[12] and [9]; the present authors, on the other hand, have already established an Mathematics Subject Classification.
Primary 47H10; Secondary 47H09.
Key words and phrases.
Kannan maps, fixed points, cyclical conditions.The first author is indebted to the
UGC (University Grants Commissions), India for awardinghim a
JRF (Junior Research Fellowship) during the tenure in which this paper was written.
1N A FIXED POINT THEOREM FOR A CYCLICAL KANNAN-TYPE MAPPING 2 analogue of Pata’s result that generalizes Kannan’s theorem instead [2]. Lettingeverything else denote the same as in [8] except for fixing a slightly more general β ≥ , we have actually shown the following: Theorem 1. [2]
If the inequality (1.3) d ( T x, T y ) ≤ − ε d ( x, T x ) + d ( y, T y ] + Λ ε α ψ ( ε )[1 + k x k + k T x k + k y k + k T y k ] β is satisfied ∀ ε ∈ [0 , and ∀ x, y ∈ X , then T possesses a unique fixed point x ∗ = T x ∗ ( x ∗ ∈ X ) . In this article, we want to utilize theorem 1 to bridge the gap by providingthe only remaining missing link, i.e. a fixed point theorem for cyclical contractivemappings in the sense of both Kannan and Pata.2.
The Main Result
Let us start by recalling a definition which has its roots in [6]; we shall make useof a succinct version of this as furnished in [5]:
Definition 2. [5] Let X be a non-empty set, m ∈ N , and T : X → X a map.Then we say that S mi =1 A i (where ∅ 6 = A i ⊂ X ∀ i ∈ { , , . . . , m } ) is a cyclicrepresentation of X with respect to T iff the following two conditions hold.(1) X = S mi =1 A i ;(2) T ( A i ) ⊂ A i +1 for ≤ i ≤ m − , and T ( A m ) ⊂ A .Now, let ( X, d ) be a complete metric space. We have to first assign ψ : [0 , → [0 , ∞ ] to be an increasing function that vanishes with continuity at zero. Withthis, we are ready to formulate our main result, viz.: Theorem 3.
Let Λ ≥ , α ≥ , and β ≥ be fixed constants. If A , . . . , A m arenon-empty closed subsets of X with Y = S mi =1 A i , and if T : Y → Y is such a mapthat S mi =1 A i is a cyclic representation of Y with respect to T , then, provided theinequality (2.1) d ( T x, T y ) ≤ − ε d ( x, T x ) + d ( y, T y ) + Λ ε α ψ ( ε )[1 + k x k + k T x k + k y k + k T y k ] β is satisfied ∀ ε ∈ [0 , and ∀ x ∈ A i , y ∈ A i +1 (where A m +1 = A and, as in [8] , k x k : = d ( x, x ) , ∀ x ∈ Y , for an arbitrarily chosen x ∈ Y —a sort of “zero” of thespace Y ), T has a unique fixed point x ∗ ∈ T mi =1 A i .Remark . Since we can always redefine Λ to keep (2.1) valid no matter what initial x ∈ X we choose, we are in no way restricting ourselves by choosing that x asour “zero” instead of a generic x ∈ X [8]. Proofs.
For the sake of brevity and clarity both, we shall henceforth exploit thefollowing notation when j > m : A j : = A i ,where i ≡ j (mod m ) and ≤ i ≤ m .Let’s begin by choosing our zero from A , i.e., we fix x ∈ A . Starting from x ,we then introduce the sequence of Picard iterates x n = T x n − = T n − x ( n ≥ . N A FIXED POINT THEOREM FOR A CYCLICAL KANNAN-TYPE MAPPING 3
Also, let c n : = k x n k ( n ∈ N ) . With the assumption that x n = x n +1 ∀ n ∈ N , (2.1) gives us d ( x n +1 , x n ) = d ( T x n , T x n − ) ≤
12 [ d ( x n +1 , x n ) + d ( x n , x n − )] if we consider the case where ε = 0 . But this means that ≤ d ( x n +1 , x n ) ≤ d ( x n , x n − ) ≤ · · ·≤ d ( x , x ) (2.2) = c , whence the next result, i.e. our first lemma, is delivered: Lemma 5. { c n } is bounded.Proof. Let n ∈ N . We assume that n ≡ k (mod m ) (1 ≤ k ≤ m ) . Since x k − ∈ A k − and x k − ∈ A k − , using (2.1) with ε = 0 , c n = d ( x n , x )= [ d ( x , x ) + d ( x , x ) + · · · + d ( x k − , x k − )] + d ( x k − , x n ) ≤ ( k − c + d ( T x k − , T x k − ) ≤ ( k − c + 12 [ d ( x k − , x k − ) + d ( x n , x n − )] ≤ ( k − c + 12 ( c + c )= ( k − c . And hence we have our proof. (cid:3)
Remark . One finds in [1] an attempt to prove the boundedness of an analogoussequence c n (the notations in play there and in the present article are virtually thesame) using the cyclic contractive condition from its main theorem (vide inequality(2.1) from theorem 2.4 in [1]) on two points x ( ∈ A ) and x n ( ∈ A n ) . But thisinequality as well as our own (2.1) can only be applied to points that are members of consecutive sets A i and A i +1 for some i ∈ { , . . . , m } according to their respectiveapplicative restrictions, both of which stem from the very definition of cyclicalconditions given in [6]. x n being the general n -th term of the sequence { x n } is ina general set A n , and, following the notational convention agreed upon in both [1]and this article, A n = A l , where l ≡ n (mod m ) and ≤ l ≤ m . Since the index l need not either be succeeding or be preceding the index in general, x ( ∈ A ) and x n ( ∈ A l ) need not necessarily be members of consecutive sets as well. Hence thejustifiability of using the cyclic criterion on them is lost, and suitable adjustmentshave to be made in the structure of the proving argument. This is precisely whatwe have endeavoured to do in our proof above.To return to our central domain of discourse, next we need another: N A FIXED POINT THEOREM FOR A CYCLICAL KANNAN-TYPE MAPPING 4
Lemma 7. lim n →∞ d ( x n +1 , x n ) = 0 .Proof. (2.2) assures that we end up with a sequence, viz. { d ( x n +1 , x n ) } , that isboth monotonically decreasing and bounded below, and, therefore, lim n →∞ d ( x n +1 , x n ) = inf n ∈ N d ( x n +1 , x n ) = r (say) ≥ . But, from (2.1), r ≤ d ( x n +1 , x n )= d ( T x n , T x n − ) ≤ − ε d ( x n +1 , x n ) + d ( x n , x n − )] + Λ ε α ψ ( ε )(1 + c n +1 + 2 c n + c n − ) β ≤ − ε d ( x n +1 , x n ) + d ( x n , x n − )] + Kεψ ( ε ) for some K ≥ . (By virtue of lemma 5, it is ensured that K does not depend on n .) Letting n → ∞ , r ≤ − ε r + r ) + Kεψ ( ε )= ⇒ r ≤ Kψ ( ε ) ∀ ε ∈ (0 , ⇒ r = 0 . Therefore, lim n →∞ d ( x n +1 , x n ) = 0 . (cid:3) With this, we are now in a position to derive:
Lemma 8. { x n } is a Cauchy sequence.Proof. This proof has the same generic character as the one given in [6]. Wesuppose, first, that ∃ ρ > such that, given any N ∈ N , ∃ n > p ≥ N with n − p ≡ m ) and d ( x n , x p ) ≥ ρ > . Clearly, x n − and x p − lie in different but consecutively labelled sets A i and A i +1 for some i ∈ { , . . . , m } . Then, from (2.1), ∀ ε ∈ [0 , , d ( x n , x p ) ≤ − ε d ( x n , x n − ) + d ( x p , x p − )] + Λ ε α ψ ( ε )(1 + c n + c n − + c p + c p − ) β ≤ − ε d ( x n , x n − ) + d ( x p , x p − )] + Cε α ψ ( ε ) , where, to be precise, C = sup j ∈ N Λ(1 + 4 c j ) β < ∞ (on account of lemma 5) thistime. If we let n, p → ∞ with n − p ≡ m ) , then lemma 7 gives us that < ρ ≤ d ( x n , x p ) → as ε → , which is, again, contrary to what we had supposed earlier.Therefore, we can safely state that, given ε > , ∃ N ∈ N such that(2.3) d ( x n , x p ) ≤ εm whenever n, p ≥ N and n − p ≡ m ) . N A FIXED POINT THEOREM FOR A CYCLICAL KANNAN-TYPE MAPPING 5
Again, by lemma 7 it is possible to choose M ∈ N so that d ( x n +1 , x n ) ≤ εm if n ≥ M . If we now let n, p ≥ max { N, M } with n > p , then ∃ r ∈ { , , . . . , m } such that n − p ≡ r (mod m ) . Thus n − p + i ≡ m ) , where i = m − r + 1 . And, bringing into play (2.3), d ( x n , x p ) ≤ d ( p , x n + i ) + [ d ( x n + i , x n + i − ) + · · · + d ( x n +1 , x n )] ≤ ε. This proves that { x n } is Cauchy. (cid:3) Now, looking at Y = S i A i , a complete metric space on its own, we can concludestraightaway that { x n } , a Cauchy sequence in it, converges to a point y ∈ Y .But { x n } has infinitely many terms in each A i , i ∈ { , . . . , m } , and each A i isa closed subset of Y . Therefore, y ∈ A i ∀ i = ⇒ y ∈ m \ i =1 A i = ⇒ m \ i =1 A i = ∅ . Moreover, T mi =1 A i is, just as well, a complete metric space per se . Thus, consideringthe restricted mapping U : = T ↾ T A i : \ A i → \ A i , we notice that it satisfies the criterion to be a Kannan-type generalized map alreadyproven by us to have a unique fixed point x ∗ ∈ T A i by virtue of theorem 1. (cid:3) Remark . We have to minutely peruse a certain nuance here for rigour’s sake:the moment we know that T A i = ∅ , we can choose an arbitrary y ∈ T A i toserve as its zero, and the restriction of T to T A i can still be made to satisfy (amodified form of) (2.1) insofar as Λ can be appropriately revised as per remark 4;this renders the employment of theorem 1 in the proof vindicated.3. Some Conclusions
Following the terminology of [11], we can actually show something more, viz.:
Corollary 10. T is a Picard operator, i.e. T has a unique fixed point x ∗ ∈ T mi =1 A i ,and the sequence of Picard iterates { T n x } n ∈ N converges to x ∗ irrespective of ourinitial choice of x ∈ Y . N A FIXED POINT THEOREM FOR A CYCLICAL KANNAN-TYPE MAPPING 6
Proof.
So far, we have already shown that for a fixed x ∈ A one and only one fixedpoint x ∗ of T exists. To complete the proof, let’s first observe that the decisionto let x ∈ A at the beginning of the main proof was based partly on mereconvention and partly on an intention to develop our argument thenceforth moreor less analogously to the proof given in [1]; if we would have chosen any generic x ∈ Y instead, then, seeing as how Y = S A i , that x would have belonged to A j for some j ∈ { , . . . , m } , and our discussion thereupon would have differed only insome labellings, not in its conclusion: i.e. we would have, eventually, inferred theexistence of a unique fixed point of T in T A i .Next we want to demonstrate that the convergence of the Picard iterates headsto a fixed point of T . To this end we recall that { T n x } n ∈ N = { x n +1 } n ∈ N convergesto y ∈ T A i . Our claim is that this y itself is a fixed point of T . This we can verifysummarily:As x n ∈ A k for some k ∈ { , . . . , m } and as y ∈ T mi =1 A i ⊂ A k +1 , d ( y, T y ) ≤ d ( y, x n +1 ) + d ( x n +1 , T y )= d ( y, x n +1 ) + d ( T x n , T y ) ≤ d ( y, x n +1 ) + 12 [ d ( x n +1 , x n ) + d ( y, T y )] for every n ∈ N , using (2.1) with ε = 0 again, and, from that, d ( y, T y ) ≤ d ( y, x n +1 ) + 12 d ( x n +1 , x n ) for all n . Letting n → ∞ , d ( y, T y ) = 0 since x n +1 → y and d ( x n +1 , x n ) → . Therefore, y = T y.
As observed, the choice of the starting point x is irrelevant, and we alreadyknow that x ∗ ∈ T A i is the unique fixed point of T . So obviously, x ∗ = y, i.e. T n x → x ∗ as n → ∞ .This completes the proof. (cid:3) Remark . Trying to show their map f (corresponding to the T in the presentarticle) is a Picard operator, the authors in [1] have set out to prove that x n → x ∗ as n → ∞ too. (Again, apart from denoting the operator differently, we haven’t reallychanged much of their notation so as to make a comparison fairly self-explanatory.)The argument they have used in this follows from a technique given in [8], andthey, too, have ended up stating, to quote a portion of the concerned reasoning in[1] verbatim, d ( x n , x ∗ ) = lim p →∞ d ( x n , x n + p ) . This is where a problem arises.In [8] the convergence of the Cauchy sequence x n is established directly to itslimit x ∗ , and, therefore, the utilization of an equality like d ( x ∗ , x n ) = lim m →∞ d ( x n + m , x n ) N A FIXED POINT THEOREM FOR A CYCLICAL KANNAN-TYPE MAPPING 7 (quoted as it is from [8] this time) is perfectly justified. The cyclical setting in boththis article and [1], however, only ensures initially that x n → y for some y ∈ T A i as n → ∞ , and, as a consequence, guarantees next the existence of a unique fixedpoint x ∗ for the operator. The fact that this y turns out be a fixed point for theoperator as well (thereby rendering it equal to the unique x ∗ ) is something thatneeds to be actually proved in a separate treatment, which we believe is a taskwe’ve accomplished. [1], though, overlooks this distinction and assumes the veryfact (viz. x n → x ∗ ) it wants to prove in the proof itself, committing the fallacy of petitio principii .As a final note, let us remind ourselves of the fact that (1.3) is weaker than (1.1)(see [2]), and, in light of this we also have, as a corollary to our theorem 3 thefollowing: Corollary 12. [12, 9]
Let { A i } pi =1 be nonempty closed subsets of a complete metricspace X . Suppose that T : S pi =1 A i → S pi =1 A i is a cyclic map, i.e. it satisfies T ( A i ) ⊂ A i +1 for every i ∈ { , . . . , p } (with A p +1 = A ), such that d ( T x, T y ) ≤ α d ( x, T x ) + d ( y, T y )] for all x ∈ A i , y ∈ A i +1 ( ≤ i ≤ p ), where α ∈ (0 , is a constant. Then T has aunique fixed point x ∗ in T pi =1 A i and is a Picard operator. References
1. M. Alghamdi, A. Petrusel, and N. Shahzad,
A fixed point theorem for cyclic generalizedcontractions in metric spaces , Fixed Point Theory Appl. (2012), no. 1, 122.2. M. Chakraborty and S. K. Samanta,
A fixed point theorem for kannan-type maps in metricspaces , pre-print (2012), arXiv:1211.7331v2 [math.GN], nov 2012.3. A. Granas and J. Dugundji,
Fixed point theory , Springer Monographs in Mathematics,Springer, 2003.4. R. Kannan,
Some results on fixed points–ii , Amer. Math. Monthly (1969), 405–408.5. E. Karapinar, Fixed point theory for cyclic weak ϕ -contraction , Appl. Math. Lett. (2011),no. 6, 822–825.6. W. A. Kirk, P. S. Srinivasan, and P. Veeramani, Fixed points for mappings satisfying cyclicalcontractive conditions , Fixed Point Theory (2003), no. 1, 79–89.7. M. Păcurar and I .A. Rus, Fixed point theory for cyclic ϕ -contractions , Nonlinear Anal. (2010), no. 3, 1181–1187.8. V. Pata, A fixed point theorem in metric spaces , J. Fixed Point Theory Appl. (2011),299–305 (English).9. M. Petric and B. Zlatanov, Fixed point theorems of kannan type for cyclical contractive con-ditions , REMIA 2010, University Press "Paisii Hilendarski", Plovdiv, 2010.10. B. E. Rhoades,
A comparison of various definitions of contractive mappings , Trans. Amer.Math. Soc. (1977), no. 1977, 5.11. I. A. Rus,
Picard operators and applications , Sci. Math. Jpn. (2003), no. 1, 191–219.12. , Cyclic representations and fixed points , Ann. Tiberiu Popoviciu Semin. Funct. Equ.Approx. Convexity (2005), 171–178.13. P. V. Subrahmanyam, Completeness and fixed-points , Monatsh. Math. (1975), no. 4, 325–330. Mitropam Chakraborty, Department of Mathematics, Visva-Bharati, Santiniketan731235, India
E-mail address : [email protected] S. K. Samanta, Department of Mathematics, Visva-Bharati, Santiniketan 731235,India
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