On contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points


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1 De la Sen et al. Fixed Point Theory and Applications ( :103 DOI /s R E S E A R C H Open Access On contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points Manuel De la Sen 1*, Antonio F Roldán 2 and Ravi P Agarwal 3,4 * Correspondence: 1 Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia, Barrio Sarriena, P.O. Box 644, Bilbao, 48940, Spain Full list of author information is available at the end of the article Abstract This paper investigates some properties of cyclic fuzzy maps in metric spaces. The convergence of distances as well as that of sequences being generated as iterates defined by a class of contractive cyclic fuzzy mapping to fuzzy best proximity points of (nonnecessarily intersecting adjacent subsets of the cyclic disposal is studied. An extension is given for the case when the images of the points of a class of contractive cyclic fuzzy mappings restricted to a particular subset of the cyclic disposal are allowed to lie either in the same subset or in its next adjacent one. Keywords: fixed and best proximity points; fuzzy cyclic contractive maps; cyclic fuzzy contraction and fuzzy sets 1 Introduction Fixed point theory has received much attention in the last decades with a rapidly increasing number of related theorems on nonexpansive and on contractive mappings, variational inequalities, optimization etc. Variants and extensions have also been considered in the frameworks of probabilistic metric spaces and multivalued mappings. The wide variety of problems of cyclic and proximal contractions has also received much attention in the last years. Fixed point theory has also proved to be an important tool for the study of relevant problems in science and engineering concerning local and global stability, asymptotic stability, stabilization, convergence of trajectories and sequences to equilibrium points [1 3], dynamics switching in dynamic systems and in differential/difference equations, etc. In particular, the equilibrium problems associated with switching processes in dynamic systems [1 4] are sometimes related to cyclic mappings where the cyclic disposal contains either trajectory solutions or sequences which commute to adjacent subsets being appropriately defined. Cyclic mappings, related contractive properties, and best proximity points have been widely studied in metric spaces in both the deterministic [5 10]and the probabilistic frameworks, see, for instance, [11 14], as well the related proximaltype problems, see, for instance, [15, 16] and references therein. The concept of fuzzy contractions was proposed in [17] and has been revisited later on in [18 21] including the study the common fixed points and compatibility problems in fuzzy metric spaces, [22 28]. In particular the paper by Azam and Beg [22] is very 2015 De la Sen et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s and the source, provide a link to the Creative Commons license, and indicate if changes were made.
2 Dela Senet al. Fixed Point Theory and Applications ( :103 Page 2 of 22 illustrative and well worked out on a set of proved properties associated with a very general fuzzy contractive condition. On the other hand, a fuzzy theory related to the more general intuitionistic fuzzy metric spaces is also being developed. See, for instance, [29, 30] and references therein. See also the recent references [31, 32]. The main objective of this paper is the study of contractive fuzzy cyclic multivalued mappings and the convergence properties of sequences constructed via such maps to fuzzy bestproximitypointsandfuzzyfixedpoints.itisassumedthatthesubsetsdefiningthe cyclic disposal do not necessarily intersect. Some extended results are given for the more general case when the cyclic map can have an image within its domain instead of on its adjacent subset of the cyclic disposal. 1.1 Notation n = {1,2,...,n}; Z 0+, Z +,andz are the sets of nonnegative integers, positive integers, and integers, respectively; R 0+, R +,andr are the sets of nonnegative real numbers, positive real numbers, and real numbers, respectively. 2 Somepreliminaries Let (X, d beametricspace,withx nonempty, and let CB(X be the set of all nonempty closed bounded subsets of X.ForA, B CB(X, let the PompeiuHausdorff metric (a more appropriate name because of historical reasons than the simpler name Hausdorff metric  see the discussion in [33] H on CB(Xbegivenby ( H(A, B=max sup a A d(a, B, sup d(b, A b B for any A, B CB(X, where d(x, A=inf y A d(x, y. A fuzzy set A in X is a function from X to [0, 1] whose grade of membership of x in A is the functionvalue A(x. The αlevel set of A is denoted by [A] α defined by [A] α = { x X : A(x α } X if α (0, 1], [A] 0 = { x X : A(x>0 } X, where B denotes the closure of B. LetF(X be the collection of all fuzzy sets in a metric space (X, d. For A, B F(X, A B means A(x B(x, x X.Notealsothatifα [β,1] and β (0, 1], then [A] α [A] β.ifthereexistsα [0, 1] such that [A] α,[b] α CB(X, then define p α (A, B= inf d(x, y, D αβ (A, B=H ( [A] α,[b] β, x [A] α,y [B] α D α (A, B=D αα (A, B=H ( [A] α,[b] α for α, β [0, 1]. Note that D αβ (A, B D βα (A, B, in general, if A B, whiled αβ (A, B = D βα (B, A, and max(d αβ (A, B, D βα (A, B D α (A, B ifα [β,1] and β [0, 1]. If [A] α,
3 Dela Senet al. Fixed Point Theory and Applications ( :103 Page 3 of 22 [B] α CB(X, α [0, 1], then p(a, B = sup p α (A, B, d (A, B= sup D αβ (A, B, α [0,1] α,β [0,1] d (A, B= sup D α (A, B, α [0,1] so that d (A, B= sup α,β [0,1] H ( [A] α,[b] β d (A, B= sup H ( [A] α,[b] α. α [0,1] The notation p(x, A meansp({x}, A andafuzzyseta in a metric linear space V is said to be an approximate quantity if [A] α is compact and convex in V for each α [0, 1] and sup x V A(x=1. The collection of all the approximate quantities in a metric linear space X is denoted by W(X. T : X F(Y is a fuzzy mapping from an arbitrary set X to F(Y, which is a fuzzy subset in X Y, and the grade of membership of y in T(xisT(x(y. The notation f : X Y Z means that the domain of the function f from X to Z is restricted to the subset Y of X. The next definition characterizes pcyclic fuzzy mapping in an appropriate way to establish some results of this paper. Definition 1 Let p( 2 Z + and let X i be nonempty subsets of a nonempty abstract set X, i p,andletd be a metric on X. A mapping T : i p X i F( i p X iissaidtobe a pcyclic fuzzy mapping from i p X i to F( i p X iif (a T : i p X i X j F(X j+1 ; j p. (b There is α(x (0, 1] such that the α(xlevel set [Tx] α(x CB(X i+1 ; x X i, i p. The above definition is understood under the cyclic constraint X p+1 = X 1, and in general with X np+i = X i for any i p and any n Z 0+,sothatX j X i if j = i(mod p. Now, given the sets X i, i p, theαlevel sets in F(X i are[a] α = {x X i : A(x α}, where A(x is the grade of membership of x X i in A F(X i for any i p. Notethat for any given i p, z i X i and α(z i (0, 1], [Tz i ] α(zi = {x X i+1 : Tz i (x α(z i } X i+1 is the α(z i level set of Tz i which consists of points in X i+1 of guaranteed membership grade not being less than α(z i. If T : i p X i F( i p X iisapcyclic fuzzy mapping, since [Tz i ] α(zi is nonempty, there exists z i+1 [Tz i ] α(zi X i+1 such that [Tz i+1 ] α(zi+1 = {x X i+2 : Tz i+1 (x α(z i+1 } X i+2 is also a nonempty set for some α(z i+1 (0, 1] which is contained in the set [T 2 z i ] α(zi [T(Tz i ] α(zi for some α(z i+1 (0, 1]. Then, by recursion, there is a sequence {z i+j } i p X i satisfying z i+p [Tz i+p 1 ] α(zi+p 1 [ T p 1 ] z i+1 α(z i+1 [ T p z i ]α(z i X i for some set of real values α(z j (0, 1]; j = i, i +1,...,i + p 1; i p. Thesubsequent definition characterizes αfuzzy best proximity points of some subset X i of pcyclic fuzzy mappings.
4 Dela Senet al. Fixed Point Theory and Applications ( :103 Page 4 of 22 Definition 2 Let T : i p X i F( i p X ibeap( 2 Z + cyclic fuzzy mapping. For any given i p, apointz i X i is said to be an α(z i fuzzy best proximity point of X i through T for some α(z i (0, 1] if d(z i,[tz i ] α =D iα(zi = inf x Xi d(x,[tx] α. Remark 3 Note that D i = d(x i, X i+1 D iα(x for any x X i, i p. Notealsothatif there is z i X i such that d(z i,[tz i ] α(zi =D i = d(x i, X i+1 forsomei p then there is z i+1 [Tz i ] α(zi X i+1 such that d(z i, z i+1 =D i,sincethend i = D iα(zi.thus,z i+1 X i+1 (then z i+1 X i+1 if X i+1 is closed is a best proximity point of X i+1 to X i.also,z i X i is abestproximitypointofx i to X i+1 and an α(z i best which is also an α(z i fuzzy best proximity point of X i through T. The next result relies on the fact that αfuzzy best proximity points through cyclic selfmappings of adjacent subsets of the cyclic disposal which are not best proximity points can have a mutual distance exceeding the αfuzzy best proximity distance D iα. The subsequent result is concerned with the existence of a set of coincident best proximity points and αfuzzy best proximity points each one at each of the subsets of the cyclic disposal of the fuzzy cyclic mapping. Proposition 4 The following properties hold: (i If T : i p X i F( i p X i is a p( 2cyclic fuzzy mapping such that X i and X i+1 are closed with the constraints ( ( sup d x,[tzi ] α : x X i+2 < Di = d(x i, X i+1 D i+1 = d(x i+1, X i+2 α (0,1] for some given i p andz i X i is an α(z i fuzzy best proximity point of X i through T for some α(z i (0, 1], which is also a best proximity point of X i to X i+1 for the given i p. Then d(z i, z i+1 =d ( ([ z i,[tz i ] α(zi = d T 2 ] z i α(z i,[tz i] α(zi = Di for some z i+1 [Tz i ] α(zi X i+1 which is an α(z i+1 fuzzy fixed point of X i+1 through T and abestproximitypointofx i+1 to X i+2. (ii Assume that T : i p X i F( i p X i is a p( 2cyclic fuzzy mapping with X i being closed, D i = D, i p and there is some z i X i which is an α(z i fuzzy best proximity point of X i through T for some α(z i (0, 1], which is also a best proximity point of X i to X i+1 for some given i p. Then there exists a sequence of points: z i+j+np [Tz i+j+np 1 ] α(zi+j+np 1 [ T 2 ] z i+j+np 2 α(z i+j+np 2 [ T j+np z i ]α(z i X i+j, j p, n Z 0+, which satisfy the distance constraints: d(z i+j+np 1, z i+j+np =d ( z i+j+np 1,[Tz i+j+np 1 ] α(zi+j+np 1 = d ([ T j+np+k+1 ] z i α(z i+k+1, [ T j+np+k z i ]α(z i+k = D; j p, n, k Z0+ which are best proximity points of X i+j to X i+j+1 and α(z i+j+np fuzzy best proximity points through T for some real values α(z i+j+np (0, 1], j p, n, k Z 0+.
5 Dela Senet al. Fixed Point Theory and Applications ( :103 Page 5 of 22 Proof It follows that d(z i,[tz i ] α =D i = D iα(zi since z i X i is an α(z i fuzzy best proximity point of X i through T which is also a best proximity point of X i to X i+1 since [Tz i ] α(zi X i+1 is closed. Note that, for α(z i+1 α(z i, one gets [Tz i+1 ] α(zi+1 = { z X i+2 : Tz i+1 (z α(z i+1 } [ T 2 z i ]α(z i = { z X i+2 : T 2 z i (z α(z i }. Note also that d([t 2 z i ] α(zi,[tz i ] α(zi d(x i+1, X i+2 D i+1 and then proceed by contradiction by assuming that the above inequality is strict. Therefore, and since [Tz i ] α(zi is closed, there is z i+1 [Tz i ] α(zi X i+1 such that [Tz i+1 ] α(zi+1 [T 2 z i ] α(zi X i+2 for some α(z i+1 (0, 1] so that the following contradiction to the hypothesis D i D i+1 for the given i p is obtained since D i > sup α (0,1] (d(x,[tz i ] α :x X i+2 : D iα(zi = D i = d(z i, z i+1 =d ( z i,[tz i ] α(zi > d ( ( [Tz i ] α(zi,[tz i+1 ] α(zi+1 d [Tzi ] α(zi, [ T 2 ] z i α(z i Di+1, and then D i > D i+1, which is impossible. Thus, d([t 2 z i ] α(zi,[tz i ] α(zi =D i+1 D i = d(z i, z i+1. Since [Tz i+1 ] α(zi+1,[t 2 z i ] α(zi,[tz i ] α(zi and X i+1 are closed, if D i = D i+1, z i+1 [Tz i ] α(zi X i+1 isabestproximitypointofx i+1 to X i+2 and an α(z i+1 fuzzy best proximity point of X i+1 through T since d ( ([ z i+1,[tz i+1 ] α(zi+1 = d T 2 ] z i α(z i,[tz i] α(zi = d(zi+1, z i+2 =D i+1 for some z i+2 [Tz i+1 ] α(zi+1 [T 2 z i ] α(zi X i+2,since[tz i+1 ] α(zi+1 is closed. Property (ii follows by using the recursion z i+j+np [Tz i+j+np 1 ] α(zi+j+np 1 [T j+np z i ] α(zi X i+j, j p, n Z 0+,andD i = D, i p. Note that if for some i p, z i X i is a best proximity point of X i to X i+1 and ( α (0,1] [Tz i] α, then there exists z i+1 [Tz i ] α(zi X i+1 which is an α = α(z i (0, 1] fuzzy fixed point of X i+1 through T which is also a best proximity point of X i+1 to X i if D i = D iα(zi = d(x i, X i+1 sothat D i = d ( z i,[tz i ] α(zi = d(zi, z i+1. If ( α (0,1] [Tz i] α = and [Tz i ] 0 ( clx i+1 and D i = D i0,thend i = d(z i,[tz i ] 0 = d(z i, z i+1 forsomez i+1 [Tz i ] 0 X i+1 which is a 0fuzzy fixed point of X i+1 through T and also a best proximity point of X i+1 to X i. Definition 5 Let p( 2 Z + and let X i be nonempty subsets of a nonempty abstract set X, i p,andletd be a metric on X. A composite mapping T(= gf : i p X i F( i p X iis said to be a decomposable pcyclic fuzzy mapping from i p X i to F( i p X iif: (a f : i p X i X j X j+1 ; j p, (b g : i p X i X j F(X j ; j p, (c there is α(x (0, 1] such that the α(xlevel set [Tx] α(x CB(X i+1 ; x X i, i p.
6 Dela Senet al. Fixed Point Theory and Applications ( :103 Page 6 of 22 The above definition is an extended definition of Definition 1.Notealsothatf (X i X i+1, i p, andthat,sincet = g f with f : i p X i X j X j+1, j p and g : i p X i X j+1 F(X j+1, j p, thecyclicmapst : i p X i F( i p X i are restricted to subsets of the cyclic disposal which are of the form T : i p X i X j F(X j+1, j p. On the other hand, notice that the formalism of contractive cyclic selfmaps in uniformly convex Banach spaces requires for the distances inbetween any pairs of nonadjacent subsets to be identical. Thus, a fixed point, if any, is a confluent point of pairs of best proximity points inbetween all the intersecting adjacent subsets of the cyclic disposal. On the other hand, note that the case f : i p X i X j X j+1 and g : i p X i X j+1 F(X j+1, j p,isaparticular case of the more general above problem statement for the cyclic disposal. Now, given the sets X i, i p,wecandefinetheαlevel sets in F(X i by[a] α = {x X i : A(x α},and A(x is the grade of membership of x X i in A F(X i for any i p. The following assumption establishes that if T(= gf : i p X i F( i p X iisadecomposable pcyclic fuzzy mapping, then the 1level set of Tx intersects with fx for any x i p X i. Equivalently, there is at least a point y fx such that the grade of membership of y in Tx is unity, i.e. (Txy = 1, for any x i p X i.notethatfx X i+1 and [Tx] 1 X i+1, x X i, i p. Assumption 6 If T(= gf : i p X i F( i p X iisadecomposablepcyclic fuzzy mapping, then fx [Tx] 1, x i p X i. In other words, Assumption 6 says that any image of x X i in X i+1 through f belongs to the 1level set [Tx] 1. NotethatAssumption6 implies trivially that [Tx] 1, x i p X i. Assumption 6 is relevant for identifying best proximity points inbetween adjacent subsets through f which are also 1fuzzy best proximity points through T. Assertion 7 Assume that T : i p X i F( i p X iisadecomposablepcyclic fuzzy mapping which satisfies Assumption 6 and that x X i for some given arbitrary i p,suchthat X i is closed. Then, if x isabestproximitypointofx i to X i+1 through f (in the sense that there exists y fx such that d(x, y=d(x, X i then it is an 1fuzzy best proximity point of X i through T. Proof Since x X i is a best proximity point of X i to X i+1 through f and fx [Tx] 1, x i p X i, then there exists y fx such that D i1 = d ( x,[tx] 1 d(x, y=d(x, fx=d(x, Xi =D i D i1 D i = D i1. Then x is an 1fuzzy best proximity point of X i through T. 3 Some results on αfuzzy best proximity points The following result holds related to αfuzzy best proximity points of a pcyclic fuzzy mapping which is contractive in the sense that it is subject to a contractive condition. Theorem 8 Let p( 2 Z + and let X i be nonempty subsets of a nonempty abstract set X, i p, where (X, d is a complete metric space. Let T : i p X i F( i p X i be a pcyclic fuzzy mapping with X i being closed, i p. Assume that the following contractive condition
7 Dela Senet al. Fixed Point Theory and Applications ( :103 Page 7 of 22 holds: D α(xα(y ( [Tx], [Ty] = H ( [Tx]α(x,[Ty] α(y a 1 d ( x,[tx] α(x + a2 d ( y,[ty] α(y + a3 d(x, y+(1 a ; (1 x X i, y X i+1, i p, and some R + where a i 0 for i = 1,2,3 and 0<a = 3 i=1 a i <1. Then the following properties are fulfilled: (i Assume that the following constraints hold: 2(a 1 + a 2 +a 3 <1, = (1 a 2 a 4 (1 ρ 0, 1 a ρ 0 = 1 [ (( a1 + a 3 a1 + a 2 3 a 1/2 ] < a 1 a 2 1 a 1 a 2 1 a 1 a 2 (1 a Then lim sup n d(x n, x n+1 (1 a 1 a 2 (1 ρ 0, where x np+j [Tx np ] α(xnp X i+j 1 ; j 4 for any given initial points x 1 X i and x 2 X i+1 for any arbitrary i p. (ii For any arbitrarily given i p, asequence{x n } i p X i can be built for any given x = x 1 X i, y = x 2 [Tx 1 ] α(x1 X i+1 fulfilling x np+j [Tx np+j 1 ] α(xnp+j 1 X i+j 1 ; j p, n Z 0+, which fulfills: d(x np+j+2, x np+j+1 a 1 + a 3 d(x np+j+1, x np+j + 1 a, 1 a 2 1 a 2 j p 1 {0}, n Z 0+, (2 lim sup d(x (n+1p+j+1, x (n+1p+j, j p 1 {0}, (3 n lim sup N N n=n 0 d(x (n+1p+j+1, x (n+1p+j (M +1 <, j p 1 {0}, (4 if N N 0 M <. (iii If the constraint (1 holds with = D = min i p D i then the limit below exists: lim d(x (n+1p+j+1, x (n+1p+j =D, j p 1 {0}, (5 n so that D = D i, i p, for any sequences constructed as in Property (i. As a result, any sequence {x n } i p X i fulfilling x np+j [Tx np+j 1 ] α(xnp+j 1 X i+j 1 ; j p, n Z 0+, converges to a limit cycle consisting of a set of p best proximity points x i of the sets X i, i p, which are also α(x i 1 fuzzy best proximity points of X through T = g f : i p X i F( i p X i for some set of values α(x i 1 (0, 1] and best proximity points of the pcyclic selfmapping f : i p X i i p X i. (iv If D =0and the constraint holds with = D = min i p D i, then any sequence {x n } i p X i fulfilling x np+j [Tx np+j 1 ] α(xnp+j 1 X i+j 1 ; j p, n Z 0+, converges to x [Tx ] α(x i p X i which is a unique α = α(x fuzzy fixed point of T : i p X i F( i p X i.
8 Dela Senet al. Fixed Point Theory and Applications ( :103 Page 8 of 22 Proof Take x 1 = x X i, x 2 X i+1, for any given arbitrary i p, thereexistx 3 [Tx 1 ] α(x1, x 4 [Tx 2 ] α(x2 such that one gets from (1 d(x 3, x 4 H ( [Tx 1 ] α(x1,[tx 2 ] α(x2 a 1 d ( x 1,[Tx 1 ] α(x1 + a2 d ( x 2,[Tx 2 ] α(x2 + a3 d(x 1, x 2 +(1 a a 1 d(x 1, x 3 +a 2 d(x 2, x 4 +a 3 d(x 1, x 2 +(1 a a 1 d(x 1, x 2 +a 1 d(x 2, x 4 +a 1 d(x 4, x 3 + a 2 d(x 2, x 3 +a 2 d(x 3, x 4 +a 3 d(x 1, x 2 +(1 a =(a 1 + a 3 d(x 1, x 2 +a 2 d(x 2, x 3 +(1 a, so that d(x 3, x a 1 a 2 [ (a1 + a 3 d(x 1, x 2 +a 2 d(x 2, x 3 +(1 a ]. Then one gets by proceeding recursively for n Z + d(x np+3, x np+4 n Z 0+, 1 1 a 1 a 2 [ (a1 + a 3 d(x np+1, x np+2 +a 2 d(x np+2, x np+3 +(1 a ] ; where x np+j [Tx np ] α(xnp X i+j 1 ; j 4 for the given initial arbitrary i p.thus, [ ] [ d(x np+2, x np d(x np+3, x np+4 a 2 1 a 1 a 2 a 1 +a 3 1 a 1 a 2 ][ ] [ d(x np+1, x np+2 + d(x np+2, x np a 1 a 1 a 2 ] ; n Z +. The matrix A 0 = [ 0 1 ] a 2 a 1 +a 3 1 a 1 a 2 1 a 1 a is convergent, and then A0 n 0asn if and only 2 if its eigenvalues z 1,2 are inside the complex open unit circle, that is, if and only if z 1,2 = 1 2 [ (( a1 + a 3 a1 + a 2 3 ± +4 1 a 1 a 2 1 a 1 a 2 a 2 1 a 1 a 2 1/2 ] ( 1, 1, that is, if and only if 2(a 1 + a 2 +a 3 < 1, while the convergence abscissa of A 0 is ρ 0 = z 2 = 1 2 [ (( a1 + a 3 a1 + a a 1 a 2 1 a 1 a 2 a 2 1 a 1 a 2 1/2 ] <1. (1 a Then lim n d(x np+3, x np+4 = (1 a 1 a 2 (1 ρ 0. Since the initial conditions x 1 X i and x 2 X i+1 of the sequence are arbitrary, it follows that lim sup n d(x np+3, x np+4 (1 a (1 a 1 a 2 (1 ρ 0. On the other hand, since such conditions are taken for any arbitrary i p, (1 a lim sup n d(x n, x n+1 (1 a 1 a 2 (1 ρ 0.Property(ihasbeenproved.Ontheotherhand, since for all x X there exists α(x (0, 1] such that [Tx] α(x CB(X and since any x X is in some X i for some i p,wehave[tx] α(x CB(X i+1 forsomei p and some α(x (0, 1] for each given x X i and any arbitrary i p.asaresult,[tx] α(x and [Ty] α(y are nonempty, closed, and bounded for some existing real values α(x (0, 1] and α(y (0, 1]; x X i,
9 Dela Senet al. Fixed Point Theory and Applications ( :103 Page 9 of 22 y X i+1, i p.since[tx] α(x [Tx] β(x if 0 < α(x β(xand[tx] α(x,[tx] β(x CB(X i+1, α(x (0, 1] can be nonunique with values in (0, γ i ] (0, 1] for some γ i R + and x X i for any arbitrary i p. Iftheimageofthecorrespondencec : x α(x ( (0, 1] for some x X is such that [Tx] α(x CB(X is nonunique, then it is possible to choose a mapping m : x α(x( (0, 1] from Zermelo s axiom of choice to fix α(x (0, 1] such that [Tx] α(x CB(X. Construct a sequence {x n } i p X i as follows. x 1 = x X i for some arbitrary i p and some existing x 2 [Tx 1 ] α(x1 since [Tx 1 ] α(x1 is nonempty, closed, and bounded for some existing α(x 1 (0, 1]. Since the contractive condition (1holdsforany x X i, y X i+1, i p, thenforx 1 X i it follows that there is x 02 [Tx 1 ] α(x1 X i+1 such that d(x 1, x 02 =d(x 1,[Tx 1 ] α(x1 d(x 1, x 2 ; x 2 [Tx 1 ] α(x1 X i+1,since[tx 1 ] α(x1 is closed, thus x 02 is an α(x 1 fuzzy best proximity points of X i through T and, since d(x 2,[Tx 2 ] α(x2 H([Tx 1 ] α(x1,[tx 2 ] α(x2 for any x 2 [Tx 1 ] α(x1 X i+1,onegets d ( x 2,[Tx 2 ] α(x2 a 1 d ( x 1,[Tx 1 ] α(x1 + a2 d ( x 2,[Tx 2 ] α(x2 + a3 d(x 1, x 2 +(1 a a 1 d(x 1, x 02 +a 2 d ( x 2,[Tx 2 ] α(x2 + a3 d(x 1, x 2 +(1 a a 1 d(x 1, x 2 +a 2 d ( x 2,[Tx 2 ] α(x2 + a3 d(x 1, x 2 +(1 a a 1 d(x 1, x 2 +a 2 d ( x 2,[Tx 2 ] α(x2 + a3 d(x 1, x 2 +(1 a. (6 Then (1 a 2 d ( x 2,[Tx 2 ] α(x2 (a1 + a 3 d(x 1, x 2 +(1 a and, since [Tx 2 ] α(x2 is nonempty and closed, there is some x 3 [Tx 2 ] α(x2 X i+1 such that d(x 2, x 3 d ( a 1 + a 3 x 2,[Tx 2 ] α(x2 d(x 1, x a ; (7 1 a 2 1 a 2 and since [Tx 2 ] α(x2 is nonempty and closed, at least one x 3 may be chosen for which the first inequality of (7 becomes an equality and x 2 is an α(x 2 fuzzy best proximity points of X i+1 through T. Proceeding recursively, one sees for the arbitrary given i p, j p, n Z 0+,thatasequencein i p X i can be built fulfilling (6 for any arbitrary i p such that x 1 = x X i is chosen arbitrarily, x 2 [Tx 1 ] α(x1 X i+1,...,x np+j [Tx np+j 1 ] α(xnp+j 1 X i+j 1, x np+j+1 [Tx np+j ] α(xnp+j X i+j, x np+j+2 [Tx np+j+1 ] α(xnp+j+1 X i+j+1,...; j p, n Z 0+,and,furthermore,x n of α(x n fuzzy best proximity points of X through T for n Z +. Since, from the given hypotheses, 0 K = a 1+a 3 1 a 2 <1and0<1 K = 1 a 1 a 2 <1,sothat(1 KD = 1 a 1 a 2 if = D,onegetsfrom(7forj = p d(x (n+1p+j+2, x (n+1p+j+1 Kd(x (n+1p+j+1, x (n+1p+j +(1 K K np+j d(x 2, x 1 + ( 1 K np+j, n Z 0+, j p 1 {0}. (8
10 Dela Senet al.fixed Point Theory and Applications ( :103 Page 10 of 22 Thus, one sees from (8that(3 holds and, furthermore, N n=n 0 d(x (n+1p+j+2, x (n+1p+j+1 (N N 0 +1 K (j+n 0p 1 K (N N 0+1p 1 K p ( d(x2, x 1 ; (9 j p 1 {0},whichleadsto(4if N N 0 M <. Property (ii has been proved. On the other hand, if (1holdswith = D = min i p D i,thenonegetsfrom(3 D D j+i lim sup d(x (n+1p+j+1, x (n+1p+j D, j p 1 {0} (10 n (since D i < D = min i p D i is impossible for any i p so that the limit lim n d(x (n+1p+j+1, x (n+1p+j =D, j p 1 {0} exists so that D = D i, i p, and the sequence converges to a limit cycle consisting of a set of p best proximity points x i of the sets X i, i p, whichare also α(x i 1 fuzzy best proximity points of X through T for some set of values α(x i 1 (0, 1]. Since d(x i, X i+1 =D, i p, the elements of the limiting set {x i : i p} are also best proximity points of the corresponding sets X i, i p. On the other hand, if D =0, and since lim n d(x (n+1p+j+1, x (n+1p+j =D, j p 1 {0}, from Property (ii, we have x i = x [Tx ] α(x, i p, and i p [Tx ] α(x i p X i,sothatx is an α(x fuzzy fixed point of T : i p X i F( i p X iforsomeα (0, 1]. We also know the property [Tx] β [Tx] α for all β( α, α( α (0, 1] and some α (0, 1]. Therefore, there is a real constant: α = α ( x ( = max α (0, 1] : ( [ Tx ] β α i p so that x [Tx ] α is an α fuzzy fixed point of T : i p X i F( i p X iwhichisthe unique limit point of the built sequence. Now, assume that there are two distinct fuzzy fixed points x [Tx] α and y [Ty] β. Then one gets from (1 with = D =0,ifx [Tx] α(x and y 1 = y ( x i p X i,and since the level sets are nonempty and closed, and one gets from (1, d(x,[tx] α(x = d(y,[ty] α(y = = D =0,sothat, d(x 2, y 2 =d ( ( y 2,[Tx 1 ] α = D [Tx1 ] α,[ty 1 ] α(y1 a3 d ( x, y < d ( x, y for some x 2 = x 2 (y 2 [Tx 1 ] α(x1 and all y 2 [Ty 1 ] α(y1 with α(x 1 =α(x =α and α(y 1 = α(y =β.again,onegetsfrom(1 forsomey 2 [Ty ] α(y,sincey 1 = y,[ty 1 ] α(y1 [Ty ] α(y is closed and x [Tx] α(x, d ( x (, y 2 = d x,[ty 1 ] α(y1 ( max sup y [Ty 1 ] α(y1 = H ([ Tx ] α(x, [ Ty ] α(y d ( y, [ Tx ] α(x, sup d ( x,[ty 1 ] α(y1 x [Tx ] α(x a 1 d ( x, [ Tx ] α(x + a2 d ( y, [ Ty ] α(y + a3 d ( x, y
11 Dela Senet al.fixed Point Theory and Applications ( :103 Page 11 of 22 = a 2 d ( y 1,[Ty 1 ] α(y1 + a3 d ( x, y 1 = a 3 d ( x (, y 1 < d x, y 1 (11 and a sequence {y n } can be built with initial value y 1 = y which satisfies d ( x (, y n+1 = d x,[ty n ] α(yn ( max sup d ( y, [ Tx ] α(x, sup d ( x,[ty n ] α(yn y [Ty n ] α(yn x [Tx ] α(x a 2 d ( y n,[ty n ] α(yn + a3 d ( x, y n a 2 d(y n, y n+1 +a 3 d ( x, y n, n Z+. (12 Then one gets from (12 d ( x, y n+1 a2 d(y n, y n+1 +a 3 d ( x ( (, y n a2 d yn, x + d ( x, y n+1 + a3 d ( x, y n (a 2 + a 3 d ( y n, x + a 2 d ( x, y n+1, n Z0+ (13 d ( x, y n+1 (1 a2 1 (a 2 + a 3 d ( y n, x, n Z 0+ (14 since a 2 <1.Ifa 1 a 2 then (1 a 2 1 (a 2 + a 3 < 1, equivalently, 2a 2 + a 3 <1,sincea = a 1 + a 2 + a 3 < 1 implies that 2a 2 + a 3 <1ifa 1 a 2. On the other hand, one also sees instead of (12and(14thatasequence{x n } can be built with initial value x 1 = x which satisfies d ( y (, x n+1 = d y (,[Tx n ] α(xn = d x,[ty n ] α(yn a 1 d(x n, x n+1 +a 3 d ( y, x n, n Z+, (15 d ( x, x n+1 (1 a1 1 (a 1 + a 3 d ( x n, x, n Z +, (16 since a 1 <1.Ifa 1 a 2 then (1 a 1 1 (a 1 + a 3 < 1, equivalently, 2a 1 + a 3 <1,sincea = a 1 + a 2 + a 3 <1.Sincea < 1, it follows that ρ = min( a 2+a 3 1 a 2, a 1+a 3 1 a 1 < 1. Thus, it follows from (14 thatd(x, y n 0asn if 2a 2 + a 3 < 1. Then a sequence with initial condition y converges to x. Also, one gets from (16 d(y, x n 0asn if 2a 1 + a 3 <1.Then a sequence with initial condition x converges to y.thenx and y cannot be distinct αfuzzy fixed points. Properties (iii(iv have been proved. Remark 9 The uniqueness of the limiting fixed point for some α (0, 1] of each built sequence proved in Theorem 8(iv, which is also an α fuzzy fixed point of T, follows from the existence of the maximum α such that x [Tx ] α. On the other hand, the conclusion of Theorem 8(iv that the distances inbetween adjacent subsets are identical is not very surprising, since it has been found to be a general property for nonexpansive mappings for pcyclic nonexpansive selfmappings in metric spaces (Lemma 3.3, [5]. Remark 10 Feasible values of in the contractive condition (1 might be, for instance, ( = d (x n : n Z + =sup d [Txj ] α(xj,[tx j+1 ] α(xj+1 j Z + ( sup d [Txj ] α(xj,[tx j+1 ] α(xj+1, j Z +
12 Dela Senet al.fixed Point Theory and Applications ( :103 Page 12 of 22 or replaced with the jth iterationdependent values j = d ([Tx j ] α(xj,[tx j+1 ] α(xj+1. From the last expression one also concludes that lim sup n d(x (n+1p+j+2, x (n+1p+j+1, j p 1 {0}. Both values depend only on the sequences being constructed rather than on the metrics of the sets X i, i p. Corollary 11 Assume that the contractive condition (1 of Theorem 8 holds with = (x,[tx] α(x D α(xα(y ([Tx] α(x,[ty] α(y for all given x X i and y [Tx] α(x X i+1 for any given arbitrary i p. Then there is a (nonnecessarily unique sequence {x n } of α(x n fuzzy best proximity points of X through T with first element x X i, being subject to free choice if card X i 2, for any given i p such that the sequence of distances {d(x n, x n+1 } is nonincreasing A possible choice is a constant sequence. Proof The contractive condition (1nowbecomes D α(xα(y ( [Tx]α(x,[Ty] α(y a 1 ( a 1 d ( x,[tx] α(x + a2 d ( y,[ty] α(y + a3 d(x, y ; (17 x X i, y X i+1, i p,and,insteadof(7, one sees for any x 1 X i for any i p, that there exist (nonunique, in general, see the proof of Theorem 8 α(x 1 (0, 1], x 2 [Tx 1 ] α(x1 X i+1, α(x 2 (0, 1], and x 3 [Tx 2 ] α(x2 X i+2,since[tx 1 ] α(x1 and [Tx 2 ] α(x2 are nonempty, closed, and bounded: so that d(x 2, x 3 d ( a 1 (a 1 + a 3 x 2,[Tx 2 ] α(x2 d(x a a 1 1, x 2 =d(x 1, x 2, (18 a 2 d(x n+2, x n+1 d(x n+1, x n d(x 2, x 1, n Z 0+, (19 lim sup d(x n+1, x n d(x 1, x 2 d ( x 1,[Tx 1 ] α(x1, (20 n d(x 1, x 2 =d ( x 1,[Tx 1 ] α(x1 = Diα(x1 = inf d ( x 1,[Tx 1 ] α(x1 x 1 X i = D α(x1 α(x 2 ( [Tx1 ] α(x1,[tx 2 ] α(x2, d(x n, x n+1 =d ( x n,[tx n ] α(xn = Di+n 1,α(xn = inf d ( x n,[tx n ] α(xn x n X n+i 1 for n p, d(x (n+1p+j+2, x (n+1p+j+1 =d ( x (n+1p+j+1,[tx (n+1p+j+1 ] α(x(n+1p+j+1, (21 with x np+j+1 [Tx np+j ] α(xnp+j X i+j, j p 1 {0}, n Z 0+, for the chosen arbitrary i p such that {x n } is a sequence of α(x n fuzzy best proximity points of X through T with first element x X i,subjecttofreechoiceifcard X i 2, for any given i p, and the sequence of distances {d(x n, x n+1 } is nonincreasing. Corollary 12 A constant sequence exists which satisfies Corollary 11.
13 Dela Senet al.fixed Point Theory and Applications ( :103 Page 13 of 22 Proof It follows since elements of the sequence {x n } can be chosen so that the first inequalities of (12and(13areequalitiesforn Z + so that d(x n, x n+1 =d(x 2, x 1, n Z +. Example 13 Let X = X 1 X 2 and X 1 = {1, 2, 3}, {1}, {2}, {3} and X 1 = {0.6, 0.7, 0.8}, {0.6}, {0.7}, {0.8} be crisp sets and let T : X 1 X 2 F(X 1 X 2 be a 2cyclic fuzzy mapping defined as follows: 3/4 if t =0.8, T(1(t= 1/2 if t =0.7, 0 ift =0.6; 3/4 if t =1, T(0.8(t= 1/2 if t =2, 0 ift =3; The αlevel sets are 0 ift =0.8, T(2(t= 1/3 if t =0.7, 3/4 if t =0.6; 0 ift =1, T(0.6(t=T(0.7(t= 1/3 if t =2, 3/4 if t =3. [T1] 1/2 =[T3] 1/2 = {0.7, 0.8}, [T1] 3/4 =[T3] 3/4 = {0.8}, [T0.7] α =[T0.6] α = {3}; α (1/2, 3/4, [T2] α = {0.6}; α (1/2, 3/4, [T0.8] 1/2 = {1, 2}, [T0.8] 3/4 = {1}. Let d : X X R 0+ be the Euclidean norm, then D = d(x 1, X 2 = 0.2. Point to image set distances and Pompeiu Hausdorff distances are the following: d ( x,[tx] α =0.2; x {1, 3}, α (1/2, 3/4, d ( 2, [T2] α =1.4; α (1/2, 3/4, d ( ( 0.6, [T0.6] α = d 0.7, [T0.7]α =2.4; α (1/2, 3/4, d ( 0.8, [T0.8] α =0.2; α (1/2, 3/4, and H ( ( [T2] α,[ty] 1/2 = H [Tx]1/2,[Ty] α =2.3; x = {1, 3}, y {0.6, 0.7}, α {1/2, 3/4}, H ( ( [T2] α,[ty] 3/4 = H [Tx]3/4,[Ty] α =2.2; x = {1, 3}, y {0.6, 0.7}, α {1/2, 3/4}, H ( [T2] α,[t0.8] 3/4 =0.4; α {1/2, 3/4}, H ( [T2] α,[t0.8] 1/2 =1.4; α {1/2, 3/4}, H ( [T0.8] 3/4,[Tx] 1/2 =0.3; x = {1, 3}, H ( [T0.8] 3/4,[Tx] 3/4 =0.2; x = {1, 3}, H ( [T0.8] 1/2,[Tx] 1/2 =1.3; x = {1, 3},
14 Dela Senet al.fixed Point Theory and Applications ( :103 Page 14 of 22 H ( [T0.8] 1/2,[Tx] 3/4 =1.2; x = {1, 3}. The contractive constraint (1andTheorem8 can be applied with a 2 =0.8,a 1 + a 3 <1 a 2. Some sequences which converge to αfuzzy best proximity points of X 1 and X 2 through T, which are also best proximity points in X 1 of X 2 and conversely, are { 1, 0.8 [T1]3/4,1 [T0.8] 3/4,... }, { 1, 0.7 [T1]1/2,3 [T0.7] 3/4,0.8 [T3] 3/4,1 [T0.8] 1/2, 0.8 [T1] 3/4,1 [T0.8] 3/4,... }. 4 Further results on αfuzzy best proximity points under a more general contractive condition We now establish the following result related to a contractive condition which is more general than (1. Theorem 14 Let p( 2 Z + and let X i be nonempty subsets of a nonempty abstract set X, i p, where (X, d is a complete metric space. Let T : i p X i F( i p X i be a pcyclic fuzzy mapping with X i being closed, i p. Assume that the following contractive condition holds: H ( [Tx] α(x,[ty] α(y a1 d ( x,[tx] α(x + a2 d ( y,[ty] α(y + a3 d(x, y + a 4 d ( x,[ty] α(y + a5 d ( y,[tx] α(x +(1 a ; (22 x X i, y X i+1, i p, and some R + where a i 0 for i = 1,2,3 and 0<a = 5 i=1 a i <1.Then the following properties hold: (i For any arbitrarily given i p, j p, n Z 0+, and x = x 1 X i, a sequence {x n } i p X i can be built fulfilling {x n } i p X i for any given initial points x 1 = x X i for any given i p, x 2 [Tx 1 ] α(x1 X i+1, x 3 = y X i+1, and x 4 [Tx 3 ] α(x3 X i+2, n Z +, such that lim sup d(xn, x n+1 (1 a 2 n (1 a 2 a 4 (1 ρ, provided that θ = a 2 + a 4 +2(a 1 + a 3 +a 5 <1,where ρ = 1 ( (( a1 + a 3 + a 5 a1 + a 3 + a a 1 + a 1/ a 2 a 4 1 a 2 a 4 1 a 2 a 4 (ii If = D = D i ; i p, a 1 = a 3 =0,θ = a 2 + a 4 + a 5 <1,then a sequence {x n } i p X i can be built with initial points x 1 = x X i for any arbitrary given i p, x 2 [Tx 1 ] α(x1 X i+1, x 3 = y X i+1, and x 4 [Tx 3 ] α(x3 X i+2 which satisfies x np+1 [Tx np ] α(xnp X i, x np+2 [Tx np+1 ] α(xnp+1 X i+1, x np+3 [Tx np+2 ] α(xnp+2 X i+1, x np+4 [Tx np+2 ] α(xnp+2 X i+2,
15 Dela Senet al.fixed Point Theory and Applications ( :103 Page 15 of 22 and d(x np+1, x np+2 D, d(x np+2, x np+3 0, d(x np+3, x np+4 D, d(x np+4, x np+5 0,... as n. Thus, the sequence contains a convergent subsequence to a limit cycle consisting of a setofpbestproximitypointsx i of the sets X i, for i p, which are also α(x i 1 fuzzy best proximity points of X through T for some set of values α(x i 1 (0, 1]. If D =0then the above convergent subsequence converges to a unique α = α(x fuzzy fixed point of T : i p X i F( i p X i. Proof Since [Tx] α and [Ty] β are nonempty and closed for some α (0, γ a ] (0, 1] and β (0, γ b ] (0, 1], we can choose any points x X i and y [Tx] α(x X i+1 (then d(y,[tx] α(x = 0andd(x, y d(x,[tx] α(x. Thus, (22becomes d ( ( y,[ty] α(y H [Tx]α(x,[Ty] α(y a 1 d ( x,[tx] α(x + a2 d ( y,[ty] α(y + a3 d(x, y + a 4 d ( x,[ty] α(y +(1 a (23 d ( y,[ty] α(y (1 a2 1[ a 1 d ( x,[tx] α(x + a3 d(x, y+a 4 d ( x,[ty] α(y +(1 a ] (1 a 2 1[ (a 1 + a 3 d(x, y+a 4 d ( x,[ty] α(y +(1 a ]. (24 Further assume that d(y,[ty] α(y d(x,[ty] α(y forthechosenx, y. Thenonegetsfrom (24 the following cases: Case (a If y [Tx] α(x and d(y,[ty] α(y d(x,[ty] α(y then d ( x,[ty] α(y 1 1 a 2 a 4 [ (a1 + a 3 d(x, y+(1 a ]. (25 Case (b Assume that y [Tx] α(x and d(y,[ty] α(y d(x,[ty] α(y so that, by using those relations in the righthand side of (23, one gets d ( ( y,[ty] α(y H [Tx]α(x,[Ty] α(y a 1 d ( x,[tx] α(x +(a2 + a 4 d ( y,[ty] α(y + a3 d(x, y+(1 a ; (26 then d ( y,[ty] α(y (1 a2 a 4 1[ a 1 d ( x,[tx] α(x + a3 d(x, y+(1 a ] 1 1 a 2 a 4 [ (a1 + a 3 d(x, y+(1 a ]. (27 Joint Cases (a(b The combination of (25and(27 yields, if y [Tx] α(x, max ( d ( x,[ty] α(y, d ( y,[ty]α(y 1 1 a 2 a 4 [ (a1 + a 3 d(x, y+(1 a ]. (28
16 Dela Senet al.fixed Point Theory and Applications ( :103 Page 16 of 22 Case (c If y / [Tx] α(x then d(y,[tx] α(x 0,sothat(28 becomesvia(25 and(27 for this case that max ( d ( x,[ty] α(y, d ( y,[ty]α(y 1 1 a 2 a 4 [ (a1 + a 3 d(x, y+a 5 d ( y,[tx] α(x +(1 a ], (29 which is more general than (28. Now, construct the sequence {x n } i p X i with x 1 = x X i for some given i p, x 2 [Tx 1 ] α(x1 X i+1, x 3 = y X i+1,andx 4 [Tx 3 ] α(x3 X i+2 such that d(x 2, x 4 =d(x 2,[Tx 3 ] α(x3, which exists since [Tx 2 ] α(x2 is nonempty and closed. Thus, one gets after using the triangle inequality d(x 1, x 3 d(x 1, x 2 +d(x 2, x 3 in the righthand side of (29 d(x 3, x 4 max ( d(x 1, x 4, d(x 3, x 4 One gets from (30 1 [ (a1 + a 3 d(x 1, x 3 +a 5 d(x 2, x 3 +(1 a ] 1 a 2 a a 2 a 4 [ (a1 + a 3 d(x 1, x 2 +(a 1 + a 3 + a 5 d(x 2, x 3 +(1 a ]. (30 [ ] [ d(x 2, x d(x 3, x 4 a 1 +a 3 1 a 2 a 4 a 1 +a 3 +a 5 1 a 2 a 4 ][ ] [ d(x 1, x 2 + d(x 2, x a 1 a 2 a 4 ]. (31 Proceed recursively with (31 by defining the sequence {x n } i p X i with initial values x j,subjectto(30forj =1,2,3,4andthegivenarbitraryi p, and a twodimensional real vector by [ ] d(x n+1, x n+2 v n =, n Z +, (32 d(x n+2, x n+3 with initial value v 1 = [ d(x 1,x 2 d(x 2,x 3 ],andasquarereal2matrixa and a twodimensional vector d defined by A = [ 0 1 a 1 +a 3 1 a 2 a 4 a 1 +a 3 +a 5 1 a 2 a 4 ] [, d = 0 1 a 1 a 2 a 4 Then it is possible to write, in a similar way to (31, ]. (33 v n+1 Av n + d, n Z +, (34 such that for any x n+2 X j,thereisx n+3 [Tx n+2 ] α(xn+2 X j+1 with j = j(n, i p depending on the given arbitrary initial i p with x 1 X i and x 2 X i+1.thus,onegetsfrom(34 v np+1 2 A 2 v np 2 + d 2, n Z +, (35
17 Dela Senet al.fixed Point Theory and Applications ( :103 Page 17 of 22 where d 2 is the l 2 vector norm of d and ( Aμ 2 A 2 = sup : μ R 2 μ 1 μ 2 ( = sup Aμ 2 : μ R 2 = λ 1/2 ( max A T A μ =1 is the l 2 (or spectralmatrix norm of A (which is the induced l 2 vector norm, where λ max (A T A denotes the maximum/real eigenvalue of the (symmetric matrix A T A and the superscript T denotes matrix transposition. If λ = A 2 ( 1, 1, then A is a convergent matrix and then A n 2 0asn so that ( n 1 v n+1 2 A 2 v n 2 + d 2 A n 2 v d 2 A n 2 v d 2 ( i=0 A n i 1 2 i=n i=0 A n i 1 2 A n i 1 2 λ n v λn 1 λ d 2 v d 2 1 λ 1 a = v (1 a 2 a 4 (1 λ <, n Z +. (36 Now, note that if ρ ( 1, 1 is the dominant eigenvalue of A, thenλ ρ <1andthen, from (36, lim sup d(xn+3, x n+4 2 lim sup n n v n+1 2 (1 a (1 a 2 a 4 (1 ρ. (37 From the definition of the matrix A, its eigenvalues are the roots of its characteristic polynomial p(z=z 2 a 1+a 3 +a 5 1 a 2 a 4 z a 1+a 3 1 a 2 a 4 resulting in z 1,2 = 1 ( (( a1 + a 3 + a 5 a1 + a 3 + a 2 5 ± +4 a 1 + a 1/ a 2 a 4 1 a 2 a 4 1 a 2 a 4 and z 1,2 <1,andthenA is a convergent matrix so that (37holds,ifa 2 + a 4 +2(a 1 + a 3 + a 5 <1.Property(ihasbeenproved. Now, assume that the sets X i fulfill the constraint D = D i = d(x i, X i+1, i p, andthat the contractive condition holds with = (1 a 2 a 4 (1 ρ 1 a D D, or equivalently, with ρ = ρ = z 2 = 1 ( (( a1 + a 3 + a 5 a1 + a 3 + a a 1 + a 1/2 3 < a 2 a 4 1 a 2 a 4 1 a 2 a 4 But note from (38that ρ a 1+a 3 +a 5 1 a 2 a 4 with the equality occurring if and only if a 1 = a 3 =0. Then, if D = D i, i p, a 1 = a 3 =0,and z 1,2 z 2 <1,thenlim n v np+1 2 = D.This implies that d(x np+3, x np+2 0andd(x np+1, x np+2 D as n from (32, since for the definition of the real sequences {v n } and {x n }, x np+2 and x np+3 are both in some set X i+1, while x np+1 and x np+2,andx np+3 and x np+4 are, respectively, in the sets X i and X i+2,adjacent to X i, for any given initial points x 1 X i and x 2 X i+1 for any arbitrary i p. Thisleads directly to the proof of Property (ii. See also the proof of Theorem 8(iii(iv concerning the convergence of the sequence to a set of αfuzzy best proximity points and a unique αfixed point if D =0. (38
18 Dela Senet al.fixed Point Theory and Applications ( :103 Page 18 of 22 Example 15 Example 13 can also easily be monitored via Theorem 14 via the modified contractive constraint (22witha 4 = a 5 =0. The subsequent definition extends Definition 1 in the sense that an extended pcyclic fuzzy mapping of domain restricted X j has its image in X j X j+1 for any j p. Definition 16 Let p( 2 Z + and let X i be nonempty subsets of a nonempty abstract set X, i p,where(x, d is a metric space. A mapping T : i p X i F( i p X iissaidto be an extended pcyclic fuzzy mapping from i p X i to F( i p X iif: (a T : i p X i X j F(X j X j+1 ; j p, (b there is α(x (0, 1] such that the α(xlevel set [Tx] α(x CB(X i X i+1 ; x X i, i p. Theorem 17 Let p( 2 Z + and let X i be nonempty subsets of a nonempty abstract set Xwithdiam(X i Dandd(X i, X i+1 =D; i p, where (X, d is a complete metric space. Let T : i p X i F( i p X i be an extended pcyclic fuzzy mapping with X i being closed, i p. Assume that the following pointdependent contractive constraint extended from (1 holds: H ( [Tx] α(x,[ty] α(y a 1(x,[Tx]α(x d( x,[tx] α(x + a2(y,[ty]α(y d( y,[ty] α(y + a 3(x,y d(x, y+(1 aμ(x, yd; (39 x X i, y X i+1, i p, and some R + where a i( 0 for i =1,2,3,and μ(x, y= { 0 if (x, y X i X i+1, 1 if x, y X i (40 for any given i p. Then the following properties hold: (i For any arbitrarily given i p, asequence{x n } i p X i can be built for any given x = x 1 X i, y = x 2 [Tx 1 ] α(x1 X i+1 such that x np+j [Tx np+j 1 ] α(xnp+j 1 X i+j 1 ; j p, n Z 0+, which fulfills lim n d(x (n+1p+j+2, x (n+1p+j+1 =D; j Z 0+ (41 for any q Z 0+ if lim n ( n p+i kjm j=0 m=i l=1 [K jml] = 0, where K jml = a 1jml + a 3jml 1 a 2jml ; l k jm, j Z 0+, m p, (42 where the integer subscripts l k jm, k jm ( k < Z 0+, j( n Z 0+, m p, (43 are, respectively, the indicator of each current iteration within the set X m, the number of consecutive iterations at the jth tour over the cyclic disposal in the X m set before jumping to
19 Dela Senet al.fixed Point Theory and Applications ( :103 Page 19 of 22 the X m+1 set, the tour counter of completed tours over the cyclic disposal, and the indicator of the current X m set for the iteration. (ii If Property (i holds, then Theorem 8(iii(iv hold. Proof Define K ( a 1 (x,[tx] α(x +a 3 (x, y x, y,[tx] α(x,[ty] α(y =. (44 1 a 2 (y,[ty] α(y Note that the contractive constraint (39(40leads,in a similar way to(8in Theorem 8, to the following inequality: d(x (n+1p+q+2, x (n+1p+q+1 ( n p+i k jm j=0 m=i l=1 [K jml ] δ+i 1 m=i k jm l=1 [K n+1,ml ] ˆk n+1,δ+i l=1 [K n+1,δ+i,l ] (d(x1, x 2 D + D; n Z 0+, j p, (45 for some integer 0 δ p, where0 ˆk n+1,m = k n+1,m for i m δ + i 1,0 ˆk n+1,δ+i k n+1,δ+i, n Z 0+ is the last completed tour of the cyclic disposal, and q = q(i, n, δ, k 0m.,...,k nm, ˆk n+1,δ+i = p+i n δ+i 1 k jm + k n+1,m + ˆk n+1,δ+i, (46 j=0 m=i m=i K jml = a 1jml + a 3jml 1 a 2jml ; l k jm, j Z 0+, m p. (47 Note that k jm k < ; m p, j n {0} and k n+1,j+i k < for j δ 1 {0} if n <. Now, note also that ˆk n+1,δ+i k < if n,sincethenumberofsequence tours (tour counter through the cyclic disposal tends to infinity. It turns out that lim n d(x (n+1p+q+2, x (n+1p+q+1 =D for any q Z 0+ as n if lim n ( n p+i k jm j=0 m=i l=1 [K jml ] =0. Property (i has been proved. The proof of Property (ii is straightforward from Property (i by similar arguments to those used in the proofs of Theorem 8(iii(iv. Note that Theorem 17 does not require all the elements of the contractive sequence (42 to be less than 1. The following result is proved for the case when the built sequences are permanent on some of the subsets of the cyclic disposal. Corollary 18 The following properties hold under the assumptions of Theorem 17: (i Theorem 17(i still holds if there is a finite number n 0 of complete tours in the cyclic disposal and lim n ( ˆk n0 +1,δ+i l=1 [K n0 +1,δ+i,l] = 0 for some integer 0 δ p with the sequence {x n } being permanent at {X i+δ }. Then lim n d(x (n+1p+j+2, x (n+1p+j+1 =0; j Z 0+. (ii The sequence converges to a unique α = α(x fuzzy fixed point x of T : i p X i F( i p X i allocated in [Tx ] α(x X i+δ.
20 Dela Senet al.fixed Point Theory and Applications ( :103 Page 20 of 22 Proof Property (i follows from (45 forn n 0 < and ˆk n0 +1,δ+i. Property (ii follows from Property (i since (X, discompleteand[tx] α(x is closed for any x i p X i in a similar way to the proof of Theorem 8(iv. Example 19 Define the linear metric space (X, d withx being the union of closed real intervals X 1 =[ r 1, D/2], X 2 =[D/2, r 2 ]subjectto r 1 >3D/2 0, min(r 1, r 2 >0, max(r 1, r 2 <+, and which have mutual and PompeiuHausdorff distances d(x 1, X 2 =D and H(X 1, X 2 = D + max(r 1, r 2, respectively. Consider some extended 2cyclic fuzzy mapping T : X 1 X 2 F(X 1 X 2 whose restrictions of domain are to the two subsets X 1 and X 2 fulfill T : X 1 X 2 X 1 F(X 1 X 2, T : X 1 X 2 X 2 F(X 1. Such a map is defined in such a way that it constructs iterative sequences which are ordered with two consecutive elements x n, x n+1 [Tx n ] 1 X 1 and the next one x n+2 [Tx n+1 ] 1 X 2, and which satisfies the constraints: d(x n+1, x n ρd(x n, x n 1 ifx n, x n+1 [Tx n ] 1 X 1, d(x n+2, x n+1 Kd(x n, x n 1 +(1 KD if x n+2 [Tx n+1 ] 1 X 2, for n Z + subject to K 2 ρ < 1. Note that either K or ρ (but not both of them cannot be less than unity. Since the 1level sets are nonempty and contain the points, identified with their associated crisp sets, we can consider F(X 1 andf(x 2 as approximate quantities [22, 28], since X 1 and X 2 are compact and convex, and such crisp sets, equivalently as points, obtained from a maximum defuzzifier [34], since there exists a compact [Tx] 1 ( [Tx] α(x for any x X 1 X 2 and any α(x (0, 1]. Assume that x 1, x 2 X 1 and x 3 X 2, and then x 2n [Tx 2n 1 ] 1 X 1, x 2n+1 [Tx 2n ] 1 X 1, x 2n+2 [Tx 2n+1 ] 1 X 2 ; n Z +,and one gets d(x 2n+1, x 2n ( K 2 ρ n d(x1, x 2 + ( 1 K n D, d(x 2n+2, x 2n+1 Kd(x 2n+1, x 2n +(1 KD, lim d(x 2n+1, x 2n = lim d(x 2n+2, x 2n+1 =D. n n If x 1 X 2 and x 2, x 3 X 1, x 2n [Tx 2n 1 ] 1 X 2, x 2n+1 [Tx 2n ] 1 X 1,andx 2n+2 [Tx 2n+1 ] 1 X 1 ; n Z +, the same conclusion for the distance limits as above follows. In both cases, sequences converging to the limit cycle { 3D/2, D/2, D/2} are obtained for any initial conditions x 1, x 2 X 1 or x 1 X 2, x 2 X 1. All the sequences are not necessarily converging to the above limit cycle, in general, unless it be assumed that T : X 1 X 2 F(X 1 X 2 is such that [Tx] α(x =[Tx] 1 ; x X 1 X 2. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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