This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are nonempty, closed and convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.
Strict pseudocontractive mappings and pseudocontractive mappings in the intermediate sense formulated in the framework of Hilbert spaces have received a certain attention in the last years concerning their convergence properties and the existence of fixed points. See, for instance, [1-4] and references therein. Results about the existence of a fixed point are discussed in those papers. On the other hand, important attention has been paid during the last decades to the study of the convergence properties of distances in cyclic contractive self-mappings on p subsets of a metric space , or a Banach space . The cyclic self-mappings under study have been of standard contractive or weakly contractive types and of Meir-Keeler type. The convergence of sequences to fixed points and best proximity points of the involved sets has been investigated in the last years. See, for instance, [5-20] and references therein. It has to be noticed that every nonexpansive mapping [21,22] is a 0-strict pseudocontraction and also that strict pseudocontractions in the intermediate sense are asymptotically nonexpansive . The uniqueness of the best proximity points to which all the sequences of iterations converge is proven in  for the extension of the contractive principle for cyclic self-mappings in either uniformly convex Banach spaces (then being strictly convex and reflexive ) or in reflexive Banach spaces . The p subsets of the metric space , or the Banach space , where the cyclic self-mappings are defined, are supposed to be nonempty, convex and closed. If the involved subsets have nonempty intersections, then all best proximity points coincide, with a unique fixed point being allocated in the intersection of all the subsets, and framework can be simply given on complete metric spaces. The research in  is centered on the case of the 2-cyclic self-mapping being defined on the union of two subsets of the metric space. Those results are extended in  for Meir-Keeler cyclic contraction maps and, in general, with the -cyclic self-mapping defined on any number of subsets of the metric space with . Other recent research which has been performed in the field of cyclic maps is related to the introduction and discussion of the so-called cyclic representation of a set M, as the union of a set of nonempty sets as , with respect to an operator . Subsequently, cyclic representations have been used in  to investigate operators from M to M which are cyclic φ-contractions, where is a given comparison function, and is a metric space. The above cyclic representation has also been used in  to prove the existence of a fixed point for a self-mapping defined on a complete metric space which satisfies a cyclic weak φ-contraction. In , a characterization of best proximity points is studied for individual and pairs of non-self-mappings , where A and B are nonempty subsets of a metric space. The existence of common fixed points of self-mappings is investigated in  for a class of nonlinear integral equations, while fixed point theory is investigated in locally convex spaces and non-convex sets in [25-28]. More recently, the existence and uniqueness of best proximity points of more general cyclic contractions have been investigated in [29,30] and a study of best proximity points for generalized proximal contractions, a concept referred to non-self-mappings, has been proposed and reported in detail in . Also, the study and characterization of best proximity points for cyclic weaker Meir-Keeler contractions have been performed in  and recent contributions on the study of best proximity and proximal points can be found in [33-38] and references therein. In general, best proximity points do not fulfill the usual ‘best proximity’ condition under this framework. However, best proximity points are proven to jointly globally optimize the mappings from x to the distances and . Furthermore, a class of cyclic φ-contractions, which contains the cyclic contraction maps as a subclass, has been proposed in  in order to investigate the convergence and existence results of best proximity points in reflexive Banach spaces completing previous related results in . Also, the existence and uniqueness of best proximity points of cyclic φ-contractive self-mappings in reflexive Banach spaces have been investigated in . This paper is devoted to the convergence properties and the existence of fixed points of a generalized version of pseudocontractive, strict pseudocontractive and asymptotically pseudocontractive in the intermediate sense in the more general framework of metric spaces. The case of 2-cyclic pseudocontractive self-mappings is also considered. The combination of constants defining the contraction may be different on each of the subsets and only the product of all the constants is requested to be less than unity. It is assumed that the considered self-mapping can perform a number of iterations on each of the subsets before transferring its image to the next adjacent subset of the 2-cyclic self-mapping. The existence of a unique closed finite limiting sequence on any sequence of iterations from any initial point in the union of the subsets is proven if X is a uniformly convex Banach space and all the subsets of X are nonempty, convex and closed. Such a limiting sequence is of size (with the inequality being strict if there is at least one iteration with image in the same subset as its domain), where p of its elements (all of them if ) are best proximity points between adjacent subsets. In the case that all the subsets intersect, the above limit sequence reduces to a unique fixed point allocated within the intersection of all such subsets.
2 Asymptotic contractions and pseudocontractions in the intermediate sense in metric spaces
If H is a real Hilbert space with an inner product and a norm and A is a nonempty closed convex subset of H, then is said to be an asymptotically β-strictly pseudocontractive self-mapping in the intermediate sense for some if
for some sequence , as [1-4,23]. Such a concept was firstly introduced in . If (2.1) holds for , then is said to be an asymptotically pseudocontractive self-mapping in the intermediate sense. Finally, if as , then is asymptotically β-strictly contractive in the intermediate sense, respectively, asymptotically contractive in the intermediate sense if . If (2.1) is changed to the stronger condition
then the above concepts translate into being an asymptotically β-strictly pseudocontractive self-mapping, an asymptotically pseudocontractive self-mapping and asymptotically contractive one, respectively. Note that (2.1) is equivalent to
Note that the high-right-hand-side term of (2.3) is expanded as follows for any :
The objective of this paper is to discuss the various pseudocontractive in the intermediate sense concepts in the framework of metric spaces endowed with a homogeneous and translation-invariant metric and also to generalize them to the β-parameter to eventually be replaced with a sequence in . Now, if instead of a real Hilbert space H endowed with an inner product and a norm , we deal with any generic Banach space , then its norm induces a homogeneous and translation invariant metric defined by ; so that (2.6) takes the form
which exists since it follows from (2.7), since the metric is homogeneous and translation-invariant, that
The following result holds related to the discussion (2.7)-(2.9) in metric spaces.
Theorem 2.1Let be a metric space and consider a self-mapping . Assume that the following constraint holds:
for some parameterizing bounded real sequences , and of general terms , , satisfying the following constraints:
with and, furthermore, the following condition is satisfied:
if and only if ; as .
Then the following properties hold:
(i) for any so that is asymptotically nonexpansive.
(ii) Let be complete, be, in addition, a translation-invariant homogeneous norm and let , with being the metric-induced norm from , be a uniformly convex Banach space. Assume also that is continuous. Then any sequence ; is bounded and convergent to some point , being in general dependent onx, in some nonempty bounded, closed and convex subsetCofA, whereAis any nonempty bounded subset ofX. Also, is bounded; , ; , and is a fixed point of the restricted self-mapping ; . Furthermore,
Proof Consider two possibilities for the constraint (2.10), subject to (2.11), to hold for each given and as follows:
(A) for any , . Then one gets from (2.10)
, , where
which holds from (2.12)-(2.13) if since
as in (2.13) is equivalent to (2.16). Note that is ensured either with or with if
However, with has to be excluded because of the unboundedness or nonnegativity of the second right-hand-side term of (2.15).
(B) for some , . Then one gets from (2.10)
which holds from (2.12) and if , and
Thus, (2.15)-(2.16), with the second option in the logic disjunction being true if and only if together with (2.18)-(2.20), are equivalent to (2.12)-(2.13) by taking to be either or for each . It then follows that ; from (2.15)-(2.19) since and ; as . Thus, is asymptotically nonexpansive. Thus, Property (i) has been proven. Property (ii) is proven as follows. Consider the metric-induced norm equivalent to the translation-invariant homogeneous metric . Such a norm exists since the metric is homogeneous and translation-invariant so that norm and metric are formally equivalent. Rename and define a sequence of subsets of X. From Property (i), is bounded; if is finite, since it is bounded for any finite and, furthermore, it has a finite limit as . Thus, all the collections of subsets ; are bounded since is bounded. Define the set which is nonempty bounded, closed and convex by construction. Since is complete, is a uniformly convex Banach space and is asymptotically nonexpansive from Property (i), then it has a fixed point [1,23]. Since the restricted self-mapping is also continuous, one gets from Property (i)
Then any sequence is convergent (otherwise, the above limit would not exist contradicting Property (i)), and then bounded in C; . This also implies is bounded; , and ; , . This implies also as ; such that ; which is then a fixed point of (otherwise, the above property ; , would be contradicted). Hence, Property (ii) is proven. □
First of all, note that Property (ii) of Theorem 2.1 applies to a uniformly convex space which is also a complete metric space. Since the metric is homogeneous and translation-invariant, a norm can be induced by such a metric. Alternatively, the property could be established on any uniformly convex Banach space by taking a norm-induced metric which always exists. Conceptually similar arguments are used in later parallel results throughout the paper. Note that the proof of Theorem 2.1(i) has two parts: Case (A) refers to an asymptotically nonexpansive self-mapping which is contractive for any number of finite iteration steps and Case (B) refers to an asymptotically nonexpansive self-mapping which is allowed to be expansive for a finite number of iteration steps. It has to be pointed out concerning such a Theorem 2.1(ii) that the given conditions guarantee the existence of at least a fixed point but not its uniqueness. Therefore, the proof is outlined with the existence of a for any nonempty, bounded and closed subset A of X. Note that the set C, being in general dependent on the initial set A, is bounded, convex and closed by construction while any taken nonempty set of initial conditions is not required to be convex. However, the property that all the sequences converge to fixed points opens two potential possibilities depending on particular extra restrictions on the self-mapping , namely: (1) the fixed point is not unique so that for any (and any A in X) so that some set for some contains more than one point. In other words, as ; has not been proven although it is true that ; ; (2) there is only a fixed point in X. The following result extends Theorem 2.1 for a modification of the asymptotically nonexpansive condition (2.10).
Theorem 2.2Let be a metric space and consider the self-mapping . Assume that the constraint below holds:
for some parameterizing real sequences , and satisfying, for any ,
Then the following properties hold:
(i) so that is asymptotically nonexpansive, and then ; if
and the following limit exists:
(ii) Property (ii) of Theorem 2.1 if is complete and is a uniformly convex Banach space under the metric-induced norm .
Sketch of the proof Property (i) follows in the same way as the proof of Property (i) of Theorem 2.1 for Case (B). Using proving arguments similar to those used to prove Theorem 2.1, one proves Property (ii). □
The relevant part in Theorem 2.1 being of usefulness concerning the asymptotic pseudocontractions in the intermediate sense and the asymptotic strict contractions in the intermediate sense relies on Case (B) in the proof of Property (i) with the sequence of constants ; , and ; as , . The concepts of an asymptotic pseudocontraction and an asymptotic strict pseudocontraction in the intermediate sense motivated in Theorem 2.1 by (2.7)-(2.9), under the asymptotically nonexpansive constraints (2.10) subject to (2.11) and in Theorem 2.2 by (2.22) subject to (2.23) are revisited as follows in the context of metric spaces.
Definition 2.3 Assume that is a complete metric space with being a homogeneous translation-invariant metric. Thus, is asymptotically β-strictly pseudocontractive in the intermediate sense if
for ; and some real sequences , being, in general, dependent on the initial points, i.e., , and
Definition 2.4 is asymptotically pseudocontractive in the intermediate sense if (2.30) holds with , , , , , as and the remaining conditions as in Definition 2.3 with , and .
Definition 2.5 is asymptotically β-strictly contractive in the intermediate sense if , , ; , , as , in Definition 2.3 with , .
Definition 2.6 is asymptotically contractive in the intermediate sense if , , ; , , , and as in Definition 2.3 with , and .
Remark 2.7 Note that Definitions 2.3-2.5 lead to direct interpretations of their role in the convergence properties under the constraint (2.22), subject to (2.23), by noting the following:
(1) If is asymptotically β-strictly pseudocontractive in the intermediate sense (Definition 2.3), then the real sequence of asymptotically nonexpansive constants has a general term ; , and it converges to a limit since and as ; from (2.22) since from (2.27). Then is trivially asymptotically nonexpansive as expected.
(2) If is asymptotically pseudocontractive in the intermediate sense (Definition 2.4), then the sequence of asymptotically nonexpansive constants has the general term: ; , and it converges to a limit since , as . Then is also trivially asymptotically nonexpansive as expected. Since , note that and for any , while , as since as ; from (2.22)-(2.23).
(3) If is asymptotically β-strictly contractive in the intermediate sense (Definition 2.5), then the sequence of asymptotically contractive constants is defined by ; and as for any such that as , since . Then is an asymptotically strict contraction as expected since as ; from (2.22)-(2.23). Note that the asymptotic convergence rate is arbitrarily fast as α and β are arbitrarily close to zero, since becomes also arbitrarily close to zero, and with .
(4) If is asymptotically contractive in the intermediate sense (Definition 2.6), then the sequence of asymptotically contractive constants is defined by
with and as for some since with so that . Then is an asymptotically strict contraction as expected since as ; from (2.23). Note that if and and . Note also that if and , while if and . In the first case, the convergence to fixed points (see Theorem 2.8 below) is guaranteed to be asymptotically faster if the self-mapping is asymptotically β-strictly contractive in the intermediate sense than if it is just asymptotically contractive in the intermediate sense if , . Note also that if the sequences and are identical in both cases, then for any such that and for any such that .
(5) The above considerations could also be applied to Theorem 2.1 for the case (Case (B) in the proof of Property (i)) being asymptotically nonexpansive for the asymptotically nonexpansive condition (2.10) subject to (2.11).
The subsequent result, being supported by Theorem 2.2, relies on the concepts of asymptotically contractive and pseudocontractive self-mappings in the intermediate sense. Therefore, it is assumed that .
Theorem 2.8Let be a complete metric space endowed with a homogeneous translation-invariant metric and consider the self-mapping . Assume that is a uniformly convex Banach space endowed with a metric-induced norm from the metric . Assume that the asymptotically nonexpansive condition (2.22), subject to (2.23), holds for some parameterizing real sequences , and satisfying, for any ,
, . Then for any satisfying the conditions
Furthermore, the following properties hold:
(i) is asymptoticallyβ-strictly pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set and any given nonempty, bounded and closed subset of initial conditions if (2.29) hold with , , , and as ; , . Also, has a fixed point for any such setCif is continuous.
(ii) is asymptotically pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set and any given nonempty, bounded and closed subset of initial conditions if (2.29) hold with , , , , and as ; , . Also, has a fixed point for any such setCif is continuous.
(iii) If (2.29) hold with , , , ; and as , then is asymptoticallyβ-strictly contractive in the intermediate sense. Also, has a unique fixed point.
(iv) If (2.29) hold with , , , ; , and as , then is asymptotically strictly contractive in the intermediate sense. Also, has a unique fixed point.
Proof (i) It follows from Definition 2.3 and the fact that Theorem 2.2 holds under the particular nonexpansive condition (2.22), subject to (2.23), so that is asymptotically nonexpansive (see Remark 2.7(1)). Property (ii) follows in a similar way from Definition 2.4 (see Remark 2.7(2)). Properties (iii)-(iv) follow from Theorem 2.2 and Definitions 2.5-2.6 implying also that the asymptotically nonexpansive self-mapping is also a strict contraction, then continuous with a unique fixed point, since (see Remark 2.7(3)) and with (see Remark 2.7(4)), respectively. (The above properties could also be got from Theorem 2.1 for Case (B) of the proof of Theorem 2.1(ii) - see Remark 2.7(5).) □
Example 2.9 Consider the time-varying pth order nonlinear discrete dynamic system
for some given nonempty bounded set , where is a matrix sequence of elements with and with ; , and defines the state-sequence trajectory solution . Equation (2.13) requires the consistency constraint to calculate . However, other discrete systems being dealt with in the same way as, for instance, that obtained by replacing in (2.31) with the initial condition (and appropriate ad hoc re-definition of the mapping which generates the trajectory solution from given initial conditions) do not require such a consistency constraint. The dynamic system (2.31) is asymptotically linear if as ; . Note that for the Euclidean distance (and norm), ; . Assume that the squared spectral norm of is upper-bounded by for some parameterizing scalar sequences , and which can be dependent, in a more general case, on the state . This holds, for instance, if , where is a real positive sequence satisfying and both being potentially dependent on the state as the rest of the parameterizing sequences. Since the spectral norm equalizes the spectral radius if the matrix is symmetric, then can be taken exactly as the spectral radius of in such a case, i.e., it equalizes the absolute value of its dominant eigenvalue. We have to check the condition
provided, for instance, that the distance is the Euclidean distance, induced by the Euclidean norm, then both being coincident, and provided also that we take the metric space which holds, in particular, if
(a) , , , ; , and , , as ; . This implies that ; and as ; . Thus, is asymptotically nonexpansive being also an asymptotic strict β-pseudocontraction in the intermediate sense. This also implies that (2.31) is globally stable as it is proven as follows. Assume the contrary so that there is an infinite subsequence of which is unbounded, and then there is also an infinite subsequence which is strictly increasing. Since and as ; , one has that for , any given and some sufficiently large , , , such that and ; , . Now, take and . Then ; and any given . If , then stability holds trivially. Assume not, and there are unbounded solutions. Thus, take such that for any given , and some . Note that since is a strictly increasing real sequence implying as , which leads to a contradiction to the inequality for for some sufficiently large , then for some sufficiently large M, if such a strictly increasing sequence exists. Hence, there is no such sequence, and then no unbounded sequence for any initial condition in . As a result, for any initial condition in any given subset of (even if it is unbounded), any solution sequence of (2.31) is bounded, and then (2.31) is globally stable. The above reasoning implies that there is an infinite collection of numerable nonempty bounded closed sets , which are not necessarily connected, such that ; and any given . Assume that the set of initial conditions is bounded, convex and closed and consider the collection of convex envelopes , define constructively the closure convex set which is trivially bounded, convex and closed. Note that it is not guaranteed that is either open or closed since there is a union of infinitely many closed sets involved. Note also that the convex hull of all the convex envelopes of the collection of sets is involved to ensure that A is convex since the union of convex sets is not necessarily convex (so that is not guaranteed to be convex while A is convex). Consider now the self-mapping which defines exactly the same solution as for initial conditions in so that is identified with the restricted self-mapping from a nonempty bounded, convex and closed set to itself. Note that for the Euclidean distance is a convex metric space which is also complete since it is finite dimensional. Then and are both continuous, then is also continuous and has a fixed point in A from Theorem 2.8(i).
(b) If the self-mapping is asymptotically pseudocontractive in the intermediate sense, then the above conclusions still hold with the modification and as ; . From Remark 2.7(2), and for any . Thus the convergence is guaranteed to be faster for an asymptotic β-strict pseudocontraction in the intermediate sense than for an asymptotic pseudocontraction in the intermediate sense with a sequence such that ; with the remaining parameters and parametrical sequences being identical in both cases. If and ; are both continuous, then is continuous and has a fixed point in A from Theorem 2.8(ii).
(c) If is asymptotically β-strictly contractive in the intermediate sense, then ; so that it is asymptotically strictly contractive and has a unique fixed point from Theorem 2.8(iii).
(d) If is asymptotically contractive in the intermediate sense, ; . Thus, is an asymptotic strict contraction and has a unique fixed point from Theorem 2.8(iv).
Remark 2.10 Note that conditions like (2.32) can be tested on dynamic systems being different from (2.31) by redefining, in an appropriate way, the self-mapping which generates the solution sequence from given initial conditions. This allows to investigate the asymptotic properties of the self-mapping, the convergence of the solution to fixed points, then the system stability, etc. in a unified way for different dynamic systems. Close considerations can be discussed for different dynamic systems and convergence of the solutions generated by the different cyclic self-mappings defined on the union of several subsets to the best proximity points of each of the involved subsets.
3 Asymptotic contractions and pseudocontractions of cyclic self-mappings in the intermediate sense
Let be nonempty subsets of X. is a cyclic self-mapping if and . Assume that the asymptotically nonexpansive condition (2.10), subject to (2.11), is modified as follows:
with ; as , and that the asymptotically nonexpansive condition (2.22), subject to (2.23), is modified as follows:
with ; as , where and . If , then and Theorems 2.1, 2.2 and 2.8 hold with the replacement . Then if A and B are closed and convex, then there is a unique fixed point of in . In the following, we consider the case that so that . The subsequent result based on Theorems 2.1, 2.2 and 2.8 holds.
Theorem 3.1Let be a metric space and let be a cyclic self-mapping, i.e., and , whereAandBare nonempty subsets ofX. Define the sequence of asymptotically nonexpansive iteration-dependent constants as follows:
, provided that satisfies the constraint (3.1), subject to (3.2), and
for ( ) and for ( ) provided that satisfies the constraint (3.3) subject to (3.4) provided that the parameterizing bounded real sequences , , and of general terms , and fulfill the following constraints:
and assuming that the following limits exist:
Then, the following properties hold:
(i) satisfies (3.3) subject to (3.4)-(3.9); . Then
so that is a cyclic asymptotically nonexpansive self-mapping. If is a best proximity point ofAand is a best proximity point ofB, then and and , which are best proximity points ofAandB (not being necessarily identical toxandy), respectively if is continuous.
(ii) Property (i) also holds if satisfies (3.1) subject to (3.2), (3.7), (3.8)-(3.9) and (3.5b) provided that ; .
Proof The second condition of (2.18) now becomes under either (3.1)-(3.2) and (3.8)-(3.9)
and it now becomes under (3.3)-(3.4) and (3.8)-(3.9)
since ; since and , and and as ; . Note that (3.8) implies that there is no division by zero in (3.11). Now, assume that (3.10) holds with . From (3.8) and (3.2), , equivalently, and , which contradicts (3.5a) if so that in (3.5a) under (3.7) implies that and, since from (3.6), there is no division by zero on the right-hand side of (3.10) if .
Also, if is continuous, then so that ; , , and since and . This proves Properties (i)-(ii). □
Remark 3.2 Note that Theorem 3.1 does not guarantee the convergence of and to best proximity points if the initial points for the iterations and are not best proximity points if is not contractive.
The following result specifies Theorem 3.1 for asymptotically nonexpansive mappings with ; subject to .
Theorem 3.3Let be a metric space and let be a cyclic self-mapping which satisfies the asymptotically nonexpansive constraint (3.1), subject to (3.2), whereAandBare nonempty subsets ofX. Let the sequence of asymptotically nonexpansive iteration-dependent constants be defined by a general term under the constraints , , and . Then the subsequent properties hold:
(i) The following limits exist:
(ii) Assume, furthermore, that is complete, AandBare closed and convex and is translation-invariant and homogeneous and is uniformly convex where is the metric-induced norm. Then
, ; , and , ; , , wherezandTzare unique best proximity points inAandB, respectively. If , then is the unique fixed point of .
Proof Note from (3.9), under (3.6) and (3.7), that there is no division by zero on the right-hand side of (3.10) and if . Then one has from (3.1)-(3.2), (3.5a), (3.6) and (3.7) that
There are several possible cases as follows.
Case A: is non-increasing. Then as ; . Since , one gets (3.12).
Case B: is non-decreasing. Then either as ; or it is unbounded. Then it has a subsequence which diverges, from which a strictly increasing subsequence can be taken. But this contradicts following from (3.14) subject to the given parametrical constraints. Thus, if is non-decreasing, it cannot have a strictly increasing subsequence so that it is bounded and has a finite limit as in Case A.
Case C: has an oscillating subsequence. It is proven that such a subsequence is finite. Assume not, then if , there is an integer sequence of general term subject to such that
but the above expression is equivalent, for and which are in , but not jointly in either A or B, to
which contradicts since both sequences and