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Results on proximal and generalized weak proximal contractions including the case of iteration-dependent range sets
Fixed Point Theory and Applications volume 2014, Article number: 169 (2014)
This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms for and , or , subject to and , such that converges uniformly to T, and the distances are iteration-dependent, where , , and are non-empty subsets of X, for , where is a metric space, provided that the set-theoretic limit of the sequences of closed sets and exist as and that the countable infinite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical identification of dynamic systems.
The characterization and study of existence and uniqueness of best proximity points is an important tool in fixed point theory concerning cyclic nonexpansive mappings including the problems of (strict) contractions, asymptotic contractions, contractive and weak contractive mappings and also in related problems of proximal contractions, weak proximal contractions and approximation results and methods [1–15]. The application of the theory of fixed points in stability issues has been proved to be a very useful tool. See, for instance, [16–18] and references therein. This paper is devoted to formulating and proving some further results for more general classes of proximal contractions. The problem of proximal contractions associated with uniformly converging non-self-mappings of the form ; , where and are in general distinct, with a set-theoretic limit of the form , provided that the set-theoretic limits of the involved set exist and that the infinite unions of the involved closed sets are also closed. Further related results are obtained for generalized weak proximal and proximal contractions in metric spaces [2, 3, 19], which are subject to certain parametrical constraints on the contractive conditions. Such constraints guarantee that the implying condition of the proximal contraction holds for the proximal sequences so that it can be removed from the analysis [1–3]. Some related generalizations are also given for non-self-mappings of the form , subject to a set distance , where A and are non-empty and closed subsets of a metric space for , provided that the set-theoretic limit of the sequence of sets exists as . The properties of convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated for the different constraints and the given extension. Application examples are given related to the exact and approximate solutions of compatible and incompatible linear algebraic systems and with the parametrical identification of dynamic systems [8–11].
denotes uniform convergence to a limit T of the sequence of, in general, non-self-mappings from A to B; .
and are, respectively, the sets of non-negative and positive integer numbers and and are, respectively, the sets of non-negative and positive real numbers.
The notation stands for a sequence with n running on simplifying the more involved usual notation . A subsequence for indexing subscripts larger than (respectively, larger than or equal to) is denoted as (respectively, as ).
The symbols ¬, ∨, ∧ stand, respectively, for logic negation, disjunction, and conjunction.
2 Proximal and asymptotic proximal contractions of uniformly converging non-self-mappings
Let us establish two definitions of usefulness for the main results of this section.
Definition 2.1 Let be a pair of non-empty subsets of a metric space . A mapping is said to be:
A proximal contraction if there exists a non-negative real number such that, for all , one has
An asymptotic proximal contraction if there exists a sequence of non-negative real numbers , with ; , with () as such that, for all sequences ,
If satisfy then and are a pair of best proximity points of T in A and B, respectively. Note that if is an asymptotic proximal contraction and the sequence is such that ; is a best proximity pair, then there is a subsequence of such that the relation ; , holds for some real constant .
Some asymptotic properties of the distances between the sequences of domains and images of the sequences of non-self-mappings ; , which converge uniformly to a limit non-self-mapping , are given and proved related to the distance between the domain and image of the limit non-self-mapping.
Lemma 2.2 Let be a metric space endowed with a homogeneous translation-invariant metric . Consider also a proximal non-self-mapping and a sequence of proximal non-self-mappings T and defined, respectively, by , having non-empty images of its restrictions ; , and ; , where and are non-empty subsets of X subject to and ; such that the sets of best proximity points:
are non-empty, where and ; . Let and be proximal sequences built in such a way that , , and ; and define also the error sequence by ; . Then the following properties hold:
If then .
If then .
If , and is uniformly Lipschitzian then .
If at exponential rate , such that for some real constant , then
Assume that is non-increasing and it converges linearly to g at rate . Then
Assume that is non-increasing and converges linearly to g at rate with order . Then
Proof Note that, since is a homogeneous translation-invariant metric, , , and ; , one has via induction by using the constraints that (), (); according to
and also one gets in a similar way
and properties (i)-(ii) follow directly. Also, if then and then property (iii) is proved from property (i) since
If, furthermore, and is uniformly Lipschitzian in its definition domain then there is a positive real constant such that ; and then property (iv) follows from property (iii) since
Property (v) follows from property (i) by using
Properties (vi)-(vii) follow from property (i) with
with and for property (vi) since is non-negative and converges to zero which leads for both (vi)-(vii) to
and also one gets for and
which together with (i) yields (vii). □
It turns out that Lemma 2.2 is extendable to the condition with the replacements and . Some results on boundedness of distances from points of the domains and their images of ; , and and their asymptotic closeness to the set distance are given in the subsequent result.
Lemma 2.3 Let be a metric space. Consider two sequences and built, respectively, under the proximal non-self-mapping and under the sequence of proximal non-self-mappings of Lemma 2.2 and assume that . Then, for any given , there is such that the following properties hold:
, ; .
for some non-negative sequences of integers with ; and some real constants and , then
where , and for any given arbitrary finite .
Proof Assume that the first assertion fails. Then such that ; . As a result, since by construction and , one gets
and some , a contradiction. Then the first assertion is true.
Now, assume that the second assertion fails. Then such that ; . As a result, since by construction and , one gets
and some , a contradiction. Then the second assertion is also true and property (i) has been proved. To prove (ii) note that, since , for any given , there are for such that
for some non-negative sequences of integers with ; , where ; , ; , , , and is a non-negative real constant, which is not dependent on n, defined by . Since , one gets
On the other hand, note that
and property (ii) has been proved. □
By interchanging the positions of and in the triangle inequality of (2.18), it follows that
so that if either , since ; is bounded, or is bounded, both sequences and are bounded. On the other hand, note that the non-negative sequence of integers might imply the use of an infinite memory in the upper-bounding term of (2.11) if ; or a finite memory of such a bound if ; .
Let be a pair of non-empty subsets of a complete metric space . The set A is said to be approximatively compact with respect to the set B if every sequence such that for some has a convergent subsequence.
Theorem 2.5 Let a proximal contraction and a sequence of proximal mappings and be defined, respectively, by having non-empty images of its restrictions ; and ; , where and are subsets of X, where is a metric space, subject to and ; and the set-theoretic limits , , and of the sequences of the sets , , , , respectively, exist and are non-empty being defined in the usual way for any sequence of subsets as:
via the binary set indicator sequences , satisfy the improper set inclusion condition . Assume that the sets of best proximity points , , , and are non-empty, where and ; . Then the following properties hold:
The set-theoretic limits and exist and they are non-empty and closed if the subsets of X are all closed and is closed while it satisfies the following set inclusion constraints:
If, furthermore, exists then it is non-empty and then
The set-theoretic limits and exist and they are non-empty and closed if the subsets of X; are all closed and is closed, and then
If, furthermore, exists then it is non-empty and
Assume that is a proximal contraction and that , where ; . Then ; , is a sequence of asymptotic proximal contractions.
If the sets in sequence , for some , are closed then is closed and also(2.21)
for any sequences , satisfying ; and ; for any sequences , being built in such a way that ; and ; .
If is complete and is approximatively compact with respect to then there is a convergent subsequence where is the limit of . If is approximatively compact with respect to then there is a convergent subsequence where is the limit of . If , where and as , and the above two approximative compactness conditions hold and, furthermore, and are closed; and and (it suffices that and be closed for some ) are closed then , which is then the unique best proximity point of T in the limit set .
Proof Note that since and exist, we have
Note also that, since exists, we have since
Also, since exists, we have , which follows under a similar reasoning. On the other hand, the existence and non-emptiness of , by hypothesis, implies that exists since ; so that
From the above set inclusion conditions it also holds that
if exists. If the subsets ; are all closed and is closed then is closed for any . In order to prove that is closed, that is, an infinite intersection of unions of infinitely many closed sets is closed, it is first proved that all those unions are closed under the given assumption that is closed. Assume this not to be the case, so that there is such that is closed with not being closed and such that is closed, since it is the union of a finite number of closed sets. Then either is not closed or it is closed. In the second case, since if is not closed then cannot be closed by construction. But, if then is closed, which contradicts that it is not closed, since is closed for any . As a result being no closed for any implies that is not closed. By complete induction, it follows that is not closed for any . Thus, is not closed what contradicts that it is closed. As a result, since is non-empty and closed, any infinite union of non-empty closed sets is also non-empty and closed for any since is assumed to be closed by hypothesis and it is trivially non-empty. Then is non-empty and closed since it is the infinite intersection of infinitely many unions of non-empty closed sets (but already proved to be non-empty and closed) and there exists the set limit which is now proved to be non-empty. Proceed by contradiction by assuming that it is empty. Then there is some such that . But then, since , is non-empty and closed and all the sets in the sequence are closed we have for some so that from the definition of the non-self-mapping T which contradicts the existence of such that . It has been proved that is closed and exists and it is non-empty and closed. The proof that is similar to the above one. Now, one proves by contradiction that is non-empty if is non-empty. The limit set is non-empty since the subsets of X; are all closed and is closed under similar arguments that those used above to prove those properties for . Assume that is empty. Since is non-empty, there is such that . Since the sets in the sequence are closed, and , since , for some . Thus, , from the definition of T and since . Then is non-empty since is non-empty. On the other hand, since we have
In the same way, it follows that exists and if, in addition, exists then
Hence, property (i) has been proved. Now, build sequences and , and and in X sequences built in such a way that , , , and ; . It follows inductively by using those distance constraints that (), (); . Note that implies the following inclusion of limit sets of the sets of best proximity points and the four above limit sets are trivially non-empty. Since is a proximal contraction:
for and some real constant and as , since the limit set exists. On the other hand, if is a sequence of asymptotic proximal contractions ; then there is a real sequence with a subsequence such that
for and some and some real constant . Recall the following properties for logical assertions then to be used related to (2.24). Consider the logical propositions ; . Then
The condition (2.25) corresponds to (2.26a) with and . It is now proved by contradiction that is a sequence of asymptotic proximal contractions, that is, by assuming that (2.25) is false so that its logical negation
for and some , obtained from (2.26a)-(2.26b) versus to (2.26a), is true. Then it follows from (2.24) that for any given arbitrary , there are such that ; (since ) and (2.27) hold. Then
Since is arbitrary, , and as since the limit set exists and , since , one concludes from (2.28) that
Since and , so that , since is closed by hypothesis for some and is also closed. Then so that the possibility of Ty being undefined is excluded and then Ty is defined provided that there is no finite or infinite jump discontinuities at y. If T is continuous at y, then (2.29) leads to the contradiction . Otherwise, if T has a (finite or infinity) jump discontinuity at y, with left and right limits and (), then , again a contradiction is got. Thus, (2.27) is false so that its negation (2.22) is true. Then is a sequence of asymptotic proximal contractions which converge uniformly to the proximal contraction T. Hence, property (ii) has been proved.
Property (iii) is proved by taking into account also (2.24) and the constraints and together with the fact that, since is closed, is closed. Note that the conditions of , , being closed and and being closed can be relaxed in property (i) to closeness of the sets , , and and for some being closed while keeping the corresponding result. Assume this not to be the case. Now, take a subsequence of (non-empty closed) subsets of X such that there is such that , , since is not closed, and , the indicator variable of x in any subset in the sequence is then , since , and which contradicts . Thus, is non-empty and closed. Then one obtains (2.21)-(2.23).
On the other hand, since as and is approximatively compact with respect to for some sequence
with . From property (i), , . Then there is a subsequence since is closed and it is obvious that () so that . Assume that is empty then the best proximity point , a contradiction so that is non-empty and it is closed since it is on the boundary of which is then non-empty and closed. But (). Since is convergent all its subsequences are convergent to the same limit so that . Thus, . Under a similar reasoning, it can be proved under the second given approximative compactness condition that with being non-empty and closed. If the above both approximative compactness conditions jointly hold and then one gets from the contractive condition for the proximal non-self-mapping T that for any given :
so that and implies so that holds what implies , which has to be necessarily unique from the identity, itself. Hence, property (iv) has been proved. □
Remark 2.6 Note that the conditions of , , being closed and the infinite countable unions and being closed can be relaxed in Theorem 2.5 (property (i)) to the closeness of the sets , , and and for some being closed while keeping the corresponding result.
The following remark is of interest; it concerns a condition of validity of the assumption that a countable union of closed sets is closed used in Theorem 2.5.
Remark 2.7 It can be pointed out that a sufficient condition for the infinite countable union of closed subsets (respectively, ) of a topological space to be closed is that X has the local finiteness property, that is, each point in X has a neighborhood which intersects only finitely many of the closed sets in (respectively, in ) (, pp.29-31). This property can also be applied to a metric space since metric spaces are specializations of topological spaces where the metric is used to define the open balls of the topology. More general results guaranteeing that the infinite countable union of closed sets is closed, and equivalently that the infinite countable intersection of open sets is open, stand also for Alexandrov spaces (topological spaces under topologies which are uniquely determined by their specialization preorders) and for P-spaces (the intersection of countably many neighborhoods of each point of the space is also a neighborhood of such a point).
3 Weak proximal contractions of uniformly converging non-self-mappings
Let us establish two definitions of usefulness for the main results of this section.
Definition 3.1 Let be a pair of non-empty subsets of a metric space . A mapping is said to be:
A generalized asymptotic weak proximal contraction if there are sequences of non-negative real numbers , with ; and () as ; and , with ; and () as such that, for all sequences :(3.1)
is a strongly generalized asymptotic weak proximal contraction if:
and with ; are non-empty and ; .
for any sequences and such that (); and , are real non-negative sequences which satisfy and
The sequence of set distances converges and , where(3.2b)
Note if in Definition 3.1(2), one has the subclass of weak proximal contractions. In this case, one gets by making ; that ; , so that as , since , provided that and ; implying as and under the conditions that and converge they should necessarily converge to best proximity points. Note that Definition 3.1(1) relaxes Definition 3.1(2) and Definition 3.1(3) allows considering weak proximal contractions with sequences built from non-self-mappings which have iteration-dependent image sets.
The following results hold.
Proposition 3.2 Let A and ; be non-empty subsets of a metric space . Assume that , such that is non-empty and ; , with ; satisfies the contractive condition:
Then the following properties hold:
which is bounded for any finite if and , where , and the sequence satisfies (3.2b) which can be relaxed to (3.5) below(3.5)
If (3.3) holds, subject to (3.5), with then(3.6)
if as .
If (3.3) holds subject to (3.5) with then(3.11)(3.12)(3.13)
If and as then the limits below exist and are identical:(3.14)
If (3.3) holds, subject to (3.5), with , and then .
Proof Since we have ; . Also, if follows from (3.5) and (3.3) with that (3.4) holds since
for any given . If and then the sequence is bounded. Thus, property (i) has been proved. The relations (3.6) and (3.7) of property (ii) follow directly from (3.4) of property (i) if . On the other hand, the triangle inequality and (3.5) lead to (3.11) since
The relation (3.9) follows from (3.7) and (3.8). To prove (3.10), note that if , then for any given , there is such that for any and then, from (3.10), . Since ε is arbitrary, the limit exists and . This property and (3.8) yield directly and then property (ii) has been fully proved. To prove property (iii), note that (3.11) holds directly from ; and . Also, (3.5) leads to (3.12) by taking into account (3.7). The relation (3.13) follows from (3.7), (3.12), and the relation
Hence, property (iii) has been proved. Property (iv) is a direct consequence of property (iii) for the case when and as including its particular sub-case when and . □
Note that Proposition 3.2 is applicable to the strongly generalized asymptotic weak proximal contraction of Definition 3.1(3) which do not need the fulfilment of the implying part of the logic proposition of Definition 3.1(1)-(2) but the distances of sequences of sets satisfy (3.2b) or, at least, (3.5). The subsequent result is concerned with the existence and uniqueness of best proximity points if, in addition to the assumptions of Proposition 3.2, the set-theoretic limit of the sequence exists and is closed and approximatively compact with respect to A.
Theorem 3.3 Under all the assumptions of Proposition 3.2 and property (iii), equation (3.10), assume also that is complete, that A and ; , are non-empty subsets of X such that A is closed, is non-empty, the set-theoretic limit exists, is closed and approximatively compact with respect to A (or the weaker condition that is closed) and . Assume also that the non-self-mapping restriction , for some subset , which contains the set of best proximity points , is a strongly generalized asymptotic weak proximal contraction. Then has a unique best proximity point if .
Proof Since as , from (3.10). Since and A is closed we have . Since (→D) since , because ; and ; . Since B is approximatively compact with respect to A and , there are and a sequence , such that , then since , and such that one gets as :
, and . Since B is approximatively compact with respect to A and , the sequence , such that , has a convergent subsequence , since B is also closed and both and have the same limit . Also, and () as so that x is a best proximity point of . Note that since the limit set B exists, it is by construction the infinite union of intersections of the form so that so that there is a restriction for some which is non-empty. Assume not so that . If then is also empty which is impossible then . It is now proved by contradiction that the best proximity point is unique. Assume this not to be the case so that there are two best proximity points x, y such that there are two sequences and contained in A. Since is a strongly generalized asymptotic weak proximal contraction, one gets from the implied logic proposition of (3.1) with and and that
which fails for if . Thus, . □
Closely to Theorem 3.3, the following result can be proved.
Theorem 3.4 Let A and ; be non-empty closed subsets of a complete metric space . Assume that is a weakly generalized asymptotic weak proximal contraction, such that is non-empty and ; , with and ; . Assume also that the set limit exists, is closed and approximatively compact with respect to A, or instead, the weaker condition that is closed. Then has a unique best proximity point.
Two examples are described to the light of proximal contractions. The first one is concerned with the solution of algebraic systems which can have more or less unknown than equations and which can be compatible or not. The second one is referred to an identification problem of a discrete dynamic system whose parameters are unknown and which can be subject to unmodeled dynamics and/or exogenous noise which makes not possible, in general, an exact identification.
Example 4.1 (Moore-Penrose pseudo-inverse)
The problem of solving either exactly or approximately a linear system of algebraic equations is very important and it appears in many engineering and scientific applications. It is possible to focus it to the light of best proximity points of non-self-mappings as follows. Consider the linear algebraic system where (a real matrix of order ) and . It is known from the Rouché-Frobenius theorem from Linear Algebra that a solution exists if . The solution is unique given by if , and with the algebraic system being determined compatible. If then there are infinitely many solutions and the algebraic system is indetermined compatible being, in particular, overdetermined if and undetermined if . If then the algebraic system is incompatible. A more general setting is , where , are given and is a solution which exists if and only if . The following cases hold:
If , then , and there are infinitely many solutions of the form with being the Moore-Penrose pseudo-inverse of C. The domain of the non-self-mapping , represented by the matrix C, can be restricted to . To close a proper formalism we extend the matrices in A to matrices and those in B to by adding zero columns (if either or ) and we consider them as subsets of and consider the metric space with d being the Euclidean metric so that with the sets of best proximity points of and being:
is the set of solutions of the compatible indeterminate algebraic system.
If then the solution is unique and consist of one element which is the unique solution.
If then and the algebraic system is incompatible. By considering as a Banach space endowed with the Euclidean norm, we can check for the best solution which minimizes over if it exists. Such a solution exists in a least-squares sense and it is unique if , since is non-singular, of order p, and , and minimizes over the set of matrices so that , the sets of best proximity points of being
since and is as the above one of case (a). is the best solution of the incompatible algebraic system.
The pseudo-inverse can be calculated without inverting by the iterative process:
(so-called Ben-Israel-Cohen, or hyper-power sequence, iterative method ). It follows that as since (1) is unique; and (2) the iterative process satisfies the pseudo-inverse properties and under the replacement ; . By using the iterative process, we can also define sequences of sets and associate sequences of distances by:
and are unique, if the pseudo-inverse exists for the given initial . However, if , so that (case (a) - incompatible algebraic system) then if we use the iterative procedure:
is a non-unique set of arbitrary solutions of the incompatible algebraic system of which the best (error-norm minimizing) solution is the unique one described above. For initial conditions satisfying , for instance, , , the pseudo-inverse converges quadratically to its limit, that is, . Furthermore,
and, since the convergence of the pseudo-inverse is quadratic, there is a bounded positive sequence