The viscosity iteration process (3.1) is generalized in this section by including two more forcing terms not being directly related to the solution sequence. One of them being dependent on a nondecreasing distancevalued function related to a complete metric space while the other forcing term is governed by an external sequence . Furthermore the sum of the four terms of the scalar sequences , , and and at each sample is not necessarily unity but it is asymptotically convergent to unity.
The following generalized viscosity iterative scheme, which is a more general difference equation than (3.1), is considered in the sequel
for all for a sequence of given finite numbers with (if , then the corresponding sum is dropped off) which can be rewritten as (2.1) if ; for all (except possibly for a finite number of values of the sequence what implies ) by defining the sequence
with , where
(i) is a strongly leftregular sequence of means on , that is, . See Definition 3.5;
(ii) is a left reversible semigroup represented as Lipschitzian mappings on by .
The iterative scheme is subject to the following assumptions.
Assumption 1.
() , , and are real sequences in , is a real sequence in , and are sequences in , for all for some given and .
() , .
() .
() .
() ; for all with being a bounded real sequence satisfying and .
() is a contraction on a nonempty compact convex subset , of diameter of a Banach space , of topological dual , which is smooth, that is, its normalized duality mapping from into the family of nonempty (by the HahnBanach theorem [6, 11]), weakstar compact convex subsets of , defined by
is single valued.
() The representation of the left reversible semigroup with identity is asymptotically nonexpansive on (see Definition 3.3) with respect to , with which is strongly left regular so that it fulfils .
() .
() is a complete metric space and is a selfmapping satisfying the inequality
where , for all are continuous monotone nondecreasing functions satisfying if and only if ; for all .
() is a sequence in generated as , with and is a finite given number.
Note that Assumption 1() is stronger than the conditions imposed on the sequence in Theorem 2.1 for (2.1). However, the whole viscosity iteration is much more general than the iterative equation (2.1). Three generalizations compared to existing schemes of this class are that an extracoefficient sequence is added to the set of usual coefficient sequences and that the exact constraint for the sum of coefficients being unity for all is replaced by a limittype constraint as while during the transient such a constraint can exceed unity or be below unity at each sample (see Assumption 1(). Another generalization is the inclusion of a nonnegative term with generalized contractive mapping involving another iterative scheme evolving on another, and in general distinct, complete metric space (see Assumptions 1() and 1(). Some boundedness and convergence properties of the iterative process (4.1) are formulated and proven in the subsequent result.
Theorem 4.1.
The difference iterative scheme (4.1) and equivalently the difference equation (2.1) subject to (4.2) possess the following properties under Assumption 1.
(i). Also, and for any norm defined on the smooth Banach space and there exists a nonempty bounded compact convex set such that the solution of (4.2) is permanent in , for all and some sufficiently large finite with .
(ii) and as .

(iii)
(iv)Assume that such that each sequence element (the first closed orthant of ); for all , for some so that (4.1) is a positive viscosity iteration scheme. Then,
(iv.1) is a nonnegative sequence (i.e., all its components are nonnegative for all , for all ), denoted as ; for all .
(iv.2)Property (i) holds for and Property (ii) also holds for a limiting point .
(iv.3)Property (iii) becomes
what implies that either
or
Proof.
From (4.2) and substituting the real sequence from the constraint Assumption 1(), we have the following:
Thus,
where is an arbitrary finite sufficiently large integer, and
since the functions are continuous on with and as , [2] with being prefixed and arbitrarily small. The constants and are finite for sufficiently large since (Assumption 1(), is a contraction on (Assumption 1(), and is a selfmapping on satisfying Assumption 1(). Since , and as from Assumptions 1() and 1() and is finite, as and ; for all being arbitrarily small since is arbitrarily large. Since from Assumption 1(), is an asymptotically nonexpansive semigroup on , and , , and as :
with as . One gets from (4.12) into (4.10),
what implies that
(see [8]) since as since is in and is a strongly leftregular sequence of means on such that ; furthermore, , , , , as and as . Thus, from (4.14) and using the above technical result in [8] for difference equations of the class (2.1) (see also [2]), it follows that
since from Assumption 1() since , , and as . From (4.1),
so that
Using Assumption 1 and using (4.15) into (4.17) yield
since , , , as . Also, it follows that as from (4.15) and (4.18). Note that it has not been proven yet that the sequences and converge to a finite limit as since it has not been proven that they are bounded. Thus, the four sequences and converge asymptotically to the same finite or infinite real limit. Proceed recursively with the solution of (4.1). Thus, for a given sufficiently large finite and for all , one gets
for all , for some positive real sequences, , and satisfying and , and as with and being arbitrarily small for sufficiently large , and
for sufficiently large and a sufficiently small which exists from Assumptions 1() and 1(). Note that the sequences and may be chosen to satisfy and ; for all . Now, proceed by complete induction by assuming that for given sufficiently large and finite . Then, one gets from (4.20) that for any prescribed if
with and which always holds for sufficiently large finite since as . It has been proven by complete induction that the first part of Property (i) holds with the set being built such that for the given initial condition . For a set of initial conditions with any set convex and bounded, a common set might be defined for any initial condition of (4.1) in with a redefinition of the constant as . The second part of Property (i) follows for any norm on from the property of equivalence of norms. Furthermore, the real sequences and converge strongly to a finite limit in since they are uniformly bounded so that Property (ii) has also been proven. Property (iii) follows directly from (4.1) and Property (ii). Property (iv.) follows since is a nonnegative vector sequence provided that if what follows from simple inspection of (4.1). Properties (iv.)(iv.) follow directly from separating nonnegative positive and nonpositive terms in the righthand side of the expression in Property (iii).
The convergence properties of Theorem 4.1(ii) are now related to the limits being fixed points of the asymptotically nonexpansive semigroup which is the representation as Lipschitzian mappings on of a left reversible semigroup with identity.
Theorem 4.2.
The following properties hold.

(i)
Let be the set of fixed points of the asymptotically nonexpansive semigroup on . Then, the common strong limit of the sequences and in Theorem 4.1(ii) is a fixed point of located in and, thus, a stable equilibrium point of the iterative scheme (4.1) provided that , and then , is sufficiently large.
(ii).
Proof.

(i)
Proceed by contradiction by assuming that so that there exists such that
since , where the above two limits exist and are zero from Theorem 4.1(ii). Then, , with being nonempty since, at least one such finite fixed point exists in .
Property (ii) follows directly from Theorem 4.1(iii)(iv).
Remark 4.3.
Note that the boundedness property of Theorem 4.1(i) does not require explicitly the condition of Assumption 1() that is asymptotically nonexpansive. On the other hand, neither Theorem 4.1 nor Theorem 4.2 requires Assumption 1().
Definition 4.4 (see [8]).
Let the sequence of means be in , and let be a representation of a left reversible semigroup . Then is stable if the functions and on are also in ; for all , for all .
Definition 4.5 (see [8, 11]).
Let and be convex subsets of the Banach space , with under proper inclusion, and let be a retraction of onto . Then is said to be sunny if ; for all , for all provided that .
Definition 4.6.
is said t be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto .
It is known that if is weakly compact, is a mean on (see Definition 3.5), and is in for each , then there is a unique such that for each . Also, if is smooth, that is, the duality mapping of is single valued then a retraction of onto is sunny and nonexpansive if and only if , for all [6, 11].
Remark 4.7.
Note that Theorem 4.2 proves the convergence to a fixed point in , with being constructively proven to be nonempty by first building a sufficiently large convex compact so that the solution of the iterative scheme (4.1) is always bounded on . Note also that Theorems 4.1 and 4.2 need not the assumption of being a leftinvariant stable subspace of containing "" and to be a leftinvariant mean on , although it is assumed to be strongly left regular so that it fulfils ; for all (Assumption 1(), see Definition 3.6. However, the convergence to a unique fixed point in the set is not proven under those less stringent assumptions. Note also that Assumption 1() required by Theorem 4.1 and also by Theorem 4.2 as a result is one of the two properties associated with the stability of .
The results of Theorems 4.1 and 4.2 with further considerations by using Definitions 4.4 and 4.5 allow to obtain the convergence to a unique fixed point under more stringent conditions for the semigroup of selfmappings, as follows.
Theorem 4.8.
If Assumption 1 hold and, furthermore, is a leftinvariant stable subspace of then the sequence , generated by (4.1), converges strongly to a unique ; for all , for all , for all which is the unique solution of the variational inequality . Equivalently, where is the unique sunny nonexpansive retraction of onto .
The proof follows under similar tools as those used in [6] since is a nonempy sunny nonexpansive retract of which is unique since is nonexpansive for all .
Proof.
Let be the sequence solution generated by the particular iterative scheme resulting from (4.1) for any initial conditions when all the functions and are zeroed. It is obvious by the calculation of the recursive solution of (4.1) from (4.19) that the error from both solutions satisfies
Since the convergence of the solution to fixed points of Theorems 4.1, 4.2, and 4.8 follows also for the sequence it follows that a unique fixed point exists satisfying
where is unique since is also unique from Theorem 4.8. Assume that with . Then,
If and ; for all and the functions are zero then both fixed points are related by the constraint . Thus, consider a representation of a left reversible semigroup as Lipschitzian mappings on (see Definitions 3.2 and 3.3), a nonempty compact subset of the smooth Banach space with Lipschitz constants which is asymptotically nonexpansive. Consider the iteration scheme:
with , where
(i) is a strongly leftregular sequence of means on , that is, (the dual of ). See Definitions 3.5 and 3.6;
(ii) is a left reversible semigroup represented as Lipschitzian mappings on by .
Assumption 2.
The iterative scheme (4.27) keeps the applicable parts of Assumptions 1()–1(), 1() for the nonidentically zero parameterizing sequences , and . Assumptions 1() and 1() are modified with the replacements , , and .
Theorems 4.1 and 4.8 result in the following result for the iterative scheme (4.27) for , :
Theorem 4.9.
The following properties hold under Assumption 2.
(i); for all . Also, and for any norm defined on the smooth Banach space and there exists a nonempty bounded compact convex set such that the solution of (4.2) is permanent in , for all and some sufficiently large finite with .
(ii) and as .
(iii)
(iv)Assume that the nonempty convex subset of the smooth Banach space , which contains the sequence of means on , is such that each element ; for all , for some so that (4.1) is a positive viscosity iteration scheme (4.27). Then,
(iv.1) is a nonnegative sequence (i.e., all its components are nonnegative for all , for all ), denoted as ; for all .
(iv.2)Property (i) holds for and Property (ii) also holds for a limiting point .
(iv.3)Property (iii) becomes
what implies that either
or
(v)If, furthermore, is a leftinvariant stable subspace of , then the sequence , generated by (4.27), converges strongly to a unique ; for all , for all which is the unique solution of the variational inequality . Equivalently, where is the unique sunny nonexpansive retraction of onto . Furthermore, the unique fixed points of the iterative schemes (4.1) and (4.27) are related by
If, in addition, and and the functions are identically zero in the iterative scheme (4.1), then .
Remark 4.10.
Note that the results of Section 4 generalize those of Section 2 since the iterative process (4.1) possesses simultaneously a nonlinear contraction and a nonexpansive mapping plus terms associated to driving terms combining both external driving forces plus the contribution of a nonlinear function evaluating distances over, in general, distinct metric spaces than that generating the solution of the iteration process. Therefore, the results about fixed points in Theorem 2.1(vi)(vii) are directly included in Theorem 4.1.
Venter's theorem can be used for the convergence to the equilibrium points of the solutions of the generalized iterative schemes (4.1) and (4.27), provided they are positive, as follows.
Corollary 4.11.
Assume that
() are both contractive mappings with being compact and convex, , such that is a leftinvariant stable subspace of with being a left reversible semigroup;
() , with being compact and convex, , , and ; for all for some real constants , and if ;
() and .
Then, the sets of fixed points of the positive iteration schemes (4.1) and (4.27) contain a common stable equilibrium point which is a unique solution to the variational equations of Theorems 4.8 and 4.9; that is, and that .
Outline of Proof
The fact that the mappings are both contractive, and imply that the generated sequences , are both nonnegative and bounded for any and they have unique zero limits from Theorem 2.1(v).
The following result is obvious since if the representation is nonexpansive, contractive or asymptotically contractive (Definitions 3.3 and 3.4), then it is also asymptotically nonexpansive as a result.
Corollary 4.12.
If the representation is nonexpansive, contractive or asymptotically contractive, then Theorems 4.1, 4.2, and 4.8 still hold under Assumption 1, and Theorem 4.9 still holds under Assumption 2.