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Strong convergence and stability of Picard iteration sequences for a general class of contractive-type mappings
Fixed Point Theory and Applicationsvolume 2014, Article number: 233 (2014)
Let be a normed linear space, be a mapping of E into itself satisfying the following contractive condition: , for each , , where is a sub-additive monotone increasing function with and , , . It is shown that the Picard iteration process converges strongly to the unique fixed point of T. Furthermore, several classes of nonlinear operators studied by various authors are shown to belong to this class of mappings. Our theorem improves several recent important results. In particular, it improves a recent result of Akewe et al. (Fixed Point Theory Appl 2014:45, 2014), and a host of other results.
MSC:accretive-type mappings, pseudocontractive mappings, Picard sequence, contractive-type mappings.
1 Nonlinear operators of accretive-type and fixed points
Let K be a nonempty subset of a real normed space E. A mapping is called Lipschitz if there exists such that
If , the map T is called a contraction map, and if , T is called nonexpansive.
Let H be a real Hilbert space; a mapping is called monotone if
Let denote the topological dual space of E. A map defined by
is called the normalized duality map on E. It is well known that if is strictly convex then J is single-valued. In the sequel, single-valued normalized duality map will be denoted by j. In real Hilbert spaces, the normalized duality map is the identity map. A mapping A with domain and range in E is called accretive if, for all , the following inequality is satisfied:
As a consequence of a result of Kato , it follows from inequality (1.3) that A is accretive if, for each , there exists such that
where is the normalized duality map on E. It follows, again from inequality (1.3), that A is accretive if and only if is expansive and, consequently, its inverse exists and is nonexpansive as a mapping from into , where denotes the range of . The range of does not need to be all of E. This leads to the following definition.
Definition 1.1 An operator A is said to be m-accretive if A is accretive and the range of is all of E for some .
It can be shown that if for some , then it holds for all . The operator −Δ, where Δ denotes the Laplacian, is an m-accretive operator. Let be a convex functional on a real normed space E. The subdifferential of f, denoted by ∂f, is a map defined for each by
If , a real Hilbert space, it is easy to show that the subdifferential of f is a maximal monotone operator. Furthermore, it follows from (1.5) that if zero is in the subdifferential of f at some , then is a minimizer of f. Thus, for a convex functional f on a real Hilbert space, solving the inclusion
amounts to finding a minimizer of f. More generally, we have the inclusion
where A is a maximal monotone operator is of great interest in nonlinear operator theory.
The accretive operators were introduced independently in 1967 by Browder  and Kato . Interest in such mappings stems mainly from their firm connection with the existence theory for nonlinear equations of evolution in Banach spaces of the form
where A is an accretive map on an appropriate Banach space. At equilibrium, and solving the equation
where A is an accretive operator amounts to solving for the equilibrium points of the evolution system (1.7).
Browder converted (1.8) to a fixed point problem. He introduced an operator T defined as follows: , where A is accretive and called such a T, pseudocontractive. It is clear that fixed points of T correspond to zeros of A.
Pseudo-contractive maps are not necessarily continuous. The map defined by
is pseudocontractive but is neither nonexpansive nor continuous.
Existence of solutions of system (1.7) has been established. Browder  proved that the system is solvable if A is locally Lipschitzian and accretive on E, and utilizing the existence result for system (1.7), he proved that if A is locally Lipschitz and accretive on E, then A is m-accretive.
Martin  proved that if A is continuous and accretive on E, then A is m-accretive. Browder  further proved that if is Lipschitz and strongly accretive (i.e., there exists such that for each , there exists such that ) then A is surjective. This result was subsequently generalized by Deimling  to the continuous strongly accretive operators (see, e.g., Deimling [, Theorem 13.1]). For details of accretive and monotone operators, the reader may consult Reich [6, 7].
2 Iterative methods for solutions of certain nonlinear equations
We begin with the well-known and celebrated contraction mapping principle.
Theorem 2.1 (Contraction mapping principle)
Let be a complete metric space and be a contraction map of X into itself. Then
T has a unique fixed point, say in X;
the sequence in X defined by ,(2.1)
converges to .
Theorem 2.1 is, perhaps, the most important fixed point theorem. The sequence of the recursion formula (2.1) is called the Picard sequence.
One important (see, e.g., [, p.57]) class of nonlinear mappings generalizing the class of contraction mappings is the class of nonexpansive mappings. Readers interested in nonexpansive mappings may consult, for example, Goebel and Reich , Reich .
If K is a nonempty compact convex subset of and is a nonexpansive map, even with a unique fixed point, the Picard sequence defined by (2.1) may fail to converge to the fixed point. It suffices to take and let be a rotation of K about the origin of coordinates through a fixed angle θ, (say). It is easy to check that T is nonexpansive, zero is the unique fixed point of T and that the Picard sequence (2.1) with fails to converge to zero.
Following research efforts by Mann , Krasnoselskii , Schaefer , Ishikawa , Edelstein [15–17], Reinermann , Edelstein and O’Brian , Chidume , and a host of other authors, the following recursion formula was developed and found to be effective for approximating fixed points of nonexpansive mappings.
Let K be a nonempty convex subset of a normed space E and be a nonexpansive map. Let the sequence in K be defined by
where is a sequence in satisfying the following conditions: (i) , (ii) . If the sequence is bounded, Ishikawa  proved that the sequence is an approximate fixed point sequence in the sense that
Edelstein and O’Brian  considered the recursion formula
where T maps K into K and proved that if K is bounded, then the convergence in (2.3) is uniform.
Chidume  considered the recursion formula (2.2), introduced the concept of admissible sequence, and proved that if K is bounded, then the convergence in (2.3) is uniform for the sequence defined by (2.2).
Remark 1 We note here that the recursion formula (2.2) which is certainly cumbersome when compared with Picard iteration was developed for the class of nonexpansive maps because the simpler Picard sequence will not always converge for nonexpansive maps. Furthermore, the recursion formula (2.2) can only yield the result that the sequence defined by (2.2) satisfies (2.3). In general, it does not yield convergence of the sequence to a fixed point of T. To obtain convergence to a fixed point of T, some type of compactness condition must be imposed either on K or on the map T (e.g., T may be required to be demicompact at zero, or may be required to map closed bounded subsets of E into closed subsets of E, etc.; see, e.g., Chidume ). The recursion formula (2.2) is now generally referred to as Mann formula in the light of Mann .
An important class of mappings generalizing the class of nonexpansive mappings is the class of Lipschitz pseudocontractive maps. It is not difficult to check that every nonexpansive map is a Lipschitz pseudocontraction. We have already given an example of a pseudocontractive map which is not even continuous. All attempts to use the Mann formula, which has been successfully employed for nonexpansive mappings, to approximate a fixed point of a Lipschitz pseudocontractive map even on a compact convex domain in a real Hilbert space, proved abortive. In 1974, Ishikawa  proved the following theorem.
Theorem IS Let K be a nonempty compact convex subset of a real Hilbert space H and be a Lipschitz pseudocontractive map. Let the sequence be defined by ,
where and are real sequences satisfying the following conditions: (i) ; (ii) ; (iii) . Then converges strongly to a fixed point of T.
Remark 2 It is clear that the recursion formulas (2.5) and (2.6) of the Ishikawa scheme are more cumbersome than the Mann formula (2.2). However, since it was not known whether or not the simpler Mann sequence would always converge to fixed points of Lipschitz pseudocontractive maps, the cumbersome Ishikawa scheme was applied for this class of maps. The question of whether or not the simpler Mann sequence had actually failed for this class of maps remained open for many years. This was resolved in 2001 by Chidume and Mutangadura  who produced an example of a Lipschitz pseudocontractive map defined on a compact convex subset of with a unique fixed point for which no Mann sequence converges.
Remark 3 (a) We first observe that if we set ∀n in the recursion formula (2.6) then condition (i) in Theorem IS shows that ∀n and so (2.5) and (2.6) reduce to ∀n, so that converges to , the initial approximation which may not be a fixed point of T.
(b) Because the Ishikawa formulas were used successfully in approximating a fixed point of T in Theorem IS, several authors started studying a modification of it in which condition (i) is replaced by the condition: (i)∗ , and condition (ii) is modified accordingly. In this modification, and are independent and it is permissible to set for all n. They still called such a modified formula an Ishikawa formula. This is wrong. To see this, it suffices to set ∀n and see that the sequence obtained from the modified scheme will not converge to a fixed point of T in Theorem IS. In particular, if ∀n, the modified formula generally reduces to the Mann formula and then the example of Chidume and Mutangadura  shows that the modified formula will not converge to a fixed point of T in the setting of Theorem IS (see, e.g.,  for more comments on the Ishikawa iteration formula).
(c) The order of convergence of the Picard sequence is that of a geometric progression, that of the Mann sequence is of the form , while that of the Ishikawa sequence is of the form . Furthermore, whenever Picard sequence converges, it is preferred to the Mann sequence which itself is preferred to the Ishikawa formula whenever it converges, because the preferred recursion formula is simpler (consequently requiring less computation and therefore reducing cost of computation).
Three other iteration methods have been introduced and have successfully been employed to approximate fixed points of Lipschitz pseudocontractive mappings even in Banach spaces more general than Hilbert spaces.
Let K be a nonempty closed convex and bounded subset of a Hilbert space H. Suppose that is a pseudocontractive and Lipschitzian map with constant . For arbitrary , Schu  defined the following two-step iteration process:
where the real sequences and are in and satisfy appropriate conditions and are such that has property A (see Schu  for a definition). Schu proved that converges strongly to the unique fixed point of T nearest to w.
This result was extended by Chidume  to real Banach spaces possessing weakly sequential continuous duality maps (e.g., spaces, ).
A second iteration scheme for approximating fixed points of Lipschitz pseudocontractive mappings was implicitly introduced by Bruck  who actually applied the scheme, still in Hilbert spaces, to approximate a solution of the inclusion where A is an m-monotone operator.
Let H be a Hilbert space, be an m-monotone operator with , the range of A. For arbitrary , Bruck considered the sequence in H defined by ,
and proved that if and are bounded, then converges strongly to some , solution of , provided and are acceptably paired sequences (e.g., see  for a definition).
An example of acceptably paired sequences given in  is , , .
The ideas of sequences with property A and sequences that are acceptably paired are due to Halpern . Reich  also studied the recursion formula (2.9) for Lipschitz accretive operators on real uniformly convex Banach spaces with a duality mapping that is weakly sequentially continuous at zero.
Motivated by the papers of Abbas et al. , Chidume , Reich [26, 29, 30], Shahzad and Al-Dubiban , Chidume and Zegeye  studied the following perturbation of the Mann recurrence relation to approximate fixed points of Lipschitz pseudocontractive mappings in real Banach spaces much more general than Hilbert spaces. Let E be a real normed space, K be a nonempty convex subset of E, be a Lipschitz pseudocontractive map. For arbitrary , let the sequence be defined iteratively by
where and are real sequences in satisfying appropriate conditions. They proved the following theorem.
Theorem CZ ()
Let K be a nonempty closed convex subset of a real Banach space E. Let be a Lipschitz pseudocontractive map with constant and . Let a sequence be generated from arbitrary by
for all positive integers n, where and are real sequences in satisfying appropriate conditions.
Then as .
Remark 4 Real sequences applicable in Theorem CZ are
Remark 5 We have shown in this section that a cumbersome recurrence relation is desirable and introduced only when a simpler recurrence formula is not available for the class of mappings under consideration. It is obvious that whenever a k-step method works for any class of maps, it is trivial to construct an n-step method that will work for the same class of maps, , . Such an n-step method will, in general, require more computation time and therefore will be less efficient than the k-step method. In general, the rate of convergence of such an n-step method is at best the same as that of the k-step method. Consequently, such n-step methods serve no useful purpose and are therefore not desirable.
3 A strong convergence theorem
Akewe et al.  stated the following theorem.
Theorem AOO ([, Theorem 2.2, p.7])
Let be a normed linear space, be a self-map of E satisfying the following contractive condition:
for each , , where is a sub-additive monotone increasing function with and , , . For , let be the Kirk-multistep iterative scheme defined by
where , for each , , , .
T defined by (3.1) has a unique fixed point p;
the Kirk-multistep iterative scheme (defined by (3.2)) converges strongly to the fixed point p of T.
Remark 6 The authors of Theorem AOO did not prove (i) as claimed. The existence of a fixed point of T was not proved. What the authors showed is that if T has a fixed point, then the fixed point is unique.
Remark 7 Kirk  introduced the following one-step iterative method for approximating a fixed point of a nonexpansive map T: :
He never introduced the multistep method defined in (3.2).
Remark 8 In Theorem AOO, the summation is simply . Therefore, the contractive condition (3.1) reduces to the following:
so that, setting , we have with and , , and thus obtain the following contractive-type condition:
which is basically a compact form of (3.1).
Now, assuming the existence of a fixed point for a mapping T satisfying contractive condition (3.1) or (3.5), we prove that a Picard sequence for converges strongly to the unique fixed point of T.
Theorem 3.1 Let be a real normed space and be a map satisfying the contractive condition (3.1) or (3.5), with constant . Assume that T has a fixed point . For arbitrary , let be a sequence defined by
Then converges strongly to p.
Proof Since , put and in the contractive condition (3.1) or (3.5), to obtain
for all , where . Using formula (3.6) and inequality (3.7), we obtain
Hence, converges strongly to p. This completes the proof. □
Another general class of mappings generalizing the contraction mappings was introduced by Hardy and Rogers  as follows: Let be a complete metric space and satisfy the following contractive condition: ,
where , . Hardy and Rogers proved that T has a unique fixed point. Several authors proved fixed point theorems for mappings satisfying special cases of the contractive condition of Hardy and Rogers. We observe that if p denotes the unique fixed point of T in the theorem of Hardy and Rogers, then the following inequality holds:
Remark 9 If an operator satisfies inequality (3.7) where p is a fixed point of T, then p is necessarily unique. For assume that there exists such that . Then so that , which yields .
Remark 10 The T-stability of the Picard iterative scheme, whenever it converges, is well known (see, e.g., Ostrowski , Berinde [37, 38], Bruck , Rhoades , Harder and Hicks , Shahzad and Zegeye ).
Remark 11 In the light of Remark 5, our theorem is a significant improvement on the results of Akewe et al.  in the sense that the Picard sequence defined by (3.6) is much simpler than the multistep methods (3.2) considered in . Furthermore, the Picard sequence converges as fast as a geometric progression whereas convergence with the multistep methods considered in  is either of order or of order .
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The author declares that there are no competing interests.