Self-Adaptive Algorithms for the Split Common Fixed Point Problem of the Demimetric Mappings ()
1. Introduction
Let
and
be two real Hilbert spaces. Let
and
be two nonlinear mappings. We denote the fixed point sets of S and T by
and
, respectively. Let
be a bounded linear operator with its adjoint
. Then, we consider the following split common fixed point problem:
(1.1)
The split common fixed point problem (1.1) is a generalization of the split feasibility problem arising from signal processing and image restoration; see [1] - [7] for instance. It was first introduced and studied by Censor and Segal [8]. Note that solving (1) can be translated to solve the fixed point equation
Censor and Segal also proposed the following algorithm for directed mappings.
Algorithm 1.1 Initialization: let
be arbitrary. Iterative step: let
where
and
are two directed mappings and
with
being the spectral radius of the operator
.
Since then, there has been growing interest in the split common fixed point problem; please, see [9] - [15].
Recently, Wang [16] introduced the following new iterative algorithms for the split common fixed point problem of directed mappings.
Algorithm 1.2 Choose an arbitrary initial guess
. Assume
has been constructed. If
then stop; otherwise, continue and construct
via the formula:
where
is chosen self-adaptively as
Algorithm 1.3 Let
and start an initial guess
. Assume
has been constructed. If
then stop; otherwise, continue and construct
via the formula:
where the stepsize sequence
is chosen self-adaptively as
Wang obtained the weak and strong convergence of Algorithms 1.2 and 1.3, respectively. Inspired by the above work in the literature, Yao, et al. [17] extend Wang’s results in [16] from the directed mappings to the demicontractive mappings. Further, they construct the following two self-adaptive algorithms for solving the split common fixed point problem (1.1).
Algorithm 1.4. Initialization: let
be arbitrary. For
, assume the current iterate
has been constructed. If
then stop; otherwise, calculate the next iterate
by the following formula
where
is a positive constant and
is chosen self-adaptively as
Algorithm 1.5. Initialization: Let
be a fixed point and let
be arbitrary. Iterative step: for
, assume the current iterate
has been constructed. If
then stop; otherwise, calculate the next iterate
by the following formula
where
is a positive constant and
is chosen self-adaptively as
They also obtained the weak and strong convergence of Algorithms 1.4 and 1.5, respectively. Motivated and inspired by the work in the literature, the main purpose of this paper is to extend the results of Wang [16] and Yao, et al. [17] from the directed mappings or demicontractive mappings to the demicontractive mappings. We present two self-adaptive algorithms for solving the split common fixed point problem (1.1). Weak and strong convergence theorems are given under some mild assumptions. Our results improve essentially the corresponding results in [16] [17]. Further, some other results are also improved; see [9] - [22].
2. Preliminaries
Let C be a nonempty closed convex subset of a real Hilbert space H.
Definition 2.1. A mapping
is said to be:
1) directed if
2) β-demicontractive if there exists a constant
such that
3) k-demimetric if there exists a constant
such that
(2.1)
Clearly, (2.1) is equivalent to the following:
It is obvious that the demimetric mappings include the directed mappings and the demicontractive mappings as special cases. Furthermore, this class mapping also contains the classes of strict pseudo-contractions, firmly-quasinon expansive mappings, 2-generalized hybrid mappings and quasi-non-expansive mappings. The class of demimetric mappings is fundamental because many common types of mappings arising in optimization belong to this class, see for example [23] [24] and references therein.
Definition 2.2 A sequence
is called Fejér-monotone with respect to a given nonempty set
, if for every
,
Next we adopt the following notations:
a)
and
denote the strong and weak convergence of the sequence
, respectively;
b)
is the weak ω-limit set of the sequence
.
Recall that a mapping
is said to be contractive if there exists a constant
such that
We use
to denote the collection of mappings f verifying the above inequality. That is
Let D be a nonempty subset of C. A sequence
of mappings of C into H is said to be stable on D (see [25]) if
is a singleton for every
. It is clear that if
is stable on D, then
for all
and
.
Recall that the (nearest point or metric) projection from H onto C, denoted
, assigns to each
, the unique point
with the property
The metric projection
of H onto C is characterized by
Lemma 2.1 ( [26]) Let
be a nonempty closed convex subset in H. If the sequence
is Fejér monotone with respect to
, then we have the following conclusions:
1)
iff
;
2) the sequence
converges strongly;
3) if
, then
.
Lemma 2.2 ( [27]) Let
be a sequence of nonnegative numbers satisfying the property:
where
satisfy the restrictions:
1)
;
2)
or
.
Then,
.
Lemma 2.3 ( [23] [24]) Let E be a smooth, strictly convex and reflexive Banach space and let k be a real number with
. Let U be an k-demimetric mapping of E into itself. Then
is closed and convex.
3. Main Results
Now we study the split common fixed points problem (1) under the following hypothesis:
and
are two real Hilbert spaces;
and
are two demimetric mappings with constants
and
, respectively;
is a bounded linear operator with its adjoint operator
;
is stable on
, where
denotes the solution set of problem (1.1).
Lemma 3.1
solves problem (1) iff
.
Proof. If
solves problem (1), then
and
. Therefore, we get
. To see the converse, suppose that
. Then, we have for any
that
(3.1)
Since S and T are demimetric, we have that
(3.2)
and
(3.3)
Combining (3.1), (3.2) and (3.3), we obtain that
(3.4)
Since
, we infer that
and
by (3.4). Therefore,
solves problem (1.1). This completes the proof.
Next we construct the following self-adaptive algorithm to solve problem (1.1).
Algorithm 3.1. Initialization: let
be arbitrary. For
, assume the current iterate
has been constructed. If
then stop (in this case
solves problem (1.1) by Lemma 3.1); otherwise, calculate the next iterate
by the following formula
(3.5)
where
is a positive constant and
is chosen self adaptively as
We assume that the sequence
generated by Algorithm 3.1 is infinite. In other words, Algorithm 3.1 does not terminate in a finite number of iterations.
Theorem 3.2. Assume that S and T are demiclosed at zero. If
, then the sequence
generated by (3.5) converges weakly to a solution
(
) of problem (1.1).
Proof. Since A is linear and continuous, noticing Lemma 2.3, we see
is closed and convex. Thus we have that
is well defined.
We next prove that the sequence
is Fejér-monotone with respect to
. Letting
, we then obtain that
(3.6)
In view of Equation (3.5) and Equation (3.6), we deduce
(3.7)
This implies that the sequence
is Fejér monotone.
Next, we show that every weak cluster point of the sequence
belongs to the solution set of problem (1.1).
From the Fejér-monotonicity of
, it follows that the sequence
is bounded. Further, we deduce from (3.7) that
An induction induces that
which implies that
Observe that
(3.8)
By the demiclosedness (at zero) of S and T, we deduce immediately
. To this end, the conditions of Lemma 2.1 are all satisfied. Consequently,
. This completes the proof.
Next, we study an iteration with strong convergence for solving problem (1.1).
Algorithm 3.3 Initialization: Let
be arbitrary. Iterative step: for
, assume the current iterate
has been constructed. If
then stop (in this case
solves problem (1.1) by Lemma 3.1); otherwise, calculate the next iterate
by the following formula
(3.9)
where
is a positive constant and
is chosen self-adaptively as
Theorem 3.4 Assume that:
(C1)
;
(C2) S and T are demiclosed at zero;
(C3)
and
.
Then the sequence
generated by (3.9) converges strongly to the solution
of problem (1.1).
Proof. Putting
, we obtain from (3.7) that
(3.10)
Next, we show that the sequence
is bounded. Indeed, we obtain from (3.9) and (3.10) that
By induction, we get
which gives that the sequence
is bounded.
By virtue of (3.9), we deduce
which implies
This together with (3.10) implies that
(3.11)
Set
and
(3.12)
for all
. Returning to (3.11) to obtain
(3.13)
From (3.12), we find
It follows that
.
Next we show that
.
If
, then there exists
such that
for all
. It then follows from (3.13) that
for all
. By induction, we have
(3.14)
By taking
as
in (3.14), we have
which induces a contradiction. So,
. Thus, we can take a subsequence
such that
(3.15)
Since
is a bounded real sequence, without loss of generality, we may assume
exists. Consequently, from (3.15), the following limit also exists
It turns out that
(3.16)
Taking into consideration that
we then deduce from (3.16) that
(3.17)
It follows that any weak cluster point of
belongs to
. Observe that
By (C3) and (3.16), we derive
This means that any weak cluster point of
also belongs to
. Without loss of generality, we assume that
converges weakly to
. Hence, we obtain
due to the fact that
. Rewriting (3.13) as
and noticing Lemma 2.2, we get
as
.
Theorem 3.5 Let
and
be two demicontractive mappings with constants
and
, respectively. Then the sequence
generated by (1.1) converges strongly to the solution
of problem (3.9) under the assumption of Theorem 3.4.
4. Conclusion
In this paper, we consider a class of the split common fixed point problems. By extending results in [16] [17] from the directed mappings or the demicontractive mappings to the demimetric mappings, and a fixed point
to a sequence mappings
, we construct two self-adaptive algorithms for solving the split common fixed point problem. Further, we also establish the weak and strong convergence theorems under some certain appropriate assumptions. The results in this paper are the extension and improvement of the recent results in the literature.
Acknowledgements
This research was supported by the Key Scientific Research Projects of Higher Education Institutions in Henan Province (20A110038).