1. Introduction
It is well known that the theory of variational inequalities, introduced by Hartman and Stampacchia [
1] in the early 1960s, has played an important role in the study of physics, engineering, and economics, and it encompasses as special cases complementarity problems, optimization problems, the problem of finding zeros of operators, etc. Within the period of the past 20 years, a great deal of effort has gone into the existence and algorithms for variational inequality problems, related optimization problems, and related fixed-point problems; see, e.g., [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18] and the references therein.
Let
be a Hadamard manifold and
C be a nonempty subset of
. In 2012, Calao et al. [
19] first studied the following equilibrium problem of finding
such that
for all
, in which the function
is of equilibrium. It was proven in [
19] that there holds the existence of solutions to the issue (1) for
. Moreover, the outcomes were employed to settle mixed variational inequalities, fixed point issues, and Nash equilibrium issues in
. Using Picard’s iterative method, the authors introduced an iterative algorithm for finding a solution of issue (1). Subsequently, it was pointed out in [
20] that there hold certain gaps involving the existence argument for mixed variational inequality problems and the domain for the resolvent implicating the equilibrium issue of [
19].
Recently, the authors [
21,
22] investigated the problem of finding an element
, with
T being the nonexpansive operator,
B being the maximal monotone vector field, and
A being the continuous monotone vector field in a Hadamard manifold
. They proposed some Halpern-type and Mann-type iterative methods. Under some mild conditions, they proved that the iterative sequences fabricated in the suggested methods are strongly convergent to an element in the set of common solutions of the fixed-point problem of
T and the variational inclusion problem for
A and
B.
Assume
, where the bounded
C is of both closedness and geodesic convexity in
; let
be the resolvent of equilibrium function
for
, and let
be the inverse of the exponential map
at
. Very recently, Chang et al. [
23] considered the problem of finding an element
in which
is the fixed point set of the quasi-nonexpansive operator
,
is the equilibrium point set of
, and
is the maximal monotone vector field; for
, each continuous
is a monotone vector field, and
is the common singularity set of a quasivariational inclusion system. They proposed the following splitting iterative algorithm, that is, for any initial
, the sequence
is defined as follows:
where
, and
is the mapping defined by
and
for
. Under some mild assumptions, it was proven that the
constructed in (3) is strongly convergent to an element of
.
Inspired and aroused by the works above, this article considers the issue of seeking an element
in Hadamard manifold
, where
is the set of common fixed points of finitely many quasi-nonexpansive mappings
, and
are the same as above. We propose a triple Mann iteration method and show that the sequence fabricated in suggested algorithm is convergent to an element in the set of common solutions for issue (4). Finally, making use of the main result, we deal with the minimizing issue with a CFPP constraint and saddle-point issue with a CFPP constraint on Hadamard manifolds, respectively. Our main result improves, extends, and develops Chang et al. [
23], Theorem 3.1, in some aspects.
2. Basic Notions and Tools
To address the problem (4), one first releases certain preliminaries on geometric theory of manifolds, including some concepts and basic tools. It is worth mentioning that these can be found in many introductory books on Riemannian and differential geometry (see, e.g., [
24]).
Assume is a differentiable manifold of finite dimension. Then, the tangent space of at is denoted as . The tangent bundle of is formulated as . Naturally, it is a manifold. The inner product in is termed as a Riemannian metric in . The tensor field is termed as a Riemannian metric on if and only if, for each , the tensor is a Riemannian metric on . The associated norm with on is written as . The differentiable manifold equipped with Riemannian metric is termed as Riemannian manifold. Let the curve joining u to v (i.e., and ) be piecewise smooth. Then, the length of is defined as . Moreover, the Riemannian distance , inducing the original topology on , is formulated as minimizing this length over the set of all such curves joining v to y.
The Riemannian manifold
is of completeness if and only if, for each
, any geodesic emanating from
y is definable with respect to (w.r.t.)
t in
. The geodesic joining
v to
y in
is termed as the minimal geodesic if and only if the length of it equals to
. Riemannian manifold
endowed by Riemannian distance
d is the metric space
. From the Hopf–Rinow result [
24], it is easily known that, in case
is of completeness, each pair of elements in
is joined with the minimal geodesic. Meanwhile, the metric space
is of completeness, and each closed bounded subset is of compactness.
Suppose that Riemannian manifold is of completeness. The map at y is the exponential one formulated as for all , in which denotes the geodesic initiating from y with velocity v, that is, and . Then, for any . It is clear to check , in which 0 denotes the zero tangent vector. It is noteworthy that map is the differentiable exponential one on w.r.t. .
A complete simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard Manifold. In the rest of this paper, we always assume that is a Hadamard Manifold of finite dimension without causing confusion.
Proposition 1 (see [
24])
. For any two points , there exists a unique normalized geodesic joining to , which is actually a minimal geodesic denoted byMoreover, for each in satisfying in , there hold the following: It is easy to check that the following lemma is valid.
Lemma 1. (i) If is a geodesic joining x to y, then we have(From now on, indicates the Riemannian distance). (ii) For each and , there hold the inequalities below: The set C in is of geodesic convexity if and only if, for each , the geodesic joining y to v belongs to C. The geodesic convex hull of set is the smallest geodesic convex set containing D in , and it is written as .
In what follows, unless otherwise specified, we always assume that , where C is of both closedness and geodesic convexity in Hadamard manifold , and denotes the set of fixed points of an operator S.
The function
is termed as being of geodesic convexity if and only if, for each geodesic
(
) joining
, the
is of convexity, that is,
Assume the metric space X is complete, and let . The sequence in X is of Fejér monotonicity w.r.t. D if and only if, for each and each non-negative n, one has .
Lemma 2 (see [
25,
26])
. Assume the metric space X is complete, and let . In the case that is of Fejér monotonicity w.r.t. D, is of boundedness. In addition, in the case that has a cluster point y lying in D, is convergent to y. An operator S from C to itself is termed as being the following:
(a) Contractive if and only if
such that
(in particular, in the case of
,
S is termed as being nonexpansive);
(b) Quasi-nonexpansive if and only if
is not equal to ∅, and
(c) Firmly nonexpansive [
27] if and only if, for each
in
C, the map
formulated by
is nonincreasing.
Proposition 2 (see [
27])
. Let be a mapping. Then, the following statements are equivalent:(i) S is firmly nonexpansive;
(ii) For all and , (iii) for all , Lemma 3 (see [
23])
. Let be firmly nonexpansive, and assume is not equal to ∅. Then, for each and , there holds the inequality below: It is remarkable that, when , by Lemma 3, one obtains the relations below:
The firm nonexpansivity of the nonexpansivity of the quasinonexpansivity of S.
However, the converse relations do not hold.
An operator S from C to itself is termed as being demiclosed at zero if and only if, for each satisfying and , one has .
In what follows, one uses
to denote the collection of all single-valued vector fields
such that
, and
the domain of
A defined by
Also, one uses
to indicate the collection of all set-valued vector fields
such that
, and
is the domain of
B formulated by
.
Next, one furnishes some vital notions (see [
28]) that will be helpful to demonstrate the major outcome:
(i) A single-valued vector field
is termed as being monotone if and only if
(ii) A set-valued vector field is termed as being the following:
(a) Monotone if and only if, for each
(b) Maximal monotone if and only if it is monotone and, for the given
and
, the relation
implies
.
(iii) For given
, the resolvent of
B for
is a set-valued mapping
formulated by
Lemma 4 (see [
21])
. Given a single-valued vector field of monotonicity and a set-valued vector field of maximal monotonicity, one then has that for each , there holds the equivalence of the assertions below:(i) ;
(ii) .
In addition, we recall the resolvent of a bifunction on Hadamard manifold
introduced in Calao et al. [
19], as outlined below:
Assume
, where
C is of both closedness and geodesic convexity in
. Given a bifunction
, the resolvent of
is set-valued mapping
such that
Lemma 5 (see [
19,
20])
. Let be a bifunction satisfying the following conditions:(A1) ;
(A2) , i.e., Φ is monotone;
(A3) is upper semicontinuous for each ;
(A4) is geodesic convex and lower semicontinuous for each ;
(A5) ∃
(compact set) D in such thatOne then has that for each , the assertions listed below hold: (i) The resolvent is nonempty and single-valued;
(ii) The resolvent is firmly nonexpansive;
(iii) , i.e., the fixed point set of is the equilibrium point set of Φ;
(iv) is of both closedness and geodesic convexity.
3. Major Outcomes
Let be a Hadamard manifold of finite dimension and the nonempty be of both closedness and geodesic convexity in . Assume throughout that the following conditions hold:
is a multi-valued vector field of maximal monotonicity, is a single-valued vector field of both continuity and monotonicity for , and is the mapping defined by and for ;
is the bifunction fulfilling the hypotheses (A1)–(A5) of Lemma 5 and is the resolvent of for and is of both closedness and geodesic convexity;
is a finite family of L-Lipschitzian and demiclosed at zero quasi-nonexpansive self-mappings on C, where ;
;
For given
and
, the inequality holds
and such that the following are true:
- (i)
, and ;
- (ii)
, and .
In terms of Xu and Kim [
18], we write
for integer
with the mod function taking values in the set
, that is, if
for some integers
and
, then
if
and
if
.
Theorem 1. Let , and be chosen as in the above conditions. Given a starting arbitrarily. Let and be the sequences defined as follows:Then, we have the following: (i) The sequence converges to a solution of problem (4); (ii) In addition, if , and , then the sequence defined byconverges to a solution of problem (2); (iii) In addition, if , and (the null operator), , then the sequence defined byconverges to an element ; (iv) In addition, if for , , and , is the sequence constructed asthen converges to an element ; (v) In addition, if for , , and , is the sequence constructed asthen converges to a common fixed point of ; (vi) In addition, if , and , then the sequence defined byconverges to an element ; (vii) In addition, if , and , then the sequence defined byconverges to an element . Proof. First of all, using the conditions in Theorem 1 and Lemmas 4 and 5, one knows the following:
(i) The resolvent of function is the single-valued mapping of firm nonexpansivity such that ;
(ii) If
, then one has that
and
,
We now divide the rest of the proof into several steps.
Step 1. We claim that the sequences and all are bounded in C.
Indeed, by condition (17), for each
and
,
is a mapping of quasi-nonexpansivity. Take a fixed
arbitrarily. Then, for
, by (17) and Lemma 1 (ii), we have
Since
is quasi-nonexpansive for
, from Lemma 1 (ii), (18), and (20) we obtain
Thus, from Lemma 1 (ii), Lemma 5, (18), and (21), we have
This implies that, for all
and
,
that is,
is of Fejér monotonicity w.r.t.
. Via Lemma 2,
is of boundedness and so are the sequences
. As a result, the limit
exists for each
.
Step 2. We claim that for
,
Indeed, from (20), (21), and (23) we obtain that
Taking
and setting
, by Lemma 1 (ii), we deduce that, for each
,
By induction, one deduces that, for
,
Taking the limit in (26) as
, we obtain
Since
and
, we obtain that
Consequently,
In a similar way, from (25), we can also show that
Thus, the assertions in (24) are valid.
Step 3. We claim that, for
,
Indeed, by (18) and Lemma 1 (ii), we obatain
which hence yields
Noticing
, from (24) we have
Since each
is
L-Lipschitzian, using Lemma 1 (ii), one has
which hence leads to
Noticing
, from (24), we have
Also, since
is firmly nonexpansive, by Lemma 1 (ii), we obatain
which immediately implies that
Noticing
, from (24), we obtain
Step 4. We claim that converges to some point .
Indeed, since C is closed and geodesic convex in finite dimensional , and is of boundedness in C, one has that such that is convergent to an element in C. We first claim . Without loss of generality, we may assume for all k. Since combining (due to (24)) and guarantees for all , we deduce from (28) that . Then, the demiclosedness assumption of implies that for all j. This ensures that . Again, note that and are all nonexpansive mappings, which are also demiclosed at zero. So, it follows from (27) and (29) that . Since for and , we have . Consequently, . Since is Fejér monotone with respect to , in terms of Lemma 2, we obtain . The conclusion (i) of Theorem 1 is proved.
Step 5. We claim that there hold the conclusions (ii), (iii), (iv), (v), (vi), and (vii) in Theorem 1:
(1) Indeed, (a) if
, then we obtain that, for
and
,
(b) If
and
, then we obtain
Therefore, (18) reduces to the following iterative algorithm:
which converges to
as a solution of problem (2). The conclusion (ii) of Theorem 1 is proved.
(2) Indeed, (a) if
and
, then we obtain that, for
and
,
(b) If
, we obtain
Therefore, (18) reduces to the following iterative algorithm:
which converges to an element
. The conclusion (iii) is proved.
(3) Indeed, (a) if
and
, then we obtain that, for
and
,
and hence
(b) If
, then we have
, and hence
Therefore, (18) reduces to the following iterative algorithm:
which converges to an element
. The conclusion (iv) is proved.
(4) Indeed, if
and
, then we obtain that, for
and
,
and hence
(b) If
, then we have
, and hence
Therefore, (18) reduces to the following iterative algorithm:
which converges to
a common fixed point of
. The conclusion (v) is proved.
(5) Indeed, (a) if
, and
, then we obtain that, for
and
,
(b) If
, then we obtain
Therefore, (18) reduces to the following iterative algorithm:
which converges to an element
. The conclusion (vi) is proved.
(6) Indeed, (a) if
, and
, then we obtain that, for
and
,
(b) If
, then we obtain
Therefore, (18) reduces to the following iterative algorithm:
which converges to an element
. The conclusion (vii) is proved. □
Remark 1. Compared with Theorem 3.1 in Chang et al. [23], our Theorem 1 improves, extends, and develops it in the following aspects: (i) The problem of finding an element of in [23] is extended to develop our problem of finding an element of , where is a finite family of L-Lipschitzian () demiclosed at zero and which features quasi-nonexpansive self-mappings on C. (ii) The boundedness assumption of the nonempty closed and geodesic convex subset in [23], Theorem 3.1, is dropped by our Theorem 1, and there is only the assumption of the nonempty closed and geodesic convex subset in our Theorem 1. (iii) Because is diffeomorphic to an Euclidean space , has the same topology and differential structure as . Moreover, Hadamard manifolds and Euclidean spaces have some similar geometrical properties. Therefore, the convergence statement of the sequence in our Theorem 1 is more general.
(iv) The splitting iterative algorithm of Theorem 3.1 in [23] is developed into the triple Mann-type iteration method of this paper, i.e., the iteration steps , , and in [23], Theorem 3.1, are extended to develop the ones , , and in our Theorem 1, respectively. 4. Applicable Examples
Assume throughout that is a Hadamard manifold of finite dimension, and the nonempty is of both closedness and geodesic convexity in .
4.1. The Minimizing Problem with CFPP Constraint
Suppose that
are two proper functions of both lower semicontinuity and geodesic convexity. One first considers the minimizing problem of seeking an element
such that
where
denotes the set of solutions of the minimization issue (30), i.e.,
This matter possesses a specific scenario for linear inverse issues, and some authors have furnished its applications in compressed sensing, image reconstruction, data retrieve and machine learning; refer to [
29,
30,
31,
32].
Next, one writes
in which
. It is easily known that function
satisfies the conditions below:
(A1) ;
(A2) , i.e., is monotone;
(A3) is upper semicontinuous for each ;
(A4) is geodesic convex and lower semicontinuous for each ;
Assume that is a finite family of L-Lipschitzian () demiclosed at zero and with quasi-nonexpansive self-mappings on C. In Theorem 1, we put , and . Then, using Theorem 1 (iii), we derive the following result.
Theorem 2. Assume and are chosen as in the above conditions, and is the resolvent of Φ formulated in (16), where is of both closedness and geodesic convexity. Suppose there hold the relations below:
(i) ∃
(compact set) such that (ii) , and .
Then the sequence defined byconverges to an element . 4.2. The Saddle Point Problem (SPP) with CFPP Constraint
Given Hadamard manifolds , we consider geodesic convex subsets of , respectively. The mapping is termed as a saddle function iff the following hold:
(i) is of geodesic convexity on w.r.t. ;
(ii) is of geodesic concavity, that is, is of geodesic convexity on w.r.t. .
An element
is termed as a saddle point of
H if and only if
where
denotes the saddle point set of problem (33).
Let
be an associated multi-valued vector field with saddle function
H formulated below
Let
denote a product space. Then, it is a Hadamard manifold, and the tangent space of it at
is
(for more information, refer to Page 239 of [
24]). The relevant metric is specified below
The product of two geodesics in
and
denotes a geodesic in the product manifold
. So, for arbitrary two elements
and
in
, one has
The vector field
is termed as being monotone if and only if, for each
and
, one has
Lemma 6 (see [
33])
. Suppose that H is a saddle function on , and is the multi-valued vector field formulated in (34). Then, is of maximal monotonicity. It can be readily seen that an element is a saddle point of H if and only if is a singularity of . Next, for , let the mapping be L-Lipschitzian (), demiclosed at zero, and quasi-nonexpansive.
In Theorem 1, we put , and . By Theorem 1 (vi), we immediately obtain the assertion below.
Theorem 3. Suppose that is a saddle function, and is the corresponding vector field of maximal monotonicity. Assume is a finite family of L-Lipschitzian (), demiclosed at zero, and with quasi-nonexpansive self-mappings on C such that . Choose the initial point , and define as follows:for , with satisfying and . Then, converges to an element .