Let be a metric space and with . A geodesic path from x to y is an isometry such that . The image of a geodesic path is called a geodesic segment. A metric space X is a geodesic space if every two points of X are joined by a geodesic segment. A geodesic triangle in a geodesic space X consists of three points of X and three geodesic segments joining each pair of vertices. A comparison triangle of a geodesic triangle is the triangle in the Euclidean space such that for all .
A geodesic space X is a CAT(0) space if for each geodesic triangle in X and its comparison triangle in , the CAT(0) inequality
is satisfied by all and . The meaning of the CAT(0) inequality is that a geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. It is well-known that any complete and simply connected Riemannian manifold having non-positive sectional curvature is a CAT(0) space. Other examples of CAT(0) spaces include pre-Hilbert spaces, R-trees, Euclidean buildings. A complete CAT(0) space is called a Hadamard space.
Let C be a nonempty set and consider the following composite optimization problem: find such that
where are real-valued functions defined on C. This problem has a typical scenario in linear inverse problems, and it has applications in image reconstruction, machine learning, data recovering and compressed sensing (see  -  and the references therein).
In the case that X is a real Hilbert space or a real Banach space, problem (1) has been studied by many authors (    - ). For example, in 2019, Chang et al.  used a modified hybrid algorithm to find a minimizer for problem (1) in Banach spaces without the assumption that the potential function is Fréchet differentiable and its gradient is L-Lipschitz continuous.
Recently, many convergence results for solving optimization problems have been extended from the classical linear spaces to the setting of manifolds. For example, in 2015, Cholamjiak-Abdou-Cho  established strong convergence of the sequence to a minimizer of a convex function and to a fixed point of nonexpansive mappings in CAT(0) spaces. Also in 2019, Chang et al.  presented a new modified proximal point algorithm for solving the minimization of a convex function and the common fixed points problem for two k-strictly pseudononspreading mappings in Hadamard spaces.
Recall that a mapping is said to be
(i) nonexpansive, if
(ii) quasi-nonexpansive, if and
(iii) k-strictly pseudononspreading, if there exists a constant such that for all
(iv) demicontractive, if and there exists such that
Definition 1. An operator is said to be pseudo-contractive if
Remark 1. The interest of pseudo-contractive operators lies in their connection with monotone mappings, namely, T is a pseudo-contraction if and only if is a monotone mapping. It is well known that T is pseudo-contractive if and only if
Definition 2. An operator is said to be quasi-pseudo-contractive if and
From the above definitions, it is easy to see that the class of quasi-pseudo-contractive mappings is fundamental. It includes many kinds of nonlinear mappings such as the demicontractive mappings, the quasi-nonexpansive mappings and the k-strictly pseudononspreading with fixed points as special cases. Motivated by the researches above, we establish the convergent results to a minimizer of two convex functions and a common fixed point of quasi-pseudo-contractive mappings in Hadamard spaces. Thus our results generalize the corresponding results of Cholamjiak-Abdou-Cho , Chang et al. , Ariza-Ruiz et al. , Bačák , Dhompongsa et al. , Khan-Abbas  and many others.
2. Preliminaries and Lemmas
We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results. In 1976, Lim  introduced the concept of Δ-convergence in a general metric space. Recall that a sequence in a CAT(0) space X is said to Δ-converge to if x is the unique asymptotic center of for every subsequence of . A geodesic space is a CAT(0) space, if and only if
for all and all . Berg and Nikolaev  introduced the concept of quasilinearization as follows. Denote a pair by and call it a vector. Then quasilinearization is defined as a map defined by
for all . It is easy to see that
for all . It is proved in  that a geodesically connected metric space is a CAT(0) space if and only if it satisfies the Cauchy-Schwarz inequality:
Lemma 1.  Let X be a Hadamard space. Then for all and , we have
Definition 3.  Let C be a nonempty subset of a Hadamard space X and let be a sequence in X. Then is Fejér monotone respect to C if
Lemma 2.  Let be a sequence in a Hadamard space X and let C be a nonempty subset of X. Suppose that is Fejér monotone with respect to C and that every Δ-sequential cluster point of belongs to C. Then Δ-converges to a point in C.
Lemma 3. Let C be a nonempty closed and convex subset of a Hadamard space X and be an L-Lipschizian mapping with. Denote
If, then the following conclusions hold:
(ii) If is demiclosed at 0, then is also demiclosed at 0;
(iii) If T is quasi-pseudo-contractive, then the mapping K is quasi-nonexpansive, that is,
Proof. (i) If, it is obvious that. Conversely, if, i.e., , letting, then. Put. Then. Now we prove that. In fact, we have
Since, we have, i.e.,. This shows that . It is obvious that if and only if. The conclusion (1) is proved.
(ii) For any sequence satisfying and. Next we prove that. From conclusion (1), we only need to prove that. In fact, since T is L-Lipschizian, we get
which implies that
Since T is demiclosed at 0, we have. The conclusion (2) is proved.
(iii) Since, we have from (2)
for all. Since T is L-Lipschitzian, we get
From (2) and (3), one has
By (2) and (6), we obtain
By (5), (7) and (8), we have
Since, we deduce that. From (9), one gets
for all and. Combing (2) and (10) one has
which together with implies that
The proof is completed.
Now we consider the following problem: find a point such that
where C is a nonempty closed convex set of a Hadamard space X, are proper convex functions and is a quasi-pseudo-contractive mapping. Recall that a function is said to be convex, if for any geodesic joining, the function is convex. If we set
then the problem (11) is equivalent to the problem of finding such that
It is easy to show that the bifunction has the following properties:
(A2) F is monotone, i.e.,;
(A3) The function is convex for all;
Define a mapping by
Lemma 4. Let C be a nonempty closed convex subset of a Hadamard space X. Let F be a bifunction satisfying assumptions (A1)-(A3) and
(A4) For each and, there exists a compact subset containing a point such that whenever.
Then, the following conclusions hold:
(a) is well defined in X and is single-valued;
(b) is firmly nonexpansive restricted to C, i.e., ,
(c), where is the solution set of problem (1) (i.e., the set of minimizers of problem (1));
(d) For, one has
Proof. The result is a special case of Theorem 4 and Theorem 5 in , so we omit the proof here.
3. Δ-Convergence Theorems
We are in a position to give our main theorems. Throughout this section we assume that
(1) is a Hadamard space and C is a nonempty closed convex subset of X;
(2) are proper convex functions and the bifunction satisfies the assumption (A4);
(3) is an L-Lipschitzian and quasi-pseudo-contractive mapping with, is demiclosed at 0;
Theorem 1. Let be the same above. For any given, define the sequence as follows:
where, are sequences in with . If the solution set of problem (11) is nonempty, then the sequence Δ-converges to a point, which is a minimizer of in C and also a common fixed point of in C.
Proof. Step 1. It follows from Lemma 4 (c) that if, then . Besides, by Lemma 3 (ii) we have is demiclosed at 0.
Step 2. Next we prove that is Fejér monotone with respect to. In fact, by Lemma 4 (b), is firmly nonexpansive, then it is nonexpansive. Let, then one has
It follows from (13) and (14) that
From (13), (14) and (15) we obtain
which implies that is decreasing and bounded below. Thus the limit exists for each. It implies that is Fejér monotone with respect to. Without loss of generality, we can assume that
Therefore the sequence is bounded and so are the sequences .
Step 3. Now we prove that
In fact, it follows from (12) that
Hence in order to prove (18), it suffices to prove that. Indeed, by (16) we get
which can be rewritten as
which together with (17) implies that
Combing (15) and (17) we obtain
which together with (20) implies that
Also, by (15) we have
Then one gets
which together with (21) shows that
On the other hand, it follows from (14) that
These imply that. Thus by (19) one has that the equality (18) holds.
Step 4. In this step, we show that
In fact, it follows from (3), (13), (14) and Lemma 3 (iii) that
which together with (3), (13), (14) and Lemma 3 (iii) implies that
After simplifying and by using the condition that, one gets
which shows that
Thus by (13) and (22), we get
Furthermore, it follows form (18), (22) and (23) that
Step 5. Finally, we prove that Δ-converges to some point. Since in the second step, we have shown that is bounded in C and it is Fejér monotone with respect to. Then by Lemma 2, in order to prove Δ-converges to some point in, it suffices to show that every Δ-sequential cluster point of belongs to.
In fact, let be a Δ-sequential cluster point of, then there exits a subsequence of Δ-converging to. From (18) and (23), it follows that and. Since is nonexpansive, is demiclosed at 0. Note that and are also demiclosed at 0 by Lemma 3 (ii). Now by (24) and Lemma 3 (i), we obtain . Therefore, by Lemma 2, Δ-converges to some point in. The proof is completed.
4. Strong Convergence Theorems
Let be a Hadamard space and C be a nonempty closed convex subset of X. Recall that a mapping is said to be demi-compact, if for any bounded sequence in C such that (as), then there is a subsequence such that converges strongly (i.e., in metric topology) to some point in C.
Theorem 2. Let all the conditions in Theorem 1 be satisfied and be demi-compact restricted to C, then the sequence defined by (13) converges strongly to a point.
Proof. Indeed, since is demi-compact restricted to C, it follows from (24) that there is a subsequence such that converges strongly to some point. Since is demiclosed at 0, we have.
Moreover, it follows from (18) and (23) that and as. Since is demi-closed at 0, by (24) we have . Hence. Besides, it follows form (17) that exists. Thus we get. The proof is completed.
Theorem 3. Suppose that all the conditions in Theorem 1 are satisfied. Moreover, let be a nondecreasing function with and
then the sequence defined by (13) converges strongly to a point.
Proof. It follows form (24) and (25) that
Since is nondecreasing with and, we have
which implies that
Hence is a Cauchy sequence in C. Noting that C is closed and convex in the Hadamard space X, C is also complete. Without loss of generality, we can assume that converges strongly to some point. Then . Besides, since is quasi-nonexpansive and is nonexpansive, it is clear that is closed in C. Thus we get. The proof is completed.
5. Conclusion and Remarks
Let us conclude this paper with some open questions whose answers might largely improve the applicability of the results in this present paper.
Question. Whether or not we can improve the (A4) condition: For each and, there exists a compact subset containing a point
such that whenever, in order to obtain similar results regarding the resolvent operator?
The author would like to thank the referees for their pertinent comments and valuable suggestions.
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