_{1}

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In this paper, we prove some Δ-convergence and strong convergence results for the sequence generated by a new algorithm to a minimizer of two convex functions and a common fixed point for quasi-pseudo-contractive mappings in Hadamard spaces. Our theorems improve and generalize some recent results in the literature.

Let ( X , d ) be a metric space and x , y ∈ X with l = d ( x , y ) . A geodesic path from x to y is an isometry c : [ 0, l ] → X such that c ( 0 ) = x , c ( l ) = y . 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 Δ ( x 1 , x 2 , x 3 ) in a geodesic space X consists of three points x 1 , x 2 , x 3 of X and three geodesic segments joining each pair of vertices. A comparison triangle of a geodesic triangle Δ ( x 1 , x 2 , x 3 ) is the triangle Δ ¯ ( x 1 , x 2 , x 3 ) : = Δ ( x 1 ¯ , x 2 ¯ , x 3 ¯ ) in the Euclidean space ℝ 2 such that d ( x i , x j ) = d ℝ 2 ( x i ¯ , x j ¯ ) for all i , j = 1,2,3 .

A geodesic space X is a CAT(0) space if for each geodesic triangle Δ : = Δ ( x 1 , x 2 , x 3 ) in X and its comparison triangle Δ ¯ : = Δ ( x 1 ¯ , x 2 ¯ , x 3 ¯ ) in ℝ 2 , the CAT(0) inequality

d ( x , y ) ≤ d ℝ 2 ( x ¯ , y ¯ )

is satisfied by all x , y ∈ Δ and x ¯ , y ¯ ∈ Δ ¯ . 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 x * ∈ C such that

f ( x * ) + g ( x * ) = m i n x ∈ C { f ( x ) + g ( x ) } , (1)

where f , g 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 [

In the case that X is a real Hilbert space or a real Banach space, problem (1) has been studied by many authors ( [

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 [

Recall that a mapping T : C → C is said to be

(i) nonexpansive, if

d ( T x , T y ) ≤ d ( x , y ) , ∀ x , y ∈ C ;

(ii) quasi-nonexpansive, if F i x ( T ) ≠ ∅ and

d ( T x , x * ) ≤ d ( x , x * ) , ∀ x ∈ C , ∀ x * ∈ F i x ( T ) ;

(iii) k-strictly pseudononspreading, if there exists a constant k ∈ ( 0,1 ) such that for all x , y ∈ C

d 2 ( T x , T y ) ≤ d 2 ( x , y ) + k d 2 ( x , T x ) + k d 2 ( y , T y ) + 2 ( 1 − k ) 〈 x T ( x ) → , y T ( y ) → 〉 ;

(iv) demicontractive, if F i x ( T ) ≠ ∅ and there exists k ∈ ( 0,1 ) such that

d 2 ( T x , x * ) ≤ d 2 ( x , x * ) + k d 2 ( x , T x ) , ∀ x ∈ C , ∀ x * ∈ F i x ( T ) .

Definition 1. An operator T : C → C is said to be pseudo-contractive if

〈 T x T y → , x y → 〉 ≤ d 2 ( x , y ) , ∀ x , y ∈ C .

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 I − T is a monotone mapping. It is well known that T is pseudo-contractive if and only if

d 2 ( T x , T y ) ≤ d 2 ( x , y ) + d 2 ( ( I − T ) x , ( I − T ) y ) , ∀ x , y ∈ C .

Definition 2. An operator T : C → C is said to be quasi-pseudo-contractive if F i x ( T ) ≠ ∅ and

d 2 ( T x , x * ) ≤ d 2 ( x , x * ) + d 2 ( x , T x ) , ∀ x ∈ C , ∀ x * ∈ F i x ( T ) . (2)

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 [

We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results. In 1976, Lim [

d 2 ( ( 1 − t ) x ⊕ t y , z ) ≤ ( 1 − t ) d 2 ( x , z ) + t d 2 ( y , z ) − t ( 1 − t ) d 2 ( x , y ) (3)

for all x , y , z ∈ X and all t ∈ [ 0,1 ] . Berg and Nikolaev [

〈 a b → , c d → 〉 = 1 2 [ d 2 ( a , d ) + d 2 ( b , c ) − d 2 ( a , c ) − d 2 ( b , d ) ]

for all a , b , c , d ∈ X . It is easy to see that

〈 a b → , c d → 〉 = 〈 c d → , a b → 〉 , 〈 a b → , c d → 〉 = − 〈 b a → , c d → 〉 , 〈 a x → , c d → 〉 + 〈 x b → , c d → 〉 = 〈 a b → , c d → 〉

for all a , b , c , d , x ∈ X . It is proved in [

〈 a b → , c d → 〉 ≤ d ( a , b ) d ( c , d ) , ∀ a , b , c , d ∈ X .

Lemma 1. [

(i) d ( ( 1 − t ) x ⊕ t y , z ) ≤ ( 1 − t ) d ( x , z ) + t d ( y , z ) ;

Definition 3. [

Lemma 2. [

Lemma 3. Let C be a nonempty closed and convex subset of a Hadamard space X and

If

(ii) If

(iii) If T is quasi-pseudo-contractive, then the mapping K is quasi-nonexpansive, that is,

Proof. (i) If

Since

(ii) For any sequence

which implies that

Since T is demiclosed at 0, we have

for all

From (2) and (3), one has

By (2) and (6), we obtain

By (5), (7) and (8), we have

Since

for all

which together with

that is,

The proof is completed.

Now we consider the following problem: find a point

where C is a nonempty closed convex set of a Hadamard space X,

then the problem (11) is equivalent to the problem of finding

Define

It is easy to show that the bifunction

(A_{1})

(A_{2}) F is monotone, i.e.,

(A_{3}) The function

Define a mapping

Lemma 4. Let C be a nonempty closed convex subset of a Hadamard space X. Let F be a bifunction satisfying assumptions (A_{1})-(A_{3}) and

(A_{4}) For each

Then, the following conclusions hold:

(a)

(b)

(c)

(d) For

Proof. The result is a special case of Theorem 4 and Theorem 5 in [

We are in a position to give our main theorems. Throughout this section we assume that

(1)

(2) _{4});

(3)

(4) Denote

with

Theorem 1. Let

where

Proof. Step 1. It follows from Lemma 4 (c) that if

Step 2. Next we prove that

It follows from (13) and (14) that

From (13), (14) and (15) we obtain

which implies that

Therefore the sequence

Step 3. Now we prove that

In fact, it follows from (12) that

Hence in order to prove (18), it suffices to prove that

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

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

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

In fact, let

Let

Theorem 2. Let all the conditions in Theorem 1 be satisfied and

Proof. Indeed, since

Moreover, it follows from (18) and (23) that

Theorem 3. Suppose that all the conditions in Theorem 1 are satisfied. Moreover, let

then the sequence

Proof. It follows form (24) and (25) that

Since

which implies that

Hence

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 (A_{4}) condition: For each

The author would like to thank the referees for their pertinent comments and valuable suggestions.

The author declares that there is no conflict of interest regarding the publication of this paper.

Wan, L.L. (2020) Composite Minimization Problems in Hadamard Spaces. Journal of Applied Mathematics and Physics, 8, 597-608. https://doi.org/10.4236/jamp.2020.84046