Let (X,d) be a metric space and x,y∈X. A geodesic joining x to y is a map γ from the closed interval [0,d(x,y)]⊂ℝ to X such that γ(0)=x,γ(d(x,y))=y and d(γ(t1),γ(t2))=|t1−t2| for all t1,t2∈[0,d(x,y)]. The image of γ is called a geodesic segment joining x and y. When it is unique, this geodesic segment is denoted by [x,y] and we write αx⊕(1−α)y for the unique point z in the geodesic segment joining from x to y such that d(x,z)=(1−α)d(x,y) and d(y,z)=αd(x,y) for α∈[0,1]. The space (X,d) is said to be a geodesic metric space [2] if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x,y∈X.

A geodesic triangle △(x1,x2,x3) in a geodesic metric space (X,d) consists of three points x1,x2,x3 in X and a geodesic segment between each pair of vertices. A comparison triangle for the geodesic triangle △(x1,x2,x3) in X is a triangle △¯(x1,x2,x3):=△( x ¯ 1, x ¯ 2, x ¯ 3) in ℝ2 such that dℝ2x¯i,x¯j=d(xi,xj) for i,j∈1,2,3. Let △ be a geodesic triangle in X and △¯ be a comparison triangle for △ in ℝ2. Then △ is said to satisfy the CAT(0) inequality if for any x,y∈△ and their comparison points x¯,y¯∈△¯, the following holds,

A geodesic metric space (X,d) is said to be a CAT(0) space if all geodesic triangles satisfy the CAT(0) inequality. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space; see more details in [2]. Other examples include Euclidean spaces, Hilbert spaces, the Hilbert ball [7], ℝ-trees [17], and many others.

In 2017, Khamsi and Shukri [9] introduced the concept of CAT_p(0) spaces based on the idea that comparison triangles belong to a general Banach space instead of the Euclidean plane as follows:

Definition 1.1. Let (E,‖⋅‖) be a Banach space. A geodesic metric space (X,d) is said to be a CATE(0) space if for any geodesic triangle △ in X, there exists a comparison triangle △¯ in E such that the comparison axiom is satisfied, i.e., for all x,y∈△ and all comparison points x¯,y¯∈△¯, we have

If E=lp, for p≥1, we say X is a CAT_p(0)

It is obvious that CAT₂(0) space is exactly the classical CAT(0) space, which has been extensively studied.

Let x,y,z be in a CAT_p(0) space X, with p≥ 2, and x⊕y2 is the midpoint of the geodesic [x,y]. Then the comparison axiom implies

This inequality is the (CN_p) inequality of Khamsi and Shukri [9]. Note that the (CN_p) inequality coincides with the classical (CN) inequality [4] if p = 2. Below are some strong inequalities in a CAT_p(0) space.

Lemma 1.2.([1, 5]) Let (X,d) be a CAT_p(0) space, with p≥ 2. Then, for any x,y,z in X and α∈[0,1], we have

• (i) d(z,αx⊕(1−α)y)≤αd(z,x)+(1−α)d(z,y);

• (ii) d(z,αx⊕(1−α)y)p≤αd(z,x)p+(1−α)d(z,y)p−12p−1α(1−α)d(x,y)p.

Let C be a nonempty subset of a CAT_p(0) space (X,d). A subset C of X is said to be convex if C includes every geodesic segment joining any two of its points, that is, for any x,y∈C, we have [x,y]⊂C.

A complete CAT_p(0) space is called a p-Hadamard space. Throughout our work, we mainly focus on p-Hadamard spaces for p≥2.

Let T be a mapping of C into itself. An element x∈ C is called a fixed point of T if x = Tx. The set of all fixed points of T is denoted by F(T), that is, F(T)={x∈C:x=Tx}. A sequence {xn} in C is called approximate fixed point sequence for T (AFPS in short) if limn→∞d(xn,Txn)=0.

The famous fixed point theorem for nonexpansive mappings in Banach spaces have first studied by Browder [3]and Göhde [8] in 1965 as follows:

Theorem 1.3. Let X be a uniformly convex Banach space and C be a nonempty bounded closed convex subset of X. Then, each nonexpansive mapping T:C→C has a fixed point.

In 1972, Geobel and Kirk [6] extended their result to asymptotically nonexpansive mappings. Later in 2004, Kirk [10] obtained a similar result for complete CAT(0) spaces. In 2013, Phuengrattana and Suantai [12] extended those result to generalized asymptotically nonexpansive mappings and to complete uniformly convex metric spaces as follows:

Theorem 1.4. Let (X,d) be a complete uniformly convex metric space. Let C be a nonempty bounded closed convex subset of X and T:C→C be a generalized asymptotically nonexpansive mapping whose graph G(T)={(x,y)∈C×C:y=Tx} is closed. Then, T has a fixed point.

In this work, we extend some known existence and convergence results for generalized asymptotically nonexpansive mappings in Hadamard spaces to the case of p-Hadamard spaces, for p≥2.

Let C be a nonempty subset of a CAT_p(0) space (X,d) and T:C→C be a mapping. A mapping T is said to be generalized asymptotically nonexpansive [12, 15] if there exist sequences {kn}⊂[1,∞) and {sn}⊂[0,∞) with limn→∞kn=1, limn→∞sn=0 such that

for all x,y∈C and n∈ℕ. In the case of sn=0 for all n∈ℕ, a mapping T is called an asymptotically nonexpansive mapping. In particular, if kn=1 and sn=0 for all n∈ℕ, a mapping T reduce to a nonexpansive mapping.

Remark 2.1. If T is a generalized asymptotically nonexpansive mapping, it is know that F(T) is not necessarily closed; see [13].

Recall that ϕ:X→[0,∞) is called a type function if there exists a bounded sequence {xn} in X such that

for any x∈ X. A sequence {zn} in X is said to be a minimizing sequence of ϕ whenever

Lemma 2.2.([9]) Let (X,d) be a p-Hadamard space, with p≥ 2. Let C be a nonempty bounded closed convex subset of X and ϕ be a type function defined on C. Then any minimizing sequence of ϕ is convergent and its limit z is the unique minimum point of ϕ, i.e., ϕ(z)=inf{ϕ(x):x∈C}.

The following results are needed for proving our convergence results.

Definition 2.3.([11]) A mapping T:C→C is said to be semi-compact if C is closed and each bounded AFPS for T in C has a convergent subsequence.

Definition 2.4.([14]) A mapping T:C→C is said to satisfy the condition (I) if there exists a nondecreasing function ρ:[0,∞)→[0,∞) with ρ(0)=0 and ρ(r)>0 for all r∈(0,∞) such that

for all x∈C, where dist(x,F(T))=inf{d(x,z):z∈F(T)}.

Definition 2.5.([13]) Let {xn} be a sequence in a metric space (X,d) and F⊂X. We say that {xn} is of monotone type (I) with respect to F if there exists sequences {δn} and {γn} of nonnegative real numbers such that ∑ n=1∞δn<∞, ∑ n=1∞γn<∞, and

for all n∈ℕ and z∈F.

Lemma 2.6.([13]) Let {xn} be a sequence in a complete metric space (X,d) and F⊂X. If {xn} is of monotone type (I) with respect to F and liminfn→∞dist(xn,F)=0, then limn→∞xn=z for some z∈ X satisfying dist(xn,F). In particular, if F is closed, then z∈ F.

Lemma 2.7.([18]) Let {an}, {bn} and {cn} be sequences of nonnegative real numbers satisfying

where ∑ n=1∞bn<∞ and ∑ n=1∞cn<∞. Then limn→∞an exists.

In this section, we study the existence theorems for a generalized asymptotically nonexpansive mapping in p-Hadamard spaces.

We now state and prove our existence results.

Theorem 3.1. Let (X,d) be a p-Hadamard space, with p≥ 2. Let C be a nonempty bounded closed convex subset of X and T:C→C be a generalized asymptotically nonexpansive mapping whose graph G(T)={(x,y)∈C×C:y=Tx} is closed. Then, T has a fixed point.

Proof. Fix x∈C. Consider the type function ϕ generated by the bounded sequence {Tnx}. Let z be the minimum point of ϕ which exists by using Lemma 2.2. Therefore,

for any n,m∈ℕ. Taking n→∞, we get

for any m∈ℕ. Taking m→∞, we get

Since z is the minimum point of ϕ, it implies that limm→∞ϕ(Tmz)=inf{ϕ(x):x∈C}. Thus, {Tmz} is a minimizing sequence of ϕ. By Lemma 2.2, we obtain that Tmz→z as m→∞, and so T(Tmz)=Tm+1z→z as m→∞. By the closedness of G(T), we have Tz=z. This completes the proof.

Theorem 3.2. Let (X,d) be a p-Hadamard space, with p≥2. Let C be a nonempty bounded closed convex subset of X and T:C→C be a generalized asymptotically nonexpansive mapping whose graph G(T)={(x,y)∈C×C:y=Tx} is closed. Then, F(T) is nonempty closed and convex.

Proof. By Theorem 3.1, F(T) is nonempty. To show that F(T) is closed, we let {xn} be a sequence in F(T) such that limn→∞xn=x. By the definition of T, we have

Since limn→∞kn=1 and limn→∞sn=0, we get limn→∞d(Tnx,x)=0. That is Tnx→x as n→∞, and so T(Tⁿx)=Tⁿ⁺¹x→ x as n→∞. By the closedness of G(T), we have Tx=x. Hence x∈F(T) so that F(T) is closed.

In order to prove F(T) is convex, it is enough to prove that x⊕y2∈F(T) whenever x,y∈F(T) with x≠y. Set z=x⊕y2. By the (CN_p) inequality and the definition of T, for any n∈ℕ, we have

Since z=x⊕y2, we get that d(z,x)=12d(x,y) and d(y,x)=12d(x,y). So, we have

By limn→∞kn=1 and limn→∞sn=0, we get limn→∞d(Tnz,z)p=0. This implies that Tnz→z as n→∞, and so T(Tnz)=Tn+1z→z as n→∞. From the closedness of G(T), we have Tz = z. Therefore, F(T) is convex. This completes the proof.

Next, we show that the existence of a fixed point of a generalized asymptotically nonexpansive mapping in a p-Hadamard space is equivalent to the existence of a bounded orbit at a point.

Theorem 3.3. Let (X,d) be a p-Hadamard space, with p≥2. Let C be a nonempty closed convex subset of X and T:C→C be a generalized asymptotically nonexpansive mapping whose graph G(T)={(x,y)∈C×C:y=Tx} is closed. Then F(T)≠∅ if and only if there exists an x∈C such that {Tnx} is bounded.

Proof. The necessity is obvious. Conversely, assume that x is an element in C such that {Tnx} is bounded. Consider the type function ϕ generated by {Tnx}. By the same step of the proof as in Theorem 3.1, we can conclude that F(T)≠∅. This completes the proof.

Remark 3.4.

• (i) If T is generalized asymptotically nonexpansive, F(T) is not necessarily closed. However, if G(T) is also closed, Theorem 3.2 guarantee that F(T) is always closed.

• (ii) If T is continuous, then G(T) is always closed. Therefore, Theorems 3.1, 3.2 and 3.3 are obtained for a class of continuous generalized asymptotically nonexpansive mappings.

In this section, we study the strong convergence theorems for a generalized asymptotically nonexpansive mapping by the modified two-step iterative sequence for finding fixed points of such mapping in p-Hadamard spaces. We now introduce the modified two-step iterative sequence [16] as below:

Let C be a nonempty closed convex subset of a p-Hadamard space (X,d) and T:C→C be a generalized asymptotically nonexpansive mapping. We generate the sequence {xn} in C by x1∈C and

where {αn} and {βn} are sequences in [0,1].

Now we prove the strong convergence results.

Lemma 4.1. Let (X,d) be a p-Hadamard space, with p≥ 2. Let C be a nonempty bounded closed convex subset of X and T:C→C be a uniformly continuous generalized asymptotically nonexpansive mapping with sequences {kn}⊂[1,∞) and {sn}⊂[0,∞) such that ∑ n=1∞(kn−1)<∞ and ∑ n=1∞sn<∞. Let x1∈C and {xn} be a sequence in C defined by \eqref{two-step} where {αn} and {βn} are sequences in (0,1) such that 0<a≤αn,βn≤b<1. Then, we have the following:

• (i) there exit two sequences {δn} and {εn} of nonnegative real numbers such that ∑ n=1∞δn<∞, ∑ n=1∞εn<∞, and d(xn+1,z)≤(1+δn)d(xn,z)+εn for all n∈ℕ and z∈F(T);

• (ii) limn→∞d(xn,z) exists for all z∈F(T);

• (iii) {xn} is an AFPS for T.

Proof. (i) : By the uniform continuity of T, we have G(T) is closed. It implies from Theorem 3.1 that F(T)≠∅. Let z∈F(T). Since T is generalized asymptotically nonexpansive, by Lemma 1.2(i), we have

Since 0≤βnkn+(1−βn)≤1, we obtain

This implies that

Since 0≤βnkn+(1−βn)≤1, by (4.2), we have

where δn=2(kn−1)+(kn−1)2 and γn=(kn−1)sn+2sn. Since ∑ n=1∞(kn−1)<∞ and ∑ n=1∞sn<∞, it follows that ∑ n=1∞δn<∞ and ∑ n=1∞γn<∞. Hence, we obtain the desired result.

(ii) : By (i) and Lemma 2.7, we obtain that limn→∞d(xn,z) exists.

(iii) : By Lemma 1.2(ii) and (4.2), we have

Since kn≥1 and sn≥0, we have

Then, by (4.3), we have

This implies that

and

By limn→∞kn=1, limn→∞sn=0 and limn→∞d(xn,z) exists, we conclude that

and

By (4.4) and the uniform continuity of T, we have

By the definitions of xn+1 and y_n, we obtain

Since limn→∞kn=1, limn→∞sn=0, by (4.4), (4.5), and (4.6), we conclude that

Hence, we obtain the desired result.

Now, we prove a strong convergence theorem for a generalized asymptotically nonexpansive semi-compact mapping in p-Hadamard spaces.

Theorem 4.2. uppose that X, C, T, {xn}, {αn}, {βn} are as in Lemma 4.1. If T^m is semi-compact for some m∈ℕ, then {xn} converges strongly to a fixed point of T.

Proof. By Lemma 4.1(iii), limn→∞d(xn,Txn)=0. Fix m∈ℕ, we have

Since T is uniformly continuous, we have

That is, {xn} is an AFPS for T^m. By the semi-compactness of T^m, there exist a subsequence {xnk} of {xn} and z∈C such that limk→∞xnk=z. Again, by the uniform continuity of T, we have

Then z∈F(T). By Lemma 4.1(ii), limn→∞d(xn,z) exists, thus z is the strong limit of the sequence {xn} itself. This completes the proof.

Finally, we prove a strong convergence theorem for a generalized asymptotically nonexpansive mapping which satisfies condition (I) in p-Hadamard spaces.

Theorem 4.3. Suppose that X, C, T, {xn}, {αn}, {βn} are as in Lemma 4.1. If T satisfies condition (I), then {xn} converges strongly to a fixed point of T.

Proof. By condition (I), there exists a nondecreasing function ρ:[0,∞)→[0,∞) with ρ(0)=0 and ρ(r)>0 for all r∈(0,∞) such that

It implies by Lemma 4.1(iii) that

Then we have

By Lemma 4.1(i), we obtain that the sequence {xn} is of monotone type (I) with respect to F(T). This implies by Lemma 2.6 that the sequence {xn} converges strongly to a point z∈F(T). This completes the proof.

Remark 4.4. Any complete CAT(0) space is a 2-Hadamard space, therefore the results in this paper can be applied to any complete CAT(0) space.

The authors are thankful to the referees for careful reading and the useful comments and suggestions.