# Strong Convergence of New Two-Step Viscosity Iterative Approximation Methods for Set-Valued Nonexpansive Mappings in CAT(0) Spaces.

1. Introduction

In , the fixed point theory in CAT(0) spaces was first introduced and studied by Kirk. Further, Kirk  presented that each nonexpansive (single-valued) mapping on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. On the other hand, fixed point theory for set-valued mappings has been applied to applied sciences, game theory, and optimization theory. This promotes the rapid development of fixed point theory for single-valued (set-valued) operators in CAT(0) spaces, and it is natural and particularly meaningful to extensively study fixed point theory of set-valued operators. Particularly, some old relative works on Ishikawa iterations for multivalued mappings can be found in [2-4]. For more detail, we refer to [5-14] and the references therein.

Definition 1. Let g : X [right arrow] X be a nonlinear operator on a metric space (X, d) and G : E [right arrow] BC(X) be a set-valued operator, where E [subset] X is a nonempty subset and BC(X) is the family of nonempty bounded closed subsets of X. Then

(i) g is said to be a contraction, if there exists a constant k [member of] [0,1) such that

d (g (x), g (y)) [less than or equal to] [kappa]d (x, y) [for all]x, y [member of] X. (1)

Here, g is called nonexpansive when k = 1 in (1).

(ii) G is said to be a nonexpansive, if

H(G(x), G(y)) [less than or equal to] d(x, y), (2)

where H(x, x) is Hausdorff distance on BC(X), i.e.,

[mathematical expression not reproducible]. (3)

Recently, Shi and Chen  first considered the following Moudafi's viscosity iteration for a nonexpansive mapping g : E [right arrow] E with 0 [not equal to] Fix(g) = {x | x = g(x)} and a contraction mapping f: E [right arrow] E in CAT(0) space X:

[x.sub.[alpha]] = [alpha]f ([x.sub.[alpha]]) [direct sum] (1 - [alpha]) g ([x.sub.[alpha]]), (4)

and

[x.sub.n+1] = [[alpha].sub.n]f([x.sub.n]) [direct sum] (1 - [[alpha].sub.n]) g ([x.sub.n]), n [greater than or equal to] 1, (5)

where [alpha], [[alpha].sub.n] [member of] (0,1) and [x.sub.1] is an any given element in a nonempty closed convex subset E [subset not equal to] X. [x.sub.[alpha]] [member of] E is called unique fixed point of contraction x [??] [alpha]f(x) [direct sum] (1 - [alpha])g(x). Shi and Chen  showed that {[x.sub.[alpha]]} defined by (4) converges strongly to [mathematical expression not reproducible] in CAT(0) space (X, d) satisfies the following property P: for all x, u, [y.sub.1], [y.sub.2] [member of] X,

d (x, [m.sub.1])d(x, [y.sub.1]) [less than or equal to] d (x, [m.sub.2]) d (x, [y.sub.2])

+ d (x, u) d ([y.sub.1], [y.sub.2]), (6)

that is, an extra condition on the geometry of CAT(0) spaces is requested, where [mathematical expression not reproducible]. Further, the authors also found that the sequence {[x.sub.n]} generated by (5) converges strongly to [??] [member of] Fix(g) under some suitable conditions about {[[alpha].sub.n]}. Afterwards, based on the concept of quasilinearization introduced by Berg and Nikolaev , Wangkeeree and Preechasilp  explored strong convergence results of (4) and (5) in CAT(0) spaces without the property P and presented that the iterative processes (4) and (5) converges strongly to [??] [member of] Fix(g) such that [mathematical expression not reproducible] is the unique solution of the following variational inequality:

[mathematical expression not reproducible]. (7)

In , Panyanak and Suantai extended (4) and (5) to T being a set-valued nonexpansive mapping from E to BC(X). That is, for each [alpha] [member of] (0,1), let a set-valued contraction [G.sub.[alpha]] on E be defined by

[G.sub.[alpha]] (x) = [alpha]f (x) [direct sum] (1 - [alpha]) Tx, [for all]x [member of] E. (8)

By Nadler's  theorem, it is easy to know that there exists [x.sub.[alpha]] [member of] E such that [x.sub.[alpha]] is a fixed point of [G.sub.[alpha]], which does not have to be unique, and

[x.sub.[alpha]] [member of] [alpha]f([x.sub.[alpha]]) [direct sum] (1 - [alpha])T [x.sub.[alpha]], (9)

i.e., for each [x.sub.[alpha]], there exists [y.sub.[alpha]] [member of] T [x.sub.[alpha]] such that

[x.sub.[alpha]] = [alpha]f([x.sub.[alpha]]) [direct sum] (1 - [alpha]) [y.sub.[alpha]]. (10)

Further, when the contraction constant coefficient of f is k [member of] [0, 1/2) and {[[alpha].sub.n]} [subset] (0, 1/(2 - k)) satisfying some suitable conditions, Panyanak and Suantai  proved strong convergence of one-step viscosity approximation iteration defined by (10) or the following iterative process in CAT(0) spaces:

[x.sub.n+1] = [[alpha].sub.n]f([x.sub.n]) [direct sum] (1 - [[alpha].sub.n]) [y.sub.n], [y.sub.n] [member of] M ([x.sub.n]), (11)

and d([y.sub.n], [y.sub.n+1]) [less than or equal to] d([x.sub.n], [x.sub.n+1]) for all n [member of] N, where M is a set-valued nonexpansive operator from E to C(E), the family of nonempty compact subsets of E, f : E [right arrow] E is a contraction, and {[[alpha].sub.n]} [subset not equal to] (0,1). Moreover, Chang et al.  affirmatively answered the open question proposed by Panyanak and Suantai [16, Question 3.6]: "If k [member of] [0,1) and {[[alpha].sub.n]} [subset] (0,1) satisfying the same conditions, does {[x.sub.n]} converge to [mathematical expression not reproducible], where F(M) denotes the set of all fixed points of M.

On the other hand, Piatek  introduced and studied the following two-step viscosity iteration in complete CAT(0) spaces with the nice projection property N:

[mathematical expression not reproducible], (12)

where [x.sub.1] [member of] E is an given element and {[[alpha].sub.n]}, {[[beta].sub.n]} [subset not equal to] (0,1) satisfying some suitable conditions and the contraction coefficient of f is k [member of] [0,1/2).

Based on the ideas of Wangkeeree and Preechasilp  and Piatek  intensively, Kaewkhao et al.  omit the nice projection property N. We note that the two-step viscosity iteration (12) is also considered and studied by Chang et al.  when the property N is not satisfied and k [member of] [0, 1), which is due to the open questions in , where the property N depends on whether its metric projection onto a geodesic segment preserves points on each geodesic segment, that is, for every geodesic segment [chi] [subset] X and x, y [member of] X, m [member of] [x, y] implies [P.sub.[chi]]m [member of] [[P.sub.[chi]] x, [P.sub.[chi]] y], where [P.sub.[chi]] denotes the metric projection from X onto y. For more works on the convergence analysis of (viscosity) iteration approximation method for (split) fixed point problems, one can refer to [20-27].

Motivated and inspired mainly by Panyanak and Suantai  and Piatek  and so on, we consider the following twostep viscosity iteration for set-valued nonexpansive operator T:E [right arrow] C(E):

[mathematical expression not reproducible], (13) where E is a nonempty closed convex subset of complete CAT(0) space (X,d), [x.sub.1] [member of] E is an given element and {[[alpha].sub.n]}, {[[beta].sub.n]} [subset not equal to] (0, 1), f : E [right arrow] E is a contraction mapping, and [z.sub.n] [member of] T([x.sub.n]) satisfying d([z.sub.n], [z.sub.n+1]) [less than or equal to] d([x.sub.n], [x.sub.n+1]) for any n [member of] N.

By using the method due to Chang et al. [7, 8], the purpose of this paper is to prove some strong convergence theorems of the viscosity iteration procedure (13) in complete CAT(0) spaces. Hence, the results of Chang et al. [7, 8] and many others in the literature can be special cases of main results in this paper.

2. Preliminaries

Throughout of this paper, let (X, d) be a metric space. A geodesic path joining x [member of] X to y [member of] X (or, more briefly, a geodesic from x to y) is a map [xi] : R [contains] [0, l] [right arrow] X such that [xi](0) = x, [xi](l) = y, and d([xi](s), [xi](t)) = [absolute value of s - t] for each s, t [member of] [0, l]. In particular, [xi] is a isometry and d(x, y) = l. The image of [xi] is called a geodesic segment (or metric) joining x and y if unique is bespoke by [x, y]. The space (X, d) is called a geodesic space when every two points in X are joined by a geodesic, and X is called uniquely geodesic if there is exactly one geodesic joining x and y for any x, y [member of] X. A subset E of X is said to be convex if E includes every geodesic segment joining any two of its points. A geodesic triangle [DELTA](p, q, r) in a geodesic space (X, d) consists of three points p, q, r in X (the vertices of [DELTA]) and a choice of three geodesic segments [p, q], [q, r], [r, p] (the edge of [DELTA]) joining them. A comparison triangle for geodesic triangle [DELTA](p, q, r) in X is a triangle [bar.[DELTA]]([bar.p], [bar.q], [bar.r]) in the Euclidean plane [R.sup.2] such that

[mathematical expression not reproducible]. (14)

A point [bar.u] [member of] [[bar.p], [bar.q]] is said to be a comparison point for u [member of] [p, q] if [mathematical expression not reproducible]. Similarly, we can give the definitions of comparison points on [[bar.q], [bar.r]] and [[bar.r], [bar.p]].

Definition 2. Suppose that [DELTA] is a geodesic triangle in (X, d) and [bar.[DELTA]] is a comparison triangle for [DELTA]. A geodesic space is said to be a CAT(0) space, if all geodesic triangles of appropriate size satisfy the following comparison axiom (i.e., CAT(0) inequality):

[mathematical expression not reproducible]. (15)

Complete CAT(0) spaces are often called Hadamard spaces (see ). For other equivalent definitions and basic properties of CAT(0) spaces, one can refer to . It is well known that every CAT(0) space is uniquely geodesic and any complete, simply connected Riemannina manifold having nonpositive sectional curvature is a CAT(0) space. Other examples for CAT(0) spaces include pre-Hilbert spaces , R-trees , Euclidean buildings , and complex Hilbert ball with a hyperbolic metric  as special case.

Let E be a nonempty closed convex subset of a complete CAT(0) space (X, d). By Proposition 2.4 of , it follows that, for all x [member of] X, there exists a unique point [x.sub.0] [member of] E such that

d (x, [x.sub.0]) = inf {d (x, y) : y [member of] E}. (16)

Here, [x.sub.0] is said to be unique nearest point of x in E.

Assume that (X, d) is a CAT(0) space. For all x, y [member of] X and t [member of] [0,1], by Lemma 2.1 of Phompongsa and Panyanak , there exists a unique point z [member of] [x, y] such that

d(x, z) = (1 - t) d (x,y)

and d (y, z) = td (x, y). (17)

We shall denote by tx [direct sum] (1 - t)y the unique point z satisfying (17). Now, we give some results about CAT(0) spaces for the proof of our main results.

Lemma3 ([1,10]). Let (X, d) be a CAT(0) space. Then for each x, y, z [member of] X and [alpha] [member of] [0,1],

(i) d([alpha]x [direct sum] (1 - [alpha])y, z) [less than or equal to] [alpha]d(x, z) + (1 - [alpha])d(y, z).

(ii) [d.sup.2]([alpha]x [direct sum] (1 - [alpha])y, z) [less than or equal to] [alpha][d.sup.2](x, z) + (1 - [alpha])[d.sup.2](y, z) [alpha](1 - [alpha])[d.sup.2](x, y).

(iii) d([alpha]x [direct sum] (1 - [alpha])z, [alpha]y [direct sum](1 - [alpha])z) [less than or equal to] [alpha]d(x, y).

Lemma 4 (). Suppose that (X, d) is a CAT(0) space. Then for all x, y [member of] X and [alpha], [beta] [member of] [0,1],

[mathematical expression not reproducible]. (18)

Lemma 5 (). Assume that {[x.sub.n]} and {[y.sub.n]} are two bounded sequences in a CAT(0) space (X, d) and {[[beta].sub.n]} is a sequence in [0, 1] with 0 < [lim inf.sub.n] [[beta].sub.n] [less than or equal to] [lim sup.sub.n] [[beta].sub.n] < 1. If

[mathematical expression not reproducible]. (19)

Lemma 6 (). Suppose that nonnegative real numbers sequence {[u.sub.n]} is defined by

[u.sub.n+1] [less than or equal to] (1 - [[alpha].sub.n]) [u.sub.n] + [for all]n [greater than or equal to] 1, (20)

where {[[alpha].sub.n]} [subset] [0, 1] and {[[beta].sub.n]} [subset] R are two sequences satisfying

(i) [[summation].sup.[infinity].sub.n] [[alpha].sub.n] = [infinity]; [lim sup.sub.n[right arrow][infinity]] [[beta].sub.n] [less than or equal to] or [[summation].sup.[infinity].sub.n] [absolute value of [[alpha].sub.n] [[beta].sub.n]] < [infinity].

Then [lim.sub.n[right arrow][infinity]] {[u.sub.n]} = 0.

Lemma 7 (). Assume that E is a closed convex subset of a complete CAT(0) space (X,d). If a set-valued nonexpansive operator T : E a BC(X) satisfies endpoint condition C, i.e., F(T) [not equal to] 0 and T(x) = {x} for any x [member of] F(T) (see ), then F(T) is closed and convex.

In , Berg and Nikolaev introduced the concept of quasilinearization. Now we denote a pair (a, b) [member of] X x X by [??], which is a vector. Define the quasilinearization by a map <x, x>: (X x X) x (X x X) [right arrow] R as follows:

[mathematical expression not reproducible], (21)

One can easily know that

[mathematical expression not reproducible] (22)

for every a, b, c, [member of], x [member of] X. We say that a geodesic metric space (X, d) satisfies the Cauchy-Schwarz inequality if

[mathematical expression not reproducible]. (23)

From [15, Corollary 3], it is known that a geodesic space (X, d) is a CAT(0) space if and only if X satisfies the Cauchy-Schwarz inequality. Further, we give the following other properties of quasilinearization.

Lemma 8 (). Assume that E is a nonempty closed convex subset of a complete CAT(0) space (X, d). Then for u [member of] X and x [member of] E,

[mathematical expression not reproducible]. (24)

Lemma 9 (). For two points u and v in a CAT(0) space (X, d) and any [alpha] [member of] [0,1], letting [u.sub.[alpha]] = [alpha]u [direct sum] (1 - [alpha])v, then for all x, y [member of] X, the following results hold:

(i) [mathematical expression not reproducible];

(ii) [mathematical expression not reproducible].

Definition 10. A continuous linear functional [mu] is said to be Banach limit on [l.sub.[infinity]] if

[mathematical expression not reproducible]. (25)

Lemma 11 (). Suppose that, for real number [alpha] and all Banach limits [mu], ([[mu].sub.1], [[mu].sub.2], ...) [member of] [l.sub.[infinity]] satisfies

[mathematical expression not reproducible]. (26)

Lemma 12 (). Assume that (X, d) is a complete CAT(0) space, E c X is a nonempty closed convex subset, T : E [member of] C(E) is a set-valued nonexpansive operator satisfying endpoint condition C, and f : E [right arrow] E is a contraction with k [member of] [0,1). Then we have following results:

(i) {[x.sub.[alpha]]} generated by (10) converges strongly to [mathematical expression not reproducible].

(ii) In addition, if {[x.sub.n]} c E is a bounded sequence such that [lim.sub.n[right arrow][infinity]] dist([x.sub.n], T([x.sub.n])) = 0, where dist(a, B) is the distance from a point a e X to the set B e C(X), then for all Banach limits [mu],

[mathematical expression not reproducible]. (27)

3. Main Results

Employing the preliminaries in the previous section, now we will study the strong convergence of the new two-step viscosity iteration (13) for set-valued nonexpansive operators in complete CAT(0) spaces.

Theorem 13. Assume that (X, d) is a complete CAT(0) space, E c X is a nonempty closed convex subset, T : E [right arrow] C(E) is a set-valued nonexpansive operator satisfying endpoint condition C, and f : E [right arrow] E is contraction with k [member of] [0,1). If sequences {[[alpha].sub.n]}, {[[beta].sub.n]} [member of] (0, 1) satisfy

([L.sub.1]) [lim.sub.n[right arrow][infinity]] [[alpha].sub.n] = 0, ([L.sub.2]) [[summation].sup.[infinity].sub.n=1] [[alpha].sub.n] = [infinity], and ([L.sub.3]) 0 < [lim inf.sub.n[right arrow][infinity]] [[beta].sub.n] [less than or equal to] [lim sup.sub.n[right arrow][infinity]] < 1, then the sequence {[x.sub.n]} generated by (13) converges strongly to [??], where

[mathematical expression not reproducible], (28)

Proof. The proof shall be divided into the following four steps.

Step 1. We first prove that sequences {[x.sub.n]}, {f([x.sub.n])}, {[y.sub.n]}, and {[z.sub.n]} are bounded. In fact, setting p e F(T), then from Lemma 3, we know

[mathematical expression not reproducible], (29)

and

[mathematical expression not reproducible]. (30)

Thus, we obtain

d([x.sub.n], p) [less than or equal to] d ([x.sub.1], p), d(f(p), p)/1 - k}. (31)

Hence, {[x.sub.n]} is bounded, so is {f([x.sub.n])}. By (29), it is easy to knowthat {[y.sub.n]} is bounded. Since d([z.sub.n], p) [less than or equal to] H(T([x.sub.n], T(p)) [less than or equal to] d([x.sub.n], p), one can easily know that the sequence {[z.sub.n]} is also bounded.

Step 2. We present that [lim.sub.n[right arrow][infinity]] d([x.sub.n], [y.sub.n]) = 0, [lim.sub.n[right arrow][infinity]] dist([x.sub.n], T([x.sub.n])) = 0, [lim.sub.n[right arrow][infinity]] d([x.sub.n], [z.sub.n]) = 0, [lim.sub.n[right arrow][infinity]] d([x.sub.n], [x.sub.n+1]) = 0 [lim.sub.n[right arrow][infinity]] d([z.sub.n], [z.sub.n-1]) = 0, and [lim.sub.n[right arrow][infinity]] dist([z.sub.n], [z.sub.n], T ([z.sub.n])) = 0.

Indeed, by applying Lemmas 3 and 4, we have

[mathematical expression not reproducible] (32)

and so

[mathematical expression not reproducible] (33)

From [lim.sub.n[right arrow][infinity]] [[alpha].sub.n] = 0 and the boundedness of {[x.sub.n]}, {f([x.sub.n])}, and {[z.sub.n]}, we know

[mathematical expression not reproducible]. (34)

It follows from Lemma 5 that

[mathematical expression not reproducible]. (35)

Thus,

[mathematical expression not reproducible]. (36)

By (36), now we know that

[mathematical expression not reproducible]. (37)

Moreover,

d([x.sub.n], [x.sub.n+1]) = (1 - [[beta].sub.n]) d ([x.sub.n], [y.sub.n]) [right arrow] 0 (38)

and

d([x.sub.n], [z.sub.n+1]) [less than or equal to] d ([x.sub.n], [x.sub.n+1]) [right arrow] 0 (39)

as n [right arrow] [infinity]. By (36) and (37), we get

[mathematical expression not reproducible]. (40)

Step 3. Now, we show that

[mathematical expression not reproducible], (41)

with [??] = [P.sub.F(T) f([??]) satisfying

[mathematical expression not reproducible]. (42)

Above all, since T(x) is compact for any x [member of] E, then T(x) [member of] BC(X). It follows from Lemma 7 that F(T) is closed and convex, which implies that [P.sub.F(T)] u is well defined for any u [member of] X. By Lemma 12 (i), we know that {[x.sub.[alpha]]} generated by (10) converges strongly to [mathematical expression not reproducible]. Then by Lemma 8, we know that [??] is the unique solution of the following variational inequality:

[mathematical expression not reproducible]. (43)

Next, since {[z.sub.n]} is bounded and [lim.sub.n[right arrow][infinity]] dist([z.sub.n], T([z.sub.n])) = 0, it follows from Lemma 12 (ii) that for all Banach limits p,

[mathematical expression not reproducible], (44)

and so

[mathematical expression not reproducible]. (45)

Further, [lim.sub.n[right arrow][infinity]] d([z.sub.n], [z.sub.n+1]) = 0 implies that

[mathematical expression not reproducible]. (46)

By Lemma 11, we have

[mathematical expression not reproducible]. (47)

Step 4. [lim.sub.n[right arrow][infinity]] [x.sub.n] = x will be verified. In fact, by Lemma 3 and (13), now we know

[mathematical expression not reproducible], (48)

and

[mathematical expression not reproducible]. (49)

It follows from (21), Cauchy-Schwarz inequality, and Lemma 9 that

[mathematical expression not reproducible], (50)

From (50) and (49), we know

[mathematical expression not reproducible]. (51)

Combining (51) and (48), we get

[mathematical expression not reproducible], (52)

i.e.,

[mathematical expression not reproducible] (53)

[mathematical expression not reproducible]. (54)

Thus, from the conditions ([L.sub.1])-([L.sub.3]) and the inequality (41), it follows that [alpha]'.sub.n] [member of] (0,1), and

[mathematical expression not reproducible]. (55)

Hence, it follows from Lemma 6 that [u.sub.n] [right arrow] 0. This implies that the proof is completed.

If T [equivalent to] g is a nonexpansive single-valued operator with Fix(g) [not equal to] 0, then from Theorem 13, one can easy to obtain the following result.

Corollary 14. Suppose that f, E, and (X, d) are the same as in Theorem 13, and the conditions ([L.sub.1])-([L.sub.3]) in Theorem 13 are satisfied. If g : E [right arrow] E is a nonexpansive single-valued operator with Fix(g) [not equal to] 0, then the sequence {[x.sub.n]} generated by (12) converges strongly to [mathematical expression not reproducible] with

[mathematical expression not reproducible]. (56)

Remark 15. Corollary 14 is the corresponding results of Theorem 3.1 in .

If f [equivalent to] I, the identity operator, then by Theorem 13, now we directly have the following theorem.

Theorem 16. Assume that T, E, and (X, d) are the same as in Theorem 13, and the conditions ([L.sub.1])-([L.sub.3]) in Theorem 13 hold. Then for any given u, [x.sub.1] [member of] E, sequence {[x.sub.n]} generated by

[mathematical expression not reproducible], (57)

converges strongly to the unique nearest point [??] of u in F(T), i.e., [mathematical expression not reproducible] where [??] also satisfies

[mathematical expression not reproducible]. (58)

Remark 17. Theorems 13 and 16 also extend and improve the corresponding results of Chang et al. , Piatek , Kaewkhao et al. , Panyanak and Suantai , and many others in the literature.

4. Concluding Remarks

The purpose of this paper is to introduce and study the following new two-step viscosity iterative approximation for finding fixed points of a set-valued nonlinear mapping G : D [right arrow] C(D) and a contraction mapping g : D [right arrow] D:

[mathematical expression not reproducible], (59)

where D is a nonempty closed convex subset of a metric space E, [u.sub.1] [member of] D is an any given element and {[[alpha].sub.n]}, {[[beta].sub.n]} [subset] (0,1), and [w.sub.n] [member of] G([u.sub.n]) satisfying d([w.sub.n], [w.sub.n+1]) < d([u.sub.n], [u.sub.n+1]) for any n [member of] N.

By using the method due to Chang et al. [7, 8], Cauchy-Schwarz inequality, and Xu's inequality, we exposed strong convergence theorems of the new two-step viscosity iteration approximation (59) in complete CAT(0) spaces. The main theorems ofthis paper extend and improve the corresponding results of Chang et al. [7, 8], Piatek , Kaewkhao et al. , Panyanak and Suantai , and many others in the literature.

However, when g is a set-value contraction operator or is also nonexpansive in (59), whether can our main results be obtained? Furthermore, can our results be obtained when the iterations (13) (i.e., (10)), (12), and (57) become three-step iterations as in  or operator T is total asymptotically nonexpansive single-valued (set-valued) operator? These are still open questions to be worth further studying.

https://doi.org/10.1155/2018/1280241

Conflicts of Interest

The authors declare that there are not any conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was partially supported by the Scientific Research Project of Sichuan University of Science & Engineering (2017RCL54) and the Scientific Research Fund of Sichuan Provincial Education Department (16ZA0256).

References

 W. A. Kirk, "Geodesic geometry and fixed point theory. II," in Proceedings of the International Conference on Fixed Point Theory and Applications, pp. 113-142, Yokohama Publisher, Yokohama Japan, 2004.

 I. Beg and A. Azam, "On iteration methods for multivalued mappings," Demonstratio Mathematica, vol. 27, no. 2, pp. 493-499, 1994.

 T. Hu, J.-C. Huang, and B. E. Rhoades, "A general principle for Ishikawa iterations for multi-valued mappings," Indian Journal of Pure and Applied Mathematics, vol. 28, no. 8, pp. 1091-1098, 1997.

 Z. Liu, S. M. Kang, and J. S. Ume, "General principles for Ishikawa iterative process for multi-valued mappings," Indian Journal of Pure and Applied Mathematics, vol. 34, no. 1, pp. 157-162, 2003.

 L. Y. Shi and R. D. Chen, "Strong convergence of viscosity approximation methods for nonexpansive mappings in CAT(0) spaces," Journal of Applied Mathematics, Article ID 421050, 11 pages, 2012.

 R. Wangkeeree and P. Preechasilp, "Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces," Journal of Inequalities and Applications, vol. 2013, no. 93,15 pages, 2013.

 S.-S. Chang, L. Wang, J.-C. Yao, and L. Yang, "An affirmative answer to Panyanak and Suantai's open questions on the viscosity approximation methods for a nonexpansive multimapping in CAT(0) spaces," Journal of Nonlinear Sciences and Applications. JNSA, vol. 10, no. 5, pp. 2719-2726, 2017

 S.-S. Chang, L. Wang, G. Wang, and L. Qin, "An affirmative answer to the open questions on the viscosity approximation methods for nonexpansive mappings in CAT(0) spaces," Journal of Nonlinear Sciences and Applications. JNSA, vol. 9, no. 6, pp. 4563-4570, 2016.

 W. A. Kirk, "Fixed point theorems in CAT(0) spaces and R-trees," Fixed Point Theory and Applications, vol. 2004, no. 4, pp. 309-316, 2004.

 S. Dhompongsa and B. Panyanak, "On A-convergence theorems in CAT(0) spaces," Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2572-2579, 2008.

 P. Chaohaand A. Phon-on, "A note on fixed point sets in CAT(0) spaces," Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 983-987, 2006.

 W. Laowang and B. Panyanak, "Strong and A-convergence theorems for multivalued mappings in CAT(0) spaces," Journal of Inequalities and Applications, Art. ID 730132, 16 pages, 2009.

 S. Dhompongsa, A. Kaewkhao, and B. Panyanak, "On Kirk's strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces," Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 2, pp. 459-468, 2012.

 H. Dehghan and J. Rooin, "A characterization of metric projection in CAT(0) spaces," in Proceedings of Interational Conference on Functional Equation, Geometric Functions and Applications (ICFGA), vol. 1, pp. 41-43, Tabriz, Iran, 2012.

 I. D. Berg and I. G. Nikolaev, "Quasilinearization and curvature of Aleksandrov spaces," Geometriae Dedicata, vol. 133, pp. 195-218, 2008.

 B. Panyanak and S. Suantai, "Viscosity approximation methods for multivalued nonexpansive mappings in geodesic spaces," Fixed Point Theory and Applications, vol. 2015, no. 114, 14 pages, 2015.

 J. Nadler, "Multi-valued contraction mappings," Pacific Journal of Mathematics, vol. 30, pp. 475-488, 1969.

 B. Piatek, "Viscosity iteration in CAT(k) spaces," Numerical Functional Analysis and Optimization, vol. 34, no. 11, pp. 1245-1264, 2013.

 A. Kaewkhao, B. Panyanak, and S. Suantai, "Viscosity iteration method in CAT(0) spaces without the nice projection property," Journal of Inequalities and Applications, vol. 2015, no. 278, 9 pages, 2015.

 J. Deepho, J. Martnez-Moreno, and P. Kumam, "A viscosity of Cesaro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems," Journal of Nonlinear Sciences and Applications. JNSA, vol. 9, no. 4, pp. 1475-1496, 2016.

 R. Dewangan, B. S. Thakur, and M. Postolache, "Strong convergence of asymptotically pseudocontractive semigroup by viscosity iteration," Applied Mathematics and Computation, vol. 248, pp. 160-168, 2014.

 V. Gupta and N. Malik, "Approximation of functions by complex genuine Polya-Durrmeyer operators," Computational Methods and Function Theory, vol. 17, no. 1, pp. 3-17, 2017.

 H. He, S. Liu, and R. Chen, "Strong convergence theorems for an implicit iterative algorithm for the split common fixed point problem," Journal of Function Spaces, vol. 2016, Article ID 4093524, 7 pages, 2016.

 S. Wang, C. Hu, G. Chai, and H. Hu, "Equivalent theorems of the convergence between Ishikawa-Halpern iteration and viscosity approximation method," Applied Mathematics Letters, vol. 23, no. 6, pp. 693-699, 2010.

 Y. J. Wu, R. D. Chen, and L. Y. Shi, "Strong convergence theorems for variational inequalities and split equality problem," journal of function spaces and applications, Article ID 151906, 7 pages, 2013.

 J. Z. Xiao, J. Yan, and X. H. Zhu, "Explicit, implicit and viscosity iterations for nonexpansive cosine families in Hilbert spaces," Acta Mathematica Scientia Series A Chinese Edition, vol. 34, no. 6, pp. 1518-1531, 2014.

 D. Zhang and Z. He, "Viscosity approximation methods for hybrid iteration in Hilbert space," Mathematical Sciences Research Journal, vol. 12, no. 6, pp. 141-147, 2008.

 M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Wiley-interscience, New York, NY, USA, 2001.

 M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, Germany, 1999.

 K. S. Brown, Buildings, Springer, New York, NY, USA, 1989.

 K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.

 H. K. Xu, "An iterative approach to quadratic optimization," Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659-678, 2003.

 S. Dhompongsa, A. Kaewkhao, and B. Panyanak, "Browder's convergence theorem for multivalued mappings without endpoint condition," Topology and its Applications, vol. 159, no. 1011, pp. 2757-2763, 2012.

 N. Shioji and W. Takahashi, "Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces," Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641-3645, 1997

 G. S. Saluja and M. Postolache, "Three-step iterations for total asymptotically nonexpansive mappings in CAT(0) spaces," Filomat, vol. 31, no. 5, pp. 1317-1330, 2017

Ting-jian Xiong (iD) and Heng-you Lan (iD)

College of Mathematics and Statistics, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, China

Correspondence should be addressed to Heng-you Lan; hengyoulan@163.com

Received 3 December 2017; Accepted 24 May 2018; Published 2 July 2018

Title Annotation: Printer friendly Cite/link Email Feedback Research Article Xiong, Ting-jian; Lan, Heng-you Journal of Function Spaces 9JAPA Jan 1, 2018 5240 Stability of the Diffusion Equation with a Source. Common Fixed Point Results for Generalized [mathematical expression not reproducible] Contractive Mappings and Applications. Game theory Mathematical optimization Optimization theory