CONVERGENCE RESULTS FOR PROXIMAL POINT ALGORITHM IN COMPLETE CAT(0) SPACE FOR MULTIVALUED MAPPINGS

In this paper, we propose the modified proximal point algorithm with the process for three nearly Lipschitzian asymptotically nonexpansive mappings and multivalued mappings in CAT(0) space under certain conditions. We prove some convergence theorems for the algorithm which was introduced by Shamshad Hussain et al. [22]. A numerical example is given to illustrate the efficiency of proximal point algorithm for supporting our result.

On the otherhand, Markin [28] and Nadler [31] introduced the study of fixed points for multivalued contractions and nonexpansive mappings using the Hausdorff 2020 Mathematics Subject Classification: 47H09, 47H10, 47J25, 65K10. Received: 24-01-2021, accepted: 23-12-2020. metric. Shimizu et al. [39] proved the existence of fixed points for multivalued nonexpansive mappings in convex metric space was established by Shimizu et al. [39], i.e. he proved that every multivalued mapping T : Y → C(Y ) has a fixed point in a bounded, complete and uniformly convex metric space (Y, d), where C(Y ) is family of all compact subsets of Y . In this direction to generalize the nonlinear multivalued mappings, Kim et al. [26] introduced the nearly Lipschitzian multivalued mapping.
In 2019, Hussain et al. [22] has been introduced modified proximal point algorithm in complete CAT(0) space (Y, d) as follows : suppose that h is a convex, proper and lower semi-continuous function on Y . The modified proximal point algorithm is given by for s 1 ∈ Y and π m > 0 where z m ∈ P T (p m ), y m ∈ P T (q m ) and x m ∈ P T (r m ) for each m ∈ N . Let {a m }, {b m } and {c m } be a sequence in [0, 1] for all m ∈ N and {π m } be a sequence with π m > 0 for all m ∈ N and established some ∆-convergence theorems of the proposed algorithm to common fixed points of nonexpansive mappings including a total asymptotically nonexpansive mapping, multivalued mapping and minimizer of a convex function.
In the view of above literature, we propose the modified proximal point algorithm with the process for three nearly Lipschitzian asymptotically nonexpansive mappings and multivalued mappings in CAT(0) space under certain conditions. We prove ∆− convergence, strong and weak convergence results for the algorithm which was defined in (1) by Shamshad Hussain et al. [22]. A numerical example is given to illustrate the efficiency of proximal point algorithm for supporting our result.

Preliminaries
Throughout in this paper, we assume that is called a CAT(0) space if it is geodesically connected and every geodesic triangle in Y is atleast as thin as its comparison triangle in the Euclidean plane.
is unique geodesic joining s and r. In this paper, we can write ts ⊕ (1 − t)r for the unique point q in the geodesic segment joining s to r such that Then, 1. The asymptotic radius ofr({s m }) of {s m } is given bŷ In complete CAT(0) space, A({s m }) consists of exactly one point [15].  A subset W ⊂ Y = φ is said to be proximal if for each s ∈ Y , there exists an element r ∈ W such that d(s, r) = dist(s, W) = inf{d(s, q) : q ∈ W}.
It is well known that each weakly compact convex subset of a Banach space is proximal as well as each closed convex subset of a uniformly convex Banach space is also proximal. Many authors have been discussed fixed point in CAT(0) space (see [1,19,32]).
Let T : Y → 2 Y be a multivalued mapping. An element s ∈ Y is said to fixed point of T if s ∈ T s. Definition 2.6. A multivalued mapping T : Y → CB(Y ) is called nonexpansive, if for x, y ∈ Y and for m ∈ N , we have where the sequence {v m } in [0, ∞) such that lim m→∞ v m = 0. The infimum of constants k m in (3) is called the nearly Lipschitzian constant of T m , denoted by η(T m ).
The following example of nearly Lipschitzian mapping given by Abbas et al. [2] as follows. Similarly, we define here two nearly Lipschitzian mappings in our next two examples as follows.
Example 2.9. Assume that B : (0, ∞) → (0, ∞) is defined by Here Here Thus Recall that a function h : The Moreau-Yoshida resolvent of function h in the CAT(0) space is given by for any π > 0 and for all s ∈ Y.
(2) If h is convex, proper and lower semi-continuous function, then the set of fixed point of the resolvent associated with h coincides with the set of minimizers of h (see [7]).

Main Results
Theorem 3.1. Let (Y, d) be a complete CAT(0) space and W be a nonempty closed convex subset of Y . Let T : W → P (W) be multivalued mapping and P T be a nonexpansive mapping. Let h : Y → (−∞, ∞] be a proper convex and lower semicontinuous function and A, B, C : W → W be three nearly Lipschitzian mappings with {k m }, {v m } being nonnegative real sequences such that Proof. Since E = φ. So we can assume that t ∈ E which implies that t = At = Bt = Ct and h(t) ≤ h(r) for any r ∈ W. Thus, we have for each r ∈ W, and we have t = J πm t for each m ∈ N. Since p m = J πm s m and J πm is nonexpansive, so we have Now using (4), (5) and Lemma 2.3, we have where C = 1 − c m (1 − k m ) and G = c m k m v m . Now using (4), (6) and Lemma 2.3, we have (4), (7) and Lemma 2.3, we have Thus, from Lemma 2.14 and inequality (8), lim m→∞ d(s m , t) exists and we may assume that lim m→∞ d(s m , t) = k ≥ 0.
(1) Our results extends the results of Hussain et al. [22] in the framework of CAT(0) spaces. They established convergence theorems for different classes of generalized nonexpansive mappings including a total asymptotically nonexpansive mapping, a multivalued mapping, and a minimizer of a convex function for solving the convex minimization problem and the common fixed point problem.
(2) Our results is generalization of the results of Pakkaranang et al. [33] in the framework of CAT(0) spaces. They established convergence theorems for three asymptotically quasinonexpansive mappings involving the convex and lower semi-continuous function for solving the convex minimization problem and the common fixed point problem.

Numerical Examples
In this section, we discuss a numerical result to illustrate the convergence of the iterative algorithm (4) to support our example.
It is clear that the mapping T is nonexpansive. It is easy to check that h is a proper convex and lower semi-continuous function and consider the nearly Lipschitzian mappings A, B, C from definitions (2.8), (2.9) and (2.10), respectively and T is nonexpansive mapping with G(A) ∩ G(B) ∩ G(C) ∩ G(T ) = {2}. Suppose that a m = 15m−3 16m , b m = m+5 22m and c m = 33m−7 34m and s 1 = 50 is the initial value. We obtain the numerical results with the errors values in Table 1. From Table 1, Figure 1 and Figure 2, it is clear that the sequence {s m } converges to 1.99999 ≡ 2 which is common fixed point of solution of a minimizer of a function h, multivalued mapping T and three nearly Lipschtzian mappings A, B and C.

Conclusion
In this paper, we proved the ∆−convergence, strong and weak convergence results for the modified proximal point algorithm for three nearly Lipschitzian asymptotically nonexpansive mappings and multivalued mapping in CAT(0) space. Also, we illustrated the efficiency of modified proximal point algorithm by numerical  [22] and Pakkaranang et al. [33].