# Fixed Point Results for a Class of Monotone Nonexpansive Type Mappings in Hyperbolic Spaces.

1. Introduction

A mapping p from the set of reals R to a metric space (X, [rho]) is said to be metric embedding if [rho](p(m), p(n)) = [absolute value of m - n] for all m, n [member of] R. The image of R under a metric embedding is called a metric line. The image of a real interval [a, b] = {t [member of] R : a [less than or equal to] t [less than or equal to] b} under metric embedding is called a metric segment. Assume that (X, [rho]) has a family F of metric lines such that, for each pair u, v [member of] X (u [not equal to] v), there is a unique metric line in F which passes through u and v. This metric line determines a unique metric segment joining u and v. This segment is denoted by [u, v] and this is an isometric image of the real interval [0, [rho](u, v)]. We denote by yu [direct sum] (1 - [gamma])v the unique point w of [u, v] which satisfies

[rho] (u, w) = (1 - [gamma]) [rho] (u, v) and

[rho] (w, v) = [gamma][rho] (u, v), (1)

where [gamma] [member of] [0,1]. Such a metric space with a family of metric segments is called a convex metric space [1]. Moreover, if we have

[rho] ([gamma]u [direct sum] (1 - [gamma]) v, [gamma]w [direct sum] (1 - [gamma]) z)

[less than or equal to] [gamma]p (u, w) + (1- [gamma]) [rho] (v, z) (2)

for all u, v, w, z [member of] X, then X is said to be a hyperbolic metric space [2].

Hyperbolic spaces are more general than normed spaces and CAT(0) spaces. These spaces are nonlinear. Indeed, all normed linear and CAT(0) spaces are hyperbolic spaces (cf. [3-5]). As nonlinear examples, one can consider the Hadamard manifolds [6] and the Hilbert open unit ball equipped with the hyperbolic metric [7].

A mapping T : X [right arrow] X is said to be nonexpansive if [rho](T(u), T(v)) [less than or equal to] [rho](u, v), for all u, v [member of] X The study of existence of fixed point of nonexpansive mappings has been of great interest in nonlinear analysis. Fixed point theory of nonexpansive mappings in hyperbolic spaces has been extensively studied (cf. [2, 8-14]). Bin Dehaish and Khamsi [8] obtained a fixed point theorem for a monotone nonexpansive mapping in the setting partially ordered hyperbolic metric spaces.

On the other hand, a number of extensions and generalizations of nonexpansive mapping has been considered by many authors (cf. [15-26]). Gregus [21] considered the following class of nonexpansive type mappings. Let T : X [right arrow] X be a mapping such that, for all u, v [member of] X,

[rho] (T (u), T(v)) [less than or equal to] ap (u, v) + bp (T (u), u)

+ c[rho] (T (v), v), (3)

where a, b, and c are nonnegative constants such that a + b + c = 1. A mapping satisfying (3) is also known as Reich type nonexpansive mapping. A mapping satisfying (3) also satisfies the following condition:

[rho] (T (u),T(v)) [less than or equal to] k[rho] (T (u), u) + k[rho] (T (v), v)

+ (1 - 2k) [rho] (u, v), (4)

where k [member of] [0,1).

Suzuki [26] introduced the following class of nonexpansive type mappings.

Definition 1 (see [26]). Let Y be a nonempty subset of a Banach space X. A mapping T : Y [right arrow] Y is said to satisfy condition (C) if, for all u, v [member of] Y,

[mathematical expression not reproducible]. (5)

Notice that the class of mappings satisfying (3)-(5) properly contains the class of nonexpansive mappings and need not be continuous.

Motivated by the above fact and works of Gregus [21], Suzuki [26], and others, in this paper, we make an attempt to define a wider class of nonexpansive type mapping which properly contains nonexpansive, Reich type nonexpansive, and Suzuki type nonexpansive mappings. In particular, in Section 3, we study some existence results in partially ordered hyperbolic space for this class of nonexpansive type mapping and some illustrative nontrivial examples have also been discussed. In Section 4, we present some convergence results for an iteration algorithm due to Abbas and Nazir [27]. In Section 5, we discuss an application of our results to nonlinear integral equations.

2. Preliminaries

Let us recall the following definition which is due to Kohlenbach [11].

Definition 2 (see [11]). A triplet (X, [rho], H) is said to be a hyperbolic metric space if (X, [rho]) is a metric space and H : X x X x [0,1] [right arrow] X is a function such that, for all u, v, w, z [member of] X and [beta], [gamma] [member of] [0, 1], the following hold:

(K1) [mathematical expression not reproducible];

(K2) [mathematical expression not reproducible];

(K3) H(u, v, [beta]) = H(v, u, 1 - [beta]);

(K4) [mathematical expression not reproducible].

The set seg[u, v] := {H(u, v, [beta]); [member of] [0, 1]} is called the metric segment with endpoints u and v. Now onwards, we write H(u, v, [beta]) = (1 - [beta]) u [direct sum] [beta]v. A subset Y of X is said to be convex if (1 - [beta])u [direct sum] [beta]v [member of] Y for all u, v [member of] Y and [beta] [member of] [0,1]. When there is no ambiguity, we write (X, [rho]) for (X, [rho], H).

Let [member of] be a partially ordered set with a partial order '[less than or equal to]' and let (E, p, [less than or equal to]) be a partially ordered hyperbolic metric space. We say that u, v [member of] E are comparable whenever u [less than or equal to] v or v [less than or equal to] u. Throughout, we will assume that order intervals are closed convex subsets of hyperbolic metric space E. We denote these intervals as follows:

[a, [right arrow]) := {u [member of] E; a [less than or equal to] u} and

([less than or equal to], b] := {u [member of] E; u [less than or equal to] b}, (6)

for any a, b [member of] E (cf. [8]).

Definition 3 (see [28, 29]). Let (E, [rho]) be a hyperbolic metric space. For any a [member of] E, r > 0 and [epsilon] > 0. Set

[mathematical expression not reproducible]. (7)

We say that [member of] is uniformly convex if [delta](r, [epsilon]) > 0, for any r > 0 and [epsilon] >0.

Definition 4 (see [30]). A hyperbolic metric space (E, [rho]) is said to satisfy property (R) if, for each decreasing sequence {[F.sub.n]} of nonempty bounded closed convex subsets of E, [[intersection].sup.[infinity].sub.n=1] [F.sub.n] [not equal to] 0.

Uniformly convex hyperbolic spaces satisfy the property (R); see [8].

Definition 5 (see [8]). Let (E, [rho], [less than or equal to]) be a metric space endowed with a partial order. A mapping T : X [right arrow] X is said to be monotone

T (u) [less than or equal to] T (v) whenever u [less than or equal to] v, (8)

for all u, v [member of] E.

Definition6 (see [8]). Let (E, [rho], [less than or equal to]) be a metric space endowed with a partial order. A mapping T : X [right arrow] X is said to be monotone nonexpansive if T is monotone and

[rho](T(u), T(v)) [less than or equal to] p(u, v), (9)

for all u, v [member of] E such that u, v are comparable.

Definition 7 (see [31]). Let K be a subset of a metric space (E, [rho]). A mapping T : K [right arrow] K is said to satisfy condition (I) if there exists a nondecreasing function g : [0, [right arrow]) [right arrow] [0, [infinity]) satisfying g(0) = 0 and g(r) > 0 for all r [member of] (0, [infinity]) such that [rho](u, T(u)) [greater than or equal to] g(dist(u, F(T))) for all u [member of] K, and here dist(u, F(T)) denotes the distance of u from F(T).

Let K be a nonempty subset of a hyperbolic metric space (E, [rho]) and {[u.sub.n]} a bounded sequence in E. For each u [member of] E, define

(i) asymptotic radius of {[u.sub.n]} at u as r({[u.sub.n]}, u) := [lim sup.sub.n[right arrow][infinity]] [rho]([u.sub.n], u);

(ii) asymptotic radius of {[u.sub.n]} relative to K as

r ({[u.sub.n]}, K) := inf {r ({[u.sub.n]}, u); u [member of] K}; (10)

(iii) asymptotic centre of {[u.sub.n]} relative to K by

A ({[u.sub.n]} > K) := {u [member of] K; r ({[u.sub.n]}, u) = r ({[u.sub.n]}, K)}. (11)

Lim in [32] introduced the concept of [DELTA]-convergence in a metric space. Kirk and Panyanak in [33] used Lim's concept in CAT(0) spaces and showed that many Banach space results involving weak convergence have precise analogs in this setting.

Definition 8 (see [33]). A bounded sequence {[u.sub.n]} in [member of] is said to [DELTA]-converge to a point u [member of] E, if u is the unique asymptotic centre of every subsequence {[u.sub.n]} of {[u.sub.n]}.

Definition 9 (see [8]). Let K be a nonempty subset of a hyperbolic metric space (E, [rho]). A function [tau] : K [right arrow] [0, [infinity]) is said to be a type function, if there exists a bounded sequence {[u.sub.n]} in [member of] such that

[mathematical expression not reproducible], (12)

for any u [member of] K.

We know that every bounded sequence generates a unique type function.

Lemma 10 (see [8]). Let (E, [rho], [less than or equal to]) be a uniformly convex hyperbolic metric space and K a nonempty closed convex subset of E. Let [tau] : K [right arrow] [0, [infinity]) be a type function. Then [tau] is continuous. Moreover, there exists a unique minimum point z [member of] K such that [tau](z) = inf{[tau](u); u [member of] K}.

Now, we rephrase the concept of [delta]-convergence in hyperbolic metric spaces.

Definition 11. A bounded sequence {[u.sub.n]} in [member of] is said to [DELTA]-converge to a point z [member of] E if z is unique and the type function generated by every subsequence [mathematical expression not reproducible] attains its infimum at z.

Abbas and Nazir [27] introduced an iteration process which can be defined in the framework of hyperbolic metric spaces as follows:

[mathematical expression not reproducible], (13)

where {[[alpha].sub.n]}, {[[beta].sub.n]}, and {[[gamma].sub.n]} are real sequences in (0,1).

3. Existence Results

In this section, we define a new class of nonexpansive type mappings and present some auxiliary and an existence result (see Theorem 16 below). We also discuss a couple of illustrative examples.

Definition 12. Let (E, [rho], [less than or equal to]) be a partially ordered metric space and let T : E [right arrow] E be a monotone mapping. Then T is called monotone Reich-Suzuki type nonexpansive mapping, if there exists a k [member of] [0,1) such that

[mathematical expression not reproducible] (14)

for all u, v [member of] E such that u and v are comparable.

Lemma 13. Let (E, [rho], [less than or equal to]) be a uniformly convex partially ordered hyperbolic metric space and K a nonempty closed convex subset of E. Let T : K [right arrow] K be a monotone mapping. Let [u.sub.1] [member of] K be such that [u.sub.1] [less than or equal to] T([u.sub.1]) (T([u.sub.1]) [less than or equal to] [u.sub.1]). Then, for sequence {[u.sub.n]} defined by (13), we have

(a) [u.sub.n] [less than or equal to] T([u.sub.n]) [less than or equal to] [u.sub.n+1] (or [u.sub.n+1] [less than or equal to] T([u.sub.n]) [less than or equal to] [u.sub.n]);

(b) [u.sub.n] [less than or equal to] p (or p [less than or equal to] [u.sub.n]), provided {[u.sub.n]} [DELTA]- converges to a point p [member of] K, for all n [member of] N.

Proof. We shall use induction to prove (a). By assumption, we have [u.sub.1] [less than or equal to] T([u.sub.1]). By the convexity of order interval [[u.sub.1], T([u.sub.1])] and (13), we have

[u.sub.1] [less than or equal to] [w.sub.1] [less than or equal to] T ([u.sub.1]). (15)

As T is monotone, T([u.sub.1]) [less than or equal to] T([w.sub.1]). Again, by convexity of order interval [T([u.sub.1]), T([w.sub.1])] and (13), we have

T([u.sub.1]) [less than or equal to] [v.sub.1] [less than or equal to] T([w.sub.1]). (16)

Combining (15) and (16), we get

[u.sub.1] [less than or equal to] [w.sub.1] <T ([u.sub.1]) < [v.sub.1]. (17)

As T is monotone, T([w.sub.1]) [less than or equal to] T([v.sub.1]). Again, by convexity of order interval [T([w.sub.1]), T([v.sub.1])] and (13), we have

T([v.sub.1]) [less than or equal to] [u.sub.2] [less than or equal to] T([v.sub.1]). (18)

By (15), (16), and (18), we get

[u.sub.1] [less than or equal to] T ([u.sub.1]) [less than or equal to] T ([w.sub.1]) [less than or equal to] [u.sub.2] [less than or equal to] T ([v.sub.1]). (19)

Thus (a) is true for n = 1. Now, suppose it is true for n; that is,

[u.sub.n] [less than or equal to] T ([u.sub.n]) [less than or equal to] [u.sub.n+1]. (20)

By convexity of order interval [[u.sub.n], T([u.sub.n])] and (13), we have

[u.sub.n] [less than or equal to] [w.sub.n] [less than or equal to] T ([u.sub.n]). (21)

As T is monotone, T([u.sub.n]) [less than or equal to] T([w.sub.n]). Again, by convexity of order interval [T([u.sub.n]), T([w.sub.n])] and (13), we have

T([u.sub.n]) [less than or equal to] [v.sub.n] [less than or equal to] T ([w.sub.n]). (22)

From (21) and (22), we have

[w.sub.n] [less than or equal to] T ([u.sub.n]) [less than or equal to] [v.sub.n] [less than or equal to] T ([w.sub.n]), (23)

so T([w.sub.n]) [less than or equal to] T([v.sub.n]). By convexity of order interval [T([w.sub.n]), T([v.sub.n])], (23), and (13), we have

T([u.sub.n]) [less than or equal to] T ([w.sub.n]) [less than or equal to] [u.sub.n+1] [less than or equal to] T ([v.sub.n]). (24)

From (21), (23), and (24), we have

[u.sub.n] [less than or equal to] T ([u.sub.n]) [less than or equal to] [v.sub.n] [less than or equal to] T ([w.sub.n]) [less than or equal to] [u.sub.n+1] [less than or equal to] T ([v.sub.n]), (25)

and then T([v.sub.n]) * T([u.sub.n+1]). From (25), we have

[u.sub.n+1] [less than or equal to] T([u.sub.n+1]). (26)

By convexity of order interval [[u.sub.n+1], T([u.sub.n+1])] and (13), we have

[u.sub.n+1] [less than or equal to] [w.sub.n+1] [less than or equal to] T([u.sub.n+1]), (27)

so T([u.sub.n+1]) [less than or equal to] T([w.sub.n+1]). By convexity of order interval [T([u.sub.n+1]), T([w.sub.n+1])] and (13), we have

[u.sub.n+1] [less than or equal to] T ([u.sub.n+1]) [less than or equal to] [v.sub.n+1] [less than or equal to] T([w.sub.n+1]). (28)

By (27) and (28), [w.sub.n+1] [less than or equal to] [v.sub.n+1]. By convexity of order interval we get [T([w.sub.n+1]), T([v.sub.n+1])]. Hence, from (13), we have

T([w.sub.n+1]) [less than or equal to] [u.sub.n+2] [less than or equal to] T([v.sub.n+1]). (29)

From (28) and (29), we have

[u.sub.n+1] [less than or equal to] T ([u.sub.n+1]) [less than or equal to] T ([w.sub.n+1]) [less than or equal to] [u.sub.n+2] [less than or equal to] T ([v.sub.n+1]); (30)

that is

[u.sub.n+1] [less than or equal to] T ([u.sub.n+1]) [less than or equal to] [u.sub.n+2]. (31)

Suppose p is a [DELTA]-limit of {[u.sub.n]}. Here the sequence {[u.sub.n]} is monotone increasing and the order interval [[u.sub.m], [right arrow]) is closed and convex. We claim that p [member of] [[u.sub.m], [right arrow]) for a fixed m [member of] N. If p [not member of] [[u.sub.m], [right arrow]), then the type function generated by subsequence {[u.sub.r]} of {[u.sub.n]} defined by leaving first m-1 terms of the sequence {[u.sub.n]} will not attain an infimum at p, which is a contradiction to the assumption that p is a [DELTA]-limit of the sequence {[u.sub.n]}. This completes the proof.

Lemma 14. Let (E, [rho], [less than or equal to]) be a partially ordered hyperbolic metric space and K a nonempty subset of E. Let T : K [right arrow] K be a monotone Reich-Suzuki type nonexpansive mapping. Then, for each u, v [member of] K,

(i) [rho](T(u), [T.sup.2](u)) [less than or equal to] [rho](u, T(u));

(ii) Either (1/2)[rho](u, T(u)) [less than or equal to] [rho](u, v) or (1/2)[rho](T(u), [T.sup.2](u)) [less than or equal to] [rho](T(u), v);

(iii) Either [rho](T(u), T(v)) [less than or equal to] k[rho](T(u), u) + k[rho](v, T(v)) + (1 - 2k)[rho](u, v) or [rho]([T.sup.2] (u), T(v)) [less than or equal to] k[rho]([T.sup.2](u), T(u)) + k[rho](T(v), v) + (1 - 2k)[rho](T(u), v),

where u and v are comparable.

Proof. As (1/2)[rho](u, T(u)) [less than or equal to] [rho](u, T(u)) by definition of monotone Reich-Suzuki type nonexpansive mapping, we have

[mathematical expression not reproducible], (32)

since 1 - k > 0, so

[rho](T(u), [T.sup.2] (u)) [less than or equal to] [rho](u, T(u)). (33)

To prove (ii), arguing by contradiction, we suppose that

[mathematical expression not reproducible], (34)

By (i) and triangle inequality, we have

[mathematical expression not reproducible], (35)

which is a contradiction. Hence (ii) holds. Condition (iii) follows directly from condition (ii).

The following lemma is a consequence of the above lemma.

Lemma 15. Let (E, [rho], [less than or equal to]) be a partially ordered hyperbolic metric space and K a nonempty subset of E. Let T : K [right arrow] K be a monotone Reich-Suzuki type nonexpansive mapping. Then, for all u, v [member of] K such that u and v are comparable, we have

[rho](u, T (v)) [less than or equal to] (3 + k)/(1 - k) [rho] (u, T (u)) + [rho] (u, v). (36)

Theorem 16. Let (E, p, [less than or equal to]) be a uniformly convex partially ordered hyperbolic metric space and K a nonempty closed convex subset of E. Let T : K [right arrow] K be a monotone Reich-Suzuki type nonexpansive mapping. Assume that there exists [u.sub.1] [member of] K such that [u.sub.1] and T([u.sub.1]) are comparable. Let the sequence {[u.sub.n]} defined by (13) be bounded, and there exists a point w [member of] K such that every point of the sequence {[u.sub.n]} is comparable with w and [lim inf.sub.n[right arrow][infinity]] p(T([u.sub.n]), [u.sub.n]) = 0. Then T has a fixed point.

Proof. Suppose {[u.sub.n]} is a bounded sequence and [lim inf.sub.n[right arrow][infinity]] p(T([u.sub.n]), [u.sub.n]) = 0. Then there exists a subsequence {[u.sub.n]} of {[u.sub.n]} such that

[mathematical expression not reproducible]. (37)

By Lemma 13, we have [mathematical expression not reproducible]. Clearly, for each j [member of] N, [K.sub.j] is closed convex and w [member of] [K.sub.j] so [K.sub.j] is nonempty. Set

[mathematical expression not reproducible]. (38)

Then, [K.sub.[infinity]] is closed convex subset of K. Let u [member of] [K.sub.[infinity]]; then[mathematical expression not reproducible]. Since T is monotone, for all j [member of] N,

[mathematical expression not reproducible]. (39)

This implies that T([K.sub.[infinity]]) [subset not equal to] [K.sub.[infinity]]. Let [tau] : [K.sub.[infinity]] [right arrow] [0, [infinity]) be a type function generated by [mathematical expression not reproducible]; that is,

[mathematical expression not reproducible]. (40)

From Lemma 10, there exists a unique element z [member of] [K.sub.[infinity]] such that

[tau](z) = inf [[sigma](u);u [member of] [K.sub.[infinity]]}. (41)

By definition of type function,

[mathematical expression not reproducible]. (42)

Using Lemma 15, we get

[mathematical expression not reproducible]. (43)

By the uniqueness of minimum point this implies that T(z) = z, and hence the proof is completed.

Now, let us illustrate the following examples.

Example 17. Let E = R (the set of reals) be equipped with the usual ordering and standard norm [parallel]u[parallel] = [absolute value of u]. Let K = [0, 1] [subset] R and T : K [right arrow] K be a mapping defined by

[mathematical expression not reproducible]. (44)

Then

(1) T is not a nonexpansive mapping,

(2) T is monotone Reich-Suzuki type nonexpansive mapping.

For u = 0 and v = 1/4, we have [parallel]T(u) - T(v) [parallel] = 7/16 > 1/4 = [parallel]u - v[parallel]. Therefore T is not a nonexpansive mapping.

Now, we show that T is monotone Reich-Suzuki type nonexpansive mapping with k = 1/2. We consider the following three cases:

(i) Let u, v [member of] [0, 1/4), and we have

[mathematical expression not reproducible]. (45)

(ii) Let u, v [member of] [1/4, 1], and we have

[mathematical expression not reproducible]. (46)

(iii) Let u [member of] [0, 1/4) and v [member of] [1/4, 1], and we have

[mathematical expression not reproducible]. (47)

Therefore, T is monotone Reich-Suzuki type nonexpansive mapping with unique fixed point 2/3.

Notice that the space considered in the above example was a linear space. Now we present an example of a hyperbolic space which is not linear. Therefore it is a nontrivial example of a hyperbolic space.

Example 18. Let E = {([u.sub.1], [u.sub.2]) [member of] [R.sup.2]; [u.sub.1], [u.sub.2] > 0}. Define [rho] : E x E [right arrow] [0, [infinity]) by

[rho](u, v) = [absolute value of [u.sub.1] - [v.sub.1]] + [absolute value of [u.sub.1][u.sub.2] - [v.sub.1][v.sub.2]] (48)

for all u = ([u.sub.1], [u.sub.2]) and v = ([v.sub.1], [v.sub.2]) in E. Then it can be easily seen that (E, [rho]) is a metric space. Now, for [beta] [member of] [0, 1], define a function H: E x E x[ 0,1] [right arrow] E by

[mathematical expression not reproducible]. (49)

We show that (E, [rho], H) is a hyperbolic metric space. For w = ([u.sub.1], [u.sub.2]), v = ([v.sub.1], [v.sub.2]), z = ([z.sub.1], [z.sub.2]), and w = ([w.sub.1], [w.sub.2]) in E, consider the following:

(K1)

[mathematical expression not reproducible]. (50)

(K2)

[mathematical expression not reproducible]. (51)

(K3)

[mathematical expression not reproducible]. (52)

(K4)

[mathematical expression not reproducible]. (53)

Therefore, (E, [rho], H) is a hyperbolic metric space but not a normed linear space. Now, let us define an order on [member of] as

follows: for u = ([u.sub.1], [u.sub.2]) and v =([v.sub.1], [v.sub.2]), u < v if and only if [u.sub.1] < [v.sub.1] or [u.sub.1] = [v.sub.1] and [u.sub.2] < [v.sub.2]. Thus (E, [rho], [less than or equal to]) is an ordered hyperbolic metric space.

Let K := [1,4] x [1,4] [subset] [member of] and T : K [right arrow] K be a mapping defined by

[mathematical expression not reproducible]. (54)

First we show that T is not a nonexpansive mapping on K. Let u = (1, 1) and v = (4,4). Then

[rho](T(u), T(v)) = 7/4 > 18 = [rho](u, v). (55)

Now, we show that T is monotone Reich-Suzuki type non-expansive mapping for k = 1/2. We consider the following cases.

Case i. If u = ([u.sub.1], [u.sub.2]) and v = ([v.sub.1], [v.sub.2]) = (4, 4), then

[rho](T (u), T (v)) = 0 [less than or equal to] 1/2([rho](u, T (u)) + [rho](v, T (v))) . (56)

Case ii. If u = ([u.sub.1], [u.sub.2]) = (4, 4) and v = ([v.sub.1], [v.sub.2]) = (4, 4), then

[mathematical expression not reproducible]. (57)

Therefore T is monotone Reich-Suzuki type nonexpansive mapping. The only fixed point of T is (1,1).

4. Convergence Results

In this section, we discuss some strong convergence and [DELTA]-convergence results in a partially ordered hyperbolic space for Abbas and Nazir iteration algorithm [27]. Our results are prefaced by the following proposition and lemma.

Proposition 19. Let (E, [rho], [less than or equal to]) be a partially ordered hyperbolic metric space and K a nonempty subset of E. Let T : K [right arrow] K be a monotone Reich-Suzuki type nonexpansive mapping with a fixed point z [member of] K. Then T is quasi-nonexpansive; that is, [rho](T(u), z) [less than or equal to] [rho](u, z) for all u [member of] K and z [member of] F(T) such that u and z are comparable.

Lemma 20 (see [34]). Let (E, [rho]) be a uniformly convex hyperbolic metric space with monotone modulus of uniform convexity [delta]. Let z [member of] E and {[[alpha].sub.n]} be a sequence such that 0 < a [less than or equal to] [[alpha].sub.n] [less than or equal to] b < 1. If {[u.sub.n]} and {[v.sub.n]} are sequences in E such that [mathematical expression not reproducible]

Theorem 21. Let (E, [rho], [less than or equal to]) be a uniformly convex partially ordered hyperbolic metric space and K a nonempty closed convex subset of E. Let T : K [right arrow] K be a monotone Reich-Suzuki type nonexpansive mapping. Assume that there exists [u.sub.1] [member of] K such that [u.sub.1] and T([u.sub.1]) are comparable. Suppose F(T) is nonempty, and [u.sub.1] and z are comparable for every z [member of] F(T).Let {[u.sub.n]} be a sequence defined by (13). Then the following assertions hold:

(i) The sequence {[u.sub.n]} is bounded.

(ii) max{[rho]([u.sub.n+1], z), [rho]([v.sub.n], z), [rho]([w.sub.n], z)} [less than or equal to] [rho]([u.sub.n], z) for all n [member of] N.

(iii) [lim.sub.n[right arrow][infinity]] [rho]([u.sub.n], z) exists and [lim.sub.n[right arrow][infinity]] dist([u.sub.n], F(T)) exists, where dist(u, F(T)) denotes the distance from u to F(T).

(iv) [lim.sub.n[right arrow][infinity]] [rho](T([u.sub.n]), [u.sub.n]) = 0.

Proof. Without loss of generality, we may assume that [u.sub.1] [less than or equal to] z. Since T is monotone, T([u.sub.1]) < T(z) = z. By (13) (as in (16) and (17)), we have

[u.sub.1] [less than or equal to] [w.sub.1] [less than or equal to] T([u.sub.1]) [less than or equal to] [v.sub.1] [less than or equal to] T ([w.sub.1]). (58)

So, [w.sub.1] [less than or equal to] z. By monotonicity of T,

T ([w.sub.1]) [less than or equal to] T(z) = z, (59)

and also, from (58), we have [v.sub.1] [less than or equal to] z. By monotonicity of T,

T([v.sub.1]) [less than or equal to] T (z) = z. (60)

By (13) (as in (18)),

T([w.sub.1]) [less than or equal to] [u.sub.2] [less than or equal to] T([v.sub.1]). (61)

From (60) and (61), we have [u.sub.2] [less than or equal to] z, so, by monotonicity of T,

T ([u.sub.2]) [less than or equal to] T(z) = z. (62)

Again from (31) and (62), for n = 2, we have

[u.sub.2] [less than or equal to] T ([u.sub.2]) [less than or equal to] z. (63)

Applying the same argument, we get

[u.sub.n] [less than or equal to] T ([u.sub.n]) [less than or equal to] z. (64)

Now, by (13) and Proposition 19, we have

[mathematical expression not reproducible]. (65)

By (13), (65), and Proposition 19, we have

[mathematical expression not reproducible]. (66)

By (13), (65), (66), and Proposition 19, we have

[mathematical expression not reproducible]. (67)

Thus, the sequence {[rho]([u.sub.n], z)} is bounded and decreasing, so, [lim.sub.n[right arrow][infinity]] [rho]([u.sub.n], z) exists. For each z [member of] F(T) and n [member of] N we have [rho]([u.sub.n+1], z) [less than or equal to] [rho]([u.sub.n], z). Taking infimum over all z [member of] F(T), we get dist([u.sub.n+1], F(T)) [less than or equal to] dist([u.sub.n], F(T)) for all n [member of] N. So the sequence dist([u.sub.n], F(T)) is bounded and decreasing. Therefore [lim.sub.n[right arrow][infinity]] dist([u.sub.n], F(T)) exists. Suppose

[mathematical expression not reproducible]. (68)

From (68) and Proposition 19, we have

[mathematical expression not reproducible]. (69)

By (65) and (68), we have

[mathematical expression not reproducible]. (70)

From (70) and Proposition 19, we have

[mathematical expression not reproducible]. (71)

By (66) and (68), we have

[mathematical expression not reproducible]. (72)

By (72) and Proposition 19, we have

[mathematical expression not reproducible]. (73)

By (68) and (13), we have

[mathematical expression not reproducible]. (74)

In view of (71), (73), (74), and Lemma 20, we have

[mathematical expression not reproducible]. (75)

By (13),

[mathematical expression not reproducible]. (76)

letting n [right arrow] [infinity] and using (75), we have

[mathematical expression not reproducible]. (77)

By triangle inequality and Proposition 19, we have

[mathematical expression not reproducible]. (78)

and as n [right arrow] [infinity], we have

[mathematical expression not reproducible]. (79)

By (70) and (79), we have

[mathematical expression not reproducible]. (80)

or

[mathematical expression not reproducible]. (81)

Finally, from (68), (69), (81), and Lemma 20, we conclude that

[lim.sub.n[right arrow][infinity]] [rho](T([u.sub.n]), [u.sub.n]) = 0.

Now, we present the following main result about [DELTA]-convergence.

Theorem 22. Let (E, [rho], [less than or equal to]) be a uniformly convex partially ordered hyperbolic metric space and K a nonempty closed convex subset of E. Let T : K [right arrow] K be a monotone Reich-Suzuki type nonexpansive mapping. Assume that there exists [u.sub.1] [member of] K such that [u.sub.1] and T([u.sub.1]) are comparable and F(T) is nonempty and totally ordered. Then the sequence {[u.sub.n]} defined by (13) [DELTA]-converges to a fixed point of T.

Proof. By Theorem 21, the sequence {[u.sub.n]} is bounded. Therefore there exists a subsequence [mathematical expression not reproducible] [DELTA]-converges to some p [member of] K. By Lemma 13, we have

[mathematical expression not reproducible]. (82)

Now, we show that every [DELTA]-converges subsequence of {[u.sub.n]} has a unique [DELTA]-limit in F(T). Suppose {[u.sub.n]} has two subsequences [mathematical expression not reproducible] [DELTA]-converging to l and m, respectively By Theorem 21, [mathematical expression not reproducible] is bounded and

[mathematical expression not reproducible]. (83)

We claim that l [member of] F(T). Let [tau] : K [right arrow] [0, [infinity]) be a type function generated by [mathematical expression not reproducible]; that is,

[mathematical expression not reproducible]. (84)

By Lemma 15 and (83), we have

[mathematical expression not reproducible]. (85)

By the uniqueness of the element l and definition of [DELTA]-convergence, T(l) = l. Similarly T(m) = m. By the definition of [DELTA]-convergence and Lemma 10, we have

[mathematical expression not reproducible]. (86)

which is a contradiction, unless l = m.

Next, we present a strong convergence theorem.

Theorem 23. Let (E, [rho], [less than or equal to]) be a uniformly convex partially ordered hyperbolic metric space and K, T, and {[u.sub.n]} be the same as in Theorem 21 with F(T) [not equal to] 0. The sequence defined by (13) converges strongly to a fixed point of T if and only if [lim inf.sub.n[right arrow][infinity]] dist([u.sub.n], F(T)) = 0, provided F(T) is a totally ordered set.

Proof. Suppose that [lim inf.sub.n[right arrow][infinity]] dist([u.sub.n],F(T)) = 0. From Theorem 21, [lim.sub.n[right arrow][infinity]] dist([u.sub.n], F(T)) exists, so

[mathematical expression not reproducible]. (87)

First we show that the set F(T) is closed. For this, let {[z.sub.n]} be a sequence in F(T) converging to a point v [member of] K. Since (1/2)[rho]([z.sub.n], T([z.sub.n])) = 0 [less than or equal to] [rho]([z.sub.n], v) for all n [member of] N, we have

[mathematical expression not reproducible]. (88)

As 1 - k > 0,

[rho]([z.sub.n], T(v)) [less than or equal to] [rho]([z.sub.n], V), (89)

and hence

[mathematical expression not reproducible]. (90)

Then, {[z.sub.n]} converges strongly to T(v). This implies that T(v) = v. Therefore F(T) is a closed set. In view of (87), let [mathematical expression not reproducible] be a subsequence of sequence [mathematical expression not reproducible] for all k [greater than or equal to] 1, where {[z.sub.k]} is a sequence in F(T). By Theorem 21, we have

[mathematical expression not reproducible]. (91)

Now, by the triangle inequality and (91), we have

[mathematical expression not reproducible]. (92)

A standard argument shows that {[z.sub.k]} is a Cauchy sequence. Since F(T) is closed, {[z.sub.k]} converges to some point v [member of] F(T). Now

[mathematical expression not reproducible]. (93)

Letting [mathematical expression not reproducible] converges strongly to v. By Theorem 21, [lim.sub.n[right arrow][infinity]] [rho]([u.sub.n], v) exists. Hence {[u.sub.n]} converges strongly to v. The converse is obvious; hence the proof is completed.

Theorem 24. Let (E, [rho], [less than or equal to]) be a uniformly convex partially ordered hyperbolic metric space and, T, and {[u.sub.n]} be the same as in Theorem 21. Let T satisfy the condition (I) and F(T) = 0. Then {[u.sub.n]} converges strongly to a fixed point of T.

Proof. From Theorem 21, it follows that

[mathematical expression not reproducible]. (94)

As T satisfies condition (I), we have [rho](T([u.sub.n]), [u.sub.n]) [greater than or equal to] g(dist([u.sub.n], F(T))). From (94), we have

[mathematical expression not reproducible]. (95)

Since g : [0, [infinity]) [right arrow] [0, [infinity]) is a nondecreasing function with g(0) = 0 and g(r) > 0 for all r [member of] (0, [infinity]), therefore we have

[mathematical expression not reproducible]. (96)

Therefore all the assumptions of Theorem 23 are satisfied and {[u.sub.n]} converges strongly to a fixed point of T.

5. Applications to Nonlinear Integral Equations

In this section, we present an application of our results to nonlinear integral equations.

We give an existence theorem in an ordered Banach space. These spaces are a special case of partially ordered hyperbolic spaces.

Theorem 25. Let [member of] be an ordered Banach space and K be a nonempty compact subset of E. Let T : K [right arrow] K be a monotone Reich-Suzuki type nonexpansive mapping. Assume that there exists an a.f.p.s. {[u.sub.n]}. Then T has a fixed point.

Proof. It may be completed following Theorem 16 and [26, Theorem 4].

Consider the integral equation

[mathematical expression not reproducible]. (97)

Theorem 26. Let X = C[0,1], the space of continuous

functions on I = [0, 1] with ordered relation '[less than or equal to]' in X defined as, for all u, v [member of] X, u [less than or equal to] v if and only if u(t) [less than or equal to] v(t), [for all]t [member of] [0, 1]. Suppose X is equipped with supremum norm defined by [parallel]u - v[parallel] = [sup.sub.t[member of]I] [absolute value of u(t) - v(t)] and F is a compact subset of X. Assume that the following conditions hold:

(a) h: I [right arrow] R is continuous.

(b) f : I x F [right arrow] F is continuous, f(t, u) [greater than or equal to] 0, and there exists a constant L [greater than or equal to] 0 such that, for all u, v [member of] F,

[absolute value of f(t, u) - f(t, v)] [less than or equal to] L [absolute value of u(t) - v(f)]. (98)

(c) k : I x F [right arrow] R is continuous such that k(t, u) [greater than or equal to] 0, and [[integral].sup.1.sub.0] k(t, s)ds [less than or equal to] K for all (t, u) [member of] I x F.

(d) [lambda]KL = 1.

(e) T : F [right arrow] F is a mapping defined by

Tu (t) = h(t) + [lambda] [[integral].sup.1.sub.0] k (t, s) f (s, u (s)) ds,

t [member of] I = [0, 1], [lambda] [greater than or equal to] 0. (99)

Then, the nonlinear integral equation (97) has a solution in C[0,1], if T admits an a.f.p.s.

Proof. For u, v [member of] F such that u [less than or equal to] v, we have

[mathematical expression not reproducible]. (100)

Taking the supremum norm on both sides, we have

[parallel]T(u) - T(v)[parallel] [less than or equal to] [lambda]KL [parallel]u - v[parallel] = [parallel]u - v[parallel]. (101)

i.e., T is monotone nonexpansive. Hence, T is monotone Reich-Suzuki type nonexpansive mapping with k = 0. Therefore all the assumptions of Theorem 25 have been satisfied, so (97) has a solution in F [subset not equal to] [0, 1].

Now, let us introduce the following computational example.

Example 27. Consider the following functional integral equation:

[mathematical expression not reproducible]. (102)

It is observed that the above integral equation is a special case of (97) with

[mathematical expression not reproducible]. (102)

Now, for arbitrary u, v [member of] C(I, R) with u(t) [less than or equal to] v(t) for t [member of] I = [0, 1], we have

[mathematical expression not reproducible]. (103)

As h is continuous and for t [member of] [0, 1], we have

[mathematical expression not reproducible]. (104)

Consequently, all the conditions of Theorem 26 are satisfied with L = 1/2, K = 9/10, and [lambda] = 20/9 (LK[lambda] = 1), so there exists a solution of integral equation (102). It can be easily verified that u(t) = cos(([pi]/2)t) is a solution of nonlinear integral equation (102).

https://doi.org/10.1155/2018/5850181

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they do not have any conflicts of interest.

Acknowledgments

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Rameshwar Pandey, (1) Rajendra Pant (iD), (1) and Ahmed Al-Rawashdeh (iD) (2)

(1) Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India

(2) Department of Mathematical Sciences, UAEU, P.O. Box 15551, Al-Ain, UAE

Correspondence should be addressed to Ahmed Al-Rawashdeh; aalrawashdeh@uaeu.ac.ae

Received 1 March 2018; Revised 24 April 2018; Accepted 30 May 2018; Published 2 July 2018