New Fixed Point Theorems for Nonlinear Multivalued Maps and MT-Functions in Complete Metric Spaces

The famous Banach contraction principle (see, e.g., [1]) plays an important role in various fields of nonlinear analysis and applied mathematical analysis. Many authors investigated and established generalizations in various different directions of the Banach contraction principle in the past; see [1-22] and references therein. In 1969, Nadler [2] first proved a set-valued generalized version of the Banach contraction principle. In 1989, Mizoguchi and Takahashi [3] proved the following fixed point theorem which is a generalization of Nadler’s fixed point theorem.


Introduction
The famous Banach contraction principle (see, e.g., [1]) plays an important role in various fields of nonlinear analysis and applied mathematical analysis.Many authors investigated and established generalizations in various different directions of the Banach contraction principle in the past; see  and references therein.In 1969, Nadler [2] first proved a set-valued generalized version of the Banach contraction principle.In 1989, Mizoguchi and Takahashi [3] proved the following fixed point theorem which is a generalization of Nadler's fixed point theorem.
(A2) T further satisfies one of the following conditions: (H4) for any sequence {x n } in X with x n+1 ∈ T x n , n ∈ N and lim n→∞ x n = v, we have lim n→∞ p(x n , T v) = 0; (H5) inf{p(x, z) + p(x, T x) : x ∈ X} > 0 for every z / ∈ F(T ).
In this paper, we prove some new fixed point theorems for nonlinear multivalued maps and MT -functions in complete metric spaces which will generalize and extend some Du-Zheng's fixed point theorems.

Preliminaries
Let (X, d) be a metric space and p : X × X → [0, ∞) be any function.For each x ∈ X and A ⊆ X, let Denote by N (X) the class of all nonempty subsets of X, C(X) the family of all nonempty closed subsets of X and CB(X) the family of all nonempty closed and bounded subsets of X.A function H : CB(X)×CB(X) → [0, ∞), defined by is said to be the Hausdorff metric on CB(X) induced by the metric d on X.
A point v in X is a fixed point of a map The set of fixed points of T is denoted by F (T ).Throughout this paper, we denote by N and R, the sets of positive integers and real numbers, respectively.
It is well known that the metric d is a w-distance and any w-distance is a τ -function, but the converse is not true; see [6,10,18].
The following results are crucial in this paper.Lemma 2.1.[6, 7, 10, 18] Let (X, d) be a metric space and p : X × X → [0, ∞) be a function.Assume that p satisfies the condition (τ 3).If a sequence {x n } in X with lim n→∞ sup{p(x n , x m ) : m > n} = 0, then {x n } is a Cauchy sequence in X.

Lemma 2.2. [18]
Let A be a closed subset of a metric space (X, d) and p : X × X → [0, ∞) be any function.Suppose that p satisfies (τ 3) and there exists u ∈ X such that p(u, u) = 0. Then p(u, A) = 0 if and only if u ∈ A.
Let f be a real-valued function defined on R. For c ∈ R, we recall that lim sup ) is a nondecreasing function or a nonincreasing function, then ϕ is an MT -function.So the set of MTfunctions is a rich class.But it is worth to mention that there exist functions which are not MT -functions.
Otherwise, x 2 ∈ T x 2 then, by (A1) and ϕ(η(p( So there exists Since By the same way, there exists Continuing in this way, we can construct inductively a sequence {x n } n∈AE in X satisfying x n+1 ∈ T x n , and for each n ∈ N. By (2), {p(x n , x n+1 )} is a strictly decreasing sequence in (0, ∞).Since η is a nondecreasing function, we know that the sequence {η(p(x n , x n+1 )} is a nonincreasing sequence in [0, ∞).Since τ is an MT -function, applying Theorem 2.1, we get We put and By ( 2) and (4), we have We claim lim sup n→∞ {p(x n , x m ) : n < m} = 0. Let α n = ( r n−1 1−r )p(x 1 , x 2 ), n ∈ N.For m, n ∈ N with n < m, we have and hence Since r ∈ [ 2 3 , 1), lim n→∞ α n = 0, and from (5), we get lim sup Applying Lemma 2.1, {x n } is a Cauchy sequence in X.By the completeness of X, there exists v ∈ X such that x n → v as n → ∞.From (τ 2) and ( 5), we have Now, we verify that v ∈ F(T ).We need to consider the following 5 conditions.
Case 1.If (B 1 ) holds, since T is closed, x n ∈ T x n−1 and x n → v as n → ∞, we have v ∈ T v.
Case 2. If (B 2 ) holds, by the lower semicontinuity of f , x n → v as n → ∞ and  The proof is completed.2 In Theorem 3.1, if η is an identity map, then we obtain Du-Zheng's fixed point theorem [14].
y∈A d(x, y) and p(x, A) = inf y∈A p(x, y).