Using the generalized Caristi's fixed point theorems we
prove the existence of fixed points for self and nonself multivalued weakly w-contractive maps. Consequently, Our results either improve or generalize
the corresponding fixed point results due to Latif (2007), Bae (2003), Suzuki, and Takahashi (1996) and others.

1. Introduction

It is well known that Caristi's fixed point theorem [1] is equivalent to Ekland
variational principle [2],
which is nowadays is an important tool in nonlinear analysis. Most recently,
many authors studied and generalized Caristi's fixed point theorem to various
directions. For example, see [3–6] and references therein.

Using the concept of Hausdorff metric, Nadler Jr. [7] has proved
multivalued version of the Banach contraction principle which states that each
closed bounded valued contraction map on a complete metric space, has a fixed
point. Recently, Bae [4] introduced a notion of multivalued weakly contractive
maps and applying generalized Caristi's fixed point theorems he proved several
fixed point results for such maps in the setting of metric and Banach spaces.
Many authors have been using the Hausdorff metric to obtain fixed point results
for multivalued maps on metric spaces, but, in fact for most cases the
existence part of the results can be proved without using the concept of
Hausdorff metric.

Recently, using the concept of w-distance [8], Suzuki and Takahashi [9] introduced a notion of
multivalued weakly contractive(in short, w-contractive) maps and improved the Nadler's
fixed point result without using the concept of Hausdorff metric. Most
recently, Latif [10]
generalized the fixed point result of Suzuki and Takahashi [9, Theorem 1]. Some
interesting examples and fixed point results concerning w-distance can be found in [6, 11–15] and references therein.

In this paper, introducing a notion of multivalued
weakly w-contractive maps, we prove some fixed point
results for self and nonself multivalued maps. Our results either improve or
generalize the corresponding results due to Latif [10], Bae [4], Mizoguchi and Takahashi [16], Suzuki and Takahashi [9], Husain and Latif
[17], Kaneko [18] and many others.

2. Preliminaries

Let X be a metric space with metric d.
We use 2X to denote the collection of all nonempty
subsets of X and Cl(X) for the collection of all nonempty closed
subsets of X.
Recall that a real-valued function φ defined on X is said to be lower (upper) semicontinuous if for any
sequence {xn}⊂X with xn→x∈X imply that φ(x)≤liminfn→∞φ(xn) (φ(x)≥limsupn→∞φ(xn)).

Introducing the following notion of w–distance, Kada et al. [8] improved the Caristi's
fixed point theorem, Ekland variational principle, and Takahashi existence
theorem.

A function ω:X×X→[0,∞) is called a w-distance on X if it satisfies the following for any x,y,z∈X:

ω(x,z)≤ω(x,y)+ω(y,z);

a map ω(x,⋅):X→[0,∞) is lower semicontinuous;

for any ϵ>0, there exists δ>0 such that ω(z,x)≤δ and ω(z,y)≤δ

imply d(x,y)≤ϵ.

Note that, in general for x,y∈X, ω(x,y)≠ω(y,x) and not either of the implications ω(x,y)=0⇔x=y necessarily hold. Clearly, the metric d is a w-distance on X.
Let (Y,∥⋅∥) be a normed space. Then the functions ω1,ω2:Y×Y→[0,∞) defined by ω1(x,y)=∥y∥ and ω2(x,y)=∥x∥+∥y∥ for all x,y∈Y are w-distances [8].

Let M be a nonempty subset of X.
A multivalued map T:M→2X is called w-contractive [9] if there exist a w-distance ω on X and a constant h∈(0,1) such that for any x,y∈X and u∈T(x) there is v∈T(y) satisfyingω(u,v)≤hω(x,y).In particular, if we take ω=d,
then w-contractive map is a contractive type map
[17].

We say T is weakly w-contractive if there exists a w-distance ω on X such that for any x,y∈X and u∈T(x) there is v∈T(y) withω(u,v)≤ω(x,y)−φ(ω(x,y)),where φ is a function from [0,∞) to [0,∞) such that φ is positive on (0,∞) and φ(0)=0.

In particular, if we take φ(t)=(1−h)t for a constant h with 0<h<1, then a weakly w-contractive map is w-contractive. If we define k(t)=1−φ(t)/t for t>0 and k(0)=0,
then k is a function from (0,∞) to [0,1) with limsupr→t+k(r)<1, for every t∈[0,∞).
Also we getω(u,v)≤k(ω(x,y))ω(x,y),that is, the weakly w-contractive map is generalized w-contraction [10].

We say a multivalued map T:M→2X is w-inward if for each x∈M, T(x)⊂ w-IM(x),
where w-IM(x) is the w-inward set of M at x,
which consists all the elements z∈X such that either z=x or there exists y∈M with y≠x and ω(x,z)=ω(x,y)+ω(y,z).

In particular, if we take ω=d,
then w-inward set is known as metrically inward set
[4].

A point x∈M is called a fixed point of T:M→2X if x∈T(x) and the set of all fixed points of T is denoted by Fix(T).

In the sequel, otherwise specified, we will assume
that ψ:X→[0,∞) is lower semicontinuous function, φ:[0,∞)→[0,∞) is positive function on (0,∞) and φ(0)=0 and ω is a w-distance on X.

Using the concept of w-distance, Kada et al. [8] have generalized Caristi's
fixed point theorem as follows.Theorem 2.1.

Let (X,d) be a complete metric space. Let f:X→X be a map such that for each x∈X,ψ(f(x))+ω(x,f(x))≤ψ(x).Then, there exists xo∈X such that f(xo)=xo and ω(xo,xo)=0.

Now, we state generalized Caristi's fixed point
theorems which are variant to the results of Bae [4, Theorem 2.1 and Corollary 2.5].Theorem 2.2.

Let (X,d) be a complete metric space. Let f:X→X be a map such that for each x∈X,ω(x,f(x))≤max{c(ψ(x)),c(ψ(f(x)))}(ψ(x)−ψ(f(x))),where c:[0,∞)→(0,∞) is an upper semicontinuous function from the
right. Then, f has a fixed point x0∈X such that ω(x0,x0)=0.

Theorem 2.3.

Let (X,d) be a complete metric space. Let φ:[0,∞)→[0,∞) be lower semicontinuous function such that φ(t)>0 for t>0 andlimsupt→0+tφ(t)<∞.Let f:X→X be a map such that for each x∈X, ω(x,f(x))≤ψ(x) andφ(ω(x,f(x)))≤ψ(x)−ψ(f(x)).Then, f has a fixed point x0∈X such that ω(x0,x0)=0.

Suzuki and Takahashi [9] have proved the following
fixed point result which is an improved version of the multivalued contraction
principle due to Nadler Jr. [7].Theorem 2.4.

Let (X,d) be a complete metric space. Then each
multivalued w-contractive map T:X→Cl(X) has a fixed point.

3. Main Results

Without using the Hausdorff metric, we prove the following
fixed point result for multivalued self map.
Theorem 3.1.

Let (X,d) be a complete metric space and let T:X→Cl(X) be a weakly w-contractive map for which φ is lower semicontinuous from the right and limsupt→0+(t/φ(t))<∞.
Then T has a fixed point.

Proof.

Let G={(x,y):x∈X,y∈T(x)} be the graph of T.
Clearly, G is a closed subset of X×X.
Define a metric ρ on G byρ((x,y),(u,v))=max{d(x,u),d(y,v)}.Then (G,ρ) is a complete metric space and ρ is w-distance on G.
Now, define ψ:G→[0,∞) by ψ(x,y)=ω(x,y)=d(x,y) for all (x,y)∈G and c:[0,∞)→[0,∞) byc(t)={tφ(t),ift>0,limsupt→0+(tφ(t)),ift=0.Then ψ is lower semicontinuous and c is upper semicontinuous from the right because φ is lower semicontinuous from the right. Define p:G×G→[0,∞) byp((x,y),(u,v))=max{ψ(x,y),ρ((x,y),(u,v))}.Then p is a w-distance on G (see [14, page 47]. Now, suppose Fix(T)=∅.
Then for each (x,y)∈G,
we have x≠y.
Since y∈T(x) there is z∈T(y) such thatω(y,z)≤ω(x,y)−φ(ω(x,y)).Since (x,y),(y,z)∈G,
we havep((x,y),(y,z))=ρ((x,y),(y,z))=ω(x,y)=ψ(x,y),also, note thatp((x,y),(y,z))=ω(x,y)≤ω(x,y)φ(ω(x,y))[ω(x,y)−ω(y,z)].Define a function f:G→G by f(x,y)=(y,z),
then we getp((x,y),f(x,y))≤c(ψ(x,y))[ψ(x,y)−ψ(f(x,y))].Thus, by Theorem 2.2, f has a fixed point, which is impossible. Hence, T must has a fixed point. This completes the
proof.

As a consequence, we obtain the following recent fixed
point result of Latif [10, Theorem 2.2].

Corollary 3.2.

Let (X,d) be a complete metric space. Let T:X→Cl(X) be a map such that for any x,y∈X and u∈T(x) there is v∈T(y) withω(u,v)≤k(ω(x,y))ω(x,y),where k is function from [0,∞) to [0,1) with limsupr→t+k(r)<1, for every t∈[0,∞). Then T has a fixed point.

Proof.

Define
φ:[0,∞)→[0,∞) byφ(t)=min{t(1−k(t)),liminfr→t+r(1−k(r))}∀t≥0.Then φ(t)>0 for all t>0, φ is lower semicontinuous from the right (see
[19]). Also note thatlimsupt→0+tφ(t)<∞,and for each x,y∈X,
we haveφ(ω(x,y))≤ω(x,y)(1−k(ω(x,y))).It follows from (3.8) and (3.11) thatω(u,v)≤ω(x,y)−φ(ω(x,y)).Thus T is weakly w-contractive map for which φ is lower semicontinuous from the right and limsupt→0+(t/φ(t))<∞.
Therefore, by Theorem 3.1, T has a fixed point.

Remark 3.3.

(a) Theorem 3.1 generalizes
Theorem 2.4 of Suzuki and Takahashi [9]. Indeed, consider φ(ω(x,y))=(1−h)ω(x,y) for a constant h with 0<h<1. Theorem 3.1 also generalizes and improves the
fixed point result of Bae [4, Theorem 3.1] .

(b) Corollary 3.2 generalizes fixed point result
of Husain and Latif [17, Theorem 2.3] and improves [16, Theorem 5]. Moreover, it improves and generalizes
[18, Theorem 1].

Without using the Hausdorff metric, we prove the
following fixed point result for nonself multivalued maps with respect to w-distance.

Theorem 3.4.

Let M be a closed subset of a complete metric space (X,d) and let T:M→Cl(X) be a weakly w-contractive map for which φ is lower semicontinuous and limsupt→0+(t/φ(t))<∞.
Then T has a fixed point provided T is w-inward on M.

Proof.

Let G, ρ, p, and ψ be the same as in the proof of Theorem 3.1.
Suppose Fix(T)=∅.
Then, for each (x,y)∈G we have x≠y.
Since y∈T(x)⊂ w-IM(x) there exists u∈M with u≠x andω(x,y)=ω(x,u)+ω(u,y). Since the map T is weakly w-contractive, there exists v∈T(u) such thatω(y,v)≤ω(x,u)−φ(ω(x,u)),where φ is lower semicontinuous and limsupt→0+(t/φ(t))<∞.
From (3.13) and (3.14), we getφ(ω(x,u))≤ω(x,u)−ω(y,v)=ω(x,y)−[ω(u,y)+ω(y,v)].Thus,φ(ω(x,u))≤ω(x,y)−ω(u,v).Since (x,y),(u,v)∈G,
we haveρ((x,y),(u,v))=max{ω(x,u),ω(y,v)},and hence, we
getp((x,y),(u,v))=ω(x,u)≤ω(x,y)=ψ(x,y).Now, define a function f:G→G by f(x,y)=(u,v).
Then from (3.18) we getp((x,y),f(x,y))≤ψ(x,y),and using (3.16), we obtainφ(p((x,y),f(x,y)))≤ψ(x,y)−ψ(f(x,y)).Thus by Theorem 2.3, f has a fixed point, which is impossible. Hence,
it follows that T must has a fixed point.

Using the same method as in the proof of Corollary
3.2, we can obtain the following fixed point result for nonself generalized w-contractions.

Corollary 3.5.

Let M be a closed subset of a complete metric space (X,d) and let T:M→Cl(X) be a map satisfying inequality (3.8) for which k:[0,∞)→[0,1) is upper semicontinuous. Then T has a fixed point provided T is w-inward on M.

Remark 3.6.

(a)
Our Theorem 3.4 and Corollary 3.5 improve the results of Bae [4, Theorem 3.3 and Corollary
3.4], respectively.

(b) The analogue of all the results of this
section can be established with respect to τ-distance [20].

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