We establish some fixed point results for multivalued contraction type mappings in terms of a w-distance in a complete metric space. Our results generalize very recent results of some authors (Ćirić, 2008, 2009; Feng and Liu 2006; Klim and Wardowski 2007; and Latif and Abdou 2011).

1. Introduction

The Banach contraction principle [1] plays an important role in nonlinear analysis. Following the Banach contraction principle, Nadler [2] first initiated the study of fixed point theorems for multivalued contraction mappings and inspired by his results, the fixed point theory of multi-valued contraction has been further developed in different directions by many authors, in particular, by Reich, Mizoguchi-Takahashi, Feng-Liu, and many others.

The aim of this paper is to present results which are generalizations of the very recent results of Klim and Wardowski [3], Ćirić [4, 5], and Latif and Abdou [6], as well as of the result of Mizoguchi and Takahashi [7] and many others.

Let (X,d) be a metric space. We denote the collection of all nonempty closed bounded subsets of X by CB(X) and the collectifon of all nonempty closed subsets and all nonempty compact subsets of X by C(X) and K(X), respectively. Throughout this paper, we assume that ℝ, ℕ, ℕe, and ℕo denote the sets of all real numbers, positive integers, even positive integers, and odd positive integers, respectively. Let H be the Hausdorff metric induced by d, that is,
H(A,B)=max{supx∈Ad(x,B),supy∈Bd(y,A)},
for all A,B∈CB(X), where d(x,B)=inf{d(x,y):y∈B}. An element x∈X is said to be a fixed point of a multi-valued mapping T:X→C(X) if x∈T(x). A map f:X→ℝ is called lower semicontinuous if for any sequence {xn} in X and x∈X such that xn→x, we have f(x)≤liminfn→∞f(xn).

The following theorem is an extension of Banach contraction principle for multi-valued mappings, which was obtained by Nadler in [2].

Theorem 1.1 (see [<xref ref-type="bibr" rid="B10">2</xref>]).

Let (X,d) be a complete metric space, and let T:X→CB(X) be a multi-valued mapping. Assume that there exists r∈[0,1) such that for allx,y∈X,
H(T(x),T(y))≤rd(x,y),
then there exists z∈X such that z∈T(z).

A generalization of Theorem 1.1 was proved by Mizoguchi and Takahashi that is, in fact, a partial answer of question of Reich [8].

Theorem 1.2 (see [<xref ref-type="bibr" rid="B9">7</xref>]).

Let (X,d) be a complete metric space, and let T:X→CB(X) be a multi-valued mapping. If there exists a function φ:(0,∞)→[0,1) such that limsupr→t+φ(r)<1 for each t∈[0,∞), and if for all x,y∈X,
H(T(x),T(y))≤φ(d(x,y))d(x,y),
then there exists z∈X such that z∈T(z).

Recently, an interesting result had been obtained by Feng and Liu [9]. They extended Theorem 1.1 in another direction different from that of Mizoguchi-Takahashi's theorem.

Theorem 1.3 (see [<xref ref-type="bibr" rid="B4">9</xref>]).

Let (X,d) be a complete metric space, and let T:X→C(X) be a multi-valued mapping. If there exist constants α,β∈(0,1), β<α, such that for any x∈X there is y∈T(x) satisfying the following two conditions:

αd(x,y)≤d(x,T(x)),

d(y,T(y))≤βd(x,T(x)),

then there exists z∈X such that z∈T(z) provided the function f(x)=d(x,T(x)) is lower semicontinuous.

By using the ideas of Mizoguchi-Takahashi and Feng-Liu, Klim and Wardowski proved the following two theorems that are different from Theorem 1.2.

Theorem 1.4 (see [<xref ref-type="bibr" rid="B6">3</xref>]).

Let (X,d) be a complete metric space, and let T:X→C(X) be a multi-valued mapping. Assume that the following conditions hold:

the map f:X→ℝ, defined by f(x)=d(x,T(x)), x∈X, is lower semi-continuous;

there exists a constant α∈(0,1) and a function φ:[0,∞)→[0,α) satisfying limsupr→t+φ(r)<α, for each t∈[0,∞), and for any x∈X there is y∈T(x) satisfying
αd(x,y)≤d(x,T(x)),d(y,T(y))≤φ(d(x,y))d(x,y).
Then there exists z∈X such that z∈T(z).

Theorem 1.5 (see [<xref ref-type="bibr" rid="B6">3</xref>]).

Let (X,d) be a complete metric space, and let T:X→K(X) be a multi-valued mapping. Assume that the following conditions hold:

the map f:X→ℝ, defined by f(x)=d(x,T(x)), x∈X, is lower semi-continuous;

there exists a function φ:[0,∞)→[0,1) satisfying limsupr→t+φ(r)<1, for each t∈[0,∞), and for any x∈X there is y∈T(x) satisfying
d(x,y)=d(x,T(x)),d(y,T(y))≤φ(d(x,y))d(x,y).

Then there exists z∈X such that z∈T(z).

In [4, 5], Ćirić proved a few interesting theorems which are generalizations of the above-mentioned theorems, one of which is as follows.

Theorem 1.6 (see [<xref ref-type="bibr" rid="B2">4</xref>]).

Let (X,d) be a complete metric space, and let T:X→C(X) be a multi-valued mapping. If there exist a function φ:[0,∞)→[0,1) and a nondecreasing function ψ:[0,∞)→[α,1), α>0, such that
φ(t)<ψ(t),limsupr→t+φ(r)<limsupr→t+ψ(r),∀t∈[0,∞),
and for any x∈X there is y∈T(x) satisfying the following two conditions:
ψ(d(x,y))d(x,y)≤d(x,T(x)),d(y,T(y))≤φ(d(x,y))d(x,y),
then T has a fixed point in X provided f(x)=d(x,T(x)) is lower semi-continuous.

Very recently the fixed point theorems of Ćirić were extended by Liu et al. [10] and by Nicolae [11] in a new direction. Also Latif and Abdou [6] improve two theorems of Ćirić with respect to w-distance. The concept of w-distance was introduced by Kada et al. [12] on a metric space as follows.

Let (X,d) be a metric space, then a function v:X×X→[0,∞) is called a w-distance on X if the following axioms are satisfied:

v(x,z)≤v(x,y)+v(y,z), for any x,y,z∈X;

for any x∈X, v(x,·):X→[0,∞) is lower semi-continuous;

for any ɛ>0, there exists δ>0 such that v(z,x)≤δ and v(z,y)≤δ imply d(x,y)≤ɛ.

The metric d is a w-distance on X. One can see other examples of w-distances in [12]. One of theorems, which was proved by Latif and Abdou, is as follows.

Theorem 1.7 (see [<xref ref-type="bibr" rid="B7">6</xref>]).

Let (X,d) be a complete metric space with a w-distance v. Let T:X→C(X) be a multi-valued map. Assume that the following conditions hold:

there exist a constant α∈(0,1) and a function φ:[0,∞)→[α,1) such that limsupr→t+φ(r)<1, for each t∈[0,∞);

the map f:X→ℝ, defined by f(x)=v(x,T(x)), x∈X, is lower semi-continuous;

for any x∈X, there is y∈T(x) satisfying

φ(f(x))v(x,y)≤f(x),f(y)≤φ(f(x))v(x,y).
Then there exists z∈X such that f(z)=0. Further, if v(z,z)=0, then z∈T(z).

For the proof of the main results, we need the following crucial lemma [13].

Lemma 1.8.

Let (X,d) be a metric space, and let v be a w-distance on X. Let {xn} and {yn} be two sequences in X, let {an} and {bn} be sequences in [0,∞) converging to zero, and let z∈X. Then the following hold:

if v(xn,yn)≤an and v(xn,z)≤bn for any n∈ℕ, then {yn} converges to z;

if v(xn,xp)≤an for any n,p∈ℕ with p>n, then {xn} is a Cauchy sequence.

In the present paper, using the concept of w-distance, in the same direction which has been used in [10, 11], we introduce some new contraction conditions for multi-valued mappings in complete metric spaces and three fixed point theorems for such contractions are proved. Our results generalize Theorems 1.6 and 1.7, and many other theorems.

2. Main Results

Recall that a sequence {xn}n≥0 in X is called an orbit of T at x0∈X if xn∈T(xn-1), for all n≥1, and v(x,G)=inf{v(x,y):y∈G} for any x∈X and G∈C(X).

Now, we shall prove a theorem which is a generalization of Theorem 1.7.

Theorem 2.1.

Let (X,d) be a complete metric space with a w-distance v, and let T be a multi-valued mapping from X into C(X). Assume that the following conditions hold:

the map f:X→ℝ, defined by f(x)=v(x,T(x)), x∈X, is lower semi-continuous;

there exist the functions α:[0,∞)→(0,1], β:[0,∞)→[0,1) which satisfy
∃k,ɛ∈(0,1),tk≤α(t),for eacht∈[0,ɛ],limsupr→t+β(r)α(r)<1,for eacht∈[0,∞);

for any x∈X, there is y∈T(x) satisfying

α(f(x))v(x,y)≤f(x),f(y)≤β(f(x))v(x,y).
Then, there exists z∈X such that z∈T(z).Proof.

Let x0∈X. Then there exists x1∈T(x) such that
α(f(x0))v(x0,x1)≤f(x0),f(x1)≤β(f(x0))v(x0,x1).
Then, from (2.3), we have
f(x1)≤β(f(x0))α(f(x0))f(x0).
Continuing this process, we can define an orbit {xn} of T in X, such that
α(f(xn))v(xn,xn+1)≤f(xn),f(xn+1)≤β(f(xn))v(xn,xn+1),
for all n≥0, which imply that
f(xn+1)≤β(f(xn))α(f(xn))f(xn).
We can assume that f(xn)>0 for all n≥0, since f(xn)=0 for some n, then from (2.5), v(xn,xn+1)=f(xn+1)=0, and so for every m∈ℕ, there exists ym∈T(xn+1) such that v(xn+1,ym)≤1/m; consequently, v(xn,ym)≤v(xn,xn+1)+v(xn+1,ym)≤1/m; so by Lemma 1.8, ym→xn+1 and thus xn+1∈T(xn+1), that is, the assertion of the theorem is proved. Since β(f(x))/α(f(x))<1 for all x∈X, then we have
f(xn+1)<f(xn).
Thus {f(xn)} is a decreasing sequence of positive real numbers and hence converges to a nonnegative number θ, θ≥0. Let τ=limsupn→∞β(f(xn))/α(f(xn))<1. Then, for s=(τ+1)/2, we can take n̂0∈ℕ such that
β(f(xn))α(f(xn))<s,∀n≥n̂0.
Then,
f(xn+1)≤sn-n̂0+1f(xn̂0),∀n≥n̂0,
thus, θ=0. Therefore, by (ii) we can take ñ0∈ℕ such that
f(xn)∈[0,ɛ],f(xn)k≤α(f(xn)),∀n≥ñ0.
Now, from (2.5), (2.9), and (2.10), we have
v(xn,xn+1)≤f(xn)α(f(xn))≤f(xn)f(xn)k=f(xn)1-k≤qn-n0f(xn0)1-k,
for all n≥n0, where n0=max{n̂0,ñ0} and q=s1-k<1. Then for any p,n∈ℕ, p>n≥n0, we have
v(xn,xp)≤∑i=np-1v(xi,xi+1)≤∑i=np-1qi-n0f(xn0)1-k≤qn-n01-qf(xn0)1-k.
Hence, by Lemma 1.8, {xn} is a Cauchy sequence so there exists z∈X such that limnxn=z. Since f is lower semi-continuous, we obtain
0≤f(z)≤liminfn→∞f(xn)=0,
and thus,
f(z)=v(z,T(z))=0.
Now, we clime that z∈T(z). Notice that the function v(x,·) is lower semi-continuous for all x∈X. Since xp→z, then by (2.12),
v(xn,z)≤liminfp→∞v(xn,xp)≤qn-n01-qf(xn0)1-k,
for all n≥n0. On the other hand, from (2.14), for every n∈ℕ, there exists yn∈T(z) such that v(z,yn)≤1/n. Then, for all n≥n0, we have
v(xn,yn)≤v(xn,z)+v(z,yn)≤qn-n01-qf(xn0)1-k+1n.
Therefore, from (2.15), (2.16), we can find the sequences {an} and {bn} in [0,∞) converging to zero, such that v(xn,yn)≤an and v(xn,z)≤bn for any n∈ℕ; then by Lemma 1.8yn→z. The closedness of T(z) implies z∈T(z).

Now, we shall prove a theorem which is different from Theorem 2.1 and is a generalization of Theorem 1.6.

Theorem 2.2.

Let (X,d) be a complete metric space with a w-distance v, and let T be a multi-valued mapping from X into C(X). Assume that the following conditions hold:

the map f:X→ℝ, defined by f(x)=v(x,T(x)), x∈X, is lower semi-continuous;

there exist the functions α:[0,∞)→(0,1], β:[0,∞)→[0,1) which satisfy
∃k,ɛ∈(0,1),tk≤α(t),foreacht∈[0,ɛ],limsupr→t+β(r)α(r)<1,foreacht∈[0,∞),
and α is nondecreasing;

for any x∈X, there is y∈T(x) satisfying
α(v(x,y))v(x,y)≤f(x),f(y)≤β(v(x,y))v(x,y).
Then, there exists z∈X such that z∈T(z).

Proof.

By following the lines in the proof of Theorem 2.1, one can construct an orbit {xn}n=0∞ of T in X such that
α(v(xn,xn+1))v(xn,xn+1)≤f(xn),f(xn+1)≤β(v(xn,xn+1))v(xn,xn+1),
for all n≥0, which imply that
f(xn+1)≤β(v(xn,xn+1))α(v(xn,xn+1))f(xn).
Thus, {f(xn)} is a decreasing sequence, and so there exists θ≥0 such that f(xn)→θ. From (2.19) and (2.20), we have
v(xn+1,xn+2)≤f(xn+1)α(v(xn+1,xn+2))≤β(v(xn,xn+1))α(v(xn+1,xn+2))v(xn,xn+1),
for all n≥0. Now we clime that {v(xn,xn+1)} is a nonincreasing sequence. Suppose not. Then there exists n0≥0 such that v(xn0+1,xn0+2)>v(xn0,xn0+1). Since α is nondecreasing, then from (2.22), we get that
v(xn0,xn0+1)<v(xn0+1,xn0+2)≤β(v(xn0,xn0+1))α(v(xn0+1,xn0+2))v(xn0,xn0+1)≤β(v(xn0,xn0+1))α(v(xn0,xn0+1))v(xn0,xn0+1)<v(xn0,xn0+1),
which is a contradiction. Then, {v(xn,xn+1)} is a nonincreasing sequence and so is convergent. Now by using the same argument as in the proof of Theorem 2.1, we obtain the existence of a real number s∈(0,1) and n̂0∈ℕ such that
f(xn+1)≤sn-n̂0+1f(xn̂0),∀n≥n̂0,
thus θ=0. On the other hand, since α is nondecreasing, then by (2.19), we have
v(xn,xn+1)≤f(xn)α(v(xn,xn+1))≤f(xn)α(f(xn)).
For the rest of the proof, we can go on as in the proof of Theorem 2.1.

In the same manner, we can present the following theorem.

Theorem 2.3.

Let (X,d) be a complete metric space with a w-distance v, and let T be a multi-valued mapping from X into C(X). Assume that the following conditions hold:

the map f:X→ℝ, defined by f(x)=v(x,T(x)), x∈X, is lower semi-continuous,

there exist the functions α:[0,∞)→(0,1], β:[0,∞)→[0,1) which satisfy
∃k,ɛ∈(0,1),tk≤β(t),for eacht∈[0,ɛ],β(t)<α(t),limsupr→t+β(r)α(r)<1,for each t∈[0,∞),
and also one of α and β is nondecreasing;

for any x∈X, there is y∈T(x) satisfying

α(v(x,y))v(x,y)≤f(x),f(y)≤β(v(x,y))v(x,y).
Then, there exists z∈X such that z∈T(z).Proof.

As in the proof of Theorem 2.1, one can construct an orbit {xn}n=0∞ of T in X such that (2.19), (2.20), (2.21), and (2.22) hold. Then, {f(xn} is a decreasing sequence and so there exists θ≥0 such that f(xn)→θ. Now we clime that {v(xn,xn+1)} is a nonincreasing sequence. Suppose not. Then there exists n0≥0 such that v(xn0+1,xn0+2)>v(xn0,xn0+1). Since one of α and β is nondecreasing, it follows from (2.22) that
v(xn0,xn0+1)<v(xn0+1,xn0+2)≤β(v(xn0,xn0+1))α(v(xn0+1,xn0+2))v(xn0,xn0+1)≤max{β(v(xn0,xn0+1))α(v(xn0,xn0+1)),β(v(xn0+1,xn0+2))α(v(xn0+1,xn0+2))}v(xn0,xn0+1)<v(xn0,xn0+1),
which is a contradiction. Thus {v(xn,xn+1)} is a nonincreasing sequence and so is convergent. Now as in the proof of Theorem 2.1, we obtain the existence of a real number s∈(0,1) and n̂0∈ℕ such that
f(xn+1)≤sn-n̂0+1f(xn̂0),∀n≥n̂0,
thus θ=0. If α is nondecreasing, then from assumption (ii) and (2.19), we have
v(xn,xn+1)≤f(xn)α(v(xn,xn+1))≤f(xn)α(f(xn))≤f(xn)β(f(xn)),
and if β is nondecreasing then from (2.19) and (2.20), we have
v(xn,xn+1)≤f(xn)β(v(xn,xn+1))≤f(xn)β(f(xn)),
for all n≥0. Therefore, in all cases we have shown
v(xn,xn+1)≤f(xn)β(f(xn)),∀n≥0,
and so, by following as in the proof of Theorem 2.1, we can take n0∈ℕ such that
v(xn,xn+1)≤qn-n0f(xn0)1-k,
for all n≥n0, where q=s1-k<1. The rest of the proof is similar to that of Theorem 2.1.

Remark 2.4.

Theorem 2.1 is a generalization of Theorem 1.7. In fact, if we consider β(t)=φ(t) and α(t)=φ(t), then the assumptions of Theorem 2.1 are satisfied. Also, one can see that Theorem 2.1 generalizes Theorem 2.2 of Nicolae [11].

Remark 2.5.

Theorem 2.2 essentially generalizes Theorem 1.6. Indeed, if we consider β(t)=φ(t) and α(t)=ψ(t), then all assumptions of Theorem 2.2 are satisfied.

The following example shows that there are mappings which satisfy the assumptions of Theorem 2.1 but do not satisfy the assumptions of Theorem 1.7.

Example 2.6.

Consider xn=1/n, for n∈ℕ, and x0=0. Then X={x0,x1,x2,…} is a bounded complete subset of ℝ. Let v(x,y)=d(x,y), for all x,y∈X. Define a mapping T from X into C(X) by
T(xn)={x0ifn=0,x1ifn=1,x1ifn∈{n∈Ne:n≥2},{xn+1,xn2}ifn∈{n∈No:n>2}.
It is easy to verify that
f(xn)=v(xn,T(xn))={0ifn=0,1,1-1nifn∈{n∈Ne:n≥2},1n-1n+1ifn∈{n∈No:n>2},
is lower semi-continuous in X. Define α(t):[0,∞)→(0,1] and β(t):[0,∞)→[0,1) by
α(t)={1ift∈{0}∪[1,∞),tift∈(0,1),β(t)={tift∈[0,1),0ift∈[1,∞).
Since
β(t)α(t)={tift∈[0,1),0ift∈[1,∞),
then, we have
limsupr→t+β(r)α(r)<1,foreacht∈[0,∞).
For x=x0,x1, there exists y=x∈T(x) such that
α(f(x))v(x,y)=0=f(x),f(y)=0=β(f(x))v(x,y),
and for x=xn, n≥2, if n∈ℕe, there exists y=x1∈T(x) satisfying
α(f(x))v(x,y)=α(1-1n)(1-1n)<(1-1n)=f(x),f(y)=0<(1-1n)(1-1n)=β(f(x))v(x,y),
and, if n∈ℕo, there exists y=xn2∈T(x) satisfying
α(f(x))v(x,y)=(1n(n+1))(n-1n2)<1n(n+1)=f(x),f(y)=1n2(n2+1)<(1n(n+1))(n-1n2)=β(f(x))v(x,y).

Therefore, all assumptions of Theorem 2.1 are satisfied and x0, x1 are two fixed points of T. Let us observe that T does not satisfy the assumptions of Theorem 1.7 provided that v(x,y)=d(x,y), for all x,y∈X. Indeed, for any function φ:[0,∞)→[α,1), α∈(0,1), there exists n>2, n∈ℕo, such that for x=xn, if y=xn2∈T(x), we have
f(x)=1n(n+1)<α(n-1n2)≤φ(f(x))v(x,y),
and if y=xn+1∈T(x), we have
v(y,T(y))=1-1n+1>1n(n+1)≥φ(f(x))v(x,y),
that is, the assumptions of Theorem 1.7 are not satisfied. The next example is an application of Theorem 2.3.

Example 2.7.

Let X be as in the Example 2.6, and let v(x,y)=y, for all x,y∈X. Note that v is a w-distance on X. Define a mapping T from X into C(X) by
T(xn)={x0ifn=0,{x0,x1}ifn=1,x1ifn∈{n∈Ne:n≥2},{xn4-1,xn3}ifn∈{n∈No:n>2}.
Clearly,
f(xn)=v(xn,T(xn))={0ifn=0,1,1ifn∈{n∈Ne:n≥2},1n4-1ifn∈{n∈No:n>2}
is lower semi-continuous in X. Define α(t):[0,∞)→(0,1] and β(t):[0,∞)→[0,1) by
α(t)={1ift∈{0}∪[1,∞),t1/3ift∈(0,1),β(t)={t5/6ift∈[0,12],(12)5/6ift∈(12,∞).
Note that β(t) is nondecreasing and t5/6≤β(t), for each t∈[0,(1/2)]. Since
β(t)α(t)={t1/2ift∈[0,12],(12)5/6t-1/3ift∈(12,1),(12)5/6ift∈[1,∞),
then,
limsupr→t+β(r)α(r)<1,for eacht∈[0,∞).
For x=x0,x1, there exists y=x0∈T(x) such that
α(v(x,y))v(x,y)=0=f(x),f(y)=0=β(v(x,y))v(x,y),
and for x=xn, n≥2, if n∈ℕe, there exists y=x1∈T(x) satisfying
α(v(x,y))v(x,y)=1=f(x),f(y)=0<β(v(x,y))v(x,y),
and, if n∈ℕo, there exists y=xn3∈T(x) satisfying
α(v(x,y))v(x,y)=(1n)(1n3)<1n4-1=f(x),f(y)=1n12-1<(1n3)5/6(1n3)=β(v(x,y))v(x,y).
Then, all assumptions of Theorem 2.3 are satisfied and x0, x1 are two fixed points of T. Note that v(x1,x1)≠0.

Acknowledgment

The authors would like to thank the referees for their valuable and useful comments.

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