# Fixed Points of Integral Type Multivalued Contractive Mappings withw-Distance

Authors: Haiyan Gao; Haoyue Wang; Lili Zhang; Liangshi Zhao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Two fixed point theorems for multivalued contractive mappings involvingw-distance and inequalities of integral type are proved. Our results generalize a few results in the literature. Two illustrative examples are given.


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## Body

## 1. Introduction and Preliminaries

In 2002, Branciari [1] generalized the famous Banach contraction principle and proved the following fixed point theorem for the contractive mapping of the integral type.Theorem 1 (see [1]).
LetT be a mapping from a complete metric space X,d into itself satisfying(1)∫0dTx,Tyφtdt≤c∫0dx,yφtdt,∀x,y∈X,where c∈0,1 is a constant and φ:ℝ+⟶ℝ+ is Lebesgue integrable, summable in each compact subset of ℝ+ and ∫0εϕtdt>0 for each ε>0. Then, T has a unique fixed point a∈X and limn⟶∞Tnx=a for each x∈X.Using Hausdorff metric, Nadler [2] introduced the concept of multivalued contraction and proved a multivalued version of the well-known Banach contraction principle.Theorem 2 (see [2]).
LetX,d be a complete metric space and let T be a mapping from X into CBX, where CBX is the family of all nonempty closed and bounded subsets of X. Assume that there exists c∈0,1 such that(2)HTx,Ty≤cdx,y,∀x,y∈X.
Then,T has a fixed point.In the past decades, various fixed point theorems concerning multivalued contractive mappings have been proved. Especially, Feng and Liu [3] generalized Theorem 2 and proved a few fixed point theorems for multivalued contractive mappings without Hausdorff metric.Theorem 3 (see [3]).
LetX,d be a complete metric space and T be a multivalued mapping from X into CLX, where CLX is the family of all nonempty closed subsets of X. Assume thata1 There exist constants b,c∈0,1 with c<b such that, for any x∈X, there is y∈Tx satisfying(3)bdx,y≤dx,Tx,dy,Ty≤cdx,y.a2f:X⟶ℝ+ is lower semicontinuous, where fx=dx,Tx,∀x∈X.Then,T has a fixed point in X.In 1996, Kada et al. [4] introduced the concept of w-distance in a metric space and proved several fixed point theorems for single-valued contractive mappings under w-distance. Some other fixed point results concerning w-distance can be found in [5–9]. In 2007, Guran [5] deduced the following fixed point theorem, which is a generalization of Theorem 3.Theorem 4 (see [5]).
LetX,d be a complete metric space, T:X⟶CLX be a multivalued mapping, w:X×X⟶ℝ+ be a w-distance on X, and b∈0,1. Assume thatb1 There exists c∈0,1, with c<b, such that, for any x∈X, there is y∈Tx satisfying(4)bwx,y≤wx,Tx,wy,Ty≤cwx,y.b2fw:X⟶ℝ+ is lower semicontinuous, where fwx=wx,Tx,∀x∈X.Then,T has a fixed point in X.Motivated by the results in [1, 3–5], we prove two fixed point results for multivalued contractive mappings of integral type with respect to w-distance in complete metric spaces. The results presented in this paper improve Theorems 2–4. Two examples with uncountably many points are included.Throughout this paper, we denote byℕ the set of positive integers, ℕ0=0∪ℕ, ℝ=−∞,+∞, ℝ+=0,+∞, and(5)Φ1=ϕϕ:ℝ+⟶ℝ+is Lebesgue integrable and summable in each compact subset ofℝ+and∫0εϕtdt>0,∀ε>0,Φ2=ϕϕbelongs toΦ1and satisfies that∫0a+bϕtdt≤∫0aϕtdt+∫0bϕtdt,∀a,b∈ℝ+.Definition 1 (see [4]).
LetX,d be a metric space. A function w:X×X⟶ℝ+ is called a w-distance in X if it satisfies the following:w1wx,z≤wx,y+wy,z,∀x,y,z∈Xw2 For each x∈X, a mapping wx,⋅:X⟶ℝ+ is lower semicontinuousw3 For any ε>0, there exists δ>0 such that wz,x≤δ and wz,y≤δ imply dx,y≤εExample 1.
LetX be a normed linear space with norm ⋅, α be a positive constant, and w:X×X⟶ℝ+ be defined by(6)wx,y=yα,∀x,y∈X.
Then,w is a w-distance in X.Proof.
Letx,y,z∈X. It is clear that w2 holds and(7)wx,z=zα≤yα+zα=wx,y+wy,z,which implies w1. For each ε>0, put δ=ε/2α. If wz,x≤δ and wz,y≤δ, it follows that(8)dx,y=x−y≤x+y=w1/αz,x+w1/αz,y≤δ1/α+δ1/α=ε,which yields w3. That is, w is a w-distance in X.Example 2.
LetX=ℝ be endowed with the Euclidean metric d=⋅ and w:X×X⟶ℝ+ be defined by(9)wx,y=maxax−y,ax−y,∀x,y∈X,where a is a constant in ℝ. Then, w is a w-distance in X.Proof.
Letx,y,z∈X. It is clear that w2 holds and(10)wx,z=maxax−z,ax−z≤maxax−y+ay−z,ax−y+ay−z≤maxax−y,ax−y+maxay−z,ay−z=wx,y+wy,z,which implies w1. For each ε>0, put δ=ε/2. If wz,x≤δ and wz,y≤δ, it is easy to see that(11)dx,y=x−y≤az−x+az−y≤maxaz−x,az−x+maxaz−y,az−y=wz,x+wz,y≤δ+δ=ε,which yields w3. That is, w is a w-distance in X.
LetX,d be a metric space. For any u∈X, D⊆X, T:X⟶CLX, and w:X×X⟶ℝ+, put(12)du,D=infy∈Ddu,y,wu,D=infy∈Dwu,y,fu=du,Tu,fwu=wu,Tu.
A sequencexnn∈ℕ0 in X is called an orbit of T if xn∈Txn−1 for all n∈ℕ.Definition 2.
LetX,d be a metric space and T:X⟶CLX be a multivalued mapping. A function g:X⟶ℝ+ is said to bea1 Lower semicontinuous in X if gy≤liminfn⟶∞gyn for each y∈X and ynn∈ℕ0⊆X with limn⟶∞yn=ya2T-orbitally lower semicontinuous at z∈X if gz≤liminfn⟶∞gxn for each orbit xnn∈ℕ0 of T with limn⟶∞xn=za3T-orbitally lower semicontinuous in X if it is T-orbitally lower semicontinuous at each z∈X
Obviously, ifg is lower semicontinuous in X, then g is T-orbitally lower semicontinuous in X.
The following lemmas play important roles in this paper.Lemma 1 (see [10]).
Letφ∈Φ1 and rnn∈ℕ be a nonnegative sequence. Then, limn⟶∞∫0rnφtdt=0 if and only if limn⟶∞rn=0.Lemma 2 (see [4]).
LetX be a metric space with metric d and let w be a w-distance in X. Let xnn∈ℕ and ynn∈ℕ be sequences in X, let αnn∈ℕ and βnn∈ℕ be sequences in ℝ+ converging to 0, and let x,y,z∈X, then the following hold:a If wxn,y≤αn and wxn,z≤βn for any n∈ℕ, then y=z. In particular, if wx,y and wx,z=0, then y=z.b If wxn,yn≤αn and wxn,z≤βn for any n∈ℕ, then ynn∈ℕ converges to z.c If wxn,xm≤αn for any n,m∈ℕ with n>m, then xnn∈ℕ is a Cauchy sequence.d If wx,xn≤αn for any n∈ℕ, then xnn∈ℕ is a Cauchy sequence.Lemma 3 (see [11]).
LetX,d be a metric space, w be a w-distance on X, and D∈CLX. Suppose that there exists u∈X such that wu,u=0. Then, wu,D=0 if and only if u∈D.


## 2. Fixed Point Theorems

In this section, we establish fixed point theorems for multivalued contractive mappings (13) and (21), respectively.Theorem 5.
LetX,d be a complete metric space, w be a w-distance in X, and T:X⟶CLX satisfy that for each x∈X there is y∈Tx with(13)b∫0wx,yφtdt≤∫0wx,Txφtdt,∫0wy,Tyφtdt≤c∫0wx,yφtdt,where b and c are constants in 0,1 with c<b and φ∈Φ2.Then,c1 For each x0∈X, there exists an orbit xnn∈ℕ0 of T such that limn⟶∞xn=u for some u∈X.c2fwu=0 if fw:X⟶ℝ+ is T-orbitally lower semicontinuous at u. Moreover, u is a fixed point of T if wu,u=0.Proof.
Now, we showc1. Let x0 be an arbitrary point in X and a=c/b. It follows from (13) that there exists x1∈Tx0 such that(14)b∫0wx0,x1φtdt≤∫0wx0,Tx0φtdt,∫0wx1,Tx1φtdt≤c∫0wx0,x1φtdt.
Continuing this process, we choose easily a sequencexnn∈ℕ0 in X satisfying(15)xn+1∈Txn,b∫0wxn,xn+1φtdt≤∫0wxn,Txnφtdt,∀n∈ℕ0,∫0wxn+1,Txn+1φtdt≤c∫0wxn,xn+1φtdt,∀n∈ℕ0.
Next, we claim thatxnn∈ℕ0 is a Cauchy sequence. It follows from (15) and φ∈Φ2 that(16)0≤∫0wxn,xn+1φtdt≤a∫0wxn−1,xnφtdt≤an∫0wx0,x1φtdt,∀n∈ℕ0,0≤∫0wxn,Txnφtdt≤a∫0wxn−1,Txn−1φtdt≤an∫0wx0,Tx0φtdt,∀n∈ℕ0.
Lettingn⟶∞ in (16) and using Lemma 1, we infer that(17)limn⟶∞wxn,xn+1=0,limn⟶∞fwxn=0.
Making use of (16), w1 and φ∈Φ2, we conclude that(18)∫0wxn,xmφtdt≤∫0∑i=nm−1wxi,xi+1φtdt≤∑i=nm−1∫0wxi,xi+1φtdt≤∑i=nm−1ai∫0wx0,x1φtdt≤an1−a∫0wx0,x1φtdt,∀m,n∈ℕwithm>n,which together with Lemmas 1 and 2 yields that xnn∈ℕ0 is a Cauchy sequence. Completeness of X implies that there exists some u∈X such that(19)limn⟶∞xn=u.
Finally, we showc2. Since fw:X⟶ℝ+ is T-orbitally lower semicontinuous at u, it follows from (17) that(20)0≤fwu≤liminfn⟶∞fwxn=0,which means that fwu=0. Thus, Lemma 3 and wu,u=0 yield that u∈Tu. This completes the proof.Theorem 6.
LetX,d be a complete metric space, w be a w-distance in X, and T:X⟶CLX satisfy that for each x,y∈X×Tx, and there exists z∈Ty with(21)∫0wy,zφtdt≤c∫0wx,yφtdt,where c is a constant in 0,1 and φ∈Φ2. Then, c1 and c2 hold.Proof.
Now, we showc1. Let x0,x1 be an arbitrary point in X×Tx0. It follows from (21) that there exists x2∈Tx1 such that(22)∫0wx1,x2φtdt≤c∫0wx0,x1φtdt.
Continuing this process, we construct a sequencexnn∈ℕ0 in X satisfying(23)xn+1∈Txn,∀n∈ℕ0,∫0wxn,xn+1φtdt≤c∫0wxn−1,xnφtdt,∀n∈ℕ.
Next, we claim thatxnn∈ℕ0 is a Cauchy sequence. It follows from (23) and φ∈Φ2 that(24)0≤∫0wxn,xn+1φtdt≤cn∫0wx0,x1φtdt,∀n∈ℕ.
Lettingn⟶∞ in (24) and using Lemma 1, we infer that(25)limn⟶∞wxn,xn+1=0,which together with (23) implies that(26)0≤fwxn≤wxn,xn+1,∀n∈ℕ.
Combining (25) and (26), we conclude that(27)limn⟶∞fwxn=0.
The rest of the proof is similar to that of Theorem5 and is omitted. This completes the proof.Remark 1.
Theorem5 generalizes Theorems 2–4. The below example demonstrates that Theorem 5 generalizes indeed Theorems 2 and 3.Example 3.
LetX=0,5/4 be endowed with the Euclidean metric d=⋅, u=0, w:X×X⟶ℝ+, T:X⟶CLX, and φ:ℝ+⟶ℝ+ be defined by(28)wx,y=y,∀x,y∈X,Tx=34x2,∀x∈0,1∪1,54,34,45,x=1,c=1517,b=1617,φt=1,∀t∈ℝ+.
It is clear thatwu,u=0, φ∈Φ2, and(29)fwx=wx,Tx=34x2,∀x∈X,is continuous in X. It follows that fwx is lower semicontinuous in X. In order to verify (13), for each x∈X, we have to consider the following cases.Case 1.
x∈0,1∪1,2/3∪2/3,5/4. Clearly, there exists y=3/4x2∈Tx=3/4x2 satisfying(30)b∫0wx,yφtdt=1617⋅34x2≤34x2=∫0wx,Txφtdt,∫0wy,Tyφtdt=2764x4≤1517⋅34x2=c∫0wx,yφtdt.Case 2.
x=1. It is clear that there exists y=3/4∈Tx=3/4,4/5 such that(31)b∫0wx,yφtdt=1617⋅34≤34=∫0wx,Txφtdt,∫0wy,Tyφtdt=2764≤1517⋅34=c∫0wx,yφtdt.Case 3.
x=2/3. It follows that there exists y=1∈Tx=3/4x2 such that(32)b∫0wx,yφtdt=1617⋅1≤1=∫0wx,Txφtdt,∫0wy,Tyφtdt=34≤1517⋅1=c∫0wx,yφtdt.
Hence, (13) holds. That is, the conditions of Theorem 5 are satisfied. It follows from Theorem 5 that T has a fixed point 0∈X.
However, we cannot invoke Theorem3 to show that the mapping T has a fixed point in X. Suppose that the conditions of Theorem 3 are satisfied. Take x∗=1. For y∗∈Tx∗=3/4,4/5, we consider two possible cases as follows:Case 1:y∗=3/4. It follows that(33)2164=d34,2764=dy∗,Ty∗≤cdx∗,y∗=14c,which implies that c≥21/16>1, which is impossible because c∈0,1.Case 2:y∗=4/5. It is easy to see that(34)825=d45,1225=dy∗,Ty∗≤cdx∗,y∗=15c,which implies that c≥8/5>1, which is a contradiction because c∈0,1.
It follows from Remark 1 in [3] that Theorem 3 extends Theorem 2. Thus, Theorem 2 is not applicable in proving the existence of fixed points for the multivalued contractive mapping T in X.Remark 2.
The following example shows that Theorem6 is different from Theorem 2.Example 4.
LetX=ℝ+ be endowed with the Euclidean metric d=⋅, w:X×X⟶ℝ+, T:X⟶CLX, and φ:ℝ+⟶ℝ+ be defined by(35)wx,y=y,∀x,y∈X,Tx=0,x,∀x∈Xφt=t−1/2,∀t∈ℝ+.
Putc∈1/12,1. Clearly, φ∈Φ2, w0,0=0, X,d is a complete metric space, and(36)fwx=wx,Tx=inft:t∈0,x=0,∀x∈X,is continuous in X.
For eachx,y∈X×Tx, there exists z=y/4∈Ty with(37)∫0wy,zφtdt=y≤2cy=c∫0wx,yφtdt,that is, (21) holds. It follows from Theorem 6 that T has a fixed point 0∈X.
However, Theorem2 is useless in proving the existence of fixed points for the mapping T in X. Suppose that (2) holds. Take x∗=1 and y∗=2. It follows that(38)HTx∗,Ty∗=HT1,T2=1≤c=cd1,2=cdx∗,y∗,which is impossible because c∈0,1.



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