# Fixed Points and Stability for Integral-Type Multivalued Contractive Mappings

Authors: Zhefu An; Mengyao Li; Liangshi Zhao
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The existence and iterative approximations of fixed points concerning two classes of integral-type multivalued contractive mappings in complete metric spaces are proved, and the stability of fixed point sets relative to these multivalued contractive mappings is established. The results obtained in this article generalize and improve some known results in the literature. An illustrative example is given.


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## Body

## 1. Introduction

The famous Banach fixed point theorem has both various extensions and valuable applications in a mass of differential equations, difference equations, functional equations, matrix equations, and integral equations ([1–26]). In 2002, Branciari [3] obtained an interesting integral-type fixed point theorem for the contractive mapping of integral type, which is an integral version of the Banach contraction mapping.Theorem 1 (see [3]).
Letf be a mapping from a complete metric space X,ρ into itself satisfying
(1)∫0ρfx,fypsds≤c∫0ρx,ypsds,∀x,y∈X,where c∈0,1 is a constant and p:0,+∞⟶0,+∞ is Lebesgue integrable, summable on each compact subset of 0,+∞ and ∫0εpsds>0 for each ε>0. Then, f has a unique fixed point u∈X such that limn⟶∞fnx=u for each x∈X.Later, the researchers [1, 2, 6, 7, 9, 10, 12, 14, 17–21, 26] generalized Theorem 1 from different directions and got a lot of fixed point results for various contractive mappings of integral type.In 1969, Nadler [15] gave a multivalued analog of the Banach fixed point theorem by using the Hausdorff metric and introducing the multivalued contraction mapping, that is, he presented a nice fixed point theorem for the multivalued contraction mapping.Theorem 2 (see [10]).
LetX,ρ be a complete metric space and T:X⟶CBX be a multivalued contraction mapping, that is, there exists a constant r∈0,1 satisfying
(2)HTx,Ty≤rρx,y,∀x,y∈X.
Then,T has a fixed point in X.Czerwik [5] and Gordji et al. [8] extended Theorem 2 and proved fixed point theorems for some multivalued contractive mappings, which include (2) as special cases. The researchers [4, 11, 13, 22–24] gained fixed point theorems for several multivalued contractive mappings and studied also the stability of fixed point sets with respect to the multivalued contractive mappings. Lim [11] established the stability of fixed point sets associated with the multivalued contraction mappings in Theorem 2. Choudhury et al. [24] proved that a uniformly convergent sequence of α∗−ψ multivalued contractions has stable fixed point sets.By combining the ideas of Nadler, Branciari, and Lim, in this article, we study the existence and iterative approximations of fixed points concerning two classes of integral-type multivalued contractive mappings in complete metric spaces and present stability of fixed point sets relative to a sequence of integral-type multivalued contractive mappings. Our results generalize and unify a few results in [5, 8, 11, 15]. An example is also presented to illustrate the efficiency of our results.


## 2. Preliminaries

Throughout this paper,ℕ denotes the set of all positive integers, ℝ+=0,+∞, ℕ0=0∪ℕ and
(3)Φ1=p∣p:ℝ+⟶ℝ+isLebesgueintegrable,summableoneachcompactsubsetofℝ+and∫0εpsds>0foreachε>0,Φ2=p∣pisinΦ1and∫0a+bpsds≤∫0apsds+∫0bpsdsforeacha,b∈ℝ+.Assume thatX,ρ is a metric space, CLX stands for the family of all nonempty closed subsets of X, and CBX denotes the family of all nonempty closed bounded subsets of X. For C,D∈CLX and T, Tii∈N0:X⟶CLX, define
(4)FT=x∈X:x∈Tx,ρx,D=infy∈Dρx,y,∀x∈X,HC,D=axsupx∈Cρx,D,supy∈Dρy,C,mifthemaximumexists,+∞,otherwise,Nx,y=maxρx,y,ρx,Tx,ρy,Ty,12ρx,Ty+ρy,Tx,∀x,y∈X,Mx,y=maxρx,y,ρx,Tx,ρy,Ty,12ρx,Ty+ρy,Tx,ρx,Txρy,Ty1+ρx,y,ρx,Tyρy,Tx1+ρx,y,ρx,Txρy,Ty1+HTx,Ty,ρx,Tyρy,Tx1+HTx,Ty,1+ρx,Tx+ρy,Ty1+ρx,y+HTx,Tyρx,y,1+ρx,Ty+ρy,Tx1+ρx,y+HTx,Tyρx,y,∀x,y∈X,Nix,y=maxρx,y,ρx,Tix,ρy,Tiy,12ρx,Tiy+ρy,Tix,∀x,y∈X,i∈N0,Mix,y=maxρx,y,ρx,Tix,ρy,Tiy,12ρx,Tiy+ρy,Tix,ρx,Tixρy,Tiy1+ρx,y,ρx,Tiyρy,Tix1+ρx,y,ρx,Tixρy,Tiy1+HTix,Tiy,ρx,Tiyρy,Tix1+HTix,Tiy,1+ρx,Tix+ρy,Tiy1+ρx,y+HTix,Tiyρx,y,1+ρx,Tiy+ρy,Tix1+ρx,y+HTix,Tiyρx,y,∀x,y∈X,i∈N0.A sequencexnn∈ℕ0 in X is called an orbit of T at x0 if xn+1∈Txn for each n∈ℕ0.Lemma 3 (see [12]).
Letp∈Φ1 and rnn∈ℕ be a nonnegative sequence and limn⟶∞rn=a. Then
(5)limn⟶∞∫0rnpsds=∫0apsds.Lemma 4 (see [12]).
Letp∈Φ1 and rnn∈N be a nonnegative sequence. Then
(6)limn⟶∞∫0rnpsds=0,if and only if limn⟶∞rn=0.It follows from [13] the following.Lemma 5.
Assume thatX,ρ is a metric space and C,D∈CLX. Then, for any r>1 and a∈C, there exists b∈D such that
(7)ρa,b≤rHC,D.Lemma 6 (see [13]).
Assume thatX,ρ is a metric space. Then
(8)ρx,C−ρy,C≤ρx,y,∀x,y∈X,C∈CLX.Lemma 7.
Assume thatX,ρ is a metric space, C∈CLX, and xnn∈ℕ⊂X converges to a∈X. Then
(9)limn⟶∞ρxn,C=ρa,C.Proof.
It follows from Lemma6 that
(10)ρxn,C−ρa,C≤ρxn,a⟶0asn⟶∞,that is
(11)limn⟶∞ρxn,C=ρa,C.
This completes the proof.☐Lemma 8.
LetC⊆ℝ+ and p∈Φ1. Then
(12)supa∈C∫0apsds=∫0supCpsds,(13)infa∈C∫0apsds=∫0infCpsds.Proof. (see (12)).
LetsupC=c. It follows that
(14)a≤c,∀a∈C,and there exist a sequence ann∈ℕ in C satisfying
(15)limn⟶∞an=c.
Thus, (14) and p∈Φ1 mean that
(16)∫0apsds≤∫0cpsds,∀a∈C,which yields that
(17)supa∈C∫0apsds≤∫0cpsds.
On account of (15) and Lemma 3, we infer that
(18)limn⟶∞∫0anpsds=∫0cpsds.
Clearly, (12) follows from (17) and (18). The proof of (13) is similar to that of (12) and is omitted. This completes the proof.☐


## 3. Fixed Point Theorems and an Example

Now, we investigate the existence and iterative approximations of fixed points for the integral-type multivalued contractive mappings (19) and (42), respectively.Theorem 9.
Assume thatX,ρ is a complete metric space and T:X⟶CLX satisfies that
(19)∫01qHTx,Typsds≤q∫0Mx,ypsds,∀x,y∈X,where q is a constant in 0,1 and p∈Φ2. Then, for each x0∈X, there exists an orbit xnn∈ℕ0 of T at x0 such that it converges to some fixed point a∈X of T and
(20)∫0ρxn,apsds≤qn1−q∫0ρx0,x1psds,∀n∈ℕ.Proof.
For anyx0 in X and x1 in Tx0, Lemma 5 guarantees that
(21)ρx1,x2≤1qHTx0,Tx1forsomex2∈Tx1.
Note that(22)ρx0,x1≤Mx0,x1=maxρx0,x1,ρx0,Tx0,ρx1,Tx1,12ρx0,Tx1+ρx1,Tx0,ρx0,Tx0ρx1,Tx11+ρx0,x1,ρx0,Tx1ρx1,Tx01+ρx0,x1,ρx0,Tx0ρx1,Tx11+HTx0,Tx1,ρx0,Tx1ρx1,Tx01+HTx0,Tx1,1+ρx0,Tx0+ρx1,Tx11+ρx0,x1+HTx0,Tx1ρx0,x1,1+ρx0,Tx1+ρx1,Tx01+ρx0,x1+HTx0,Tx1ρx0,x1≤maxρx0,x1,ρx0,x1,ρx1,x2,12ρx0,x1+ρx1,Tx1,ρx0,x1ρx1,x21+ρx0,x1,0,ρx0,x1ρx1,Tx11+ρx1,Tx1,0,1+ρx0,x1+ρx1,Tx11+ρx0,x1+ρx1,Tx1ρx0,x1,1+ρx0,x1+ρx1,Tx11+ρx0,x1+ρx1,Tx1ρx0,x1≤maxρx0,x1,ρx1,x2,12ρx0,x1+ρx1,x2,ρx1,x2,ρx0,x1,ρx0,x1,ρx0,x1=maxρx0,x1,ρx1,x2,which together with (19), (21), and p∈Φ2 yields.
(23)∫0ρx1,x2psds≤∫01qHTx0,Tx1psds≤q∫0Mx0,x1psds≤q∫0maxρx0,x1,ρx1,x2psds=q∫0ρx0,x1psds,and
(24)ρx1,x2≤ρx0,x1=Mx0,x1and∫0ρx1,x2psds≤q∫0ρx0,x1psds.
Lemma5 reveals that
(25)ρx2,x3≤1qHTx1,Tx2forsomex3∈Tx2.
Notice that(26)ρx1,x2≤Mx1,x2=maxρx1,x2,ρx1,Tx1,ρx2,Tx2,12ρx1,Tx2+ρx2,Tx1,ρx1,Tx1ρx2,Tx21+ρx1,x2,ρx1,Tx2ρx2,Tx11+ρx1,x2,ρx1,Tx1ρx2,Tx21+HTx1,Tx2,ρx1,Tx2ρx2,Tx11+HTx1,Tx2,1+ρx1,Tx1+ρx2,Tx21+ρx1,x2+HTx1,Tx2ρx1,x2,1+ρx1,Tx2+ρx2,Tx11+ρx1,x2+HTx1,Tx2ρx1,x2≤maxρx1,x2,ρx1,x2,ρx2,x3,12ρx1,x2+ρx2,Tx2,ρx1,x2ρx2,x31+ρx1,x2,0,ρx1,x2ρx2,Tx21+ρx2,Tx2,0,1+ρx1,x2+ρx2,Tx21+ρx1,x2+ρx2,Tx2ρx1,x2,1+ρx1,x2+ρx2,Tx21+ρx1,x2+ρx2,Tx2ρx1,x2≤maxρx1,x2,ρx2,x3,12ρx1,x2+ρx2,x3,ρx2,x3,ρx1,x2,ρx1,x2,ρx1,x2=maxρx1,x2,ρx2,x3,which together with (19), (25), and p∈Φ2 infers
(27)∫0ρx2,x3psds≤∫01qHTx1,Tx2psds≤q∫0Mx1,x2psds≤q∫0maxρx1,x2,ρx2,x3psds=q∫0ρx1,x2psds,and
(28)ρx2,x3≤ρx1,x2=Mx1,x2and∫0ρx2,x3psds≤q∫0ρx1,x2psds.
Making use of (24) and (28), we deduce
(29)∫0ρx2,x3psds≤q∫0ρx1,x2psds≤q2∫0ρx0,x1psds.
Continuing the process, we obtain an orderxnn∈N0 of T at x0 satisfying
(30)xn∈Txn−1,ρxn,xn+1≤ρxn−1,xn=Mxn−1,xn,∀n∈N,∫0ρxnxn+1psds≤q∫0ρxn−1,xnpsds,∀n∈N.
Thus, (30), q∈0,1, and p∈Φ2 mean
(31)0≤∫0ρxn,xn+1psds≤qn∫0ρx0,x1psds,∀n∈N.
Lemma4 gives
(32)limn⟶∞ρxn,xn+1=0.
By (31) and p∈Φ2, we obtain
(33)∫0ρxn,xmpsds≤∫0ρxn,xn+1+ρxn+1,xn+2+⋯+ρxm−1,xmpsds≤∫0ρxn,xn+1psds+∫0ρxn+1,xn+2psds+⋯+∫0ρxm−1,xmpsds≤qn∫0ρx0,x1psds+qn+1∫0ρx0,x1psds+⋯+qm−1∫0ρx0,x1psds=qn1+q+⋯+qm−n−1∫0ρx0,x1psds≤qn1−q∫0ρx0,x1psds,∀m,n∈Nwithm>n.
It is clear that (33), q∈0,1, p∈Φ2, and Lemma 8 guarantee
(34)0≤∫0supρxn,xm:m>npsds=sup∫0ρxn,xmpsds:m>n≤qn1−q∫0ρx0,x1psds⟶0asn⟶∞,that is
(35)limn⟶∞∫0supρxn,xm:m>npsds=0.
Lemma4 ensures
(36)limn⟶∞supρxn,xm:m>n=0.
Hence,xnn∈N0 is a Cauchy sequence.
Completeness ofX,ρ means that there exists a point a in X with
(37)limn⟶∞xn=a.
Lettingm⟶∞ in (33) and using (37) and Lemma 3, we arrive at
(38)∫0ρxn,apsds≤qn1−q∫0ρx0,x1psds,∀n∈N,that is, (20) holds.
Next, we prove thata=Ta. From (32), (37), and Lemmas 6 and 7, we get immediately
(39)ρa,Ta≤Mxn−1,a=maxρxn−1,a,ρxn−1,Txn−1,ρa,Ta,12ρxn−1,Ta+ρa,Txn−1,ρxn−1,Txn−1ρa,Ta1+ρxn−1,a,ρxn−1,Taρa,Txn−11+ρxn−1,a,ρxn−1,Txn−1ρa,Ta1+HTxn−1,Ta,ρxn−1,Taρa,Txn−11+HTxn−1,Ta,1+ρxn−1,Txn−1+ρa,Ta1+ρxn−1,a+HTxn−1,Taρxn−1,a,1+ρxn−1,Ta+ρa,Txn−11+ρxn−1,a+HTxn−1,Taρxn−1,a≤maxρxn−1,a,ρxn−1,xn,ρa,Ta,12ρxn−1,Ta+ρa,xn,ρxn−1,xnρa,Ta1+ρxn−1,a,ρxn−1,Taρa,xn1+ρxn−1,a,ρxn−1,xnρa,Ta1+ρxn,Ta,ρxn−1,Taρa,xn1+ρxn,Ta,1+ρxn−1,xn+ρa,Ta1+ρxn−1,a+ρxn,Taρxn−1,a,1+ρxn−1,Ta+ρa,xn1+ρxn−1,a+ρxn,Taρxn−1,a⟶max0,0,ρa,Ta,12ρa,Ta,0,0,0,0,0,0=ρa,Taasn⟶∞,that is
(40)limn⟶∞Mxn−1,a=ρa,Ta.
Taking into account (19), (37), (40), and Lemmas 3 and 7, we gain
(41)∫0ρa,Tapsds=limsupn⟶∞∫0ρxn,Tapsds≤limsupn⟶∞∫0HTxn−1,Tapsds≤limsupn⟶∞∫01qHTxn−1,Tapsds≤qlimsupn⟶∞∫0Mxn−1,apsds=q∫0ρa,Tapsds,which yields ρa,Ta=0 because q∈0,1. That is, a=Ta. This completes the proof.☐Note thatNx,y≤Mx,y,∀x,y∈X. It follows from Theorem 9 the following.Theorem 10.
Assume thatX,ρ is a complete metric space and T:X⟶CLX satisfies
(42)∫01qHTx,Typsds≤q∫0Nx,ypsds,∀x,y∈X,where q is a constant in 0,1 and p∈Φ2. Then, for each x0∈X, there exists an orbit xnn∈N0 of T at x0 such that it converges to some fixed point a∈X of T and (20) holds.Remark 11.
In casept=t,∀t∈R+ and q=α, where α is a constant in 0,1, then Theorem 12 reduces to a result, which generalizes Theorems 1 and 2 in [5], Theorem 2.1 in [8], and Theorem 5 in [15]. The example below shows that Theorem 10 extends properly Theorem 5 in [15].Example 1.
LetX=0,1 be endowed with the Euclidean metric ρ=∣·∣. Let q=9/10, T:X⟶CLX, and p:R+⟶R+ be defined by
(43)Tx=110,∀x∈0,1,0,110,x=1,pt=12t+1/10,∀t∈R+.It is clear thatp∈Φ2. Let x,y∈X with y<x. In order to verify (19), we consider below two possible cases:Case 1.
0≤y<x<1. Clearly,
(44)∫01/qHTx,Typsds=0≤q∫0Mx,ypsds.Case 2.
0≤y<1 and x=1. It follows that
(45)∫01/qHTx,Typsds=∫0109×11012t+1/10dt=1990−110<0.212−0.316<0.6147<9101−110=q∫0ρx,Txpsds≤q∫0Mx,ypsds.That is, (19) holds. Thus, Theorem 10 ensures that T has a fixed point 1/10∈X. Observe that
(46)HT1,T910=H0,110,110=110≮α110=αρ1,910,∀α∈0,1,which means that Theorem 5 in [15] cannot be used to show the existence of fixed points of T.


## 4. On Stability of Fixed Point Sets

Now, we discuss the stability of fixed point sets for the integral-type multivalued contractive mappings (19) and (42), respectively. Put K=supx∈XHT1x,T2x.Theorem 12.
Assume thatX,ρ is a complete metric space and T1,T2:X⟶CLX satisfy
(47)∫01/qHTix,Tiypsds≤q∫0Mix,ypsds,∀x,y∈X,i∈1,2,where q is a constant in 0,1 and p∈Φ2. Then, ∫0HFT1,FT2psds≤1/1−q∫01/qKpsds.Proof.
Without loss of generality, we assume thatK<+∞. Note that Theorem 9 yields FTi≠∅ for i∈1,2. Put x0 in FT1. Lemma 5 guarantees
(48)ρx0,x1≤1qHT1x0,T2x0≤1qKforsomex1∈T2x0,x2∈T2x1,(49)ρx1,x2≤1qHT2x0,T2x1,(50)ρx0,x1≤M2x0,x1=maxρx0,x1,ρx0,T2x0,ρx1,T2x1,12ρx0,T2x1+ρx1,T2x0,ρx0,T2x0ρx1,T2x11+ρx0,x1,ρx0,T2x1ρx1,T2x01+ρx0,x1,ρx0,T2x0ρx1,T2x11+HT2x0,T2x1,ρx0,T2x1ρx1,T2x01+HT2x0,T2x1,1+ρx0,T2x0+ρx1,T2x11+ρx0,x1+HT2x0,T2x1ρx0,x1,1+ρx0,T2x1+ρx1,T2x01+ρx0,x1+HT2x0,T2x1ρx0,x1≤maxρx0,x1,ρx0,x1,ρx1,x2,12ρx0,x1+ρx1,T2x1,ρx0,x1ρx1,x21+ρx0,x1,0,ρx0,x1ρx1,T2x11+ρx1,T2x1,0,1+ρx0,x1+ρx1,T2x11+ρx0,x1+ρx1,T2x1ρx0,x1,1+ρx0,x1+ρx1,T2x11+ρx0,x1+ρx1,T2x1ρx0,x1≤maxρx0,x1,ρx1,x2,12ρx0,x1+ρx1,x2,ρx1,x2,ρx0,x1,ρx0,x1,ρx0,x1=maxρx0,x1,ρx1,x2,which together with (47) and (49) and p∈Φ2 implies
(51)∫0ρx2,x3psds≤∫01qHT2x1,T2x2psds≤q∫0M2x1,x2psds≤q∫0maxρx1,x2,ρx2,x3psds=q∫0ρx1,x2psds,and
(52)ρx1,x2≤ρx0,x1=M2x0,x1and∫0ρx1,x2psds≤q∫0ρx0,x1psds.
Lemma5 ensures with
(53)ρx2,x3≤1qHT2x1,T2x2forsomex3∈T2x2.
Note that(54)ρx1,x2≤M2x1,x2=maxρx1,x2,ρx1,T2x1,ρx2,T2x2,12ρx1,T2x2+ρx2,T2x1,ρx1,T2x1ρx2,T2x21+ρx1,x2,ρx1,T2x2ρx2,T2x11+ρx1,x2,ρx1,T2x1ρx2,T2x21+HT2x1,T2x2,ρx1,T2x2ρx2,T2x11+HT2x1,T2x2,1+ρx1,T2x1+ρx2,T2x21+ρx1,x2+HT2x1,T2x2ρx1,x2,1+ρx1,T2x2+ρx2,T2x11+ρx1,x2+HT2x1,T2x2ρx1,x2≤maxρx1,x2,ρx1,x2,ρx2,x3,12ρx1,x2+ρx2,T2x2,ρx1,x2ρx2,x31+ρx1,x2,0,ρx1,x2ρx2,T2x21+ρx2,Tx2,0,1+ρx1,x2+ρx2,T2x21+ρx1,x2+ρx2,T2x2ρx1,x2,1+ρx1,x2+ρx2,T2x21+ρx1,x2+ρx2,T2x2ρx1,x2,≤maxρx1,x2,ρx2,x3,12ρx1,x2+ρx2,x3,ρx2,x3,ρx1,x2,ρx1,x2,ρx1,x2=maxρx1,x2,ρx2,x3,which together with (47), (53), and p∈Φ2 gives
(55)∫0ρx2,x3psds≤∫01qHT2x1,T2x2psds≤q∫0M2x1,x2psds≤q∫0maxρx1,x2,ρx2,x3psds=q∫0ρx1,x2psds,and
(56)ρx2,x3≤ρx1,x2=M2x1,x2and∫0ρx2,x3psds≤q∫0ρx1,x2psds.
From (52) and (56), we deduce that
(57)∫0ρx2,x3psds≤q∫0ρx1,x2psds≤q2∫0ρx0,x1psds.
Continuing the process, we obtain an orderxnn∈N0 of T2 at x0 satisfying (48) and
(58)xn∈T2xn−1,ρxn,xn+1≤1qHT2xn−1,T2xn,ρxn,xn+1≤ρxn−1,xn=M2xn−1,xn,∫0ρxn,xn+1psds≤q∫0ρxn−1,xnpsds,∀n∈N.
In light of (58), q∈0,1, and p∈Φ2, we have
(59)0≤∫0ρxn,xn+1psds≤qn∫0ρx0,x1psds,∀n∈N,(60)limn⟶∞∫0ρxn,xn+1psds=0.
Combining (60) and Lemma 4, we get
(61)limn⟶∞ρxn,xn+1=0.
By (59) and φ∈Φ2, we infer
(62)∫0ρxn,xmpsds≤∫0ρxn,xn+1+ρxn,xn+2+⋯+ρxm−1,xmpsds≤∫0ρxn,xn+1psds+∫0ρxn+1,xn+2psds+⋯+∫0ρxm−1,xmpsds≤qn∫0ρx0,x1psds+qn+1∫0ρx0,x1psds+⋯+qm−1∫0ρx0,x1psds=qn1+q+⋯+qm−n−1∫0ρx0,x1psds≤qn1−q∫0ρx0,x1psds,∀m,n∈Nwithm>n.
It follows from (62), q∈0,1, φ∈Φ2, and Lemma 8 that
(63)0≤∫0supρxn,xm:m>npsds=sup∫0ρxn,xmpsds:m>n≤qn1−q∫0ρx0,x1psds⟶0asn⟶∞,that is
(64)limn⟶∞∫0supρxn,xm:m>npsds=0.which together with Lemma 4 means
(65)limn⟶∞supρxn,xm:m>n=0,that is, xnn∈N0 is a Cauchy sequence.
Completeness ofX,ρ guarantees
(66)limn⟶∞xn=aforsomea∈X.
Lettingm⟶∞ in (62) and making use of (66) and Lemma 3, we deduce
(67)∫0ρxn,apsds≤qn1−q∫0ρx0,x1psds,∀n∈N.
Observe that (61), (66), and Lemma 7 ensure
(68)ρa,T2a≤M2xn−1,a=maxρxn−1,a,ρxn−1,T2xn−1,ρa,T2a,12ρxn−1,T2a+ρa,T2xn−1,ρxn−1,T2xn−1ρa,T2a1+ρxn−1,a,ρxn−1,T2aρa,T2xn−11+ρxn−1,a,ρxn−1,T2xn−1ρa,T2a1+HT2xn−1,T2a,ρxn−1,T2aρa,T2xn−11+HT2xn−1,T2a,1+ρxn−1,T2xn−1+ρa,T2a1+ρxn−1,a+HT2xn−1,T2aρxn−1,a,1+ρxn−1,T2a+ρa,T2xn−11+ρxn−1,a+HT2xn−1,T2aρxn−1,a≤maxρxn−1,a,ρxn−1,xn,ρa,T2a,12ρxn−1,T2a+ρa,xn,ρxn−1,xnρa,T2a1+ρxn−1,a,ρxn−1,T2aρa,xn1+ρxn−1,a,ρxn−1,xnρa,T2a1+ρxn,T2a,ρxn−1,T2aρa,xn1+ρxn,T2a,1+ρxn−1,xn+ρa,T2a1+ρxn−1,a+ρxn,T2aρxn−1,a,1+ρxn−1,T2a+ρa,xn1+ρxn−1,a+ρxn,T2aρxn−1,a⟶max0,0,ρa,T2a,12ρa,T2a,0,0,0,0,0,0=ρa,T2aasn⟶∞,that is
(69)limn⟶∞M2xn−1,a=ρa,T2a.
By virtue of (47), (66), (69), p∈Φ2, and Lemmas 3 and 7, we have
(70)∫0ρa,T2apsds=limsupn⟶∞∫0ρxn,T2apsds≤limsupn⟶∞∫0HT2xn−1,T2apsds≤limsupn⟶∞∫01qHT2xn−1,T2apsds≤qlimsupn⟶∞∫0M2xn−1,apsds=q∫0ρa,T2apsds,which means ρa,T2a=0 because q∈0,1, that is, a∈FT2.
Taking advantage of (48), (59), (67), φ∈Φ2, and Lemma 4, we get
(71)∫0ρx0,apsds≤∫0ρx0,x1+ρx1,x2+⋯+ρxn,xn+1+ρxn+1,apsds≤∫0ρx0,x1psds+∫0ρx1,x2psds+⋯+∫0ρxn,xn+1psds+∫0ρxn+1,apsds≤∫0ρx0,x1psds+q∫0ρx0,x1psds+⋯+qn∫0ρx0,x1psds+qn+11−q∫0ρx0,x1psds=1−qn+1+qn+11−q∫0ρx0,x1psds≤11−q∫01qHT1x0,T2x0psds≤11−q∫01qKpsds,∀n∈N.
It follows from Lemma8 that
(72)∫0ρx0,FT2psds=infy∈FT2∫0ρx0,ypsds≤∫0ρx0,apsds≤11−q∫01qKpsds,∀x0∈F0T1,and
(73)∫0supρx,FT2:x∈FT1psds=supx∈FT1∫0ρx,FT2psds≤11−q∫01qKpsds.
Reversing the roles ofT1 and T2, we also conclude
(74)∫0supρy,FT1:y∈FT2psds≤11−q∫01qKpsds.
Using (73), (74), Lemma 8, and φ∈Φ2, we obtain
(75)∫0HFT1,FT2psds=∫0maxsupρx,FT2:x∈FT1,supρy,FT1:y∈FT2psds=max∫0supρx,FT2:x∈FT1psds,∫0supρy,FT1:y∈FT2psds≤11−q∫01qKpsds.
This completes the proof.☐Theorem 13.
Assume thatX,ρ is a complete metric space and Tii∈N0:X⟶CLX satisfy
(76)∫01qHTix,Tiypsds≤q∫0Mix,ypsds,∀x,y∈X,i∈N0,where q is a constant in 0,1 and p∈Φ2. Assume that
(77)limi⟶∞HTix,T0x=0uniformlyforallx∈X.
Then,limi⟶∞HFTi,FT0=0.Proof.
Letε>0. It follows from Theorem 12.34 in [27] and p∈Φ2 that there exists δ>0 with
(78)∫Apsds<1−qεforeveryboundedsubsetA⊂R+withmA≤δ,where mA is the Lebesgue measure of A. (77) guarantees that there exists N∈N satisfying
(79)supx∈XHTix,T0x<qδ,∀i≥N.
By means of (78), (79), p∈Φ2, and Theorem 12, we conclude
(80)0≤∫0HFTi,FT0psds≤11−q∫01qsupx∈XHTix,T0xpsds≤11−q∫0δpsds<ε,∀i≥N,which implies that
(81)limi⟶∞∫0HFTi,FT0psds=0.
Thus,limi⟶∞HFTi,FT0=0 follows from (81) and Lemma 4. This completes the proof.☐Theorems12 and 13 infer immediately the following.Theorem 14.
Assume thatX,ρ is a complete metric space and T1,T2:X⟶CLX satisfy that
(82)∫01qHTix,Tiypsds≤q∫0Nix,ypsds,∀x,y∈X,i∈1,2,where q is a constant in 0,1 and p∈Φ2. Then, ∫0HFT1,FT2psds≤1/1−q∫01/qKpsds.Theorem 15.
Assume thatX,ρ is a complete metric space and Tii∈N0:X⟶CLX satisfy (77) and
(83)∫01qHTix,Tiypsds≤q∫0Nix,ypsds,∀x,y∈X,i∈N0,where q is a constant in 0,1 and p∈Φ2. Then, limi⟶∞HFTi,FT0=0.Remark 16.
Theorems14 and 15 extend, respectively, Lemma 1 and Theorem 1 in [11].


## 5. Conclusion

In this paper, we introduce two classes of integral-type multivalued contractive mappings, which include some known multivalued contractive mappings as special cases, and prove the existence, iterative approximations, and stability of fixed points for these integral-type multivalued contractive mappings under certain conditions. Our results extend several known results in the literature.



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*Source: 5542414-2021-10-14.xml*