# On Some Fixed Point Results forH+-Type Contraction Mappings in Metric Spaces

Authors: Hemant Kumar Pathak; Rosana Rodríguez-López
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We prove some fixed point theorems forH+-type multivalued  contractive mappings in the setting of Banach spaces and metric spaces. The results provided allow recovering different well-known results.


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## Body

## 1. Introduction

The interest in the study of fixed point theory in the frame of multivalued mappings by using the Hausdorff metric has its origin in the contributions of Markin [1] and Nadler [2], which have lead to interesting achievements on this topic with reference to both theoretical results and applications (see the monograph by Singh et al. on fixed point theory [3] and the references therein).Starting from the work of Nadler [2], many authors have contributed to the huge development of fixed point theory for set-valued contractions. We mention some interesting contributions, such as those made by Reich [4, 5], Lami Dozo [6], Singh [7], Lim [8], Kaneko [9], Mizoguchi and Takahashi [10], Dhompongsa et al. [11], Feng and Liu [12], Klim and Wardowski [13], Suzuki [14], and Pathak and Shahzad [15, 16] (see also the monographs by Goebel and Kirk [17], Petrusel [18], and so forth). We also refer to some recent works on this topic, for instance, those by Hasanzade Asl et al. [19], Samet et al. [20], and Kumam and Sintunavarat [21].In this paper, we provide an extension of some fixed point results to the context of multivaluedH+-type φ-contraction mappings, by establishing a common frame which allows obtaining as corollaries some well-known fixed point results.


## 2. Preliminaries

For(X,·) a normed linear space and K a nonempty subset of X, we denote by 2K, CB(K), and K(K) the collection of all nonempty subsets of K, the collection of all nonempty closed and bounded subsets of K, and the collection of all compact subsets of K, respectively.ForA,B∈CB(X), we define the following mappings: (1)HA,B=max⁡ρA,B,ρB,A,H+A,B=12ρA,B+ρB,A,where ρ(A,B)=supx∈Adist(x,B) and dist(x,B)=infy∈Bx-y. It is well known that the mapping H is a metric on CB(X) called theHausdorff metric induced by the norm in X. From Proposition 2.1 [16], we also know that H+ is a metric on CB(X). Moreover, H and H+ are equivalent metrics [22], since 1/2H(A,B)≤H+(A,B)≤H(A,B). By the results in the monograph by Kuratowski [22], we deduce that (CB(X),H+) is complete provided that (X,d) is complete (d denotes the metric induced by the norm ·) and K(X) is a closed subspace of (CB(X),H+) (see Theorem 2.6 [16]). The mapping H+:CB(X)×CB(X)→R is also continuous and satisfies the following properties (see Proposition 2.2 [16]): (i)
H+(λA,λB)=λH+(A,B), for any λ∈C and A,B∈CB(X);(ii)
H+(A+a,B+a)=H+(A,B), for any A,B∈CB(X) and a∈X.Although we have introduced the definitions ofH and H+ for normed linear spaces, we can also do it for metric spaces (X,d) just by using the metric d instead of the norm; this will be enough for our purpose.The notions of multivalued contraction andH+-contraction mapping are essential to this work and we include it here for completeness.Definition 1.
One says that a set-valued mappingT:X→CB(X) is a multivalued k-contraction mapping if there exists a fixed real number k,  0<k<1, such that (2)HTx,Ty≤kx-y,∀x,y∈X.Definition 2 (see [7]).
Let(X,d) be a metric space. A multivalued map T:X→CB(X) is called H+-contraction if the following conditions hold: (C1)
there existsk in (0,1) such that (3)H+Tx,Ty≤kdx,y,for everyx,y∈X,(C2)
for everyx in X,y in T(x) and ε>0, there exists z in T(y) such that (4)dy,z≤H+Ty,Tx+ε.
Another important concept is the notion of fixed point for a multivalued map.Definition 3.
An elementx∈X is said to be a fixed point of a multivalued map T:K⊆X→2X if x∈T(x). One denotes by F(T) the set of fixed points of T.Concerning the existence of fixed points for multivalued contractions, a highly relevant result was provided by Nadler (see [2]).Theorem 4 (Theorem 5 [2]).
Let(X,d) be a complete metric space and T:X→CB(X) a multivalued contraction mapping. Then T has a fixed point.Pathak and Shahzad [16] have given a generalization of Theorem 4 under a condition weaker than multivalued contractivity by using the metric H+, as recalled below.Theorem 5 (Theorem 3.2 [16]).
Let(X,d) be a complete metric space. Every H+-type multivalued contraction mapping T:X→CB(X) with Lipschitz constant k<1 has a fixed point.In the next section, we obtain some fixed point results for a more general class of multivaluedH+-type contractions. We provide a common structure which allows obtaining as particular cases some well-known fixed point results.


## 3. Fixed Point Results for MultivaluedH+-Type Weak φ-Contraction Mappings

In this section, we present the main results, in which the existence of fixed points is deduced for multivaluedH+-type weak φ-Lipschitz and φ-contraction mappings, as introduced below.Definition 6.
Let(X,d) be a metric space. A multivalued map T:X→CB(X) is called an H+-type multivalued weak φ-Lipschitz mapping if condition (C2) in Definition 2 holds and there exist M:(R+)5→R+ and φ:R+→R+ such that (5)H+Tx,Ty≤φMdx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx,for everyx,y∈X.If, moreover, there exists k∈(0,1) such that φ(t)≤kt, for every t≥0, we say that T is an H+-type multivalued weak φ-contraction.Remark 7.
In Definition6, if we take(6)φt=kt,withk∈0,1,Mr1,r2,r3,r4,r5=r1,we get the notion of H+-type multivalued contraction mapping.
On the other hand, if we take(7)φt=kt,withk∈0,1,Mr1,r2,r3,r4,r5=max⁡r1,r2,r3,r4+r52,then we obtain the notion of H+-type multivalued weak contractive mapping (see [23, Definition 3.3]), since inequality (5) is reduced to (8)H+Tx,Ty≤kmax⁡dx,y,dx,Tx,dy,Ty,dx,Ty+dy,Tx2,for everyx,y∈X.
Finally, if we take(9)φt=kt,withk∈0,1,Mr1,r2,r3,r4,r5=max⁡r1,r2,r3,r4,r5,then we obtain the notion of H+-type multivalued quasi-contraction mapping (compare inequality with [23, equation (2.3)]) (10)H+Tx,Ty≤kmax⁡dx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx,for everyx,y∈X.We start this study by providing a fixed point result forH+-type multivalued weak φ-Lipschitz mappings. This result (Theorem 8) will find an extension below in Theorem 18 and further in Theorem 21, both stated for partial H+-type multivalued weak φ-Lipschitz mappings. It is possible to state first these results and, then, obtain Theorem 8 as a corollary. However, by considering Theorem 8 first, we can focus our attention on the properties of function M, considering a general expression which immediately connects with some well-known fixed point results. Therefore, we present the main results in several steps:(s.i)
First, in Theorem8, we consider a general expression for function M, while for function φ it is required that φ(t)≤kt, t≥0.(s.ii)
Then, in Theorem18, we extend this result to partial H+-type multivalued weak φ-Lipschitz mappings.(s.iii)
In Theorem21, the same type of variability is allowed for function M and we explore some other possible general expressions for function φ, also for partial H+-type multivalued weak φ-Lipschitz mappings.(s.iv)
We include some considerations concerningH+-type weak φ-Lipschitz mappings with respect to a binary relation (including the case of partial orderings).(s.v)
We complete this study by proving Theorem32, which considers different restrictions for the selection of functions M and φ.Theorem 8.
Let(X,d) be a complete metric space. Let T:X→CB(X) be an H+-type multivalued weak φ-Lipschitz mapping such that the functions φ, M satisfy the following: (i)
There existsk>0 such that φ(t)≤kt, for every t≥0.(ii)
For everyi=2,3,4,5, the function M(r1,r2,r3,r4,r5) is monotonically increasing in the variable ri, provided that the other variables rj, j≠i, remain fixed.(iii)
For eachr>0 fixed, M(r1,r2,r,r4,r5) is continuous at (r1,r2,r4,r5)=(0,0,r,0).(iv)
There exists a functionN:R+×R+→R+ such that
(iv-a)
M(r,r,s,r+s,0)≤N(r,s), for every r,s∈R+,(iv-b)
N(·,s) is monotonically increasing for each s∈R+ fixed,(iv-c)
N(r,·) is monotonically increasing for each r∈R+ fixed,(iv-d)
there existsτ>0 such that N(r,r)≤τr, for every r>0,(iv-e)
there existsR≥0 such that N(0,s)≤Rs, for every s>0.(v)
The constantsk in (i) and τ, R in (iv) are such that k<1/max⁡τ,2R.Then T has a fixed point.Proof.
We denote byψ:X×X→(R+)5 the mapping defined as (11)ψx,y≔dx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx.By (5) and (i), we get (12)H+Tx,Ty≤kMψx,y,for everyx,y∈X.We construct a sequence in X in lines similar to [16, Theorem 3.2] or [23, Theorem 3.4] as follows. Let ε>0 be given and take x0∈X to be arbitrary. We fix an element x1 in Tx0. Now, from property (C2), it is possible to choose x2∈Tx1 such that d(x1,x2)≤H+(Tx0,Tx1)+ε. At this step, we could choose ε depending on x0 and x1. In general, for n∈N, if xn is chosen, then we can select xn+1∈Txn such that (13)dxn,xn+1≤H+Txn-1,Txn+ε.At this step, we could choose ε depending on xn-1 and xn. Note that if H+(Txn-1,Txn)=0 for some n, then Txn-1=Txn and the proof is complete. By (v), we can take A,B>0 such that A-Bk>0 and L≔A-Bk+k<1/max{τ,2R}. In fact, if we set on each step ε=A-BkM(ψ(xn-1,xn))>0, then, by (13), we obtain, for every n∈N, that (14)dxn,xn+1≤H+Txn-1,Txn+A-BkMψxn-1,xn,which implies, for every n∈N, that(15)dxn,xn+1≤kMψxn-1,xn+A-Bk·Mψxn-1,xn=A-Bk+k·Mψxn-1,xn=LMdxn-1,xn,dxn-1,Txn-1,dxn,Txn,dxn-1,Txn,dxn,Txn-1=LMdxn-1,xn,dxn-1,Txn-1,dxn,Txn,dxn-1,Txn,0≤LMdxn-1,xn,dxn-1,xn,dxn,xn+1,dxn-1,xn+1,0≤LMdxn-1,xn,dxn-1,xn,dxn,xn+1,dxn-1,xn+dxn,xn+1,0≤LNdxn-1,xn,dxn,xn+1,where we have used (ii) and (iv-a). Hence, (16)dxn,xn+1≤LNdxn-1,xn,dxn,xn+1,∀n∈N.If xn+1=xn for some n∈N, then xn=xn+1∈Txn, so that xn is a fixed point of T and the proof is concluded. Suppose, therefore, that d(xn,xn+1)>0, for every n∈N.
Now, assuming thatd(xn-1,xn)<d(xn,xn+1), for some n∈N, by (16) and (iv-b), we get (17)dxn,xn+1≤LNdxn,xn+1,dxn,xn+1,∀n∈Nand, by (iv-d), (18)dxn,xn+1≤Lτdxn,xn+1,∀n∈Nwhich is a contradiction since d(xn,xn+1)>0 and Lτ∈(0,1) (due to inequality L<1/max⁡{τ,2R}≤1/τ). Hence, we have proved that d(xn-1,xn)≥d(xn,xn+1), for every n∈N, so that, by (16) and (iv-c), we have (19)dxn,xn+1≤LNdxn-1,xn,dxn,xn+1≤LNdxn-1,xn,dxn-1,xn,∀n∈N.By (iv-d), we obtain (20)dxn,xn+1≤Lτdxn-1,xn,∀n∈N.Repeating the same procedure n-times, we get d(xn,xn+1)≤(Lτ)nd(x0,x1), for all n∈N, where Lτ<1 by hypothesis (v). Hence, {xn} is a Cauchy sequence since, for n,m∈N with n<m,(21)dxn,xm≤∑i=nm-1dxi,xi+1≤∑i=nm-1Lτidx0,x1=Lτn-Lτm1-Lτdx0,x1=Lτn1-Lτ1-Lτm-ndx0,x1.
Therefore, by the complete character ofX, there exists u∈X such that limn→∞⁡xn=u. Suppose that d(u,Tu)>0. Then(22)12ρTxn,Tu+ρTu,Txn=H+Txn,Tu≤kMψxn,u=kMdxn,u,dxn,Txn,du,Tu,dxn,Tu,du,Txn≤kMdxn,u,dxn,xn+1,du,Tu,dxn,Tu,du,xn+1.We note that d(xn,Tu)≤d(xn,u)+d(u,Tu) and d(u,Tu)≤d(u,xn)+d(xn,Tu), so that d(u,Tu)-d(u,xn)≤d(xn,Tu)≤d(xn,u)+d(u,Tu), which implies that d(xn,Tu)→d(u,Tu) as n→∞.
Hence, fromd(xn,u)→0, d(xn,xn+1)→0, and d(xn,Tu)→d(u,Tu), as n→∞, and using hypothesis (iii), we get from (22) (23)12liminfn→∞⁡ρTxn,Tu+ρTu,Txn≤kM0,0,du,Tu,du,Tu,0.Moreover, by (iv-a) and (iv-e), we obtain (24)12liminfn→∞⁡ρTxn,Tu+ρTu,Txn≤kN0,du,Tu≤kRdu,Tu.Using that (25)liminfn→∞⁡ρTxn,Tu+ρTu,Txn≤2kRdu,Tu,we deduce that (26)liminfn→∞⁡ρTxn,Tu≤liminfn→∞⁡ρTxn,Tu+liminfn→∞⁡ρTu,Txn≤liminfn→∞⁡ρTxn,Tu+ρTu,Txn≤2kRdu,Tu.Since(27)du,Tu≤du,xn+1+dxn+1,Tu≤ρTxn,Tu+dxn+1,uand limn→∞⁡d(xn+1,u)=0, it follows that(28)du,Tu≤liminfn→∞⁡ρTxn,Tu+dxn+1,u=liminfn→∞⁡ρTxn,Tu+limn→∞⁡dxn+1,u≤2kRdu,Tu.Hence, if R=0, then d(u,Tu)=0. On the other hand, if R>0, the assumption d(u,Tu)>0 leads to a contradiction by virtue of hypothesis (v) (2kR<2R/max⁡τ,2R≤1), which implies that d(u,Tu)=0. Finally, since Tu is closed, it is proved that u∈Tu; that is, u is a fixed point of T.Remark 9.
Note that ifmax⁡τ,2R>1, condition (v) in Theorem 8 implies that k<1, so that in this case the H+-type multivalued weak φ-Lipschitz mapping T in Theorem 8 is in fact an H+-type multivalued weak φ-contraction. In this sense, both functions φ and M may contribute to the contractivity of the multivalued mapping through condition (v) which establishes a relation among the constants involved.Corollary 10.
If one takesφ(t)≔kt, with k∈(0,1), and M(r1,r2,r3,r4,r5)≔r1, then M is monotonically increasing in all the variables and continuous. Besides, for N(r,s)≔r, one has (29)Mr,r,s,r+s,0=r=Nr,s,foreveryr,s∈R+,Nr,sismonotonicallyincreasinginrfors∈R+fixed,Nr,sismonotonicallyincreasinginsforr∈R+fixed,Nr,r=r,thusiv-dholdsforτ=1N0,s=0,foreverys>0,sothativ-eholdsforR=0,max⁡τ,2R=1,sothatvisfulfilledfor0<k<1.In this setting of H+-type multivalued contraction mappings, we have, as a corollary of Theorem 8, Theorem 3.2 [16] (see Theorem 5).In the context ofH+-type multivalued weak contractive mappings, we obtain the following corollary which corrects Theorem 3.4 [23] [see the proof of Theorem 3.4 [23], where it was assumed that d(u,Tu)=1/2(d(u,Tu)+d(u,Tu))≤1/2(ρ(Txn,Tu)+ρ(Tu,Txn))+d(u,xn+1)].Corollary 11.
Let(X,d) be a complete metric space and T:X→CB(X) an H+-type multivalued weak contractive mapping with 0<k<1/2. Then T has a fixed point.Proof.
If we takeφ(t)≔kt, with k∈(0,1/2), and (30)Mr1,r2,r3,r4,r5≔max⁡r1,r2,r3,r4+r52,M is monotonically increasing in all the variables and continuous. Choosing N(r,s)≔max⁡r,s, we have(31)Mr,r,s,r+s,0=max⁡r,r,s,r+s2=max⁡r,s=Nr,s,for everyr,s∈R+,N(r,s) is monotonically increasing in each variable, N(r,r)=max⁡r,r=r, and thus (iv-d) holds for τ=1, N(0,s)=max⁡0,s=s, for every s>0, so that (iv-e) holds for R=1 and max⁡τ,2R=2, so that (v) is fulfilled for the choice 0<k<1/2. Hence Theorem 8 applies.In the context ofH+-type multivalued quasi-contraction mappings, we obtain the following corollary which coincides with Theorem 3.6 [23].Corollary 12.
Let(X,d) be a complete metric space and T:X→CB(X) an H+-type multivalued quasi-contraction mapping with 0<k<1/2. Then T has a fixed point.Proof.
Takingφ(t)≔kt, with k∈(0,1/2) and(32)Mr1,r2,r3,r4,r5≔max⁡r1,r2,r3,r4,r5,we have that M is monotonically increasing in all the variables and also continuous. Taking N(r,s)≔r+s, we get (33)Mr,r,s,r+s,0=max⁡r,r,s,r+s,0=r+s=Nr,s,for everyr,s∈R+,N(r,s) is monotonically increasing in each variable for the other fixed, N(r,r)=r+r=2r, and thus (iv-d) holds for τ=2, N(0,s)=s, for every s>0, so that (iv-e) holds for R=1 and max⁡τ,2R=2, so that (v) is fulfilled for 0<k<1/2.On the other hand, if a result similar to Theorem8 was established for H-multivalued contractions, we would have the following theorem.Theorem 13.
Let(X,d) be a complete metric space. Let T:X→CB(X) be an H-type multivalued weak φ-Lipschitz mapping, that is, such that there exist M:(R+)5→R+ and φ:R+→R+ with (34)HTx,Ty≤φMdx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx,foreveryx,y∈X.Suppose that the functions φ, M satisfy conditions (i)–(iv) in Theorem 8 and(v∗)
the constantsk in (i) and τ, R in (iv) are such that k<1/max⁡τ,R.Then T has a fixed point.Proof.
Identical to the proof of Theorem8, except the last part, we have (35)liminfn→∞⁡ρTxn,Tu≤liminfn→∞⁡HTxn,Tu≤kN0,du,Tu≤kRdu,Tu.Since(36)du,Tu≤du,xn+1+dxn+1,Tu≤ρTxn,Tu+dxn+1,uand limn→∞⁡d(xn+1,u)=0, then(37)du,Tu≤liminfn→∞⁡ρTxn,Tu+dxn+1,u=liminfn→∞⁡ρTxn,Tu≤kRdu,Tu.If R=0, then d(u,Tu)=0. If R>0, d(u,Tu)>0 leads to a contradiction in the previous inequality due to (v∗) (kR<1), which implies that d(u,Tu)=0 and the proof is complete.Remark 14.
As a consequence of Theorem13, using the Hausdorff metric H, the restriction required on k is (r.i)
for multivalued contractions0<k<1,(r.ii)
for multivalued weak contractive mappings0<k<1,(r.iii)
for multivalued quasi-contraction mappings0<k<1/2.Remark 15.
In Theorems8 and 13, conditions (iv-b), (iv-c), and (iv-d) can be removed, adding the property that(iv-d∗)
there existsθ>0 such that N(r,s)≤θ(r+s), for every r,s>0,while condition (v) (resp., (v∗)) has to be replaced by the fact that (v∗∗)
the constantsk in (i) and R, θ in (iv) are such that k<1/max⁡2θ,2R (for H+) or k<1/max⁡2θ,R (for H).
This comes from the following ideas: from (v∗∗), following the proof of Theorem 8, we can take A,B>0 such that A-Bk>0 and L≔A-Bk+k<1/max⁡2θ,2R (for the Hausdorff metric H, we choose them in such a way that L≔A-Bk+k<1/max⁡2θ,R). Supposing that d(xn,xn+1)>0, for every n∈N, if we do not have monotonicity of N, from (16) and (iv-d∗), we get (38)dxn,xn+1≤Lθdxn-1,xn+dxn,xn+1,∀n∈Nso that, using Lθ<1, (39)dxn,xn+1≤Lθ1-Lθdxn-1,xn,∀n∈N.Repeating the same procedure n-times, we get d(xn,xn+1)≤Knd(x0,x1), for all n∈N, where K=Lθ/1-Lθ<1 by hypothesis (v∗∗). The rest of the proof is valid.Now, following the lines in [21] for q-set-valued quasi-contractions, we give the following definition.Definition 16.
Let(X,d) be a metric space and let α:X×X→[0,∞) be given. A multivalued map T:X→CB(X) is called a partial H+-type multivalued weak φ-Lipschitz mapping if condition (C2) in Definition 2 holds and there exist M:(R+)5→R+ and φ:R+→R+ such that(40)H+Tx,Ty≤φMdx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx,for everyx,y∈X×Xwithαx,y≥1.If, moreover, there exists k∈(0,1) such that φ(t)≤kt, for every t≥0, we say that T is a partial H+-type multivalued weak φ-contraction.Remark 17.
Takingφ(t)=kt, with k∈(0,1), and (41)Mr1,r2,r3,r4,r5=max⁡r1,r2,r3,r4,r5,then we obtain the notion of partial H+-type q-set-valued quasi-contraction (compare with [21, Definition 3.1]).In relation with Definitions 2.21, 3.1 and Theorem 3.2 [21], we can establish the following result.Theorem 18.
Let(X,d) be a complete metric space, let α:X×X→[0,∞) be given, and let T:X→CB(X) be a partial H+-type multivalued weak φ-Lipschitz mapping such that (h1)
T is α-admissible; that is, for each x∈X and y∈Tx with α(x,y)≥1, one has α(y,z)≥1, for every z∈Ty,(h2)
there existx0∈X and x1∈Tx0 such that α(x0,x1)≥1,(h3)
if{xn} is a sequence in X such that α(xn,xn+1)≥1, for all n∈N, and there exists x∈X such that xn→x as n→∞, then α(xn,x)≥1, for every n,(h4)
the functionsφ, M are such that conditions (i), (ii), (iii), (iv) [(a)–(e)], and (v) in Theorem 8 hold.
ThenT has a fixed point in X.Proof.
We proceed similarly to the proof of Theorem8 and also Theorem 3.2 [21]. By (40) and (i), we get (42)H+Tx,Ty≤kMψx,y,for everyx,y∈X×Xwithαx,y≥1.We take ε>0 given. We start the sequence with the terms x0 and x1∈Tx0 and then, using (C2), there exists x2∈Tx1 such that d(x1,x2)≤H+(Tx0,Tx1)+ε. Note that since T is α-admissible, we have α(x1,x2)≥1. In general, for n∈N, if xn∈Txn-1 is chosen such that α(xn-1,xn)≥1, then we can select xn+1∈Txn such that (43)dxn,xn+1≤H+Txn-1,Txn+εand we also have α(xn,xn+1)≥1. Again, ε could have been chosen at each step depending on xn-1 and xn. For this, we remark that if H+(Txn-1,Txn)=0 for some n, the proof is concluded, so that we can assume that M(ψ(xn-1,xn))>0, for every n and take on each step ε=A-BkM(ψ(xn-1,xn))>0, where A,B>0 are fixed, by (v), in such a way that A-Bk>0 and L≔A-Bk+k<1/max⁡τ,2R. Then, by (43), we get, for every n∈N, that (44)dxn,xn+1≤H+Txn-1,Txn+A-BkMψxn-1,xn≤LMψxn-1,xnwhich implies, for every n∈N, that(45)dxn,xn+1≤LMψxn-1,xn≤LMdxn-1,xn,dxn-1,xn,dxn,xn+1,dxn-1,xn+dxn,xn+1,0≤LNdxn-1,xn,dxn,xn+1,using (ii) and (iv-a). As in the proof of Theorem 8, we can suppose that d(xn,xn+1)>0, for every n∈N. Similarly to the proof of Theorem 8, we deduce that d(xn-1,xn)≥d(xn,xn+1), for every n∈N and (46)dxn,xn+1≤Lτdxn-1,xn,∀n∈N,so that {xn} is a Cauchy sequence and, by the completeness of X, there exists u∈X such that limn→∞⁡xn=u.
Finally, ifd(u,Tu)>0, inequality (22) holds; since α(xn,xn+1)≥1, for all n∈N, and xn→u as n→∞, then, by hypotheses, α(xn,u)≥1, for every n, so that (47)H+Txn,Tu≤kMψxn,u,for everyn.Similarly to the proof of Theorem 8, we get that (48)du,Tu≤du,xn+1+dxn+1,Tu≤ρTxn,Tu+dxn+1,u;hence(49)du,Tu≤2kRdu,Tu,so that if R=0, then d(u,Tu)=0 and if R>0, the assumption d(u,Tu)>0 leads to a contradiction again and the proof is complete.The previous result allows formulating a fixed point result forH+-type partial quasi-contraction mappings while, in [21], the Pompeiu-Hausdorff metric is used in b-metric spaces. The procedure in metric spaces could be adapted to b-metric spaces in the lines of [21].Remark 19.
In Theorem18, if we consider the Pompeiu-Hausdorff metric H, then condition (v) can be relaxed to hypothesis (v∗) in the statement of Theorem 13.
In particular, for partialq-set-valued quasi-contractions, φt≔kt, k∈(0,1/2), and M(r1,r2,r3,r4,r5)≔max⁡r1,r2,r3,r4,r5, we obtain, as a corollary, the assertion of Theorem 3.2 [21] for metric spaces.Remark 20.
Inequality (40) is trivially valid if one of the following conditions hold (see Corollaries 3.4–3.6 [21]): (r.i)
αx,yH+Tx,Ty≤φMψx,y, for every x,y∈X;(r.ii)
(H+(Tx,Ty)+ε)αx,y≤φ(M(ψ(x,y)))+ε, for every x,y∈X, where ε≥1;(r.iii)
(α(x,y)-1+ε)H+(Tx,Ty)≤εφ(M(ψ(x,y))), for every x,y∈X, where ε>1.Theorem 21.
Let(X,d) be a complete metric space, let α:X×X→[0,∞) be given, and let T:X→CB(X) be a partial H+-type multivalued weak φ-Lipschitz mapping such that conditions (h1), (h2), and (h3) hold. Suppose also that the functions φ, M satisfy the following:(A1)
φ is monotonically increasing and φ(0)=0.(A2)
Ifzn→z as n→∞, then liminfn→∞⁡φ(zn)≤φ(z).(A3)
Conditions (ii), (iii), and (iv) [(a)–(e)] in Theorem8 hold.(A4)
ForR given in (iv-e), one has that 2φ(Rz)<z, for every z>0.(A5)
There existsS>0 such that the operator LS≔φ+SI (where I is the identity mapping on X) satisfies the following properties:
(A5.i)
LS(τz)<z, for every z>0;(A5.ii)
LS(τz)≤τLS(z), for every z>0;(A5.iii)
∑n=1∞‍τnLSn(z)<∞, for every z>0, where LSn≔LS∘⋯∘nLS.
ThenT has a fixed point in X.Proof.
We proceed similarly to the proof of Theorem18. We take x0 and x1∈Tx0 given by (h2), with α(x0,x1)≥1; then a sequence is obtained by using (C2) and α-admissibility. The process is summarized as follows: for n∈N, if xn∈Txn-1 is chosen such that α(xn-1,xn)≥1, then, by (C2), we can select xn+1∈Txn such that(50)dxn,xn+1≤H+Txn-1,Txn+ε,and we also have α(xn,xn+1)≥1.
We remark that, due to the monotonically increasing character ofφ, the mapping LS is always strictly increasing. Note that if H+(Txn-1,Txn)=0 for some n, the proof is concluded, so that we can assume that M(ψ(xn-1,xn))>0, for every n, so that we can take, on each step, εn=SM(ψ(xn-1,xn))>0. Then, for every n∈N, by (ii) and (iv-a), (51)dxn,xn+1≤φMψxn-1,xn+SMψxn-1,xn=LSMψxn-1,xn≤LSNdxn-1,xn,dxn,xn+1.If xn=xn+1, for some n, then xn is a fixed point of T, so we assume that xn≠xn+1, for every n∈N.
Assuming thatd(xn-1,xn)<d(xn,xn+1), for some n∈N, by (iv-b) and (iv-d), we obtain (52)dxn,xn+1≤LSNdxn,xn+1,dxn,xn+1≤LSτdxn,xn+1,∀n∈N,which is a contradiction due to (A5) since d(xn,xn+1)>0. This proves that d(xn-1,xn)≥d(xn,xn+1), for every n∈N. Therefore, by (iv-c), we get(53)dxn,xn+1≤LSNdxn-1,xn,dxn,xn+1≤LSNdxn-1,xn,dxn-1,xn≤LSτdxn-1,xn≤τLSdxn-1,xn,∀n∈N.This implies, from (A5) and the property that LS(z)>0 for every z>0, that(54)dxn,xn+1≤τLSdxn-1,xn≤τLSτLSdxn-2,xn-1≤τ2LS2dxn-2,xn-1≤τnLSndx0,x1,∀n∈N.
To check that{xn} is a Cauchy sequence, we observe that, for n,m∈N with n<m, we get (55)dxn,xm≤∑i=nm-1dxi,xi+1≤∑i=nm-1τiLSidx0,x1⟶0,asn⟶∞.Since X is complete, there exists u∈X such that limn→∞xn=u. If d(u,Tu)>0, using that α(xn,xn+1)≥1, for all n∈N, and xn→u as n→∞, then, by (h3), we have α(xn,u)≥1, for every n; hence (56)H+Txn,Tu≤φMψxn,u,for everynso that(57)liminfn→∞⁡ρTxn,Tu+ρTu,Txn≤2liminfn→∞⁡φMψxn,u.Therefore, similarly to inequality (22), we have, by the monotonicity of φ,(58)φMψxn,u=φMdxn,u,dxn,Txn,du,Tu,dxn,Tu,du,Txn≤φMdxn,u,dxn,xn+1,du,Tu,dxn,Tu,du,xn+1.Analogously to the proof of Theorem 8 and using (iii),(59)limn→∞⁡Mdxn,u,dxn,xn+1,du,Tu,dxn,Tu,du,xn+1=M0,0,du,Tu,du,Tu,0.By (A2), we have(60)liminfn→∞⁡φMψxn,u≤liminfn→∞⁡φMdxn,u,dxn,xn+1,du,Tu,dxn,Tu,du,xn+1≤φM0,0,du,Tu,du,Tu,0.Hence, from this inequality and (57), by (iv-a) and (iv-e), we get(61)liminfn→∞⁡ρTxn,Tu+ρTu,Txn≤2liminfn→∞⁡φMψxn,u≤2φM0,0,du,Tu,du,Tu,0≤2φN0,du,Tu≤2φRdu,Tu.Similarly to the proof of Theorem 8, we have d(u,Tu)≤2φRd(u,Tu). Therefore, if R=0, then d(u,Tu)≤φ(0)=0 and the proof is concluded. If R>0, we get to a contradiction due to d(u,Tu)>0 and (A4). Hence, d(u,Tu)=0 and the proof is complete, since Tu is closed, so that u∈Tu.Remark 22.
Theorem8 is a particular case of Theorem 21. Indeed, if we take φ(t)=kt, where k>0, and assuming that condition (v) holds, then we have the following: (r.i)
φ is monotonically increasing and φ(0)=0; hence (A1) holds.(r.ii)
φ is continuous (so that (A2) holds).(r.iii)
Condition (A4) is valid since2φ(Rz)<z, for every z>0 (this is true since it is equivalent to k<1/2R).(r.iv)
Concerning (A5), since we assume thatkτ<1, then we can take S>0 such that (k+S)τ<1 and the operator LS≔φ+SI is monotonically increasing and
(r.a)
LS(τz)=k+Sτz<z, for every z>0;(r.b)
LS(τz)=k+Sτz=τk+Sz=τLS(z), for every z>0;(r.c)
∑n=1∞‍τnLSn(z)=z∑n=1∞‍(kτ)n<∞, for every z>0.It is also obvious that ifT is an H+-type multivalued weak φ-Lipschitz mapping and there exists k>0 such that φ(t)≤kt, for t≥0, then T is also an H+-type multivalued weak φ~-Lipschitz mapping for φ~(t)=kt.Remark 23.
In the proof of Theorems18 and 21, it is easy to observe that a proper combination of condition (C2) with the α-admissible character of the mapping, (h1), would allow relaxing slightly the definition of partial H+-type multivalued weak φ-Lipschitz mappings. Indeed, in these theorems, it is possible to replace these hypotheses ((C2) and α-admissibility of T) by the following: (2∗)
for everyx∈X, y∈Tx with α(x,y)≥1 and ε>0, there exists z∈Ty such that d(y,z)≤H+(Ty,Tx)+ε and α(y,z)≥1.If we consider self-mappingsT:X→X, a procedure similar to Theorem 2.3 [19] gives the following result.Theorem 24.
Let(X,d) be a complete metric space, let α:X×X→[0,∞) be given, and let T:X→X be such that (t.i)
for eachx∈X with α(x,Tx)≥1, one has α(Tx,T(Tx))≥1,(t.ii)
there existsx0∈X such that α(x0,Tx0)≥1,
and there existM:(R+)5→R+ and φ:R+→R+ with (62)dTx,Ty≤φMdx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx,foreveryx,y∈X×Xwithαx,y≥1.Suppose also that (h3) holds and that the functions φ, M satisfy the following conditions: (I)
φ is monotonically increasing, M is increasing in the fourth variable, and conditions (iv) [(a)–(d)] in Theorem 8 hold, where τ≤1.(II)
Consider∑n=1∞‍φn(t)<+∞, for every t>0.(III)
One of the following conditions holds:
(III.i)
Forr3>0 fixed, if r1,r2,r5 are small enough and r4 is close enough to r3, then M(r1,r2,r3,r4,r5)≤r3.(III.ii)
(A2) is satisfied and conditions (iii) and (iv-e) in Theorem8 hold, where R≤1.
ThenT has a fixed point in X.Proof.
We takex0∈X with α(x0,Tx0)≥1 and define the sequence {xn} such that xn+1=Txn,  n∈N. If xn=xn+1, for some n, then we have a fixed point of T; thus we assume that xn≠xn+1, for every n (i.e., d(xn,xn+1)>0, for every n∈N). Using the hypotheses, it is easy to prove that α(xn,xn+1)≥1, for every n. Then, for each n∈N, by the nondecreasing character of φ and M (in the fourth variable) and (iv-a),(63)dxn,xn+1=dTxn-1,Txn≤φMψxn-1,xn=φMdxn-1,xn,dxn-1,Txn-1,dxn,Txn,dxn-1,Txn,dxn,Txn-1≤φMdxn-1,xn,dxn-1,xn,dxn,xn+1,dxn-1,xn+dxn,xn+1,0≤φNdxn-1,xn,dxn,xn+1.If d(xn-1,xn)<d(xn,xn+1), for some n∈N, by the monotonicity of φ, (iv-b), (iv-d), τ≤1, and (II), we get(64)dxn,xn+1≤φNdxn,xn+1,dxn,xn+1≤φτdxn,xn+1≤φdxn,xn+1<dxn,xn+1,which is a contradiction. Then, d(xn-1,xn)≥d(xn,xn+1), for every n∈N; hence, by the monotonicity of φ, (iv-c), (iv-d), and τ≤1,(65)dxn,xn+1≤φNdxn-1,xn,dxn-1,xn≤φτdxn-1,xn≤φdxn-1,xn.Using (II), the sequence {xn} is a Cauchy sequence. Then there exists u such that xn→u as n→∞. By (h3), α(xn,u)≥1, for every n. Suppose that d(u,Tu)>0; then(66)du,Tu≤du,Txn+dTxn,Tu≤du,xn+1+φMψxn,u=dxn+1,u+φMdxn,u,dxn,xn+1,du,Tu,dxn,Tu,du,xn+1.In case (III.i), for n being large enough, we have (67)du,Tu≤dxn+1,u+φdu,Tu,so that d(u,Tu)≤φd(u,Tu)<d(u,Tu), which is a contradiction; hence Tu=u and the proof is complete. On the other hand, in case (III.ii), by the monotonicity of φ and (II), we have(68)du,Tu≤liminfn→∞⁡φMdxn,u,dxn,xn+1,du,Tu,dxn,Tu,du,xn+1≤φlimn→∞⁡Mdxn,u,dxn,xn+1,du,Tu,dxn,Tu,du,xn+1≤φM0,0,du,Tu,du,Tu,0≤φN0,du,Tu≤φRdu,Tu≤φdu,Tu<du,Tu,a contradiction again and the proof is finished.Remark 25.
IfM(r1,r2,r3,r4,r5)≔max⁡r1,r2,r3,r4+r5/2 in Theorem 24, then we have Theorem 2.3 [19]. Indeed, all the conditions are satisfied (see the proof of Corollary 11) and, for r3>0 fixed, if r1,r2,r5 are small enough and r4 is close enough to r3, then M(r1,r2,r3,r4,r5)=r3; hence (III.i) is also fullfilled.Remark 26.
Certain conditions in Theorem24 also extend some hypotheses in [20].As indicated in [21], a function α can be defined in connection with a binary relation R in X (which could be, e.g., a partial ordering in X). Thus, Theorem 4.7 [21] can be extended to H+-type multivalued weak φ-Lipschitz mappings with respect to R, which are defined as follows. Here, we only consider the case of metric spaces, but b-metric spaces could be considered accordingly.Definition 27.
Let(X,d) be a metric space and R a binary relation on X. A multivalued map T:X→CB(X) is called an H+-type multivalued weak φ-Lipschitz mapping with respect to R if condition (C2) in Definition 2 holds and there exist M:(R+)5→R+ and φ:R+→R+ such that(69)H+Tx,Ty≤φMdx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx,for everyx,y∈XwithxRy.If, moreover, there exists k∈(0,1) such that φ(t)≤kt, for every t≥0, one says that T is an H+-type multivalued weak φ-contraction with respect to R.Theorem 28.
Let(X,d) be a complete metric space, R a binary relation on X, and T:X→CB(X) an H+-type multivalued weak φ-Lipschitz mapping with respect to R such that (H1)
T is weakly preserving, that is, for each x∈X and y∈Tx with xRy, one has yRz, for every z∈Ty,(H2)
there existx0∈X and x1∈Tx0 such that x0Rx1,(H3)
if{xn} is a sequence in X such that xnRxn+1, for all n∈N, and there exists x∈X such that xn→x as n→∞, then xnRx, for every n,(H4)
one of the following conditions holds:
(H4-i)
the functionsφ, M satisfy conditions (i), (ii), (iii), (iv) [(a)–(e)], and (v) in Theorem 8 or(H4-ii)
the functionsφ, M satisfy (A1)–(A5).
ThenT has a fixed point in X.Remark 29.
The condition of weakly preservance ofT, (H1), is fulfilled if the following condition holds: (70)ifxRy,thenTx≤RTy,where, given A,B⊆X, we define(71)A≤RBif,  for eacha∈A,b∈B,we haveaRb.Remark 30.
In the proof of Theorem28, we can relax slightly the definition of H+-type multivalued weak φ-Lipschitz mappings with respect to R, by replacing hypotheses (C2) and weakly preservance of T, (H1), by the following combination of both: (2∗∗)
for everyx∈X, y∈Tx with xRy and ε>0, there exists z∈Ty such that d(y,z)≤H+(Ty,Tx)+ε and yRz.For self-mappingsT:X→X, we have the following corollary of Theorem 24.Corollary 31.
Let(X,d) be a complete metric space, R a binary relation on X, and T:X→X such that (c.i)
for eachx,y∈X with xRy, one has TxRTy,(c.ii)
there existsx0∈X such that x0RTx0,
and there existM:(R+)5→R+ and φ:R+→R+ such that (72)dTx,Ty≤φMdx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx,foreveryx,y∈XwithxRy.Suppose also that (H3) holds and that the functions φ, M satisfy conditions (I), (II), and (III) in Theorem 24.
ThenT has a fixed point in X.In the previous result, if the metric space admits a partial ordering⪯, then the relation R can be chosen as ⪯, so that the conditions in Corollary 31 reduce to those in Corollary 2.4 [19].Also in the lines of [19], we consider the family Φ of functions defined as follows: φ∈Φ if and only if φ:0,∞→0,∞ is nondecreasing and such that(73)∑n=1∞φnz<∞,∀z>0.It is clear that if φ∈Φ, then φ(t)<t, for every t>0 (see [19] and the references therein).We have the following result, where condition (C2) is removed.Theorem 32.
Let(X,d) be a complete metric space, let α:X×X→[0,∞) be given, and let T:X→CB(X) be a multifunction such that conditions (h1), (h2), and (h3) hold. Suppose, further, that there exist M:(R+)5→R+ and φ:R+→R+ such that(74)H+Tx,Ty≤φMdx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx,foreveryx,y∈X×Xwithαx,y≥1,where the functions φ, M satisfy that(a1)
φ∈Φ is strictly increasing and φ(0)=0,(a2)
there existsβ>0 such that M(ψ(x,y))≤βd(x,y), for every x≠y, where β≤1/2,(a3)
iflimn→∞xn=x, then liminfn→∞M(ψ(xn,x))=0.
ThenT has a fixed point in X.Proof.
We proceed similarly to the proof of Theorem 2.1 [19]. We include it here for completeness. We take x0∈X and x1 in Tx0 with α(x0,x1)≥1 given by hypothesis. The proof is finished if x0=x1 or if x1∈Tx1. Then, we assume that x0≠x1∉Tx1 and take ε0>1 arbitrarily fixed. Then, by the definition of H+ and inequality (74),(75)0<12dx1,Tx1≤12ρTx0,Tx1+ρTx1,Tx0=H+Tx0,Tx1≤φMψx0,x1<ε0φMψx0,x1,so that we can choose x2∈Tx1 such that 0<1/2d(x1,x2)<ε0φ(M(ψ(x0,x1))) (x1≠x2 due to x1∉Tx1). By α-admissibility, we have that α(x1,x2)≥1. Moreover, since φ(0)=0, then z0≔M(ψ(x0,x1))>0 and 0<1/2d(x1,x2)<ε0φ(z0). Since φ is strictly increasing, we have φ(1/2d(x1,x2))<φ(ε0φ(z0)). Next, we take ε1=φ(ε0φ(z0))/φ(1/2d(x1,x2))>1. Now, if x2∈Tx2, the proof is finished, so we assume that x2∉Tx2 and, similarly, we get (76)0<12dx2,Tx2≤H+Tx1,Tx2≤φMψx1,x2<ε1φMψx1,x2;then there exists x3∈Tx2 (x3≠x2) such that (77)0<12dx2,x3<ε1φMψx1,x2≤ε1φβdx1,x2≤ε1φ12dx1,x2=φε0φz0.Besides, α(x2,x3)≥1.
Using thatφ is strictly increasing, we have φ1/2d(x2,x3)<φ2(ε0φ(z0)), and we can take ε2=φ2(ε0φ(z0))/φ(1/2d(x2,x3))>1. Again, if x3∈Tx3, the proof is finished, so we assume that x3∉Tx3 and, similarly, we get(78)0<12dx3,Tx3≤H+Tx2,Tx3≤φMψx2,x3<ε2φMψx2,x3,and then there exists x4∈Tx3 (x4≠x3) such that (79)0<12dx3,x4<ε2φMψx2,x3≤ε2φβdx2,x3≤ε2φ12dx2,x3=φ2ε0φz0.Besides, α(x3,x4)≥1.
In general, forn∈N, if xn∈Txn-1 (xn≠xn-1) is chosen such that α(xn-1,xn)≥1 and φ1/2d(xn-1,xn)<φn-1(ε0φ(z0)), we can take εn-1=φn-1(ε0φ(z0))/φ(1/2d(xn-1,xn))>1. If xn∈Txn, the proof is finished, so we assume that xn∉Txn and, similarly, we get (80)0<12dxn,Txn≤H+Txn-1,Txn≤φMψxn-1,xn<εn-1φMψxn-1,xn,and then there exists xn+1∈Txn (xn+1≠xn) such that (81)0<12dxn,xn+1<εn-1φMψxn-1,xn≤εn-1φβdxn-1,xn≤εn-1φ12dxn-1,xn=φn-1ε0φz0and α(xn,xn+1)≥1.
Hence, we can take a sequence{xn}⊂X with xn+1∈Txn, xn+1≠xn, α(xn,xn+1)≥1, and 0<1/2d(xn,xn+1)<φn-1(ε0φ(z0)) for every n∈N.
This allows proving that{xn} is a Cauchy sequence since, for n,m∈N with n<m, we get (82)dxn,xm≤∑i=nm-1dxi,xi+1≤2∑i=nm-1φi-1ε0φz0,which tends to 0 as n→∞ since φ∈Φ.
Now, by completeness ofX, there exists u∈X such that limn→∞xn=u. By hypothesis, α(xn,u)≥1, for every n. Suppose that d(u,Tu)>0. Then (83)du,Tu≤du,xn+1+dxn+1,Tu≤ρTxn,Tu+du,xn+1≤2H+Txn,Tu+du,xn+1≤2φMψxn,u+du,xn+1,for every n. Since φ(z)<z, for all z, then, for every n∈N, (84)0≤φMψxn,u≤Mψxn,uso that the hypothesis liminfn→∞M(ψ(xn,u))=0 implies that liminfn→∞φ(M(ψ(xn,u)))=0. This, joint to d(u,Tu)-d(u,xn+1)≤2φ(M(ψ(xn,u))) and the convergence of {xn}, implies that u∈Tu.Remark 33.
Note that condition (74) coincides with (40). In Theorem 32, if we consider the Hausdorff distance H in (74) and β≤1, the conclusion holds.Remark 34.
If we takeMr1,r2,r3,r4,r5≔r1, then Mψx,y=dx,y and if limn→∞xn=x, then liminfn→∞Mψxn,x=liminfn→∞dxn,x=0. Hence, for the Hausdorff distance H, the existence of fixed point follows, such as in Theorem 2.1 [19], since β=1. On the other hand, for the metric H+, if we consider Mr1,r2,r3,r4,r5≔1/2r1, then the condition is valid for β=1/2 and Theorem 32 applies.Theorem 35.
Let(X,d) be a complete metric space, R a binary relation on X, and T:X→CB(X) such that conditions (H1), (H2), and (H3) hold. Suppose that there exist M:(R+)5→R+ and φ:R+→R+ such that(85)H+Tx,Ty≤φMdx,y,dx,Tx,dy,Ty,dx,Ty,dy,Tx,foreveryx,y∈XwithxRy,where the functions φ, M satisfy conditions (a1), (a2), and (a3) in Theorem 32. Then T has a fixed point in X.Remark 36.
Conditions (69) and (85) are the same.



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