# Fixed Points of Proinov Type Multivalued Mappings on Quasimetric Spaces

Authors: Erdal Karapinar; Andreea Fulga; Seher Sultan Yeşilkaya
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we obtain new results which have not been encountered before in the literature, in multivalued quasimetric spaces, inspired by Proinov type contractions. We use admissible function as proving theorems. We also give an example that supports our theorems.


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## Body

## 1. Introduction and Preliminaries

Fixed point theory has become an important research topic after the famous mathematician Banach’s definition of the metric fixed point [1]. Many theoretical and applied studies have been done on fixed point theory. In the 21st century, the fixed point is still a popular and dynamic research topic. The concept of metric space, which forms the basis of the fixed point theory, is generalized by many researchers and new spaces (b-metric, quasimetric, partial metric, fuzzy metric, etc.) are introduced. One of the important generalizations is quasimetric space proved in 1931 as follows.Definition 1 (see [2–4]).
LetX≠∅. A function q:X×X⟶ℝ0+is a quasimetric on X if it satisfies the following:
(1)qt,u=qu,t=0⇔t=u,qt,w≤qt,u+qu,w,
for allt,u,w∈X in this case, the pair X,q is a quasimetric space.Letq be a quasimetric on X, and the set Bqt,e=w∈X:qt,w<e. Thus, the family Bqt,e:t∈X,e>0 forms a base for a T0 topology τq on X. Moreover, if A is a subset of X, we denote by clqA the closure of A with respect to T0 topology; we say that the subset A is τq-closed if it is closed with respect to τq.A sequencetr in a quasimetric space converges to t∈X, (in τq) if and only if qt,tr⟶0 as r⟶∞. Moreover, we say that the sequence tr is
(1)
left-Cauchy if for everye>0 there exists re∈ℕ such that qtr,tm<e, whenever re≤r≤m(2)
right-Cauchy if for everye>0 there exists re∈ℕ such that qtm,tr<e, whenever re≤r≤mThereupon, a quasimetric space is called to be left (resp., right) complete if every left (resp., right) Cauchy sequence converges (to respectτq) (see, e.g., [5, 6, 40, 41]).Nadler [7] is the first who introduced the framework for multivalued contraction mappings. The author proved the important theorem generalized Banach principle using the Hausdorff metric for multivalued mappings. After the proof of Nadler theorem, the theory of multivalued contraction mappings attracted great attention and is used in various branches of mathematics. Multivalent mappings in different spaces are introduced. One of them is multivalued mapping introduced in quasimetric-spaces by Shoaib [8] (see also [9, 10]).LetX,q be a quasimetric space. We shall denote by PX the set of all nonempty subsets of X, by ClqX the set of all nonempty closed bounded subsets of X, and let KqX be the set of all compact subsets of X.Definition 2.
LetX≠∅ and Z:X⟶PX be a multivalued map. A point t∈X is said to be a fixed point of Z if t∈Zt.
The set of the fixed point of a mappingZ is denoted by FZ.Lemma3 is an important condition in the following main results.Lemma 3 (see [8]).
LetA and B be nonempty closed bounded subsets of a quasimetric space X,q and let δ>1. Then, for all t∈A, there exists u∈B such that qt,u≤δHqA,B.Nadler [7] stated that if A,B∈KX in the metric spaces it is also provided for δ≥1. With similarly thinking, the following lemma can be written.Lemma 4.
LetA and B be nonempty compact subsets of a quasimetric space X,q, and let δ≥1. Then, for all t∈A, there exists u∈B such that qt,u≤δHqA,B.Many researchers have stated different studies on well-known quasimetric spaces, see e.g., [11–13]. In recent years, Alqahtani et al. [14] introduced a new generalization in quasimetric spaces and defined Δ-symmetric quasimetric spaces. This definition is as follows.Definition 5 (see [14]).
Assume thatX,q is a quasimetric space. If there exists a positive real number Δ>0 such that
(2)qt,u≤Δ·qu,t,for all t,u∈X, then, the pair X,q is called a Δ-symmetric quasimetric space.To simplify the notations, in the following, we will mark byX,qΔ a Δ-symmetric quasimetric space.It is clear that ifΔ=1, thus X,q1 becomes a metric space.Definition 6 (see [8]).
LetX,qΔ and A,B∈PX. A function Hq:PX×PX⟶0,∞, defined by
(3)HqA,B=maxsupt∈Aqt,B,supu∈BqA,u,where qt,A=infu∈Aqt,u and qA,t=infu∈Aqu,t), satisfies all the axioms of quasimetric and is known as the Hausdorff quasimetric induced by the quasimetric q.Example 7.
Letℝ,d be a metric space and a function q:ℝ×ℝ⟶ℝ+, where
(4)qt,u=3dt,u,ift≥u,dt,u,otherwise..Then,X,q is a 3-symmetric quasimetric space, but it is not a metric space.In the following, we shall collect some main properties of aΔ-symmetric quasimetric space.Lemma 8 (see [15]).
LetX,qΔ, tr be a sequence in X and t∈X. Then,
(i)
tr is right-Cauchy ⇔tr is left-Cauchy ⇔tr is Cauchy(ii)
ifur is a sequence in X and qtr,ur⟶0 then qur,tr⟶0Recall the notion ofα-admissibility introduced in [16, 17].Definition 9.
A mapZ:X⟶X is defined α-admissible if for every t,u∈X, we have
(5)αt,u≥1⇒αZt,Zu≥1,where α:X×X⟶0,∞ is an offered function.Some authors [18–21] introduced by slightly modifying this definition.Definition 10.
LetX,qΔ and w:X×X⟶0,∞. A multivalued mapping Z:X⟶ClqX is called to be strictly ∗-triangular-admissible on X if the following conditions are satisfied:
(wt) for each t,u,v∈X, wt,u>1 and wu,v>1 implies wt,v>1
(wa) for each t,u∈X, wt,u>1 implies w∗Zt,Zu>1
wherew∗Zt,Zu=infwx,y:x∈Zt,y∈Zu.Definition 11.
LetX,q be a Δ-symmetric quasimetric space, and let w:X×X⟶0,∞. The space X,q is said to be strictly w∗-regular if for any sequence tr⊂X such that wtr,tr+1>1 for all r∈ℕ and tr⟶t as r⟶∞, we have wtr,t>1 for all r∈ℕ.In recent years, researchers working on the fixed point theory seem to focus on introducing new contractions in known spaces. These new contractions are also accepted by many researchers and there are important studies, for example,F-contraction ([22–26]), θ-contraction [27], and interpolation contraction [28].In 2020, Proinov [29] introduced new and interesting contractions in metric spaces. Proinov proved that several fixed point results (Wardowski [22]; Jleli and Samet [27]) observed in recent years are the result of Skof’s fixed point theorem [30], and he introduced a very general fixed point theorem containing the main result of Skof.Theorem 12 (see [29]).
LetX,d be a complete metric space and Z:X⟶X a map which satisfies the contractive type condition:
(6)ψdZt,Zu≤φdt,uforallt,u∈XwithdZt,Zu>0,where ψ,φ:ℝ+⟶ℝ are two functions such that
(i)
φm<ψm for all m>0(ii)
ψ is nondecreasing(iii)
limsupm⟶ε+φm<ψε+ for each m>0
Hence,Z has a unique fixed point w∈X and Zrt0⟶w for all t0∈X, as r⟶∞.There are several studies using Proinov’s contractions; some interesting ones are as follows: Alqahtani et al. [31] proposed the Proinov type mappings by involving certain rational expression in dislocated b-metrics. Alqahtani et al. [32] introduced the common fixed point of Proinov type contraction via simulation function. Roldán López de Hierro et al. [33] examined multiparametric contractions in b-metric spaces, inspired by Proinov results. Alghamdi et al. [34], on the other hand, introduced a new type of contraction using admissible mappings, inspired by Proinov and E-contraction.Besides these, Karapnar et al. [35] combined contractions of Proinov [29] and Górnicki [36] in complete metric spaces and proved new fixed point theorems using admissible functions. Later, Ahmed and Fulga [37] generalized the Górnicki-Proinov type contraction to quasimetric spaces. Erdal et al. [38] published the notion of α,β,ψ,ϕ-interpolative contraction using a combine of interpolative contractions, Proinov type contractions, and ample spectrum contraction. Roldán López de Hierro et al. [39] proposed a new class of contractions in non-Archimedean fuzzy metric spaces, based on the Proinov fixed point results.


## 2. Main Results

Let us now give an important lemma that we will use in our main results.Lemma 13 (see [37]).
Lettr be a sequence on X,qΔ such that limr⟶∞qtr,tr+1=0. If the sequence tr is not left-Cauchy sequence thus there exists an e>0 and two subsequences tml, trl of tr such that
(7)limk⟶∞qtrk+1,tmk+1=limk⟶∞qtrk,tmk=e+.Proof.
Supposing that the sequencetr is not left-Cauchy, we can find e>0 and the sequences of positive integers nl, rl, with l≤rl<nl for every l≥0, such that
(8)qtrl,tnl>2e.On the other hand, sincelimr⟶∞qtr,tr+1=0, we can find r0≥1 such that
(9)qtr,tr+1<e2a,for every r≥r0, where a=max1,Δ. Moreover, since the space is supposed to be Δ symmetric,
(10)qtr+1,tr≤Δqtr,tr+1<e2,for every r≥r0. Therefore,
(11)2e<qtrl,tnl≤qtrl,trl+1+qtrl+1,tnl≤qtrl,trl+1+qtrl+1,tnl+1+qtnl+1,tnl<e2+qtrl+1,tnl+1+e2a≤e+qtrl+1,tnl+1,for every l≥r0. Consequently, we have
(12)qtrl+1,tnl+1>e,for every l≥r0. Now, let ml be the smallest positive integer, greater than nl, such that
(13)qtrl+1,tml+1>e,qtrl,tml>e.Thus, we have either(14)qtrl,tml−1≤e,(15)orqtrl+1,tml≤e.In the case of the first inequality holds,(16)e<qtrl,tml≤qtrl,tml−1+qtml−1,tml≤e+qtml−1,tml,and letting l⟶∞, we get liml⟶∞qtrl,tml=e+. Similarly, in case of the second inequality holds, we can consider
(17)e<qtrl,tml≤qtrl,trl+1+qtrl+1,tml≤qtrl,trl+1+e,so, we also obtain liml⟶∞qtrl,tml=e+. Now, by the triangle inequality, and taking into account the above considerations, we have
(18)e<qtrl+1,tml+1≤qtrl,trl+1+qtrl+1,tml+1+qtml+1,tml≤qtrl,trl+1+qtrl+1,tml+1+Δ·qtml,tml+1,and as l⟶∞, we get
(19)liml⟶∞qtrl+1,tml+1=e+.We will give multivaluedw,ψ,φ-contractive mappings.Definition 14.
LetX,qΔ, be a Δ-symmetric quasimetric space, a mapping w:X×X⟶0,∞ and Z:X⟶CBX be a multivalued operator. We say that Z is a multivalued w,ψ,φ-contractive mapping if there exist two functions ψ,φ:0,∞⟶ℝ such that
(20)ψwt,uHqZt,Zu≤φqt,u,for every t,u∈X with wt,u>1 and HqZt,Zu>0.Theorem 15.
LetX,qΔ be a complete Δ-symmetric quasimetric space, and Z:X⟶CBX be a multivalued w,ψ,φ-contractive mapping. Assume that following conditions are satisfied:
(K1) Z is strictly ∗-admissible and there exist t0∈X and t1∈Zt0 such that wt0,t1>1
(K2) if tr is a sequence in X such that wtr,tr+1>1 for all r∈ℕ and tr⟶t as r⟶∞, we have wtr,t>1
(K3) ψ is nondecreasing and φv<ψv for all v>0
(K4) limsupv⟶j+φv<ψj+ for all j>0
Therefore,Z has a fixed point in X.Proof.
Lett0 be an arbitrary point in X and t1∈X such that qt0,Zt0=qt0,t1 and qZt0,t0=qt1,t0. Let now t2∈Zt1 be such that qt1,Zt1=qt1,t2 and qZt1,t1=qt2,t1. Continuing in this way, we can build the sequence tr of points in X, such that tr+1∈Ztr, with qtr,Ztr=qtr,tr+1 and qZtr,tr=qtr+1,tr, for r∈ℕ0. Moreover, by condition K1, we have that there exist t0∈X and t1∈Zt0 such that wt0,t1>1. Supposing that r0≠r1, if r1∈Zr1, we get that t1 is a fixed point of Z. Then, let t1∉Zt1. As Z is a strictly ∗-admissible map, we have that ∗Zt0,Zt1>1. Thus, there exists t2∈Zt1 such that wt1,t2>1 which implies ∗Zt1,Zt2>1. By continuing this process, we can construct a sequence tr in X such that tr+1∈Ztr where tr+1≠tr for every r≥0 (as otherwise, if tr∈Ztr, thus tr is a fixed point of Z) and wtr,tr+1>1. Therefore, HqZtr−1,Ztr>0. From Lemma 3 with wtr,tr+1>1, we obtain
(21)qtr,tr+1≤wtr−1,trHqZtr−1,Ztr,for each r≥1. Keeping in mind K3 and (20) and we get
(22)ψqtr,tr+1≤ψwtr−1,trHqZtr−1,Ztr≤φqtr−1,tr.By hypothesisK3, we have
(23)ψqtr,tr+1≤φqtr−1,tr<ψqtr−1,tr.Thus, sinceψ is a nondecreasing map, qtr,tr+1<qtr−1,tr for each r≥1. So, the sequence qtr−1,tr is positively decreasing. Then, there exists G≥0 such that limr⟶∞qtr−1,tr=G+.Assuming thatG>0 on account of (23), we get a contradiction to supposition K4 as follows:
(24)ψG+=limr⟶∞ψqtr,tr+1≤limr⟶∞supφqtr−1,tr≤limv⟶G+supφv.Therefore,G=0, as a result, limr⟶∞qtr−1,tr=0.We prove that the sequencetr is left-Cauchy. Let us suppose by contradiction that the sequence tr is not left-Cauchy. Thus, by using Lemma 13, there exist e>0 and two subsequences trk,tmk, (tmk>trk≥k,) of tr such that (7) is fulfilled. From (7), we conclude that qtrk+1,tmk+1>ε and since the mapping Z is strictly triungular admissible, wtrk,tmk>1 for every k≥1. Substituting t=trk and u=tmk in (7), we obtain for each k≥1,
(25)ψqtrk+1,tmk+1≤ψwtrk,tmkHqZtrk,Ztmk≤φqtrk,tmk,then,
(26)ψqtrk+1,tmk+1≤φqtrk,tmk<ψqtrk,tmk,for any k≥1, so that is qtrk+1,tmk+1<qtrk,tmk Because of limk⟶∞qtrk+1,tmk+1=ε+, we obtain limk⟶∞qtrk,tmk=ε+. Therefore, we can write
(27)ψε+=limk⟶∞ψqtrk+1,tmk+1≤limk⟶∞supφqtrk,tmk≤limγ⟶ε+φγ,which contradicts the supposition K4; then, tr is left-Cauchy sequence in X,q, so that it is Cauchy sequence using Lemma 8. Therefore, the sequence tr is Cauchy in the complete Δ-symmetric quasimetric space and so converges to limit t∗∈X. Now, we consider the following cases.Case 1.
Ifqtr+1,Zt∗=0 for some r∈ℕ, so by triangle inequality of Δ-symmetric quasimetric space
(28)qt∗,Zt∗≤qt∗,tr+1+qtr+1,Zt∗<qt∗,tr+1,and thus, letting r⟶∞, we conclude that qt∗,Zt∗≤0, that is,
qt∗,Zt∗=0. As Zt∗ is closed, we obtain t∗∈Zt∗.Case 2.
On the contrary, ifqtr+1,Zt∗>0 for every r∈ℕ from K2, we have wtr,t∗>1 for all r∈ℕ. We claim that qt∗,Zt∗=0. Supposing, on the contrary, qt∗,Zt∗>0, there exists r∈ℕ such that qtr,Zt∗>0. Therefore, we obtain
(29)ψqtr+1,Zt∗≤ψwtr,t∗HqZtr,Zt∗≤φqtr,t∗<ψqtr,t∗.Taking into account the conditionK3, we get qtr+1,Zt∗<qtr,t∗. Passing to limit as r⟶∞, we obtain qt∗,Zt∗<0. Therefore,qt∗,Zt∗=0, as Zt∗ is closed, t∗∈Zt∗.Example 16.
LetX=0,∞ be endowed with the 2-symmetric quasimetric q:X×X⟶0,+∞, where
(30)qt,u=2t−u,ift≥u,u−t,otherwise,and a mapping Z:X⟶CBX, defined as
(31)Zt=0,t8,ift∈0,1,2,3,otherwise.We choose two functionsψ,φ:0,∞⟶ℝ with ψ is nondecreasing, and φm<ψm for all m>0 where ψm=m and φm=m/2. Let also
(32)wt,u=2,ift,u∈0,1,0,otherwise.We check thatZ is a multivalued w,ψ,φ-contractive mapping of (20). Actually, if taking into account the way the function w is defined, we have consider the case u,t∈0,1.Let then,t,u∈0,1, u≥t. We get
(33)q0,Zu=inf0,u8=0,q0,Zt=inf0,t8=0,(34)qt8,Zu=infu20−t8,2t8−u8,qu8,Zt=inft20−u8,2t8−u8,HqZt,Zu=maxsupt∈Ztqt,Zu,supu∈Zuqu,Zt=maxsupt∈Ztinfut4,t4−u4,supu∈Zuinftu4,u4−t4=t4−u4.So, we obtain(35)ψwt,uHqZt,Zu=2t4−u4=t2−u2≤t−u=φqt,u.Therefore, (20) fulfilled. Further, all other cases are satisfying, from wu,t=0. Consequently, by Theorem 15, map Z has a fixed point, this being t=0.Definition 17.
LetX,qΔw:X×X⟶0,∞ and Z:X⟶CBX be a multivalued operator. Z is said to be a multivalued C′iric′ type w,ψ,φ-contractive mapping if there exist two functions ψ,φ:0,∞⟶ℝ such that
(36)ψwt,uHqZt,Zu≤φΘt,u,for every t,u∈X with wt,u>1 and HqZt,Zu>0 where
(37)Θt,u=maxqt,u,qt,Zt,qu,Zu,qt,Zu+qZt,u2.Theorem 18.
LetX,q be a complete Δ-symmetric quasimetric space, w:X×X⟶ℝ+\0 and Z:X⟶KX be a multivalued C′iric′ type w,ψ,φ-contractive mapping. Assume that following conditions are satisfied:
(K1) Z is strictly ∗-triangular-admissible and there exists t0∈X and t1∈Zt0 such that wt0,t1>1
(K2) if tr is a sequence in X such that wtr,tr+1>1 for all r∈ℕ and tr⟶t as r⟶∞, we have wtr,t>1
(K3) ψ is nondecreasing, and φv<ψv for all v>0
(K4) limsupv⟶j+φv<ψj+ for all j>0
Therefore,Z has a fixed point in X.Proof.
By conditionK1, and following the lines of the proof of the previous theorem, we have that wtr,tm>1, for every m>r≥1. Moreover, HqZtr−1,Ztr>0 and from Lemma 3 with wtr,tr+1>1, we obtain
(38)qtr,tr+1≤wtr−1,trHqZtr−1,Ztr,for each r≥1. Keeping in mind K3 and (36), we get
(39)ψqtr,tr+1≤ψwtr−1,trHqZtr−1,Ztr≤φΘtr−1,tr.
AsZt is closed for every t∈X, we get that tr∈Ztr−1 such that qtr−1,tr=qtr−1,Ztr−1,
(40)ψqtr,tr+1≤φΘtr−1,tr=φmaxqtr−1,tr,qtr−1,Ztr−1,qtr,Ztr,qtr−1,Ztr+qZtr−1,tr2=φmaxqtr−1,tr,qtr,tr+1,for every r≥1.Ifmaxqtr−1,tr,qtr,tr+1=qtr,tr+1 so ψqtr,tr+1≤φqtr,tr+1, from assumption K3, this is a contradiction. Hence, we obtain qtr−1,tr>qtr,tr+1, and
(41)ψqtr,tr+1≤φqtr−1,tr.Similarly, again usingK3, we get
(42)ψqtr,tr+1≤φqtr−1,tr<ψqtr−1,tr.But, the functionψ is nondecreasing map, so that we get qtr,tr+1<qtr−1,tr for all r≥1. Therefore, the sequence qtr−1,tr is positively decreasing, and then, there exists G≥0 such that limr⟶∞qtr−1,tr=G+. If G>0, from (42), we obtain
(43)ψG+=limr⟶∞ψqtr,tr+1≤limr⟶∞supφqtr−1,tr≤limρ⟶G+supφρ,which contradictions K4. Therefore, G=0 and, as a result,
(44)limr⟶∞qtr−1,tr=0.We claim thattr is Cauchy sequence. Let us assume by contradiction that the sequence tr is not left-Cauchy. Then, by Lemma 13, we can find e>0 and two subsequences trk,tmk, (with mk>rk≥k) of tr such that (7) holds. Thereupon, we have that wtrk,tmk>1 for all mk>rk>k≥1. Letting t=trk and u=tmk in (9), we get
(45)ψqtrk+1,tmk+1≤ψwtrk,tmkHqZtrk,Ztmk≤φΘtrk,tmk,for every k≥1, where
(46)Θtrk,tmk=maxqtrk,tmk,qtrk,Ztrk,qtmk,Ztmk,qtrk,Ztmk+qZtrk,tmk2.Keeping in mind the way the sequencetr was define, let trk+1∈Ztrk and tmk+1∈Ztmk. Thus,
(47)qtrk,tmk≤Θtrk,tmk=maxqtrk,tmk,qtrk,trk+1,qtmk,tmk+1qtrk,tmk+1+qtrk+1,tmk2≤maxqtrk,tmk,qtrk,trk+1,qtmk,tmk+1,qtrk,trk+1+qtrk+1,tmk+1+qtrk+1,trk+qtrk,tmk2.Lettingk⟶∞ in the above inequality, and taking into account (44), respectively (7), we get
(48)limk⟶∞Θtrk,tmk=e+.Moreover, since the functionψ is nondecreasing, taking the limit superior when k⟶∞ in (45) we get
(49)ψε+=limk⟶∞ψqtrk+1,tmk+1≤limsupk⟶∞φΘtrk,tmk≤limsupρ⟶ε+φρ,which contradicts the supposition K4; then, tr is left Cauchy sequence in X,q, so that it is Cauchy sequence using Lemma 8. Therefore, the sequence tr is Cauchy in the complete Δ-symmetric quasimetric space and so converges to a point t∗∈X. Now, we consider following cases:Case 1.
Ifqtr+1,Zt∗=0 for some r∈ℕ, so by triangle inequality of Δ-symmetric quasimetric space
(50)qt∗,Zt∗≤qt∗,tr+1+qtr+1,Zt∗<qt∗,tr+1,and thus, letting r⟶∞, we conclude that qt∗,Zt∗≤0, that is,
(51)qt∗,Zt∗=0.AsZt∗isclosed,weobtaint∗∈Zt∗.Case 2.
If we suppose the contrary, that is,qtr+1,Zt∗=0 for any r, from K2 we know that wtr,t∗>1 for all r∈ℕ. We assert that qt∗,Zt∗=0. Suppose, on the contrary, qt∗,Zt∗>0. Thus, there exists r∈ℕ such that qtr,Zt∗>0 for every r. Using (36), we obtain
(52)ψqtr+1,Zt∗≤ψwtr,t∗HqZtr,Zt∗≤φΘtr,t∗<ψΘtr,t∗,where
(53)Θtr,t∗=maxqtr,t∗,qtr,Ztr,qt∗,Zt∗,qtr,Zt∗+qZtr,t∗2=maxqtr,t∗,qtr,tr+1,qt∗,Zt∗,qtr,Zt∗+qtr+1,t∗2.Taking into account the conditionK3, we get qtr+1,Zt∗<Θt∗,Zt∗. Passing to limit as r⟶∞, we obtain qt∗,Zt∗<qt∗,Zt∗ a contradiction, then qt∗,Zt∗=0. As Zt∗ is compact, t∗∈Zt∗.Corollary 19.
LetX,q be a Δ-symmetric quasimetric space and Z:X⟶KX be a multivalued mapping satisfying the condition:
(54)ψHqZt,Zu<φqt,u,for every t,u∈X, where the functions ψ,φ:0,∞⟶ℝ and HqZt,Zu>0. The map Z admits a fixed point in X provided that following conditions hold:
(K1) ψ is nondecreasing, and φv<ψv for all v>0
(K2) limsupv⟶j+φv<ψj+ for all j>0Lettingφa=δψa, in Corollary 19, we obtain the following result.Corollary 20.
LetX,q be a Δ-symmetric quasimetric space and Z:X⟶KX be a multivalued mapping satisfying the condition:
(55)ψHqZt,Zu<δψqt,u,for every t,u∈X, where the functions ψ,φ:0,∞⟶ℝ and HqZt,Zu>0. The map Z admits a fixed point in X provided that following conditions hold:
(K1) ψ is nondecreasing, and φv<ψv for all v>0;
(K2) limsupv⟶j+φv<ψj+ for all j>0.Corollary 21.
LetX,q be a Δ-symmetric quasimetric space and Z:X⟶KX be a multivalued mapping satisfying the condition:
(56)ψHqZt,Zu<φΘt,u,for every t,u∈X and HqZt,Zu>0, where the functions ψ,φ:0,∞⟶ℝ and
(57)Θt,u=maxqt,u,qt,Zt,qu,Zu,qt,Zu+qZt,u2.The mapZ admits a fixed point in X provided that following conditions:(K1) ψ is nondecreasing, and φv<ψv for all v>0(K2) limsupv⟶j+φv<ψj+ for all j>0Takingφa=δψa, in Corollary 21, we get the following result.Corollary 22.
LetX,q be a Δ-symmetric quasimetric space and Z:X⟶KX be a multivalued mapping satisfying the condition:
(58)ψwt,uHqZt,Zu≤δψΘt,u,for every t,u∈X and HqZt,Zu>0, where the functions ψ,φ:0,∞⟶ℝ and
(59)Θt,u=maxqt,u,qt,Zt,qu,Zu,qt,Zu+qZt,u2.The mapZ admits a fixed point in X provided that following conditions hold:(K1) ψ is nondecreasing, and φv<ψv for all v>0(K2) limsupv⟶j+φv<ψj+ for all j>0


## 3. Conclusion

In this paper, we expand the very interesting results of Proinov [29] in several ways: First, we involve a more general form of the function by considering multivalued mapping. Secondly, we refine the structure of the considered abstract space with Δ-symmetric quasimetric space. Indeed, quasimetric space is one of the novel extensions of metric space. Besides, Δ-symmetric quasimetric space is more reasonable to work since almost all quasimetric space form Δ-symmetric quasimetric spaces. There are still rooms for the fixed point results in the context of Δ-symmetric quasimetric spaces.



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*Source: 7197541-2022-05-25.xml*