# Fixed Point Results for Multivalued Mappings with Applications

Authors: Arshad Khan; Muhammad Sarwar; Farhan Khan; Habes Alsamir; Hasanen A. Hammad
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this manuscript, using the concept of multivalued contractions, some new Banach- and Caristi-type fixed point results are established in the context of metric spaces. For the reliability of the presented results, some examples and applications to Volterra integral type inclusion are also studied. The established results unify and generalize some existing results from the literature.


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## Body

## 1. Introduction and Preliminaries

Volterra integral equations appear in different scientific applications such as in the spread of epidemics, semiconductor devices, and population dynamics. Also, the dynamics of multispan uniform continuous beams subjected to a moving load is one of the best applications of Volterra integral equations. A universal method for finding sorption uptake curves of fluid multicomponent mixtures in porous solid at variable and constant concentration of mixture components on the basis of the Volterra integral equation has been proposed. The fixed point theory for multivalued mappings is a serious subject of set-valued analysis. In Banach spaces, several well-known fixed point theorems of single-valued mappings such as Banach and Schauder have been extended to multivalued mappings. There are a lot of applications for multivalued mappings such as optimal control theory, differential inclusions, game theory, and many branches in physics.The Banach contraction principle is used in a variety of fields of mathematics. This technique has many applications in studying the existence of solutions for nonlinear Volterra integral equations, and nonlinear integrodifferential equations in Banach spaces.Recently, it has been widely spread. For example, in abstract spaces, the Fredholm integral equation introduced by Fredholm [1] and the solutions of Fredholm- and Volterra-type integral equations have been discussed analytically by Rezan et al. [2–4], Rus [5], Aydi et al. [6], Karapinar et al. [7], and Hammad and De la Sen [8, 9] and numerically by Panda et al. [10, 11] and Berenguer et al. [12].Throughout this paper, the symbolsNM, CM, CBM, and CPM refer to the family of all nonempty, nonempty closed, nonempty closed bounded, and nonempty compact subset of M, respectively.For multivalued contraction mappings, we have the following.Definition 1 (see [13]).
Suppose thatM,d is a metric space. Denote by CBM, the set of all nonempty closed bounded subsets of M. The Pompeiu-Hausdorff metric PHM:CBM×CBM⟶0,∞ induced by the distance d is formulated as follows: for all A,B∈CBM,(1)PHMA,B=maxsups∈Bds,A,supt∈Adt,B,where ds,A=inft∈Ads,t. Also, the pair CBM,PHM is called a generalized Hausdorff distance induced by d.In 1969, the stipulation of Banach in single-valued mappings was modified to multivalued mappings by Nadler [13] as follows.Theorem 2.
LetM,d be a complete metric space (CMS) and T be a multivalued mapping on M so that Ts is a nonempty closed, bounded subset of M. If for each s∈M there is c∈01 so that
(2)PHMTs,Tt≤cds,t,∀s,t∈M,then, T has a fixed point in M.In the literature, via abstract spaces, some authors obtained nice fixed point results for contractive mappings under certain conditions, for example, Hussain et al. [14] prove the existence of several fixed point results in ordinary and partially order metric spaces by studying the notion of Geraghty-type contractive mapping via simulation function along with C-class function. In integral type, Branciari [15] introduced some common fixed point results under general contractive conditions.In Meir-Keeler type, Agarwal et al. [16] obtained some exciting fixed point results. In Menger probabilistic metric spaces, Chauhan et al. [17] discussed a hybrid coincidence and common fixed point theorem under a strict contractive condition with an application. Via the notion of α-admissible mapping, the existence of fixed point theorems under w-distance mappings with an application is presented by Kutbi and Sintunavarat [18], and the others concerned with studying the notion of multivalued mapping and its contributions in fixed point theory such as Nadler [13], Ali and Kamran [19], Aubin and Siegel [20], Covitz and Nadler [21], Hot [22], and Ali et al. [23].Du and Karapinar [24] introduce the concept of a Caristi-type cyclic map and present a new convergence theorem and a best proximity point theorem for Caristi-type cyclic maps. Petrusel and Sînta˘marian [25] obtained a new result in the link of single-valued and multivalued Caristi-type mappings. Hussain et al. [26] introduce the notion of Suzuki-type multivalued contraction with simulation functions and then set up some new fixed point and data dependence results for these types of contraction mappings. Karapinar [27] used lower semicontinuous mappings to generalize Caristi-Kirk’s fixed point theorem on partial metric spaces. Abdeljawad and Karapinar [28] generalize Cristi-Kirik’s fixed point theorem to Cone metric spaces using Cone-valued lower semicontinuous maps. The relation between Caristi’s result and its restriction to the function verifying Caristi’s stipulations with continuous real functions is explained by Jachymski [29]. Khojasteh et al. [30] introduce the idea that many known fixed point theorems can easily be derived from the Caristi theorem. Also, Karapinar et al. [31] proposed a new fixed point theorem which is inspired from both Caristi and Banach.In 2013, Ali et al. [32] initiated the idea of generalized α∗-admissible mappings. Via this concept, fixed point consequences to generalized Mizoguchi’s fixed point theorem are derived. For single-valued mappings, Caristi [33] introduced an important theorem in fixed point field and called it the “Caristi fixed point theorem.” This theorem was generalized to multivalued mappings in Banach spaces by Feng and Liu [34] as follows:Theorem 3.
Assume thatM,d is a CMS and T:M⟶NM is a multivalued mapping so that Ts is a closed subset of M, if there is a constant c∈0,1 so that
(3)dt,Tt≤cds,t,for all s∈M and t∈Ibs, where
(4)Ibs=t∈Ts:bds,t≤ds,Ts,b∈0,1,then, there is a fixed point of T in M with c<b and ds,Ts is lower semicontinuous.Theorem 4.
Assume thatM,d is a CMS and T:M⟶CM is a multivalued mapping so that Ts is a nonempty subset of M. Let ϕ:M⟶ℝ be a lower semicontinuous and bounded from below function and p:0,∞⟶0,∞ be a nondecreasing, continuous, and subadditive function so that p−10=0. If for each s∈X, there is an element t∈Ts verifying
(5)pds,t≤ϕs−ϕt,then, the mapping T has a fixed point on M.Recently, Isik et al. [35] generalized Banach contraction and Caristi’s fixed point theorem for a single-valued map as the following:Theorem 5.
LetM,d be a CMS and T:M⟶M be a self-continuous mapping, if there is the mapping φ:0,∞⟶0,∞ so that lims⟶0+φs=0,φ0=0, and
(6)dTs,Tt≤φds,t−φdTs,Tt,∀s,t∈M,then, there is a unique fixed point of T.Theorem 6.
Assume thatM,d be a CMS and T:M⟶M is a self-mapping. Let W be a set of mappings f:ℝ⟶0,∞ so that the hypotheses in the following hold:
(1)
f is continuous and strictly increasing(2)
For each sequencean⊆ℝ+,limn⟶∞an=0 if and only if limn⟶∞fan=1(3)
For alla,b∈ℝ,fa+b≤fafb
If the functionϕ:M⟶R is lower semicontinuous and bounded from as follows so that
(7)fds,Ts≤fϕsfϕTs,∀s∈M,then, T has a fixed point.Definition 7 (see [36]).
A functionp:0,∞⟶0,∞ is called subadditive if pa+b≤pa+pb for every a,b∈0,∞.Definition 8 (see [13]).
Suppose thatT:M⟶NM be a multivalued mapping, a point s∈M is called a fixed point of T if s∈TM.Definition 9 (see [37]).
LetM,d be a metric space A and B be nonempty subsets of M. Then, a set-valued mapping T:A∪B⟶A∪B is called a set-valued cyclic map if TA⊂B and TB⊂A, where TA=∪Tx:x∈A.The lemma in the following is very useful in the sequel.Lemma 10 (Zorn’s lemma).
LetS be a partially order set. If every totally ordered subset of S has an upper bound then S contains a maximal element.Lemma 11 (see [10]).
Assume thatA and B are closed and bounded subset of M and let a∈A. For each positive η>0, then there is b∈B so that da,b≤PHMA,B+η. Moreover, if B is a compact then there is b∈B so that da,b≤PHMA,B.The focus of this work is extending Theorems3, 4, 5, and 6 for multivalued mappings via generalized contractive conditions. An example and application for the existence of solution of Volterra integral inclusion are also given.


## 2. Fixed Point Results

We begin this section with the first main result.Theorem 12.
LetM,d be a complete metric space and T:M⟶CM be a multivalued mapping. If there is a nondecreasing mapping φ:0∞⟶0∞ so that limt⟶0+φt=0,φ0=0 and for all s∈M, there is t∈Ibs, so that
(8)dt,Tt≤φds,t−φdt,Ttb,where Ibs=t∈Ts:bds,t≤ds,Ts,b∈0,1 and ds,Ts is lower semicontinuous; then, T has a fixed point in M.Proof.
Lets∘ be an arbitrary element in M. Since Ibs is nonempty for any b∈0,1, we can build a sequence sn with sn+1∈Ibsn for each n∈ℕ∪0 so that
(9)dsn+1,Tsn+1≤φdsn,sn+1−φ1bdsn+1,Tsn+1.
Sincesn+2∈Ibs+1, then, we can write
(10)dsn+1,sn+2≤1bdsn+1,Tsn+1.
Because the mappingφ is nondecreasing, so by (9), we have
(11)0≤dsn+1,Tsn+1≤φdsn,sn+1−φdsn+1,sn+2.
This implies that the sequenceφdsn,sn+1 is nonincreasing. Since φ is bounded as follows, then there is k∈ℝ so that φdsn,sn+1⟶k as n⟶∞.
Now, form,n∈ℕ with m>n, we get
(12)dsn,sm≤∑i=nm−1dsi,si+1≤1b∑i=nm−1dsi,Tsi≤1b∑i=nm−1φdsi−1,si−φdsi,si+1≤1bφdsn−1,sn−φdsm−1,sm.
Passing the limit in the above inequality asn,m⟶∞, one can write
(13)limn,m⟶∞dsn,sm=0,this proves that sn is a Cauchy sequence. The completeness of M leads to there is s∈M so that limn⟶∞dsn,s=0. Since dsn,Tsn is lower semicontinuous, decreasing, φ is nondecreasing and by (9), we obtain that
(14)0≤dsn+1,Tsn+1=φdsn,sn+1−φ1bdsn+1,Tsn+1=φ1bdsn,Tsn−φ1bdsn+1,Tsn+1,or, equivalently,
(15)dsn+1,Tsn+1<dsn,Tsn,this implies
(16)dsn+1,Tsn+1<dsn,Tsn,therefore, dsn,Tsn converges to zero. Again, because dsn,Tsn is lower semicontinuous, then, we have
(17)0≤ds,Ts≤limn⟶∞dsn,Tsn=0.
Hence,ds,Ts=0; also, the closed property of Ts implies that s∈Ts.☐Remark 13.
Theorem12 is more general than Theorem 3, because if T verifies the stipulation of Theorem 3, then for each s∈M, there is t∈Ibs so that
(18)dt,Tt≤cds,t≤c1+c/b−c/bds,t,this insinuate that
(19)1+cb−cbdt,Tt≤cds,t,this equivalent to
(20)1−cbdt,Tt+cbdt,Tt≤cds,t,yields
(21)dt,Tt≤cds,t1−c/b−c1−c/bdt,Ttb.
Settingφp=cp/1−c/b, we have
(22)dt,Tt≤φds,t−φdt,Ttb.Moreover,ds,Ts is lower semicontinuous; therefore, T has a fixed point by Theorem 12.If we putφp=p2 for all p≥0 in Theorem 12, we get the result as follows.Corollary 14.
Suppose thatM,d is a CMS and T:M⟶CM is a multivalued mapping. If for any s∈M there is t∈Ibs so that
(23)dt,Tt≤ds,t2−dt,Ttb2,then, there exists a fixed point of T.Theorem 15.
LetM,d be a CMS and T:M⟶CPM be a multivalued mapping, where CPM is a compact subset of M. If there exists a nondecreasing mapping φ:0,∞⟶0,∞ so that limt⟶0φt=0,φ0=0, and
(24)PHMTs,Tt≤φds,t−φPHMTs,Tt,∀s,t∈M,then, there is a fixed point of T in M.Proof.
Considers∘ is an arbitrary point of M. According to Lemma 11, construct a sequence sn with sn+1∈Tsn for all n∈N so that
(25)0≤dsn,sn+1≤PHMTsn−1,Tsn≤φdsn−1,sn−φPHMTsn−1,Tsn≤φdsn−1,sn−φdsn,sn+1,this implies that φdsn,sn+1 is a nonincreasing sequence, as φ is bounded from the following so there is r∈ℝ so that limn⟶∞φdsn,sn+1=r.
To show thatsn is a Cauchy sequence, let m,n∈N with m>n and applying Lemma 11, we have
(26)dsn,sm≤∑i=nm−1dsi,si+1≤∑i=nm−1PHMTsi−1,Tsi≤∑i=nm−1φdsi−1,si−φdsi,si+1=φdsn−1,sn−φdsm−1,sm.
Asn,m⟶∞, we get
(27)limn,m⟶∞dsn,sm=0.
Thus,sn is a Cauchy sequence. The completeness of M leads to there is s∘∈M so that limn⟶∞sn=s∘. Thus, the sequence Tsn converges to Ts∘. Since sn+1∈Tsn for each n∈N, this implies that s∘∈Ts∘. This finishes the proof.☐Example 1.
Let(28)M=si=ii+12i∈N,ds,t=s−t.
It is obvious that the pairM,d is complete metric space. Define a multivalued mapping T by
(29)Ts=s1,ifs=s1∧h∈Mg,s1,s2,⋯,si−1,ifs=siandi>1.Now, we verify that the multivalued mappingT satisfied the condition of Theorem 15 with φp=p.expp. Note that PHMTsj,Tsi>0 iff i=1andj>2 or j>i>1 so we have the following two cases.Case 1.
Ifj>2 and i=1, we have
(30)PHMTsj,Ts11+ePHMTsj,Ts1dsj,s1edsj,s1≤PHMTsj,Ts12ePHMTsj,Ts1dsj,s1edsj,s1=sj−1−s12esj−1−s1sj−s1esj−s1=sj−1−s12esj−1−sjsj−s1=j2−j−22e−jj2−j−2<2e−j<2e−1<1.
That is,(31)PHMTsj,Ts11+ePHMTsj,Ts1dsj,s1edsj,s1<1,which implies that
(32)PHMTsj,Ts1≤dTsj,Ts1edsj,s1−dTsj,Ts1edTsj,Ts1.Case 2.
Ifj>i>1, then, we have
(33)PHMTsj,Tsi1+ePHMTsj,Tsidsj,siedsj,si≤PHMTsj,Tsi2ePHMTsj,Tsidsj,siedsj,si=sj−1−si−12esj−1−si−1sj−siesj−si=sj−1−si−12esj−1−si−1−sj+sisj−si=j+i−12ei−jj+i+1≤2ei−j≤2e−1<1,that is,
(34)PHMTsj,Tsi1+ePHMTsj,Tsidsj,siedsj,si≤1,therefore,
(35)PHMTsj,Tsi<dsj,siedsj,si−PHMTsj,TsiePHMTsj,Tsi.Hence, the condition of Theorem15 is satisfied. Also, s1=1 is a fixed point of T.Example 2.
Suppose thatM=0,∞,ds,t=s−t for all s,t∈M. It is obvious that the pair M,d is a CMS. Define a multivalued mapping T by
(36)Ts=s2,then, we have
(37)PHMTs,Tt=12s−t=12ds,t,(38)d2s,t−PHM2Ts,Tt=d2s,t−14d2s,t=34d2s,t,it follows from (37) and (38) that
(39)PHMTs,Tt≤d2s,t−PHM2Ts,Tt,∀s,t∈M.Hence,T satisfies the condition of Theorem 15 with φp=p2.Remark 16.
Theorem15 upgrades the Nadler fixed point result in a finite-dimensional space. Indeed, if T holds condition of Nadler’s theorem, then for s,t∈M, we have
(40)PHMTs,Tt≤cds,t≤cds,t1+c−c,or, equivalently,
(41)1−cPHMTs,Tt+cPHMTs,Tt≤cds,t,yields
(42)PHMTs,Tt≤c1−cds,t−c1−cPHMTs,Tt.Takingφp=cp/1−√c, we get
(43)PHMTs,Tt≤φds,t−φPHMTs,Tt,moreover, for each s∈M,Ts∈CBM but M is a finite-dimensional space; therefore, Ts is compact, and hence, by Theorem 15, there is a fixed point of T.LetΩ be the set of all mappings f:R⟶0∞ having the same stipulations (1)–(3) of Theorem 6. Note: by property (2), of Theorem 6, we have fa=1 if and only if a=0.Example 3.
All the functions in the following belong toΩ.(i)
f1s=1+tanhs(ii)
f2s=es(iii)
f3s=1+ln1+s,ifs∈0,∞es,ifs∈−∞,0
The lemmas in the following help us to supplement the theoretical results.Lemma 17.
Assume thatM,d is a CMS and T:M⟶NM is a multivalued mapping. Let ϕ:M⟶R be a function defined in Theorem 6 and f∈Ω. Define the relation “≤” on M so that
(44)s≤t⇔fds,t≤fϕsfϕt,then, the relation “≤” is a partial order on M and M is a partial order space.Proof.
Sincefa=1 iff a=0, then, we have
(45)fds,s=f0=1=fϕsfϕs,this shows that s≤s.
Ifs≤t and t≤s, then
(46)fds,t≤fϕsfϕt,fdt,s≤fϕtfϕs,because dt,s=ds,t; thus, fdt,s=1. Moreover, fa=1 iff a=0; therefore, dt,s=0; this implies that s=t.
Finally, ifs≤t and t≤u, then
(47)fds,t≤fϕsfϕt,andfdt,u≤fϕtfϕu,it follows from fa+b≤fa.fb that
(48)fds,u≤fds,t+dt,u≤fds,t.fdt,u≤fϕsfϕu.
From the above results, we conclude that “≤” is a partial order on M.☐Lemma 18.
Let the pairM,d be a CMS and T:M⟶NM be a multivalued mapping. Consider ϕ:M⟶R be a function defined in Theorem 6, f∈Ω and η:0,∞⟶0,∞ is a nondecreasing, continuous, and subadditive function that ensures that η−10=0. Define the relation “≤” on M by
(49)s≤t⇔fηds,t≤fϕsfϕt,then, the relation “≤” is a partial on M and M is a partial order space.Proof.
Sincefa=1 iff a=0 and η−10=0, then, we get
(50)fηds,s=fη0=f0=1=fϕsfϕs,this implies that s≤s.
Ifs≤t and t≤s, then
(51)fηds,t≤fϕsfϕt,fηdt,s≤fϕtfϕs.
Sincedt,s=ds,t, thus fdt,s=1. Moreover, fa=1 iff a=0 and η−10=0; therefore, dt,s=0, i.e., s=t.
Again, ifs≤t and t≤u, then
(52)fηds,t≤fϕsfϕt,fηdt,u≤fϕtfϕu,it follows from fa+b≤fa.fb, and η is nondecreasing and subadditive that
(53)fηds,u≤fηds,t+dt,u≤fηds,t+ηdt,u≤fηds,t.fηdt,u≤fϕsfϕu,and this completes the required. So, “≤” is a partial order on M.☐Now, we can state and prove the next main theorems.Theorem 19.
LetM,d be a CMS, T:M⟶NM be a multivalued mapping, ϕ:M⟶R be a function defined as Theorem 6, and f∈Ω. If for all s∈M, there is t∈Ts so that
(54)fds,t≤fϕsfϕt,then, the mapping T has a fixed point in M.Proof.
SinceM is a partial order space, then we need to prove only that M has a maximal element. Suppose that sαα∈I is increasing sequence in M, that is, for α,β∈I with α≤β then sα≤sβ from (54), we have ϕsαα∈I is decreasing, since ϕ is bounded as follows then infα∈Iϕsα holds. Assume that αn is increasing sequence in I that ensures
(55)limn⟶∞ϕsαn=infα∈Iϕsα=r.
Now, form,n∈ℕ with m≥n, then αn≤αm and sαn≤sαm.
Applying (54), we get
(56)fdsαn,sαm=f∑i=nm−1dsi,si+1≤Πi=nm−1fdsi,si+1≤Πi=nm−1fϕsifϕsi+1=fϕsnfϕsm.
Asm,n⟶∞ in the above inequality, we have limn,m⟶∞fdsαn,sαm=1. Since f is continuous so, we obtain
(57)limn,m⟶∞dsαn,sαm=0.
Hence,sαn is a Cauchy sequence in M. Since M is complete then there exists s∈M so that
sαn⟶s as n⟶∞. Thus, s is an upper bound for sαn for each n≥1. Now, we want to show that s is also an upper bound for sα. Suppose that β∈I with sαn≤sβ for each n≥1. Then, by (54), we obtain that ϕsβ≤ϕsαn for each n≥1. Also, from (54), we have
(58)ϕsβ=infα∈Iϕsα=r.
Since(59)fdsαn,sβ≤fϕsαnfϕsβ,then, by taking the limit as n⟶∞, we get limn⟶∞sαn=sβ. Hence, for each α∈I, there is n≥1 so that sα≤sαn, also sαn≤s, yields sα≤s for each α∈I. Thus, s is an upper bound for sαα∈I. By Zorn’s lemma, M has a maximal element s∘. The condition t∘∈Ts∘ implies that
(60)fds∘,t∘≤fϕs∘fϕt∘,this shows that s∘≤t∘. Since s∘ is maximal, therefore, s∘=t∘. Therefore, s∘∈Ts∘. This ends the proof.☐Theorem 20.
LetM,d be a CMS, T:M⟶NM be a multivalued mapping, ϕ:M⟶R be a function defined as Theorem 6, f∈Ω, and η:0,∞⟶0,∞ be a nondecreasing continuous subadditive function such that η−10=0. If for any s∈M, there is t∈Ts so that
(61)fηds,t≤fϕsfϕt,then, there is a fixed point of T in M.Proof.
By the same manner of the proof of Theorem19, we can easily show that there is a maximal point s∘ of partial order space M, and by hypothesis, there is t∘∈Ts∘ that ensures
(62)fηds∘,t∘≤fϕs∘fϕt∘.
This implies thats∘≤t∘. As s∘ is a maximal element of M, therefore, s∘=t∘; hence, s∘∈Ts∘. This completes the proof.☐


## 3. Supportive Application

Here, we use Theorem15 with φp=p2 to determine the existence of a solution to the Volterra-type integral inclusion of the following form:
(63)sq∈∫aqWq,j,sjdj+gq,q∈a,b,where W:a,b×a,b×ℝ⟶Pcvℝ, Pcvℝ refers to the class of nonempty compact and convex subset of ℝ and g:a,b⟶ℝ is a continuous function.LetM=Ca,b,ℝ be the space of all continuous real-valued functions on a,b. Define the distance
(64)ds,t=supq∈a,bsq−tq,for all s∈M. It is clear that the pair M,d is a metric space.We shall consider Problem (63) under the hypotheses as follows:
(1)
W:a,b×a,b×ℝ⟶Pcvℝ and g:a,b⟶ℝ are continuous functions, as well as W is compact and convex(2)
There is a continuous functionl:M⟶0∞ so that(65)PHMWq,j,sj,Wq,j,tj≤lj−1+1+4sj−tj22,for each q,j∈a,b and s,t∈M(3)
We havesupt∈a,b∫atljdj≤1Now, our main theorem in this part becomes valid for showing.Theorem 21.
Under the assumptions (1)–(3), Problem (63) has a solution on M.Proof.
Define the multivalued mappingT:Ca,b,R⟶CPCa,b,R by
(66)Tsq=u∈M:u∈∫aqWq,j,sjdj+gq,q∈a,b.☐The unique solution of Problem (63) is equivalent to find a fixed point of T in M.It is obvious thatTs is compact because for any s∈M,W.,.,s is lower semicontinuous. For clarification, let Ws=Wq,j,sj, for each q,j∈a,b. Thus, Ws:a,b×a,b⟶Pcvℝ, by Michael selection theorem [38], there is a continuous function ms:a,b×a,b⟶ℝ so that msq,j∈Wsq,j. It follows that ∫aqmsq,jdj+gq∈Tsq; this implies that Ts is nonempty. Hence, it is compact.Proof.
Suppose thats,u∈M such that u∈Ts. Then, for each q,j∈a,b,msq,j∈Wsq,j and
(67)uq=∫aqmsq,jdj+gq,∀q∈a,b.
Now, by condition (1), there existvtq,j∈Wtq,j such that
(68)msq,j−vtq,j≤lj−1+1+4sj−tr22,∀q,j∈a,b.
Let us define the multivalued operatorO by
(69)Oq,j=Wtq,j∩w∈ℝ:msq,s−w≤qj−1+1+4sj−tj22,∀q,j∈a,b.
SinceT is lower semicontinuous, then there exist mt:a,b×a,b⟶R such that for all q,j∈a,b,mtq,j∈Wtq,j. Thus, for any r∈M,(70)rq=∫aqmtq,jdj+gq∈∫aqWq,j,tjdj+gq,q∈a,b.
By condition (3), we can write(71)uq−rq≤∫aqmsq,j−mtq,jdj≤∫aqlj−1+1+4sj−tj22dj≤∫aqlj−1+1+4sj−tj22dj=−1+1+4sj−tj22∫aqljdj≤−1+1+4sj−tj22sup∫aqljdj≤−1+1+4sj−tj22.
This implies that(72)duq,rq≤−1+1+4dsj,tj22.
Now, by exchanging the rule ofs and t, we have
(73)PHMTs,Tt≤−1+1+4dsj,tj22,∀s,t∈M,yielding
(74)1+2PHMTs,Tt2≤1+4ds,t2is equivalent to
(75)PHMTs,Tt+PHMTs,Tt2≤ds,t2.
Therefore,(76)PHMTs,Tt≤ds,t2−PHMTs,Tt2,∀s,t∈M.
Therefore, all the conditions of Theorem15 are fulfilled. Therefore, the operator T has a fixed point which is a solution to Problem (63).☐



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*Source: 9921728-2021-08-24.xml*