M N M N N N M N M M M M N N M N N H H M N M N H M N F -CONTRACTIONS OPTIMIZATIONTHROUGHBESTPROXIMITYPOINTSFORMULTIVALUED

Vol. 22 (2021), No. 1, pp. 143–151 DOI: 10.18514/MMN.2021.3355

OPTIMIZATION THROUGH BEST PROXIMITY POINTS FOR MULTIVALUEDF-CONTRACTIONS

PRADIP DEBNATH Received 18 May, 2020

Abstract. Best proximity point theorems ensure the existence of an approximate optimal solution to the equations of the typef(x) =xwhenfis not a self-map and a solution of the same does not necessarily exist. Best proximity points theorems, therefore, serve as a powerful tool in the theory of optimization and approximation. The aim of this article is to consider a global optimization problem in the context of best proximity points in a complete metric space. We establish an existence of best proximity result for multivalued mappings using Wardowski’s technique.

2010Mathematics Subject Classification: 47H10; 54H25; 54E50

Keywords: best proximity point, fixed point,F-contraction, complete metric space, multivalued map, optimization

1. INTRODUCTION ANDPRELIMINARIES

Nadler [9] defined a Hausdorff concept by considering the distance between two arbitrary sets as follows.

Let(Ω,η)be a complete metric space (in short, MS) and letCB(Ω)be the family of all nonempty closed and bounded subsets of the nonempty set Ω. For

M

^,

N

^∈ CB(Ω), define the map

H

^{:CB(Ω)×CB(Ω)→[0,∞)by}

H

⁽

M

^,

N

^{) =max{sup}

ξ∈N

∆(ξ,

M

^),sup

δ∈M

∆(δ,

N

^)},

where∆(δ,

N

^{) =inf}ξ∈N η(δ,ξ). Then(CB(Ω),

H

⁾is an MS induced byη.

Let

M

^,

N

be any two nonempty subsets of the MS(Ω,η). The following notations will be used throughout:

M

0={µ∈

M

^{:η(µ,ν) =η(}

M

^,

N

^{)for someν∈}

N

^},

N

0={ν∈

N

^{:η(µ,ν) =η(}

M

^,

N

^{)for someµ∈}

M

^}, whereη(

M

^,

N

^{) =inf{η(µ,ν):µ∈}

M

^,ν∈

N

^}.

This research is supported by UGC (Ministry of HRD, Govt. of India) through UGC-BSR Start-Up Grant vide letter No. F.30-452/2018(BSR) dated 12 Feb 2019.

© 2021 Miskolc University Press

For

M

^,

N

^∈CB(Ω), we have

η(

M

^,

N

^)≤

H

⁽

M

^,

N

^).

We say thatµ∈

M

is a best proximity point (in short, BPP) of the multivalued map Γ:

M

^→CB(

N

^{) if∆(µ,Γµ) =η(}

M

^,

N

^{). υ∈Ω}is said to be a fixed point of the multivalued mapΓ:Ω→CB(Ω)ifυ∈Γυ.

Remark1.

(1) In the MS(CB(Ω),

H

^),υ∈Ωis a fixed point ofΓif and only if∆(υ,Γυ) =0.

(2) Ifη(

M

^,

N

^{) =}0, then a fixed point and a BPP are identical.

(3) The metric function η:Ω×Ω→[0,∞) is continuous in the sense that if {υn},{ξn}are two sequences inΩwith(υn,ξn)→(υ,ξ)for someυ,ξ∈Ω, asn→∞, thenη(υn,ξn)→η(υ,ξ)asn→∞. The function∆is continuous in the sense that ifυn→υ asn→∞, then ∆(υn,

M

^)→∆(υ,

M

^)asn→∞ for any

M

^⊆Ω.

The following Lemmas are noteworthy.

Lemma 1([2,4]). Let(Ω,η)be an MS and

M

^,

N

^∈CB(Ω). Then (1) ∆(µ,

N

^{)≤η(µ,γ)for anyγ∈}

N

^{and µ∈Ω;}

(2) ∆(µ,

N

^)≤

H

⁽

M

^,

N

^{)for any µ∈}

M

^.

Lemma 2([9]). Let

M

^,

N

^{∈CB(Ω) and letυ∈}

M

, then for any r>0, there existsξ∈

N

^{such that}

η(υ,ξ)≤

H

⁽

M

^,

N

^{) +r.}

But we may not have anyξ∈

N

^{such that}

η(υ,ξ)≤

H

⁽

M

^,

N

^).

Further, when

N

is compact, there existsξ∈Ωsuch thatη(υ,ξ)≤

H

⁽

M

^,

N

^).

The concept of

H

-continuity for multivalued maps is listed next.

Definition 1([5]). Let(Ω,η)be an MS. We say that a multivalued mapΓ:Ω→ CB(Ω) is

H

-continuous at a point µ₀, if for each sequence {µ_n} ⊂Ω, such that

n→∞limη(µn,µ₀) =0, we have lim

n→∞

H

^(Γµn,Γµ0) =0 (i.e., ifµn→µ₀, then Γµn→Γµ₀ asn→∞).

Definition 2([9]). LetΓ:Ω→CB(Ω)be a multivalued map. We say thatΓis a multivalued contraction if

H

^{(Γµ,Γν)≤λη(µ,ν)for allµ,ν∈Ω, whereλ∈[0,1).}

Remark2.

(1) IfΓis

H

-continuous on every point of

M

^⊆Ω, then it is said to be continuous on

M

^.

(2) A multivalued contractionΓis

H

-continuous.

In 2012, Wardowski [16] defined the concept ofF-contraction as follows.

Definition 3. LetF:(0,+∞)→(−∞,+∞)be a function which satisfies the fol- lowing:

(F1) Fis strictly increasing;

(F2) For each sequence{u_n}_n∈

N⊂(0,+∞),

n→+∞lim u_n=0 if and only if lim

n→+∞F(un) =−∞;

(F3) There ist∈(0,1)such that lim

u→0⁺u^tF(u) =0.

Let

F

denote the class of all such functionsF. If(Ω,η) is an MS, then a self-map T :Ω→Ωis said to be anF−contraction if there existτ>0,F ∈

F

, such that for allµ,ν∈Ω,

η(T µ,Tν)>0⇒τ+F(η(T µ,Tν))≤F(η(µ,ν)).

MultivaluedF-contractions were defined by Altun et al. [1] as follows.

Definition 4([1]). Let(Ω,η)be an MS. A multivalued mapΓ:Ω→CB(Ω)is said to be a multivaluedF-contraction (MVFC, in short) if there existτ>0 andF∈

F

such that

τ+F(

H

^{(Γµ,Γν))≤F(η(µ,ν)) (1.1)} for allµ,ν∈ΩwithΓµ6=Γν.

Remark3. An MVFC is

H

-continuous.

We can find the concept of P-property in [12], whereas the notion of weak P property was defined by Zhang et al. [18].

Definition 5([12]). Let(Ω,η)be an MS and

M

^,

N

be two non-empty subsets of Ωsuch that

M

06=φ. The pair (

M

^,

N

⁾is said to have theP-property if and only if η(µ₁,ν₁) =η(

M

^,

N

^{) =η(µ}2,ν₂)impliesη(µ₁,µ₂) =η(ν₁,ν₂), whereµ₁,µ₂∈

M

0

andν₁,ν₂∈

N

^.

Definition 6([18]). Let(Ω,η)be an MS and

M

^,

N

be two non-empty subsets of Ωsuch that

M

06=φ. The pair(

M

^,

N

⁾is said to have the weakP-property if and only ifη(µ1,ν1) =η(

M

^,

N

^{) =η(µ}2,ν2)impliesη(µ1,µ₂)≤η(ν1,ν2), whereµ₁,µ₂∈

M

0

andν₁,ν₂∈

N

0.

BPP theorems forF-contractive non-self mappings were studied by Omidvari et al.

[11] with the help ofP-property. Later, Nazari [10] investigated BPPs for a particular type of generalized multivalued contractions by using the weakP-property.

Srivastava et al. [13,14] presented Krasnosel’skii type hybrid fixed point theorems and found their very interesting applications to fractional integral equations. Xu et al. [17] proved Schwarz lemma that involves boundary fixed point. Very recently, Debnath and Srivastava [6] investigated common BPPs for multivalued contractive pairs of mappings in connection with global optimization. Debnath and Srivastava [7] also proved new extensions of Kannan’s and Reich’s theorems in the context

of multivalued mappings using Wardowski’s technique. Further, a very significant application of fixed points ofF(ψ,ϕ)-contractions to fractional differential equations was recently provided by Srivastava et al. [15].

In this paper, we introduce a best proximity result for multivalued mappings with the help of F-contraction and the weak P property. Also we provide an example where theP-property is not satisfied but the weakP-property holds.

2. BEST PROXIMITY POINT FORMVFC

In this section, with the help of the notion of F-contraction, we show that an MVFC satisfying certain conditions admits a BPP.

Theorem 1. Let(Ω,η) be a complete MS and

M

^,

N

be two non-empty closed subsets ofΩsuch that

M

06=φand that the pair(

M

^,

N

⁾has the weak P-property.

SupposeΓ:

M

^→CB(

N

⁾be a MVFC such thatΓµ is compact for each µ∈

M

^and Γµ⊆

N

0for all µ∈

M

0. ThenΓhas a BPP.

Proof. Fixµ₀∈

M

0and chooseν₀∈Γµ₀⊆

N

0. By the definition of

N

0, we can selectµ₁∈

M

0such that

η(µ₁,ν₀) =η(

M

^,

N

^{). (2.1)}

Ifν0∈Γµ₁, then

η(

M

^,

N

^)≤∆(µ1,Γµ₁)≤η(µ₁,ν₀) =η(

M

^,

N

^).

Thusη(

M

^,

N

^{) =∆(µ}1,Γµ₁), i.e.,µ₁is a BPP ofΓ. Therefore, assume thatν0∈/Γµ₁. SinceΓµ₁is compact, by Lemma2, there existsν₁∈Γµ₁such that

0<η(ν₀,ν₁)≤

H

^(Γµ0,Γµ₁).

SinceFis strictly increasing, the last inequality implies that F(η(ν₀,ν₁))≤F(

H

^(Γµ0,Γµ₁))

≤F(η(µ₀,µ₁))−τ. (2.2) Sinceν₁∈Γµ₁⊆

N

0, there existsµ₂∈

M

0such that

η(µ2,ν1) =η(

M

^,

N

^{). (2.3)}

From (2.1) and (2.3) and using weakP−property , we have that

η(µ₁,µ₂)≤η(ν₀,ν₁). (2.4) From (2.2) and (2.4), we have

F(η(µ₁,µ₂))≤F(η(ν₀,ν₁))≤F(η(µ₀,µ₁))−τ. (2.5) Ifν₁∈Γµ₂, then

η(

M

^,

N

^)≤∆(µ2,Γµ2)≤η(µ2,ν1) =η(

M

^,

N

^).

Thusη(

M

^,

N

^{) =∆(µ}2,Γµ₂), i.e.,µ₁is a BPP ofΓ. So, assume thatν₁∈/Γµ₂.

SinceΓµ₂is compact, by Lemma2, there existsν₂∈Γµ₂such that 0<η(ν₁,ν₂)≤

H

^(Γµ1,Γµ₂).

Using the fact thatF is strictly increasing, we have that F(η(ν₁,ν₂))≤F(

H

^(Γµ1,Γµ₂))

≤F(η(µ₁,µ₂))−τ

≤F(η(µ₀,µ₁))−2τ(using2.5).

Sinceν₂∈Γµ₂⊆

N

0, there existsµ₃∈

M

0such that

η(µ3,ν2) =η(

M

^,

N

^{). (2.6)}

From (2.5) and (2.6) and using weak propertyP, we have that

η(µ₂,µ₃)≤η(ν₁,ν₂). (2.7) From (2.6) and (2.7), we have

F(η(µ2,µ₃))≤F(η(ν1,ν2))≤F(η(µ0,µ₁))−2τ. (2.8) Continuing in this manner, we obtain two sequences{µ_n} and{ν_n} in

M

0 and

N

0

respectively, satisfying (B1) νn∈Γµn⊆

N

0,

(B2) η(µ_n+1,νn) =η(

M

^,

N

^),

(B3) F(η(µn,µn+1))≤F(η(νn−1,νn))≤F(η(µ0,µ₁))−nτ, for eachn=0,1,2, . . ..

Putαn=η(µn,µ_n+1)for eachn=0,1,2, . . .. Taking limit on both sides of(B3)as n→∞, we have

n→∞limF(αn) =−∞.

Using(F2), we obtain

n→∞limαn=0. (2.9)

Using(F3), there existsk∈(0,1)such that

α^k_nF(αn)→0 asn→∞. (2.10)

From(B3), for eachn∈N, we have that

F(αn)−F(α₀)≤ −nτ.

This implies

α^k_nF(αn)−α^k_nF(α₀)≤ −nα^k_nτ≤0. (2.11) Lettingn→∞in (2.11) and using (2.9), (2.10), we obtain

n→∞limnα^k_n=0.

Thus there existsn₀∈Nsuch thatnα^k_n≤1 for alln≥n₀, i.e.,αn≤ ¹

n¹k

for alln≥n₀.

Letm,n∈Nwithm>n≥n₀. Then η(µm,µn)≤

m−1

∑

i=n

η(µi,µi+1) =

m−1 i=n

∑

αi

≤

∞

∑

i=n

αi≤

∞

∑

i=n

1 i^1k . Since the series ∑^∞i=n 1

i¹k

is convergent for k∈ (0,1), we have η(µm,µn) → 0 as m,n→∞. Hence {µ_n} is Cauchy in

M

0⊆

M

^{. Since(Ω,η)} is complete and

M

is closed, we have lim

n→∞µ_n=θfor someθ∈

M

^. SinceΓis

H

-continuous (for it is an MVFC), we have

n→∞lim

H

^(Γµn,Γθ) =0. (2.12) Exactly in the similar manner as above, using(B3), we can prove that{ν_n}is Cauchy in

N

^{and since}

N

is closed, there existsξ∈Bsuch that lim

n→∞νn=ξ.

Sinceη(µn+1,νn) =η(

M

^,

N

^{)for alln∈}N, we have

n→∞limη(µ_n+1,νn) =η(θ,ξ) =η(

M

^,

N

^).

We claim thatξ∈Γθ. Indeed, sinceνn∈Γµnfor alln∈N, we have

n→∞lim∆(νn,Γθ)≤ lim

n→∞

H

^{(Γµn,Γθ) =0.}

Therefore,∆(ξ,Γθ) =0. SinceΓθis closed, we haveξ∈Γθ.

Now,

η(

M

^,

N

^{)≤∆(θ,Γθ)≤η(θ,ξ) =η(}

M

^,

N

^).

Hence∆(θ,Γθ) =η(

M

^,

N

^{), i.e.,θ}is a BPP ofΓ.

A Geraghty type [8] result follows as a consequence of our previous theorem. Let

G

be the class of functions g:[0,∞)→[0,1) satisfying the condition: g(ξn)→1 implies ξn→0. An example of such a map is g(ξ) = (1+ξ)⁻¹ for all ξ>0 and g(0)∈[0,1).

Definition 7. Let

M

^,

N

be two non-empty subsets of a MS (Ω,η). A multival- ued mapΓ:

M

^→CB(

N

⁾is said to be a multivalued Geraghty-typeF-contraction (MVGFC, in short) if there existτ>0,F∈

F

^andg∈

G

^{such that}

τ+F(

H

^{(Γµ,Γν))≤g(η(µ,ν))·F(η(µ,ν)) (2.13)} for allµ,ν∈ΩwithΓµ6=Γν.

Corollary 1. Let(Ω,η) be a complete MS and

M

^,

N

be two non-empty closed subsets ofΩsuch that

M

06=φand that the pair(

M

^,

N

⁾satisfies the weak P-property.

SupposeΓ:

M

^→CB(

N

⁾be a MVGFC such thatΓµ is compact for each µ∈

M

^and Γµ⊆

N

0for all µ∈

M

0. ThenΓhas a BPP.

Proof. Sinceg(t)∈[0,1)for allt∈[0,∞), from (2.13), we have that

τ+F(

H

^{(Γµ,Γν))≤F(η(µ,ν)) (2.14)} for allµ,ν∈

M

^{withΓµ6=Γν. Thus,Γ}is an MVFC and hence from Theorem1it

follows thatΓhas a BPP.

Remark4. Corollary1extends the results due to Caballero et al. [3] and Zhang et al. [18] to their multivalued analogues usingF-contraction.

Next, we provide some examples in support of our main result.

Example1. ConsiderΩ=Rwith usual metricη(µ,ν) =|µ−ν|for allµ,ν∈Ω.

Let

M

^{= [5,6]and}

N

= [−6,−5]. Thenη(

M

^,

N

^{) =10 and}

M

0={5},

N

0={−5}.

Define the multivalued mapΓ:

M

^→CB(

N

^{)such that} Γµ= [−µ−5

2 ,−5] for allµ∈[5,6].

ThereforeΓ(5) ={−5}(i.e.,Γµ⊆

N

0for allµ∈

M

0).

We claim thatΓis a MVFC. Let

H

^(Γµ,Γν)>0. Then we have

H

^{(Γµ,Γν) =}

H

^([−µ−5

2 ,−5],[−ν−5

2 ,−5])

=|(−µ−5

2 )−(−ν−5 2 )|

=|ν−µ|

2

=η(µ,ν) 2

<η(µ,ν).

From the last inequality, we have that ln(

H

^{(Γµ,Γν))<}ln(η(µ,ν)), and further, τ+ln(

H

^{(Γµ,Γν)) ≤}ln(η(µ,ν)), for any τ ∈ (0,ln 2]. Therefore, we have that τ+F(

H

^{(Γµ,Γν))≤F(η(µ,}ν)), for anyτ∈(0,ln 2], whereF(t) =lnt,t>0.

Finally, it is easy to check that(

M

^,

N

⁾satisfies weakP-property. Thus, all condi- tions of Theorem1are satisfied and we observe thatµ=5 is a BPP ofΓ.

In fact, in Example1, the pair(

M

^,

N

^{) satisfies}P-property (and hence the weak P-property as well). Next, we present an example in which the pair(

M

^,

N

^)satisfies only the weakP-property but not theP-property.

Example2. ConsiderΩ=R²with the Euclidean metricη.

Let

M

^={(−5,0),(0,1),(5,0)}and

N

^{={(µ,ν):ν=2+p2−µ2,µ∈[−√2,√2]}.} Thenη(

M

^,

N

^{) =√3 and}

M

0={(0,1)},

N

0={(√

2,2),(−√ 2,2)}. Define the multivalued mapΓ:

M

^→CB(

N

^{)such that}

Γ(−5,0) ={(−√

2,2),(−1,3)},Γ(0,1) ={(√

2,2)},Γ(5,0) ={(√

2,2),(1,3)}.

It is easy to check thatΓis a MVFC withτ=ln 2 andF(t) =lnt,t>0.

Finally, we observe that η((0,1),(

√

2,2)) =η((0,1),(−√

2,2)) =√

3=η(

M

^,

N

^), but

η((0,1),(0,1)) =0<η((

√

2,2),(−√

2,2)) =2

√ 2.

Thus, (

M

^,

N

⁾ satisfies weak P-property, but not the P-property. Therefore, all conditions of Theorem1are satisfied and since∆((0,1),Γ(0,1)) =√

3=η(

M

^,

N

^), we conclude that(0,1)is a BPP ofΓ.

3. CONCLUSION

We have proved our main result with a strong condition that images of the MVFC are compact sets. Relaxation of this compactness criterion is a suggested future work. We have shown the non-triviality of the assumption of the weak P-property by presenting an example which does not satisfy theP-property but satisfies only the weakP-property. The results due to Caballero et al. [3] and Zhang et al. [18] are also extended to their multivalued analogues as a consequence of our results.

ACKNOWLEDGEMENT

The author expresses his hearty gratitude to the learned referees for their construct- ive comments which have improved the manuscript considerably.

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Author’s address

Pradip Debnath

Department of Applied Science and Humanities, Assam University, Silchar, Cachar, Assam - 788011, India

E-mail address:debnath.pradip@yahoo.com