Zamfirescu mappings under Pata-type condition: results and application to an integral equation Текст научной статьи по специальности «Математика»

doi https://doi.org/10.5937/vojtehg72-48937

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Zamfirescu mappings under Pata-type condition: results and application to an integral equation

Deep Chanda, Yumnam Rohenb, Nicola Fabianoc 5

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a National Institute of Technology Manipur, it

Department of Mathematics, Imphal, Manipur, Republic of India, e-mail: [email protected], ot

ORCID iD: © https://orcid.org/0000-0002-9274-620X b Manipur University, Department of Mathematics; National Institute of Technology Manipur, o

Department of Mathematics, Imphal, Manipur, Republic of India, e-mail: [email protected], ORCID iD: ©https://orcid.org/0000-0002-1859-4332 c University of Belgrade, "Vinca" Institute of Nuclear Sciences - National Institute of the Republic of Serbia, Belgrade, Republic of Serbia, e-mail: [email protected], corresponding author, ORCID iD: ©https://orcid.org/0000-0003-1645-2071

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FIELD: mathematics o

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ARTICLE TYPE: original scientific paper Abstract:

Introduction/purpose: Pata-type and Zamfirescu mappings are extended £ beyond metric spaces.

Methods: The concept of Pata-type Zamfirescu mapping within the framework of S-metric spaces is employed. S

Results: A series of corresponding outcomes has been established. Furthermore, the obtained results are employed to solve an integral equation.

Conclusions: S-Pata type and Zamfirescu mappings have unique fixed points.

Key words: Pata-type contraction, Zamfirescu mapping, S-Metric space, n

fixed point. ro"

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ACKNOWLEDGMENT: The first author Deep Chand gratefully acknowledges the financial support of q the University Grants Commission (UGC), India, under the Senior Research Fellowship (SRF) scheme, grant number 1174/(CSIRNETJUNE2019) §

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Introduction

In 1922, Banach (Banach, 1922) established that every contraction mapping on a complete metric space possesses a unique fixed point. This > theorem is commonly referred to as the Banach fixed point theorem.

Since Banach's theorem was proven, various other types of mappings have been demonstrated to possess the same fixed point property. Among

Su these mappings are Kannan-type mappings and Chatterjea-type map-

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pings, both of which were introduced in the 1960s. These mappings hold

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o significance because they enable the existence of fixed points even for

< non-continuous mappings.

° In 1972, Zamfirescu (Zamfirescu, 1972) introduced a generalized cono traction mapping that further extended the class of mappings for which the

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fixed point property can be guaranteed. Zamfirescu's results generalized the work of several other mathematicians, including Kannan (Kannan, fC 1968) and Chatterjea (Chatterjea, 1972).

In the paper, ^ represents the set of ascending functions ^ : [0,1] ^ co [0, +rc>), where ^ exhibits continuity at 0 and starts at ^(0) = 0.

Definition 1. (Zamfirescu, 1972) If (V, d) is a metric space, a function r >o V ^ V is referred to as a Zamfirescu mapping if it satisfies the following condition for any points ti and 9 in V and real numbers a, b, and c within the interval [0,1):

o d(r(ti), r(9)) <

ty max {ad(ti, 9), §[d(ti, r(ti)) + d(6, r(9))], f [d(ti, r(0)) + d(6, r(ti))]}

In 2011, Pata (Pata, 2011) introduced an improved version of the classical Banach Principle. This enhancement enables the identification of fixed points for mappings that lack strict contraction properties, instead relying on approximate contraction characteristics.

Theorem 1. (Pata, 2011) If (X, d) is a metric space that is complete and fixed constants A > 0, a > 1 and / which lies in the interval [0, a]. If the mapping r : X ^ X fulfills the subsequent inequality for each e e [0,1]

and all ti,9 e X,

d(r§, r9) < (1 - e)d(ti, 9) + Aea[1 + \\ti\\ +

then r possesses a unique fixed point ft* e X, and the sequence {rraft0} exhibits convergence towards ft* for any given initial element ft0 e X. 1

Definition 2. (Jacob et al., 2018) If (V, d) is a metric space, a mapping r : V ^ V is considered as a Pata-type Zamfirescu mapping if, for all ft and 0 in V, and for every e e [0,1], it holds the following inequality with ^ in

d(r(ft), r(d)) < (1 - e)M(ft,9)+Aea^(e)[1 + \\ft\\ + ||0|| + \\rtf\\ + \\re\\]

ß

where M(ft,0) = max Id(ft,d), d(e>r(e), and

A > 0, a > 1, ß e [0, a] are constants.

Example 1. (Sedghi et al., 2012) If V is a nonempty set equipped with an ordinary metric d on V, then the following are S-metrics on V.

• S(ft, 0,5) = d(ft, 5) + d(0,5)

• S(ft, 0,5) = d(ft, 0) + d(0, 5) + d(5, ft)

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Many other authors have previously employed the Pata-type condition to derive novel fixed point outcomes (Kadelburg & Radenovic, 2014; ° Özgür & Ta§, 2021; Kadelburg & Radenovic, 2016; Karapinaretal., 2020a; Saleem et al., 2020; Karapinar et al., 2020b; Aktay & Özdemir, 2022; Ya-haya et al., 2023; Roy et al., 2024). |

Further, in 2018 Jacob et al. (Jacob et al., 2018) defined Pata type Zamfirescu mapping and generalized the results of (Chatterjea, 1972; Pata, ® 2011).

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In 2012, Sedghi et al. (Sedghi et al., 2012) presented the notion of an £ S-metric space.

Definition 3. (Sedghi et al., 2012) If V is a nonempty set, an S-metric is a function S : V x V x V ^ [0, +rc>) on V that meets the requirements for every ft, 0,5, a eV as follows: ¡?

(i) S(ft, 0, 5) > 0,

(ii) S(ft, 0,5) = 0 if and only if ft = 0 = 5,

(iii) S(ft, 0,5) < S(ft, ft, a) + S(0,0, a) + S(5,5, a). SS The term "S-metric space" refers to the pair (V, S). I

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Lemma 1. (Sedghi et al., 2012) If the space (V,S) is an S-metric space, then as a result, there is for all ti, 9 eV,

S(ti, ti, 9) = S(9,9,ti)

Lemma 2. (Sedghi et al., 2012) In an S-metric space (V, S), for all ti,9,5 e V, there is

S(ti, ti, 9) < 2S(ti, ti, 5) + S(9,9,5)

and

S(ti, ti, 9) < 2S(ti, ti, 5) + S(5,5,9)

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w Definition 4. (Sedghi et al., 2012) If (V, S) is an S-metric space, >- (i) a sequence {tin} in V converges to ti if and only if S(tin, tin, ti) ^ 0

<t as n ^ This convergence is denoted as tin = ti,

(ii) a sequence {tin} in V is referred to as a Cauchy sequence if, for given e > 0 there is a n0 e N such that S(tin, tin, tim) < e for every n,m > no, and

^ (iii) the space (V, S) is defined as complete when it satisfies the condi-

s? tion that every Cauchy sequence in V converges.

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Several other authors have contributed to the field of S-metric spaces, establishing numerous results within this framework as well as in various extended spaces related to S-metric spaces. Some of these pertinent ^ works can be found in the references (Sedghi etal., 2012; Chand & Rohen, 2023; Ozgur & Ta§, 2023; Priyobarta et al., 2022).

Inspired by the findings outlined above, this paper introduces the concept of S-Pata type Zamfirescu mappings and demonstrates the existence of fixed point results within the framework of S-metric spaces. Notably, these results extend the findings from a previous study (Jacob et al., 2018). Furthermore, the best proximity point theorem has been established and findings applied to the context of integral equations.

Main results

In this section, the aim is to demonstrate the existence and uniqueness of fixed points for S-Pata type Zamfirescu mappings. Consider an S-metric

space denoted as (V, S). Throughout this discussion, the norm of an element ti is represented as \\ti\\ = S(ti0, ti0, ti), where ti0 is a chosen element 1 in .

s(r§,re,rs) < (i - e)M(0,e,s) + AeXe)[i+ y 0 y + y e y + y s y + y r0 y + y re y + y rs r

where

s(0,0, r0) + s(e, e, re) + s(s, s, rs)

m(0,e,s) = m^ s(0,e,s)

M(0,0,e) = max] s(0,0,e),

Lemma 3. (Alghamdiet al., 2021) Consideran s-metric space (V,s). If the sequence {0n} in V which is not Cauchy with s(0n, 0n, 0n+1) = 0.

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Definition 5. In an S-metric space (V, S), a mapping r : V ^ V is referred § to as an S-Pata type Zamfirescu mapping if it satisfies the following inequality for all ti,9,5 eV, ^ e and every e e [0,1]:

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s(0,0, re) + s (e,e, rs) + s(s,s, r0)

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A ^ 0, a ^ 1,3 e [0, a] are the constants.

Note: From the above definition, it can easily be deduced that

S(rti, rti, r9) < (1 -e)M(ti, ti, 9) +AeQ^(e)[1 + 2\\ti\\ + \\9\\ + 2\\rti\\ + \\r9\\]^ where

2S(ti,ti, rti) + S(9,9, r9) g

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s(0,0, r0) + s(0,0, re) + s (e, e, r0) i i

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A ^ 0, a ^ 1,3 e [0, a] are the constants. e

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The following lemma is a key ingredient in the proof of this work. This lemma will be used to prove the main results. The proof of the lemma in a q metric space can be found in (Alghamdi et al., 2021). Here, it is extended it to the framework of an S-metric space.

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Then, two sub-sequences {fink} and {fimk} of {fin} exist for any e > 0 such that

lim 5($nk+i,$nk+i,#mk+1)= e+ (1)

g lim S($nk ,Kk ,$mk )= lim S{§nk+1,Kk + 1 ,$mk ) =

CM fc^+TO fc^+TO

lim S^nk ,$nk,$mk+i) = e (2)

k^+TO

01 yy

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8 Proof.Since, {fin} is not a Cauchy sequence and

< limn^+TO S(fin,fin,fin+1) = 0, one can get e > 0 and N0 > 1 such that for any N > N0 there is m,n> N with m > n and

S(fin+1 ,fin+1,fim+i) > e and S(fin+1,fin+1,fin) < e.

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^ By selecting the smallest m > n so that S^^ ,fin+1, fim+1) > e holds,

it is concluded that for each N > N0 there exists m,n> N such that

S(fin+1,fin+1, fim+1) > e and S(fin+1,fin+1,fim) < e.

<5 Thus, one can construct two sub-sequences {fink} and {fimk} of {fin} such o that

S(Vnk+1, $nk+1,$mk+1) > e and Sfik+1, ) < e.

These inequalities, along with the triangular inequality, lead to

e < S(fink+1,-&Uk+1,-&mk + l) < S(-&nk + 1,-&nk+1,-&mk ) + 2S(fimk ,#mk ,fimk + 1)

< e + 2S(fimk ,fimk ,fimk+1).

fa By means of the Sandwich Theorem, one arrives at (1). Furthermore, there exists

S (fink+1,tink + 1,timk + 1)-2S(timk + 1 ,timk+1,#mk ) < S(fink+1,fink + 1 ,fimk ) < e

which implies the second limit of (2). From the subsequent two inequalities,

S(fink+1,#nk + 1,#mk ) - 2S(fink ,fink ,#nk+1) < S (fink ,fink ,fimk ) < e + 2S (fink ,fink ,fink+1),

e - 2S(fink , fink ,fink + 1) <S(fink , fink ,fimk+1) < S(fink+1,fi nk+1,fimk + 1) +

2S(fink ,fink ,fink+1) from these inequalities one will get the required limits. □

Theorem 2. Let (V, S) be a complete S-metric space. If r : V ^ V is an S-Pata type Zamfirescu mapping, then r possesses one and only one fixed 1 point.

S(tin,0n,0n+l) <

\ S(0 1 0 1 0 ) S(#n-1 ) +

f S (0 0 0 ) ) + S(&n,#n,#n+1)

Qi„<i „<i „<i \ c' m„v ) S (0n-1, 0n-1, 0n), 3 ,

S (0n, 0n, 0n+1) < ma^ S(#n-1,$n-1,'&n) + S(#n-1,'&n-1 ,#n + Q

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J S(0 1 0 1 0 ) 2S(^n-1,^n-1,^n)+S(^n,^n,^n+1) S (0n, 0n, 0n+1) < ma^ n ' ^¿-J-1 ,^n)+S(^n ^n^n+1 ) '

Now, considering S(0n-1,0n-1,0n) < S(0n, 0n,0n+1), one gets

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Proof. Assume 0O is any element within the set V. Create a sequence ° with the definitions: 0n+1 = r(0n) and cn = S(0n,0n,00). In order to demonstrate that the sequence S(0n+1,0n, 0n) is non increasing, consider setting e = 0, which leads to the following result ^

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S(0 0 0 ) ^ 2S(0n-1,0n-1,0n)+ S(0n,0n,0n+1) ^

S(0n,0n,0n+1) < -3--"V

2S(0n, 0n, 0n+1) < 2S(0n-1, 0n-1,0n) + S(0n, 0n, 0n+1) S(0n, 0n, 0n+1) < S(0n-1,0n-1, 0n)

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This results in a contradiction. Therefore, it can be deduced that

S (0n, 0n, 0n+1) < S (0n-1,0n-1,0n) <•••< S (00, 00, 01) = C1.

Claiml: cn is bounded. Define I

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Cn = S(00, 00, 0n) < 2S(0o, 00, 01) + S(0n, 0n, 01) ~

< 2S(00, 00, 01) + 2S(0n, 0n, 0n+1) + S(0n+1, 0n+1, 01) j

< 2C1 +2C1 + S(0n+1,0n+1,01) |

< 2C1 +2C1 + S(01,01,0n+1) 5

< 4C1 + S(T00,T00,T0n)

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S($0 $0 $ ) S($o,$o,$i)+S($o,$0,$i)+S(-&n

< 4ci + (1 - e) ma^ u' S^Jo,vi)+s(v0,v0,vn+i)+s(#n,vn,vi)

+ Asa^(s)[1 + |

< 4ci + (1 - s)-

J0,Vl)+S(V0,V0,Vn +

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- + Asa^(s)[1 + ||$o|| + ll$oll + ||$n|| + ll$ill + ||$iN + ll$r+illf

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| f S($0,$0,$n), 2S(*0,*0,*l)+S(*n,

OR max< S(-âo A A )+2S(iïo A ,&n+i)+2S(&o,#o,#n)+S(-&o A A )

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| +Asa^(s)[1 + 2ll$ill + ||$J + (N$iN +

^ < 4ci + (1 - s)max{cn,ci, } + Asa^(s)[1 + 3N$iN + 2N$ Ntf

0 4

1 < 4ci + (1 - s) max{cn, ci,^ + cn} + Asa^(s)[1 + 3ci + 2c„]

> < 4ci + (1 - s) max{cn, ci, — + cn} + Asa^(s)[1 + 3ci + 2cn]

oi 4 <

3ci -, , n , o _ , „ _ 1/3

If there is a subsequence cni ^

3ci

3 cni < 4ci + (1 - Si)^ + cni) + As*^(si)[1 + 3ci + 2cnz]a

S2 3d

o cni - (1 - Si)cni < 4ci + (1 - Si) + Asf^(sz)[1 + 3ci + 2nY

for some A,B> 0. The choice ei = leads to the inequality,

w Sicni < A + Bsa^(si)c:

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& 1 < B(1 + A)a^(Si) ^ 0,

which is a contradiction. Hence, {cn} is a bounded sequence.

As S($n-i,$n-i,$n) is a non-increasing sequence and it has a lower bound of 0, it follows that

lim S($n,$n,$n-i) = d > 0 S($n,$n,$n+i)= S(T$n-i, r$n-i, r$n)

< (1 - S)-

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( S (ft 1 ft 1 ft ) 2S(fl„-i ,V„-i ,fl„)+S(fl„,fl„,fl„+i ) '

< (1 - e) max s(—-i,&„-u&„)+2s(^„-i,#„-i,^„)+s(^„,^„,^n+i) '

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+ Aea^(e)[1 + 2 ||ftn-1|| + 3||ftn|| + ||ftn+1|

( S(ft ft ft ) 2S(fl„-i ,-&„-i ,ti„)+S(ti„,ti„,ti„+i) ^

< (1 - e)ma^ n-1' 3-(X-^„-i,fl„)+S(0„,^„+i) 7+ K#(e)

5 < S(ftmk, ftmk,ftnk) = S(rftmfc-1, rftmk-i, rftnk-i)

S(ftmk-1, ftmk-1, ftnk-1),

< (1 - e) ma^ S(^mk-i ,^mk-i,^mk )+S(^mk-i,^mk-i,^mk )+S($„k - i ,^„k-i ,^„k )

+ Ke^(e) < (1 — e) max + Ke^(e) .

(ftmk-1,ftmk-1, ftnk ) + S (ftnk-1,ftnk-1,ftnk ),

\ S(ft 1 ft 1 ft ) S(fl„-i,V„-i,0„)+S(d„-i,0„-i,d„)+S(0„,d„,0„+i) I maM ' S(^„-i,^„-i,fl„)+S(fl„-i,fl„-i^„+Q+S(#„^„^„) ' f 1

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+ Ae>(e)[1 + ||ftn-1|| + Wftn-1W + WftnW + WftnW + Wftn W + Wftn+1W1^ ~

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Now, the limit as n approaches infinity is considered, one obtains d < K^(e), and consequently, one gets d = 0. H

Claim2. The sequence {ftn} is a type of Cauchy sequence. <5

Assuming that {ftn} is not Cauchy, one can apply Lemma 3 to conclude that there exists a subsequence {ftnk} and another subsequence {ftmk} of {ftn} where nk > mk > k, such that

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S($mk - i ,$mk - i ,$mk ) + S(#mk - i ,$mk - i ,$„k )+S($„k - i ,&„k-i,$mk ) | "O

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2S(#mk_i,#mk-i,imk )+S(^„k-i ,v„k-i,0„k) I E

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S($mk-i ,$mk-i,tfmk )+S(&mk-i,&mk-i,&„k )+S($„k -i,&„k -i,timk ) | CO

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As k approaches infinity, one obtains 5 < K0(e). Consequently, it follows that 5 = 0, which presents a contradiction. Therefore, it can d be asserted that {ftn} is indeed a Cauchy sequence. Considering that V is a complete S-metric space, it can be concluded that there exists an element ft within V such that the sequence ftn converges to ft. Now, for all n in the natural numbers, and when e is set to zero, one obtains

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S(0, 0, r0) < 2S(0, 0, 0n+i) + S(r0, r0, 0n+i)

S(0,0,0n),

< 2S(0,0,0n+i) + max { WM+WM+WnSM,

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S(0,0, 0n),

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v Allowing n to approach infinity, the aforementioned inequality follows as:

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5(0,0,r0) <-S(0,0,r0) >- 3

* ^ S(0,0, r0) = 0 Hence, 0 is a fixed point of r as S(r0, r0,0) = 0, one obtains r0 = 0.

« To establish the uniqueness of the fixed point, assume that both 0,9 e o V, are the fixed points of r. One obtains

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(S(0, 0} 9), S(i9,i9,r9)+S(i9,i9,r9)+S(e,e,re) S (r0, r0, r9) < (1 - £)maxj ' S(&,&,r'&)+S('&,'&,r0)+S(0,0,r'&) +

Ke^(e)

Consequently, one obtains S(0,0,O) < K0(e), which leads to the conclusion that 0 = 9. Hence, it can be asserted that r possesses a unique fixed point in V. □

Example 2. Let V = R. Define a function S : V x V x V equipped by

S(0,9,5) = {|0 - 9\ + \9 - 5\ + \5 - 0|} for all 0,9,5 e V. Then (V, S) is a complete metric space.

If one defines a self mapping r on V by r0 = 10, then r satisfies the conditions of Theorem 2. For all 0,9,5 eV one obtains

333 33 33 33

S(r0,r9,r5) = S(80, 89, 85) = {\80 - 89\ + 189 - 85\ + 185 - 80\}

33 = 8 sup{\0 - 9\ + \9 - 5\ + \5 - 0\} = 8S(0,9,5)

< (1 - e)S(V, 6,5) + (- - (1 - e))S(V, 6,5) §

8 ' 3 8

< (1 - e)M(V, 6,5)+ -(1 - -(1 - e))[S(V, V, Vo) + S(6,6, Vo) + S(5,5, Vo)] ^

8 3 ^

—

< (1 - e)M(V,6,5) + -e3p|| + ||6|| +

_(r9 rO _ f aS(0,e,5),b[^^H^^H^A^],

In that case, r possesses a unique fixed point within the space V. Proof.Considering d = max{a, b, c}, one obtains that

(s{pie,5), s(,WT0)+s(,ofi,TO)+s(,WS) ) S(Ttf, TO, r^ < d ma^ ' s(#,#,re)+s(8,8,rs)+sxs,s,r#) 'f

d

ma^ ||V|| + 1161 + ||5||,

< (1 - e)M (V,6) + ded [1 + H#H + ^H + ||5|| + Hm + ||r6|| + m

< (1 - e)M(V, 6) + dee— [1 + IV + ||6|| + M + ||rV|| + m +

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< (1 - e)M(V, 6,5) + 3e2e3 [1 + m + ||6|| + M + ||rV|| + Hm + h™" ® 8

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for all e e [0,1]. This implies that r is an S-Pata type Zamfirescu mapping for a = 2, / = 1, A = 8 and ^(e) = e2. Furthermore, it can be affirmed that ® V = 0 is indeed the unique fixed point of r in V, as asserted by Theorem 2.

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Corollary 1. In a complete S-metric space (V, S), if the mapping T : V ^V is a Zamfirescu mapping that meets the following inequality criteria for all

•&,O,5 eV and a,b,c e [0,1), b

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' S(V 6 5) S(v,v,rv)+S(e,e,re)+S(s,s,rs) 1

' S(&,&,r0)+S(0,0,rS)+S(S,S,r&) ( E

Js(V,6,5), S(w,m+S(8,e,m+S(ë,ë,rë) «

+ (d +e - ^ max j ' S($,$,re)+SX0,0,rs)+S(s,s,r$) ' f I

< (1 - e)M(V,6) + d( 1 + e - 1

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Therefore, by Theorem 2 with A = d, ^(e) = e d and a = / = 1, it follows that r has a unique fixed point. □

Corollary 2. Considera complete S-metric space (V, S), and let r : V ^ V > be a mapping. If for all X e [0,1), r satisfies the following inequalities

s(t§, re, rs) < xs(0,0,5) (3)

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yy OR

=3 5(r0, r0, r5) < A [S(0,0, r0) + S(d,d, r0) + S(5,5, r5)] (4)

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OR

< \

° S(r0, r0, r5) < -[S(0,0, r0) + S(0,0, r5) + S(5,5, r0)] (5)

x 4

¡^ Then, r has a unique fixed point in V. i-

Proof. In Corollary 1, if one takes

(1). a = A, b = c = 0, inequality (3) is obtained

(2). b = A, a = c = 0, inequality (4) is obtained

(3). c = A, a = b = 0, inequality (5) is obtained « further steps followed from Corollary 1. □

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* Existence of the best proximity point for S-Pata type proximal contraction

o This section introduces a new type of proximal mappings called S-Pata q type proximal mappings. It is then proved that these mappings have the property of having the best proximity points. Let A and B be subsets of a complete S-metric space (V, S). The distance between two sets is denoted by D(A, B) and defined by

D(A, B) = inf{S(0,0,0): 0 e A and 0 e B}.

The notation A0 is used to represent the subset of A defined as follows:

Ao = {0 e A : S(0,0,0) = D(A, B), for some 0 e B} Likewise, B0 is the subset of B defined as follows:

Bo = {0 e B : S(0,0,0) = D(A, B), for some 0 e A}

Throughout the section, the assumption that both A0 and B0 are closed sets is maintained.

594

S(u,v,w) < (1 - e)S(0,0,5) +Ae>(e)[1 + \\0\\ + \\0\\ +

S (0ra, 0n, 0n+i) < S (0n-l,0n-l ,0n) < ... S (0o, 0o, 0i) = ci

Definition 6. A mapping r : A ^ B is said to be an S-Pata type proximal contraction of type-I if, for all 0,0,5 e A, ^ e and for any e e [0,1], it 1 satisfies the following inequality:

oo LT)

cp

o ro

where S(u, u, r(0)) = S(v, v, r(0)) = S(w, w, r(5)) = D(A, B) and A > 0, ® a > 1, 3 e [0, a] are arbitrary constants.

<u

ro

Definition 7. A mapping r : A ^ B is said to be an S-Pata type proximal contraction of type-II if, for all 0,0,5 e A, ^ e and for any e e [0,1], it o satisfies the following inequality:

S(u,u,v) < (1 - e)S(0,0,0) +Ae°^(e)[1 + 2\\0\\ +

where S(u, u, r(0)) = S(v, v, r(0)) = D(A, B) and A > 0, a > 1, /3 e [0, a] are arbitrary constants.

Theorem 3. In a complete S-metric space (V, S), consider non-empty closed subsets A and B. Suppose that there exists a mapping r : A ^ B that is an S-Pata type proximal contraction, and additionally, it holds that r(A0) c B0. Then r possesses one and only one best proximity point <u within the set A. ^

ro '

ro CL

cp <0

<0 tn

tn <u

o o

Proof.Let 00 be an arbitrary element in A0. Then r(00) e B0 and so there exists an element 0i e A0 such that S(0i,0i,r(00)) = S(A,A,B). Similarly, define 0n+i e A0 such that S(0n+i,0n+i,r(0n)) = S(A,A,B) and « cn = S(0n, 0n, 00). Then, one gets

S(0n, 0n, 0n+i) < (1 - e)S(0n-i, 0n-i, 0n) + Ae>(e)[1 + 2\\0n\\ + \0n-i\^ E

o

CO

<u

where e e [0,1].

Taking e = 0 in the above inequality it follows that {S(0n, 0n, 0n+i)} is a nonincraesing sequence. Therefore, ro"

<u

d

"O c ro .c

O

(M

o >

X O LU

>o

X LU I—

O z

o

Now, one shows that {cn} is bounded:

c„ = S (Vo,Vo,Vn)

< 2S (Vo, Vo, Vi) + 2S (Vn, Vn, Vn+i) + S (Vi,Vi,Vn+i )

< 4ci + (1 - e)S(Vo,Vo,Vn)+Aea^(e)[1 + 2HVoll + ||Vn||]^ (6)

CM

™ < 4ci + (1 - e)cn + Aea^(e)[1 + cnY

yy < a + (1 - e)cn + bea^(e)can

Q1

g where a,b> 0 are constants. Accordingly,

<

o ecn < a + bea^(e)c',

If there is a subsequence cni — +œ, the choice e = ei = (1 + a)/cn, leads to the contradiction

o .1

< 1 < b(1 + a)a^(ei) — 0

Hence, {cn} is a bounded sequence.

Let limn^+rc, S(-dn-i,$n-i,$n) = d. Since S(tin-i,tin-i,tin) is non in-OT creasing,

o S(tin, tin, tin+i) < (1 - £)S($n-i,#n-i, Vn) + Aea4>(e)[1 + 2\\tin-i\\ +

n|

< (1 - e)S(Vn-i,Vn-i, Vn) + Ke^(e)

As the limit as n approaches infinity is considered, it is deduced that d < K^(e) implies d = 0. One now makes the assertion that {Vn} is a Cauchy sequence.

In order to avoid contradiction, assume that {Vn} is not a Cauchy sequence. In such a case, according to Lemma 3, there must exist subsequences {Vnk} and {Vmk} of {Vn} with nk > mk > k such that

d < S(Vnk,Vnk,Vmk)

< (1 - e)S(Vnk-i, Vnk-i, Vmk-i) + Ke^(e)

< (1 - e)[2S (Vnk-i,Vnk -i, Vmk ) + S (Vmk-i,Vmk-i,Vnk )] + Ke^(e).

As k tends towards infinity, one reaches the conclusion that d < K^(e), which leads to a contradiction. Therefore, it can be affirmed that {Vn} is indeed a Cauchy sequence. Given that V is a complete space, there

exists an element 0 in V such that {0n} converges to 0.

Now, consider S(u,u, r(0)) = D(A,B) and set e to zero. Then, for all n e N, one obtains

S(u, u, 0) < 2S(u, u, 0n+i) + S(0,0, 0n+i)

< 2S(0,0,0n) + S(0,0,0n+i)

S(0,0,0) < (1 - e)S(0,0,0) + Ke^(e)

Hence, one obtains S(0,0,0) < K0(e), which implies 0 = 0. Thus, r has unique proximity point 0 in A. □

Corollary 3. Consider a complete metric space (V, S), and let r : V ^ V be a mapping that adheres to the following inequality:

Proof. The proof can be derived directly from the previous theorem when

A = B. □

fore = 0 S(u, u, v) < S (ê', ê', 0'),

for e e (0,1] S(u, u, v) < (1 - e)S (ê', ê', 0') +

e^(e)[1 + 2\\ê' y + \\0f\\],

o

cn oo LT) S± Œ

O

co

ΠCD

This leads to the conclusion that S (u, u, ê) = 0. Consequently, it can be established that ê serves as a proximity point of r.

In order to demonstrate the uniqueness of the proximity point, suppose, o for the sake of contradiction, that r has two distinct proximity points ê and 0 in V. One obtains

p p

co

co

CO

CO CD

S(r0, r0, r5) < (1 - e)S(0,0,5) + Ae>(e)[1 + \\0\\ + \\0\\ + \\5\\]V »

§

Then, there exists a unique fixed point for r.

CO CL

CD "O

CO

Example 3. Let V = R2 under 1-norm and (V, S) be an S-metric space with g

a metric defined by S(0,0,5) = i{\0-5| + \0-5|}. Consider A = {(0,a)|a e |

[0,1]} and B = {(1,6)\6 e [0,1]} then D(A,B) = 1. Define f : A ^ B as 3 f (0,0) = (1, f), A = a = / = 1, and for 5 e (0, i)

{ ) = , 2 ifs e [M) I

w = i 1 ifs e [5,1], <5

Id

One needs to demonstrate that, for all e e [0,1], f satisfies the inequality o of S-Pata type proximal contraction,

o >

CM o CM

o

where S(u, u, f (ti')) = S(v, v, f (O)) = D(A, B) = 1.

The following inequality shows that f satisfies the first case when e = 0.

£ For all ti' = (0, ti), O' = (0, O) e A.

ti2 ti2 O2 S(u,u,v) = S ((0,^4), (0,ti4), (0,O4))

ti2 O2

, =l(0, T)+(0, T)l 3 = ti2 - O2l = (ti_ + °_){ti_ - O)

o \ 4 4 1 ^2 2n2 2'

,.ti O.

< <---

o < 1 2 2 1

< S(ti',ti',O')

1 e — 1

< (1 - e)S(ti', ti', O') + -(1 + — )[2S(ti'0, ti', ti') + S(ti0, O', O')]

The following inequalities demonstrate that f satisfies the second case when e e (0,1]:

For e e (0,5) and for all ti',O' e A

ti2 ti2 O2 ti2 O2 S(u, u, v) = S((0, -), (0, -), (0, 4)) = \T - — |

3 < -S(ti',ti',O')

CD - 2 V ' ' 7

S2 1

>o = (1 - e)S(ti', ti', O') + (- + (e - 1))S(ti', ti', O')

2 2 X LU

§ - .....'"•'■2 ,

I

P < (1 - e)S(ti',ti',O') + 2e2[2\\ti'" + \\O\\]

< (1 - e)S(ti', ti', O') + e^(e)[1 + 2\\ti'\\ + \\O'\\].

For e e [5,1] and for all ti', O' e A,

ti2 ti2 O2 ti2 O2 S(u,u,v) = S((0, -), (0, -), (0, 4)) = \T - — |

< S(ti',ti',O')

= (1 - e)S(ti', ti', O') + eS(ti', ti', O') = (1 - e)S(ti', ti', O') + e[2S(ti0, ti', ti') + S(ti0, O', O')] = (1 - e)S(ti', ti', O') + e^(e)[1 + 2\\ti'\\ + \\O'\\].

Hence, f fits the criteria of being an S-Pata type proximal contraction, and as a result, there exists a best proximity point (0,0) within the set A.

598

Application to the integral equation §

In this section, an application of the main result is explored by describing a solution of the integral equation.

Consider M = C(I, R), which represents the set of all continuous func- § tions on I = [0,1], equipped with the metric S(0,9,5) = sup{|0(s) - 9(s)| + |9(s) - 5(s)| + |5(s) - 0(s)|, s € I} for all 0,9,5 € M. It is worth noting that ® (M,S) forms a complete S-metric space. In this context, one delves into | the investigation of the integral equation

r(0)(s) = %(s) + / K(s, u)n(u, 0(u))du, s € I. (8)

Jo

Theorem 4. Equation (8) has at least one solution in M, if for all u € I and a,b € R, there is

^(u, a) - n(u, b^ < 1(|a - ra| + ^ - rb| + ^ - ra|), for all u € I, followed by the assumed inequality

ci

s£l JO

where 5 is a fixed constant.

sup / K(s,u)du = 5 < 1.

s&I JO

is a fixed point of r. Now let 0,9 € M, on account of the above inequalities, one finds that

|r(0)(s) - r(9)(s)| =

/0

< / K(s,u)du ■ |n(u,0(u)) - n(u,9(u))| Jo

K(s, u)[n(u, 0(u)) - n(u, 9(u))]du

CO oo LT) £± cp

cd

co o

0(s) = x(s) + [ K(s,u)n(u,0(u))du, s € I, (7)

o

where, the functions n : I x R ^ R and % : I ^ R are continuous, and || K : I x I ^ R+ is a function satisfying K(s, ) € Ll(I) for all s € I. One examines the mapping r : M ^ M, which is defined as follows: ro

CO

CO CD

o o

CD

cp

CO CL

CD "D

CO OT C

cp

CO

E ^

o

CO CD

Proof. Note that finding a solution of (7) is equivalent to finding 0* € M that e

N

CO

CD

d

"O c CO .c O

1

(M r1 1

<u

< K(s,u)du •-(|«(s) - r«(s)| + |«(u) - r(0)(u)\ + \d(u) - r(«)(u)|) Jo 4

* ^ 5(r«, r«, rn = s '5<«•«■r«> + rg) +sw>

4

< S • M(«,«,g) ,

o >

(M o (M

c¿ ÜÜ

Oí ,

3 k 4

g Hence, it is derived that

( S(« g s) S(#,#,r(#))+S(0,0,r(0))+S(6,6,r(6)) ' where M(«,d,s) = ma^ ' 's(¿,^,r(e))+s(e,e,r¿)3+s(¿,¿,r^) '

0 S(r«, r«, rg) < (1 - e)M(«, «, d) + (S + (e - 1))M(«, «, d)

1 / e — 1

DC ÍS(«,«,g), S(0,0,r(0))+S(fl,fl,r(fl))+S(fl,fl,r(fl)),

< (1 - e)M(«,«,g) + M 1 +

< (1 - e)M(«, «, g) + Se1 max j\\«\\ + \\«\\ + \\e\\, 4M+2M+MM+Mä }

< (1 - e)M(«, «, g) + See1 "1[1 + 2\\«\\ + \\g\\ + 2\\r«\\ + \\rg\\]

cd

for all u e I and «,g e M. Hence, r is an S-Pata type Zamfirescu mapping with A = S, a = 1, ß = 1 and ^(e) = e~. Therefore, one deduces the w existence of «* e M such that «* = r«* that is «* is a solution of integral § equation (7). □

o

fy References

Aktay, M. & Özdemir, M. 2022. On (a, ^)-weak Pata contractions. MANAS Journal of Engineering, 10(2), pp. 228-240. Available at: https://doi.org/10.51354/mjen.1085695.

Alghamdi, M.A., Gulyaz-Ozyurt, S. & Fulga, A. 2021. Fixed points of Proinov contractions. Symmetry, 13(6), art.number:962. Available at: https://doi.org/10.3390/sym13060962.

Banach, S. 1922. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta mathematicae, 3, pp. 133-181. Available at: https://doi.org/10.4064/fm-3-1-133-181.

Chand, D. & Rohen, Y. 2023. Fixed Points of (as-ßs-^-Contractive Mappings in S-Metric Spaces. Nonlinear Functional Analysis and Applications, 28(2), pp. 571-587. Available at: https://doi.org/10.22771/nfaa.2023.28.02.15.

600

Chatterjea, S. 1972. Fixed-point theorems. Dokladi na Bolgarskata Akademiya o na Naukite, 25(6), p. 727. ®

oo

Jacob, G.K., Khan, M.S., Park, C. & Yun, S. 2018. On Generalized Pata Type Contractions. Mathematics, 6(2), art.number:25. Available at: ^

https://doi.org/10.3390/math6020025. |

Kadelburg, Z. & Radenovic, S. 2014. Fixed Point and Tripled Fixed Point Theorems under Pata-Type Conditions in Ordered Metric Spaces. International Journal of Analysis and Applications, 6(1), pp. 113-122 [online]. Available at: https://www.etamaths.com/index.php/ijaa/article/view/392 [Accessed: 7 February 2024].

Kadelburg, Z. & Radenovic, S. 2016. Fixed point theorems under Pata-type o conditions in metric spaces. Journal of the Egyptian Mathematical Society, 24(1), pp. 77-82. Available at: https://doi.org/10.1016/jJoems.2014.09.001.

Kannan, R. 1968. Some results on fixed points. Bulletin of the Calcutta Math- || ematical Society, 60, pp. 71-76. ®

Karapinar, E., Fulga, A. & Aydi, H. 2020a. Study on Pata E-contractions. <5 Advances in Difference Equations, 2020, art.number:539. Available at: https://doi.org/10.1186/s13662-020-02992-4.

Karapinar, E., Fulga, A. & Rakocevic, V. 2020b. A discussion on a Pata type contraction via iterate at a point. Filomat, 34(4), pp. 1061-1066. Available at: https://doi.org/10.2298/FIL2004061K.

Ozgur, N. & Ta§, N. 2021. Pata Zamfirescu Type Fixed-Disc Results with a i Proximal Application. Bulletin of the Malaysian Mathematical Sciences Society, & 44, pp. 2049-2061. Available at: https://doi.org/10.1007/s40840-020-01048-w. ^

Ozgur, N. & Ta§, N. 2023. On S-metric spaces with some topological aspects. Electronic Journal of Mathematical Analysis and Applications, 11(2), pp. 1-8. Available at: https://doi.org/10.21608/ejmaa.2023.206319.1029.

Pata, V. 2011. A fixed point theorem in metric spaces. Journal of Fixed Point Theory and Applications, 10(2), pp. 299-305. Available at: https://doi.org/10.1007/s11784-011-0060-1.

Priyobarta, N., Rohen, Y., Thounaojam, S. & Radenovic, S. 2022. Some remarks on «-admissibility in S-metric spaces. Journal of Inequalities and Applications, 2022, art.number:34. Available at: I https://doi.org/10.1186/s13660-022-02767-3. N

Roy, S., Chakraborty, P., Ghosh, S., Saha, P. & Choudhury, B.S. 2024. Investigation of a fixed point problem for Pata-type contractions with respect to w- d distance. The Journal of Analysis, 32, pp. 125-136. Available at: "g

https://doi.org/10.1007/s41478-023-00612-4. «

Saleem, N., Abbas, M., Bin-Mohsin, B. & Radenovic, S. 2020. Pata type best ° proximity point results in metric spaces. Miskolc Mathematical Notes, 21(1), pp. 367-386. Available at: https://doi.org/10.18514/MMN.2020.2764.

ro

ro

CO

CL

CO

<u

^ Sedghi, S., Shobe, N. & Aliouche, A. 2012. A generalization of fixed point

theorems in S'-metric spaces. Matematicki vesnik, 64(3), pp. 258-266 [online]. Available at: http://www.vesnik.math.rs/landing.php?p=mv123.cap&name=mv12 309 [Accessed: 7 February 2024].

Yahaya, S., Shagari, M.S. & Ali, T.A. 2023. Multivalued hybrid contraction that involves Jaggi and Pata-type inequalities. Mathematical Foundations of Computing. Available at: https://doi.org/10.3934/mfc.2023045. c¿ Zamfirescu, T. 1972. Fix point theorems in metric spaces. Archiv der Mathe-

É matik, 23. Available at: https://doi.org/10.1007/BF01304884.

O o

Mapeos de Zamfirescu bajo condición tipo Pata: resultados y o aplicación a Ecuación integral

g Deep Chanda, Yumnan Rohenba, Nicola Fabiano0

LU a

> Matemáticas, Imphal, Manipur, República de la India

Instituto Nacional de Tecnología Manipur, Departamento de

¡ft b Universidad de Manipur, Departamento de Matemáticas,

Imphal, Manipur, República de la India

Universidad de Belgrado, Instituto de Ciencias Nucleares 'Vinca" ■ Instituto Nacional de la República de Serbia, (A Belgrado, República de Serbia, autor de correspondencia

^ CAMPO: matemáticas

s¿ TIPO DE ARTÍCULO: artículo científico original

>o

Resumen:

x

^ Introducción/objetivo: Las asignaciones tipo Pata y Zamfirescu

o se extienden más allá de los espacios métricos.

o Métodos: Se emplea el concepto de mapeo de Zamfirescu tipo

Pata en el marco de espacios S-métricos. Resultados: Se han establecido una serie de resultados correspondientes. Además, los resultados obtenidos se emplean para resolver una ecuación integral.

Conclusión: Los mapeos de tipo S-Pata y Zamfirescu tienen puntos fijos únicos.

Palabras claves: contracción tipo Pata, mapeo de Zamfirescu, espacio S-métrico, punto fijo.

Отображения Замфиреску в условиях типа Пата: результаты и применение в итегральных уравнениях

Дип Чанда, Юмнам Роэнба, Никола Фабианов

точка.

о

а Национальный технологический институт Манипура, о

Департамент математики, г Имфал, Республика Индия I

б Университет Манипура, Департамент математики, г. Имфал, Республика Индия

в Белградский университет, Институт ядерных исследований «Винча» - Институт государственного значения для Республики пз

Сербия, г Белград, Республика Сербия, корреспондент о-

РУБРИКА ГРНТИ: 27.25.17 Метрическая теория функций, 27.39.15 Линейные пространства, снабженные топологией порядком и другими структурами ВИД СТАТЬИ: оригинальная научная статья

Резюме:

Введение/цель: Отображения типа Паты и Замфиреску о. выходят за пределы метрических пространств. <я

"О

Методы: В статье применяется концепция отображе- §

(Л

о

ния Замфиреску типа Паты в рамках S-метрических пространств.

Результаты: Был подтвержден ряд соответствующих результатов. Полученные результаты были использованы в решении интегральных уравнений.

Выводы: Отображения типа S-Pata и Замфиреску имеют | ф

уникальные неподвижные точки. §

■

го

Ключевые слова: сжатие типа Паты, отображение типа го Замфиреску, S-метрическое пространство, неподвижная

(Я ф

о о

(Я

СР

резултати и примена на интегралну ^дначину е

Замфиреску пресликава^а под условима типа Пата:

Дип Чанда, Jумнам Руинба, Никола Фабиано и

б,а 1—1/ н;г\пп

ф

Национални институт за технологи]у Манипура, Одсек математике, Имфал, Манипур, Република Инди]а го

б Универзитет Манипура, Одсек математике,

ОБЛАСТ: математика

КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад

N

Имфал, Манипур, Република Инди]а

Универзитет у Београду, Институт за нуклеарне науке "Винча" - а

Национални институт Републике Срби]е, Београд, Република "о

Срби]а, аутор за преписку <5

о

а

в

(M

о >

(M о (M

Сажетак:

Увод/цил>: Пресликаваъа типа Пата и Замфиреску су про-

ширена изван метричких простора.

Методе: Применен jе концепт Замфиреску пресликаваъа

типа Пата у оквиру S-метричких простора.

Резултати: Утвр^ен jе низ одговараjуfiих исхода. Затим

су се доб^ени резултати користили за решаваъе инте-

ш гралнеjедначине.

з Закъучак: Пресликава^а типа S-Пата и Замфиреску има]у

§ jединствене непокретне тачке.

< Къучне речи: контракци}а типа Пата, пресликавак>е типа

Замфиреску, S-метрички простор, непокретна тачка.

х ш

>_ Paper received on: 27.01.2024. ^ Manuscript corrections submitted on: 07.06.2024. Paper accepted for publishing on: 08.06.2024.

© 2024 The Authors. Published by Vojnotehnicki glasnik/Military Technical Courier (http://vtg.mod.gov.rs, http://BTr.M0.ynp.cp6}. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ^ (http://creativecommons.org/licenses/by/3.0/rs/).

CD

S2 >o z

X Ш I—

о

о >