SOME NEW OBSERVATIONS ON FIXED POINT RESULTS IN RECTANGULAR METRIC SPACES WITH APPLICATIONS TO CHEMICAL SCIENCES Текст научной статьи по специальности «Математика»

SOME NEW OBSERVATIONS ON FIXED POINT RESULTS IN RECTANGULAR METRIC SPACES WITH APPLICATIONS s TO CHEMICAL SCIENCES

o Mudasir Younisa, Nicola Fabianob, Zaid M. Fadailc

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or Zoran D. Mitrovicd, Stojan N. Radenovice

a University Institute of Technology - RGPV, g Department of Applied Mathematics, Bhopal, M.P, India,

0 e-mail: mudasiryouniscuk@gmail.com, ^ ORCID iD: https://orcid.org/0000-0001-5499-4272 ° b Independent researcher, Rome, Italy,

1 e-mail: nicola.fabiano@gmail.com, ft ORCID iD: https://orcid.org/0000-0003-1645-2071

: Thamar University, Faculty of Education, Department of Mathematical Science, Thamar, Republic of Yemen, e-mail: zaid.fadail@tu.edu.ye, ORCID iD: https://orcid.org/0000-0003-0899-0056 d University of Banja Luka, Faculty of Electrical Engineering, Banja Luka, Republic of Srpska, Bosnia and Herzegovina, ^ e-mail: zoran.mitrovic@etf.unibl.org, corresponding author,

^ ORCID iD: https://orcid.org/0000-0001-9993-9082

e University of Belgrade, Faculty of Mechanical Engineering, Department of Mathematics, Belgrade, Republic of Serbia, x e-mail: radens@beotel.rs,

w ORCID iD: https://orcid.org/0000-0001-8254-6688

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DOI: 10.5937/vojtehg69-29517; https://doi.org/10.5937/vojtehg69-29517 FIELD: Mathematics

ARTICLE TYPE: Original scientific paper

Abstract:

Introduction/purpose: This paper considers, generalizes and improves recent results on fixed points in rectangular metric spaces. The aim of this paper is to provide much simpler and shorter proofs of some new results in rectangular metric spaces.

Methods: Some standard methods from the fixed point theory in generalized metric spaces are used.

Results: The obtained results improve the well-known results in the literature. The new approach has proved that the Picard sequence is Cauchy in rectangular metric spaces. The obtained results are used to prove the existence of solutions

to some nonlinear problems related to chemical sciences. Finally, an open ques- ° tion is given for generalized contractile mappings in rectangular metric spaces. °°

Key words: fixed point, rectangular metric space, contractive map, Green function.

Introduction and Preliminaries

Conclusions: New results are given for fixed points in rectangular metric spaces with application to some problems in chemical sciences. i

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It is well known that the Banach contraction principle (Banach, 1922) is J one of the most important and attractive results in nonlinear analysis and mathematical analysis in general. The whole fixed point theory is a significant subject in different fields: geometry, differential equations, informatics, physics, economics, engineering, and many others. After solutions are | guaranteed, numerical methodology is established to obtain the approximated solution. The fixed point of functions depends heavily on considered spaces defined using intuitive axioms. In particular, variants of generalized metric spaces are proposed, e.g. partial metric space, b-metric, partial b-metric, extended b-metric, rectangular metric, rectangular b-metric, G-metric, Gb-metric, S-metric, Sb-metric, cone metric, cone b-metric, fuzzy metric, fuzzy b-metric, probabilistic metric, etc. For more details on all variants of generalized metric spaces, see (Budhia et al, 2017), (Collaco & Silva, 1997).

In this paper, we will discuss some results recently established in (Alsu-lami et al, 2015) and (Budhia et al, 2017). Firstly, we give the basic notion of a rectangular metric space (g.m.s or RMS by some authors). £

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Definition 1. Let X be a nonempty set and let dr : X x X ^ [0, satisfy the following conditions: for all x,y e X and all distinct u,v e X each of them different from x and y.

(i) dr (x, y) = 0 if and only if x = y,

(ii) dr (x, y) = dr (y, x),

(iii) dr (x, y) < dr (x, u) + dr (u, v) + dr (v, y) (quadrilateral inequality). ® Then the function dr is called a rectangular metric and the pair (X, dr) is J called a rectangular metric space (RMS for short).

Notice that the definitions of convergence and Cauchyness of the sequences in rectangular metric spaces are the same as the ones found in

9

the standard metric spaces. Also, a rectangular metric space (X, dr) is complete if each Cauchy sequence in it is convergent. Samet et al. (Samet et oj al, 2012) introduced the concept of a - ^-contractive mappings and proved the fixed point theorems for such mappings. In (Karapinar, 2014), Karap-inar gave contractive conditions to obtain the existence and uniqueness of a fixed point of a - ^ contraction mappings in rectangular metric spaces. Salimi etal. (Salimi etal, 2013) introduced modified a - ^ contractive mappings and obtained some fixed point theorems in a complete metric space. Alsulami etal. (Alsulami etal, 2015) established some fixed point theorems u for a - ^-rational type contractive mappings in a rectangular metric space. Let ^ be the family of all functions ^ : [0, +rc>) ^ [0, +rc>) such that ^

is nondecreasing and ^ ^n (t) < for each t > 0. Obviously, if ^ e

n= 1

then ^ (t) < t for each t > 0.

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Definition 2. (Salimi et al, 2013) Let T be a self mapping on a metric space (X, dr) and let a, n : X x X ^ [0, +rc>) be two functions. It is called an a-admissible mapping with respect to n if a (x, y) > n (x, y) implies that

a (Tx, Ty) > n (Tx, Ty) for all x, y e X.

^ If n (x, y) = 1 for all x, y e X, then T is called an a-admissible mapping.

0 It is called a triangular a-admissible mapping if for all x, y, z e X holds:

1 (a (x, y) > 1 and a (y, z) > 1) implies a (x, z) > 1. Otherwise, a rectangular metric space (X, dr) is a-regular with respect

to n if for any sequence in X such that a (xn,xn+1) > n (xn,xn+1) for all n e N and xn ^ x as n ^ then a (xn, x) > n (xn, x). ty For more details on a triangular a-admissible mapping, see (Karapinar et al, 2013), pages 1 and 2. In this paper, we will use the following result:

Lemma 1. (Karapinaretal, 2013), Lemma 7. Let T be a triangular a-admissible mapping. Assume that there exists x0 e X such that a (x0, Tx0) > 1. Define the sequence {xn} by xn = Tnx0. Then

a (xm, xn) > 1 for all m, n e NU {0} with m < n. In (Budhia et al, 2017), the authors proved the following result:

Theorem 1. Let (X, dr) be a Hausdorff and complete rectangular metric space, and let T : X ^ X be an a-admissible mapping with respect to

10

n. Assume that there exists a continuous function ^ e ^ such that

x,y e X,a (x, y) > n (x, y) implies dr (Tx, Ty) < ^ (M (x, y))

M (x, y) = dr (x, y), dr (x, Tx), dr (y, Ty) , dr (x, Tx) dr (y Ty)

1 + dr (x, y)

dr (x, Tx) dr (y, Ty) 1 1 + dr (Tx, Ty) j

Also, suppose that the following assertions hold:

Then T has a periodic point a e X and if a (a, Ta) > n (a, Ta) holds for each periodic point, then T has a fixed point. Moreover, if for all x,y e F (T), we have a (x, y) > n (x, y), then the fixed point is unique.

Taking n (x, y) = 1 for x,y e X, the authors obtained the following corollary:

dr (x, Tx) dr (y, Ty) 1 1 + dr (Tx, Ty) j

Also, suppose that the following assertions hold:

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1. there exists x0 e X such that a (x0,Tx0) > n (x0,Tx0), ^

2. for all x,y,z e X, (a (x,y) > n (x,y) and a (y,z) > n (y,z)) implies » a (x, z) > n (x, z),

3. either T is continuous or X is a-regular with respect to n.

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Corollary 1. Let (X,dr) be a Hausdorff and complete rectangular metric space, and let T : X ^ X be an a-admissible mapping. Assume that J there exists a continuous function ^ e ^ such that

x,y e X, a (x, y) > 1 implies dr (Tx, Ty) < ^ (M (x, y)) m

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where

M (x, y) = max{ dr (x, y), dr (x, Tx), dr (y, Ty), dr ^^ dr (y)Ty), S

1 + dr (x, y) E

1. there exists x0 e X such that a (x0, Tx0) > 1, | 2. for all x, y, z e X (a (x, y) > 1 and a (y, z) > 1) implies a (x, z) > 1,

a>~ 3. either T is continuous or (X, dr) is a-regular.

Then T has a periodic point a e X and if a (a, Ta) > 1 holds T has a S fixed point. Moreover, if for all x, y e F (T), we have a (x, y) > 1, then the fixed point is unique.

Further, taking a (x, y) = 1 for x, y e X authors obtained the following corollary:

< Corollary 2. Let (X, dr) be a Hausdorff and complete rectangular metric space, and let T : X ^ X be an a-admissible mapping. Assume that o there exists a continuous function ^ e ^ such that

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x, y g X, 1 > n (x, y) implies dr (Tx, Ty) < ^ (M (x, y))

where

dr (x, Tx) dr (y, Ty)

M (x, y) = max < dr (x, y), dr (x, Tx), dr (y, Ty)

1 + dr (x, y)

dr (x, Tx) dr (y, Ty) | 1 + dr (Tx, Ty) J-

Also, suppose that the following assertions hold:

^ 1. there exists x0 e X such that 1 > n (x0, Tx0),

2. for all x, y, z e X (1 > n (x, y) and 1 > n (y, z)) implies 1 > n (x, z), For ^ (t) = kt, 0 < k < 1 then the authors obtained

Corollary 3. Let (X, dr) be a Hausdorff and complete rectangular metric space, and let T : X ^ X be an a-admissible mapping with respect to n. Assume that

x, y e X, a (x, y) > n (x, y) implies dr (Tx, Ty) < kM (x, y),

where

M (x, y) = max | dr (x, y), dr (x, Tx), dr (y, Ty) , d dx ,

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dr (x, Tx) dr (y,Ty)\ 8

1 + dr (Tx, Ty) )• '

Also, suppose that the following assertions hold:

{dr {xn(k) + l, xm(k)-l) } , {dr (xn(k) + 1,xm(k)+1) } . Lemma 3. Let {xn+1 }nGNU{0} = {Txn}n£NU{0} = {Tnx0}n€NU{0} ,T°x0 =

x0 be a Picard sequence in a rectangular metric space (X, dr) induced by the mapping T : X ^ X and the initial point x0 e X. If dr (xn,xn+1) < dr (xn-1,xn) for all n e N then xn = xm whenever n = m.

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1. there exists x0 e X such that a (x0,Tx0) > n (x0,Tx0),

2. for all x,y,z e X (a (x,y) > n (x,y) and a (y,z) > n (y,z)) implies a (x, z) > n (x, z), £

3. either T is continuous or (X, dr) is a-regular.

Then T has a periodic point a e X and if a (a,Ta) > n (a,Ta) holds, T has a fixed point. Moreover, if for all x,y e F (T), we have a (x, y) > n (x, y), then the fixed point is unique.

The following two lemmas are a rectangular metric space modification of a result which is well known in the metric space, see, e.g, (Radenovic et al, 2012), Lemma 2.1. Many known proofs of fixed point results in rectangular £ metric spaces become much more straightforward and shorter using both lemmas. Also, in the proofs of the main results in this paper, we will use both lemmas: ^

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Lemma 2. (Kadelburg & Radenovic, 2014a), (Kadelburg & Radenovic, 2014b) Let (X,dr) be a rectangular metric space and let {xn} be a sequence in it with distinct elements (xn = xm for n = m). Suppose that

dr (xn, xn+1) and dr (xn, xn+2) tend to 0 as n ^ and that {xn} is not a £ Cauchy sequence. Then there exists e > 0 and two sequences {m (k)} and § {n (k)} of positive integers such that n (k) > m (k) > k and the following sequences tend to e as k ^ :

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{dr (xn(k),xm(k))} , {dr (xn(k)+1,xm(k))} , {dr {xn(k), xm(k)-^} , $

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Proof.Let xn = xm for some n, m e N with n < m. Then xn+1 = Txn = Txm = xm+1. Further, we get

dr (xn, xn+1) = dr (xTO, x^+1) < dr (xTO-1, xTO ) < ... < dr (xn,xn+1),

which is a contradiction. □

In some proofs, we will also use the following interesting as well as significant result in the context of rectangular metric spaces:

Proposition 1. (Kirk & Shahzad, 2014), Proposition 3. Suppose that {qn} is a Cauchy sequence in a rectangular metric space (X, dr) and suppose

dr (qn, q) = 0. Then dr (qn,p) = dr (q,p) for all p e X. In

particular, {qn} does not converge to p if p = q.

Main results

In this section, we generalize and improve Theorem 2 and all its corollaries. The obtained generalizations extend the result in several directions. Namely, we will use only one function a : X x X ^ [0, +rc>) instead of two a and n as in (Budhia et al, 2017), Definition 2.3. and Definition 3.1. This is possible according to the (Mohammadi & Rezapour, 2013), Page 2, after Theorem 1.2. Note that we assume neither that the rectangular metric space is Hausdorff, nor that the mapping dr is continuous.

The authors (Alsulami et al, 2015), page 6, line 6+, say that the sequence {xn} in a rectangular metric space (X, dr) is a Cauchy if limn^+^ dr (xn,xn+k) = 0, for all k e N. However, it is well know that this claim is dubious. Therefore, we also improve the proof that the sequence {xn} is Cauchy

Our first new result in this paper is the following:

Theorem 2. Let (X, dr) be a complete rectangular metric space and let T : X ^ X be a triangular a-admissible mapping. Assume that there exists continuous function ^ e ^ such that

x, y e X, a (x, y) > 1 implies dr (Tx, Ty) < ^ (M (x, y)), (1)

where

M (x, y) = max < dr (x, y), dr (x, Tx), dr (y, Ty),

dr (x, Tx) dr (y, Ty) 1 + dr (x, y)

dr (x, Tx) dr (y,Ty)\ 8

1 + dr (Tx, Ty) J. '

Also, suppose that the following assertions hold:

1. there exists x0 e X such that a (x0,Tx0) > 1,

2. either T is continuous or (X, d) a-regular.

Then T has a fixed point. Moreover, if

for all x,y e F (T) implies a (x, y) > 1, then the fixed point is unique. Proof. Given x0 e X such that

a (x0,Tx0) > 1 . (2)

Define a sequence {xn} in X by xn = Txn-1 = Tnx0 for all n e N. If xk+1 = xk for some k e N, then Txk = xk, i.e., xk is a fixed point of T and the proof is finished. From now on, suppose that xn = xn+1 for all n e Nu {0}. Using (2) and the fact that T is an a-admissible mapping, we have

a (x1, x2) = a (Tx0,Tx1) > 1.

By induction, we get

In the first step, we will show that the sequence {dr (xn, xn+1)} is nonin-creasing and dr (xn,xn+1) ^ 0 as n ^ From (1), recall that

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a (xn,xn+1) > 1 for all n e Nu {0} . ®

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dr (xn-1, xn ) dr (xn,xn+1) dr (x n 1 , xn ) dr (x n, xn+1) \ 1 + dr (xn-1 ,xn) , 1 + dr (xn,xn+1) J < max {dr (x,n-1,x,n) ,dr (x,n,x,n+1)} .

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Now from (3) follows

dr (x„,xra+i) < ^ (max {dr (xn-1,x„), dr (xn,xn+i)}). (4)

If max{dr (xn-1,xn) , dr (xn,xn+1)} = dr (xn, xn+1), we get a contradiction. Indeed, (4) implies

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dr (xn, xn+1) < ^ (dr (xn, xn+1)) < dr (xn, xn+1).

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g Therefore, we get that dr (xn, xn+1) < dr (xn-1 ,xn). This means that

° there exists dr (xn, xn+1) = dr > 0. If dr > 0, then from (3) follows

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< ^ (max {d*, d* }) < d*

which is a contradiction. Hence limn^+œ dr (xn,xn+1) = 0.

Further, we will also show that limn^+œ dr (xn, xn+2) = 0. Firstly, we have that a (xn-1,xn) > 1, i.e., a (xn-1,xn+1) > 1, because T is a triangular fj a-admissible mapping. Therefore,

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dr (xn, xn+2) = dr (Txn-1, Txn+1) < ^ (M (xn-1, xn+1))

M (xn-1, xn+1) = max {dr (xn-1, xn+1), dr (xn-1,xn) ,dr (xn+1, xn+2)

^ dr (xn-1, xn) dr (xn+1, xn+2) dr (xn-1, xn) dr (xn+1, xn+2) 1

1 + dr (xn-1, xn+1) , 1 + dr (xn, xn+2) J

Since d:(x'1+X(rX)!:(1X++:)n+2) < dr (xn-1, xn) dr (xn+1, xn+2) and (S^2- < +r (xn-1, xn) dr (xn+1, xn+2) we get that

M (xn-1, xn+1) < max {dr (xn-1, xn+1), dr (xn-1,x„), dr (xn+1, xn+2), dr (xn-1, xn) dr (xn+1, xn+2)}

that is,

M (xn-1, xn+1) < max { dr (xn-1, xn+1), dr (xn-1,x„) , d^ (x„_1,x„)}

16

< max {dr (xn-1, xn+1) ,dr (x,n-1,xn)} . 8

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The last relation follows from the fact that dr (xn-1, xn) ^ 0 as n ^ g.

Hence, for some n1 e N, we have that

dr (xn,xn+2) < max {dr (xn-i ,xn+i) ,dr (xn-i,xn)} ,

\ f 0 ■ 00 ■ 0 1

^ M [xn(k),xm(k)) =maxl e, 0,0^ ——,—— }

v ^ " y 1 + e 1+ ej

Now, taking in (5) the limit as k ^ follows

e < ^ (e) < e,

dr (x*,Tx*) < dr (x*,xn) + dr (xn,xn+i) + dr (xn+i,Tx*),

whenever n > n2, taking the limit, we obtain dr (x*, Tx*) = 0, i.e. x* = Tx*, which is a contradiction.

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whenever n > n1. Since, dr (xn-1,xn) ^ 0 as n ^ it is not hard to check that also dr (xn, xn+2) ^ 0 as n ^ +rc>.

In order to prove that the sequence {xn} is a Cauchy one, we use Lemma 6. Namely, since according to Lemma 1, a{xn(k),xm(k)) > 1 if m (k) < n (k), then, by putting in (1) x = xn(k),y = xm(k), we obtain £

dr {xn(k)+1,xm(k) + ^ < ^ (M (xn(k),xm(k))) , (5) co

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where §■

M {xn(k),xm(k)) = °

= max{dr (xn(k),xm(k)) ,dr (xn(k),xn(k)+1) ,dr (xm(k),xm(k)+1) ,

dr ixn(k),xn(k)+1) dr (xm(k),xm(k)+0 dr {xn(k), xn(k)+1) dr {xm(k), xm(k)+\) 1+ dr (xn(k),xm(k)) ' 1 + dr (xn(k) + 1,xm(k)+1)

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which is a contradiction. The sequence {xn} is hence a Cauchy one. Since (X, dr) is a complete rectangular metric space, there exists a point x* e X such that xn ^ x* as n ^ +œ. If T is continuous, we get that xn+l = Txn ^ | Tx* as n ^ +m. Let Tx* = x*. Since dr (xn,xn+i) < dr (xn-i, xn) for all oo n e Nu {0}, then, according to Lemma 7, we have that all xn are distinct. 0 Therefore, there exists n2 e N such that x*,Tx* e {xn}n>n2. Further, by (iii) follows: e

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In the case that (X, dr) is a—regular, we get the following: Since a (xn, x*) > 1 for all n e N, then from (1) follows

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dr (Txn,Tx*) < ^ (M (xn,x*)) , (6)

where

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o tinuity of the function we get dr (x*,Tx*) < ^ (dr (x*,Tx*)) < dr * m if x* = Tx*., which is a contradiction. Hence, x* is a fixed point of T. £ Now, we show that the fixed point is unique if a (x, y) > 1 whenever

x,y g F (T). Indeed, in this case, by contractive condition (1), for such possible fixed points x, y we have

1 + dr (xn,x*) ' 1 + dr (xn+1,Tx*) j

By taking in (6) the limit as n ^ and by using Proposition 8 and the continuity of the function we get dr (x*,Tx*) < ^ (dr (x*,Tx*)) < dr (x*,Tx*)

dr (x,y) = dr (Tx, Ty) < ^ (M (x,y)), (7)

where

dr (x, Tx) dr (y, Ty) 1 + dr (x, y) ,

J M (x, y) = max <{ dr (x, y), dr (x, Tx), dr (y, Ty)

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dr (x, Tx) dr (y, Ty) 1 1 + dr (Tx, Ty) j

f w ^ 0 ■ 0 0 ■ 0 1 , . .

= ma^ dr (x, y), 0, 0, ^^, 1+0 f = dr (x,y). Hence, (7) becomes

dr (x, y) < ^ (dr (x, y)) < dr (x, y), which is a contradiction. The proof of Theorem 9 is complete. □

Remark 1. In the proof of case 2 on Page 96, the authors used the fact that the rectangular metric dr (see condition (3.12)) is continuous, which is not given in the formulation of (Budhia et al, 2017), Theorem 3.2.

18

By putting in (1) instead of M (x, y), one of the following sets

{dr (x,y)} , max {dr (x,y), dr (x, Tx) ,dr (y, Ty)} , max ; dr (x, Tx) dr (y, Ty) dr (x, Tx) dr (y, Ty) \

Then T has a fixed point. Moreover, if for all x,y e F (T), we have a (x, y) > 1, then the fixed point is unique.

Corollary 6. Let (X, dr) be a complete rectangular metric space and let T : X ^ X be a triangular a-admissible mapping. Assume that there exists a

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immediately follows as a consequence of Theorem 9. °

Corollary 4. Let (X, dr) be a complete rectangular metric space and let T : X ^ X be a triangular a-admissible mapping. Assume that there exists a continuous function ^ e ^ such that %

x,y e X, a (x,y) > 1 implies dr (Tx,Ty) < ^ (dr (x,y)). (8) $ Also, suppose that the following assertions hold:

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1. there exists x0 e X such that a (x0,Tx0) > 1,

2. either T is continuous or (X, dr) is a-regular.

Then T has a fixed point. Moreover, if for all x,y e F (T), we have a (x, y) > 1, then the fixed point is unique.

Corollary 5. Let (X, dr) be a complete rectangular metric space and let T : X ^ X be a triangular a-admissible mapping. Assume that there exists a continuous function ^ e ^ such that for x,y e X,

a (x, y) > 1 yields dr (Tx, Ty) < ^ (max {dr (x, y), dr (x, Tx), dr (y, Ty)}).

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2. either T is continuous or (X, dr) is a-regular.

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continuous function ^ e ^ such that for x, y e X, a (x, y) > 1

yields d (Tx,Ty) < , (max f ^rix+fl^' 1

(10)

Also, suppose that the following assertions hold:

1. there exists x0 e X such that a (x0, Tx0) > 1,

2. either T is continuous or (X, dr) is a-regular.

Then T has a fixed point. Moreover, if for all x, y e F (T), we have a (x, y) > 1, then the fixed point is unique.

In the book (Ciric, 2003), Ciric collected various contractive mappings in the usual metric spaces, see also (Rhoades, 1977) and (Collaco & Silva, 1997). The next three contractive conditions are well known in the existing literature:

• Ciric 1: Ciric's generalized contraction of first order: there exists k1 e [0,1) such that for all x, y e X holds:

w™ L , d (x,Tx) + d (y, Ty) d (x,Ty) + d (y, Tx) 1 d (Tx, Ty) < k1 max < d (x, y),-^-,-^-f .

(11)

• Ciric 2: Ciric's generalized contraction of second order: there exists k2 e [0,1) such that for all x, y e X holds:

d (Tx, Ty) < k2 max (d (x, y) , d (x, Tx), d (y, Ty), d(x-Ty) + d (y Tx)

2 (12)

In both cases, (X, d) is a metric space, T : X ^ X is a given self-mapping of the set X.

In (Ciric, 2003), Ciric introduced one of the most generalized contractive conditions (so-called quasicontraction) in the context of a metric space as follows:

• Ciric 3: The self-mapping T : X ^ X on a metric space (X, d) is called a quasicontraction (in the sense of Ciric) if there exists k3 e [0,1) such that for all x, y e X holds:

d (Tx, Ty) < k3 max {d (x, y), d (x, Tx), d (y, Ty), d (x, Ty), d (y, Tx)} .

(13)

Since,

d (x, Tx) + d (y, Ty) 2

and

< max {d (x, Tx), d (y, Ty)}

maxjdr (x,y), 2 ' 2

\ i/ rr1 \ a t t \ dr (x,Ty)+ dr (y,Tx) maxj dr (x, y), dr (x, Tx), dr (y, Ty),---

max {dr (x, y), dr (x, Tx), dr (y, Ty), dr (x, Ty), dr (y, Tx)} .

An open problem

A suggestion for further research - it is logical to ask the following question:

Problem 0.1. Let T be a modified triangular a-admissible mapping defined on a complete rectangular metric space (X, dr ) such that T is continuous or (X, dr) is a-regular. Show that T has a fixed point.

o cn I

oo s±

CL

'o

CO

d (x,Ty)+ d (y,Tx) < max {d x Ty) , d (y, Tx)} J

it follows that (11) implies (12) and (12) implies (13).

In (Ciric, 2003), Ciric proved the following result: o

ro o

Theorem 3. Each quasicontraction T on a complete metric space (X, d) g-has a unique fixed point (say) z. Moreover, for all x e X, the sequence {Tnx}+=0, T0x = x converges to the fixed point z as n ^

Now we can formulate the following notion and one open question:

Definition 3. Let (X, dr) be a rectangular metric space and let a : X x X ^ [0, +rc>) be a mapping. The mapping T : X ^ X is said to be a modified triangular a-admissible mapping if there exists a continuous function ^ e ^ such that

x,y e X, a (x,y) > 1 implies dr (Tx, Ty) < ^ (M (x,y)), (14) where M (x, y) is one of the sets:

dr (x, Tx) + dr (y, Ty) dr (x, Ty) + dr (y, Tx)'

U1

<u o ro cp

V)

o

<u E

JÈ

13

cS

o <u

CO CD

o cp ~o

CD X

CO £ CD CO .Q

o

CD

E o OT

Applications

In this section, we will focus on the applicability of the obtained results. An application to chemical sciences

Consider a diffusing substance placed in an absorbing medium between parallel walls such that ¿^¿2 are the stipulated concentrations at walls. Moreover, let Q(r) be the given source density and £(r) be the known absorption coefficient. Then the concentration x(r) of the substance under the aforementioned hypothesis governs the following boundary value problem

-k" + £(r)K = Q(r) ; r e [0,1] = I (1)

k(0) = ¿1, k(0) = ¿2,

Problem (1) is equivalent to the succeeding integral equation

K(r) = ¿1 + (¿2 - ¿1)r + / 8(r, ro) (fi(ro) - £(ro)K(ro)) , r e [0,1], (2)

0

where 6(r, ro) : [0,1] x R ^ R is the Green's function which is continuous and is given by

f , ( r(1 - ro) 0 < r < ro < 1, G(r,ro)n «7(1 - r) 0 < ro < r < 1 (3)

Suppose that C(I, R) = X is the space of all real valued continuous functions defined on I and let X be endowed with the rectangular b-metric dr defined by

dr(k, k*) = ||(k - k*)||,

where ||k|| = sup{|K(r)| : r e I}. Obviously (X,dr) is a complete rectangular metric space.

Let the operator £ : X ^ X be defined by

£K(r) = K(r) = ¿1 + (¿2 - ¿1)r + [ 8(r, ro) (Q(ro) - £(ro)x(ro)).

0

Then k* is a unique solution of (2) if and only if it is a fixed point of £. The subsequent Theorem is furnished for the assertion of the existence of a fixed point of £.

Also S is triangular a-admissible.

|S k(t) — S к*(т)|

= / Q(t, w) (Q(w) — S(w)k(w)) dm — h(r) J 0

— / Q(t,w)(Q(w) — S(w)k*(w)) dm

Jo

< / Q(t,w) l(n(m) — S(w)k(w)) — (Q(m) — S(m)x*(m))| dm 0

, o

r 1 <u

= 0(t,w) |S(w)k(w) — S(w)k*(w)| dm Jo

< 0(t,w) |k(w) — к*(w)| dm J0

< [ 9(t,w)\(k(w) — K*(m))\\dm

0J

<\\(k — k*)\\ sup Q(r,w)dm. re[o,i] Jo

Since /0 Q(t, m)dm = r—r2 and so supre[0 g /0 Q(t, m)dm = g. Hence for all k, k* g X, we obtain

8 _ 8 where

M(k, k*) = max {dr (k, k*) , dr (k, Tk) , dr (k*,Tk*)

o cn I

oc3 s±

CL

Theorem 4. Consider problem (2) and suppose that there exists p > 0 and a continuous function S (w) : I ^ R such that the following assertion holds:

a(x(w), K*(w)) > 1 0 < |S(w)k(w) — S(w)k*(w)| < k*(w) — k(w).

Then the integral equation (2) and, consequently, the boundary value problem (1) governing the concentration of the diffusing substance has a unique

solution in X. -2

(0 a o

Proof. Clearly, for k e X and r e I, the mapping S : X ^ X is well defined. ro

<u o (0

E <u .c o

o "ci

L

ro

U1

<u o ro

L

V)

o

-I—»

<u E

JÈ

13

cs

tn <13

o

L

TD

<13

0 £

^ II

re[o,i] ./o

£

0 Q(r, w)dw = ^—r- and so supr€[01] /0 Q(r, w)dw = 1. $

o

<13

, „ dr (k, k*) M (k, k*) § dr (^ k,z, k ) < —--- < —^--, oo

dr (x, Tx) dr (x*,Tx*) dr (x, Tx) dr (x*,Tx*)

1 + dr (x, x*) , ÏTdrjTx/TK*) Taking M(x, x*)) = 8, we obtain

dr(Tx,Tx*) < ^M(x, x*)^j

Hence, all the hypotheses of Theorem 2 are contented. We conclude that S has a unique fixed point x in X, which guarantees that the integral equation (2) has a unique solution and, consequently, the boundary value problem (1) has a unique solution. □

Application to a class of integral equations for an unknown function

We present the application of the existence of a fixed point for a generalized contraction to the following class of integral equations for an unknown function u:

u(t)= g(t)+ I x(t,z)f(z,u(z))dz, t G [a,b],

J a

(4)

where f : [a,b] x R ^ R, K : [a, b] x [a, b] ^ [0, to), g : [a,b] ^ R are the given continuous functions.

Let X be the set C[a, b] of real continuous functions defined on [a, b] and let dr: X x X ^ [0, to) be equipped with the metric defined by

dr(u,v) = sup lu(t) — v(t)l. (5)

a<t<b

One can easily verify that (X, dr) is a complete rectangular metric space. Let the self map T : X ^ X be defined by

Tu(t) = g(t) + I x(t, z)f (z, u(z))dz, t e [a, b], (6)

a

then u is a fixed point of T if and only it is a solution of (4). Also, we can easily check that T is triangular a-admissible. Now, we formulate the following subsequent theorem to show the existence of a solution of the underlying integral equation.

CiD

Theorem 5. Assume that the following assumptions hold:

(1) suPa<t<b ¡a lx(t,z)ldz < b-a;

(2) Suppose that for all x,y e R,

a(x(t),y(t)) > l=^lf(z,x) — f(z,y)l < 1 lx(t) — y(t)l

2'

Then integral equation (4) has a solution.

Proof. Employing conditions (1) - (2) along with inequality (4), we have

dr(Tui,TU2) = sup lTui(t) - TU2(t)l

a<t<b

= sup

a<t<b

= sup

atb

g(t)+ ! x(t,z)f (z,Ui(z))dz - (g(t)+ Î x(t,z)f (z,u2(z))dz

aa

x(t, z)f (z, ui(z)) - x(t, z)f (z, u2(z)))dz

dz

< sup l Î lx(t,z)ldz. Î f (z,ui(z)) - f (z,u2(z))

a< t< b a a

= { sup [ \x(t,z)\dz}.{ I f (z,ui(z)) - f (z,u2(z))

a<t<b a a

= { sup [ \x(t,z)\dz}.{ I f (z,ui(z)) - f (z,u2(z))

a<t<b a a

<{b-a, M10ui(z) - u

<

2(b - a) J a a<t<b

= \ sup lui(t) - u2 (t) \

2a t b

sup \ui(t) - u2(t)\dz

i.e dr(Tui,Tu2) = dr(ui,u2)) < M(u^'u2). Which amounts to say that

dr(Tui,Tu2) <

M (ui,u2)

where

M(ui, u2) = max{dr (ui,u2), dr (ui,Tui), dr (u2,Tu2) ,

o cn I

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CL

<d 'o

V)

E <u .c o

o '

(0 c o

■-4—'

ro o

"Œ

cp ro

Ul

<u o ro cp

V)

o

-I—»

<u E

JÈ

13

cs

o <u

tn <u

o cp ~o

<u

X

ro £ <u

V) .Q

o

<u E o OT

b

b

2

dr (ul ,Tu{) dr (u2, Tu2) dr (ul ,Tu\) dr (u2, Tu2) 1 1 + dr (ul,u2) , 1 + dr (Tul,Tu2) j'

Taking M(u-\_, u2)) = 2, the above inequality turns into

dr(Tui,Tu2) < ^(m(ui,u2)

Thus, all the hypotheses Theorem 2 are satisfied and we conclude that T has a unique fixed point x* in X, which amounts to say that integral equation (4) has a unique solution which belongs to X = C[a, b]. □

References

Alsulami, H.H., Chandok, S., Taoudi, M-A. & Erhan, I.M. 2015. Some fixed point theorems for (a, yJ-rational type contractive mappings. Fixed Point Theory and Applications, 2015(art.number:97). Available at: https://doi.org/10.1186/s13663-0 15-0332-3.

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

Budhia, L., Kir, M., Gopal, D. & Kiziltunc, H. 2017. New fixed point results in rectangular metric space and application to fractional calculus. Tbilisi Mathematical Journal, 10(1), pp.91-104 [online]. Available at: http://tcms.org.ge/Journals/TMJ /Volume10/Volume10_1/Abstract/abstract10_1_6.html [Accessed: 15 November 2020].

Collaco, P. & Silva, J.C.E. 1997. A complete comparison of 25 contraction conditions. Nonlinear Analysis: Theory, Methods & Applications, 30(1), pp.471-476. Available at: https://doi.org/10.1016/S0362-546X(97)00353-2.

Ciric, Lj. 2003. Some Recent Results in Metrical Fixed Point Theory. Belgrade: University of Belgrade.

Kadelburg, Z. & Radenovic, S. 2014a. Fixed point results in generalized metric spaces without Hausdorff property. Mathematical Sciences, 8(art.number:125). Available at: https://doi.org/10.1007/s40096-014-0125-6.

Kadelburg, Z. & Radenovic, S. 2014b. On generalized metric spaces: A survey. TWMS Journal of Pure and Applied Mathematics, 5(1), pp.3-13 [online]. Available at: http://www.twmsj.az/Files/Contents%20V.5,%20%20N.1,%20%202014/pp.3-13.pdf [Accessed: 15 November 2020].

Karapinar, E. 2014. Discusion on a-y Contractions on Generalized Metric Spaces. Abstract and Applied Analysis, 2014(art.ID:962784). Available at: http: //dx.doi.org/10.1155/2014/962784.

Karapinar, E., Kummam, P. & Salimi, P. 2013. On a-^-Meir-Keeler contractive ° mappings. Fixed Point Theory and Applications, art.number:94. Available at: https:

//doi.org/10.1186/1687-1812-2013-94.

Kirk, W.A. & Shahzad, N. 2014. Fixed Point Theory in Distance Spaces. Springer International Publishing Switzerland. ISBN: 978-3-319-10927-5.

Mohammadi, B. & Rezapour, Sh. 2013. On modified a-^-contractions. Journal e of Advanced Mathematical Studies, 6(2), pp.162-166 [online]. Available at: https:

//www.fairpartners.ro/upload_poze_documente/files/volumul%206, %20no.%202/1 2

4_Mohammadi.pdf [Accessed: 15 November 2020]. |

Radenovic, S., Kadelburg, Z., Jandrlic, D. & Jandrlic, A. 2012. Some results io

on weakly contractive maps. Bulletin of the Iranian Mathematical Society, 38(3), -fc

pp.625-645 [online]. Available at: http://bims.iranjournals.ir/article_229.html [Acce- ro ssed: 15 March 2018].

Rhoades, B.E. 1977. A comparison of various definitions of contractive mapp- <g

ings. Trans. Amer. Math. Soc., 226, pp.257-290. Available at: https://doi.org/10.1 £

090/S0002-9947-1977-0433430-4. £

o

Samet, B., Vetro, C. & Vetro, P. 2012. Fixed point theorem for a-^-contractive ~

type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4), e

pp.2154-2165. Available at: https://doi.org/10.1016/j.na.2011.10.014. |

Salimi, P., Latif, A. & Hussain, N. 2013. Modified a-^-contractive mappings with

applications. Fixed Point Theory and Applications, art.number:151. Available at: tj

https://doi.org/10.1186/1687-1812-2013-151. 0

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ПРИМЕЧАНИЯ К НЕКОТОРЫМ РЕЗУЛЬТАТАМ В Ф

ОБЛАСТИ НЕПОДВИЖНЫХ ТОЧЕК В ПРЯМОУГОЛЬНЫХ 5

МЕТРИЧЕСКИХ ПРОСТРАНСТВАХ, С ИХ ПРИМЕНЕНИЕМ £

В ХИМИИ х

Мудасир Йоуниса, Никола Фабиано6, Заид М. Фадаилв, Зоран Д. Митровичг, Стоян Н. Раденовичд Университетский технологический институт - РГПВ, кафедра прикладной математики, г Бхопал, М.П, Индия

6 независимый исследователь, г. Рим, Италия в Дамарский университет, педагогический факультет,

кафедра математических наук, г Дамар, Йеменская Республика г Университет в г. Баня-Лука, электротехнический факультет, |

г Баня-Лука, Республика Сербская, Босния и Герцеговина, корреспондент,

д Белградский университет, Машиностроительный факультет, г Белград, Республика Сербия

РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА:

27.25.17 Метрическая теория функций, 27.33.00 Интегральные уравнения, 27.39.29 Приближенные методы

функционального анализа ВИД СТАТЬИ: оригинальная научная статья

Резюме:

Введение/цель: В данной статье обсуждаются, суммируются и улучшаются недавние результаты о неподвижных точках в прямоугольных метрических пространствах. Целью данной статьи является представление гораздо более простых и коротких доказательств некоторых новых результатов в области прямоугольных метрических пространств.

Методы: В статье применены стандартные методы теории неподвижной точки в обобщенных метрических пространствах.

Результаты: Полученные результаты данного исследования улучшают известные результаты существующей литературы. Благодаря новому подходу доказана последовательность Коши-Пикара в прямоугольных метрических пространствах. Полученные результаты также используются для доказательства экзистенциальных решений некоторых нелинейных задач, относящихся к химическим наукам. В конце статьи задается открытый вопрос в связи с обобщенными сжатыми отображениями в прямоугольных метрических пространствах.

Выводы: В статье приведены новые результаты, касающиеся теории неподвижных точек в прямоугольных метрических пространствах, примененные в решении некоторых проблем в области химических наук.

Ключевые слова: неподвижная точка, прямоугольное метрическое пространство, сжатое отображение, функция Гоина.

НАПОМЕНЕ О НЕКИМ РЕЗУЛТАТИМА О НЕПОКРЕТНИМ "

ТАЧКАМА У ПРАВОУГАОНИМ МЕТРИЧКИМ ПРОСТОРИМА °°

САПРИМЕНАМАУХЕМШИ а

Мудасир иоунис3, Никола Фабиано6, Заид М. Фадаилв, о Зоран Д. Митрови^1", Сто]ан Н. Раденови^д

Универзитетски институт за технологи]у - РГПВ, ю

Оде^е^е за приме^ену математику, Бопал, М. П, Индша °

о

6 независни истраживач, Рим, Итали]а ю

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Оде^е^е за математичке науке, Тамар, Република иемен о

г Универзитет у Ба^о] Луци, Електротехнички факултет, & Ба^а Лука, Република Српска, Босна и Херцеговина, аутор за преписку

д Универзитет у Београду, Машински факултет, $

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ОБЛАСТ: математика .У

ВРСТА ЧЛАНКА: оригинални научни рад [=

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Увод/цил: У овом раду се разматра]у, уопштава]у и побол>-шава}у недавни резултати о непокретним тачкама у окви-ру правоугаоних метричких простора. Циъ овог рада }е да пружи много]едноставнще и краЯе доказе о неким новим ре-

зултатима у правоугаоним метричким просторима. <д

пресликавак>а у правоугаоним метричким просторима.

Закъучак: Дати су нови резултати за непокретне тачке у правоугаоним метричким просторима са применом на неке проблеме у хеми}ским наукама.

Къучне речи: непокретна тачка, правоугаони метрички простор, контрактивно пресликаваъе, Гринова функци]а.

Методе: Користе се стандардне методе из теори]е непокретне тачке у генерализованим метричким просторима. &

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Резултати: Доби}ени резултати поболшава}у добро по- § знате резултате у литератури. КористеПи нови приступ доказу}е се да }е Пикаров низ Коши]ев у оквиру правоугаоних метричких простора. Доби}ени резултати користе се за доказ егзистенци}е решена неких нелинеарних проблема ко\и се примеру у хеми}ским наукама. На кра}у се да}е

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the terms and conditions of the Creative Commons Attribution license

a; (http://creativecommons.org/licenses/by/3.0/rs/). ш

© 2021 Авторы. Опубликовано в "Военно-технический вестник / Vojnotehnicki glasnik / Military

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