REVISITING AND REVAMPING SOME NOVEL RESULTS IN F-METRIC SPACES Текст научной статьи по специальности «Математика»

REVISITING AND REVAMPING SOME NOVEL RESULTS IN F-METRIC SPACES

Zoran D. Mitrovica, Mudasir Younisb, Miloje D. Rajovicc

a University of Banja Luka, Faculty of Electrical Engineering, Banja Luka, Republic of Srpska, Bosnia and Herzegovina, e-mail: [email protected], corresponding author, ORCID iD: https://orcid.org/0000-0001-9993-9082

b University Institute of Technology-RGPV, Department of Applied Mathematics, Bhopal, M.P, India, e-mail: [email protected], ORCID iD: https://orcid.org/0000-0001-5499-4272

c University of Kragujevac, Faculty of Mechanical Engineering, Kraljevo, Republic of Serbia, e-mail: [email protected], ORCID iD: https://orcid.org/0000-0002-7574-3832

DOI: 10.5937/vojtehg69-29615; https://doi.org/10.5937/vojtehg69-29615 FIELD: Mathematics

ARTICLE TYPE: Original scientific paper Abstract:

Introduction/purpose: This article establishes several new contractive conditions in the context of so-called F-metric spaces. The main purpose was to generalize, extend, improve, complement, unify and enrich the already published results in the existing literature. We used only the property (F1) of Wardowski as well as one well-known lemma for the proof that Picard sequence is an F-Cauchy in the framework of F-metric space.

Methods: Fixed point metric theory methods were used.

Results: New results are enunciated concerning the F-contraction of two mappings S and T in the context of F— complete F-metric spaces.

Conclusions: The obtained results represent sharp and significant improvements of some recently published ones. At the end of the paper, an example is given, claiming that the results presented in this paper are proper generalizations of recent developments.

Key words: F-metric space, F-contraction, fixed point.

Introduction and preliminaries

It is exactly one hundred years since S. Banach (Banach, 1922) proved the famous principle of contraction in his doctoral dissertation. Since then, many researchers have been trying to generalize that significant result in

many directions. In one direction, new classes of metric spaces were created and the renowned results were extended to these spaces. Among them, b-metric and F-metric spaces stand out. The former ones were introduced by Bakhtin (Bakhtin, 1989) and Czerwik (Czerwik, 1993) and the latter were recently introduced by Jleli and Samet (Jleli & Samet, 2018). Not that these two cases of spaces are intangible. Namely, there is a b-metric space that is not F-metric, and vie versa, there is an F-metric that is not b-metric. Note that convergence, Cauchyness and completeness of both types of spaces are defined for ordinary metric spaces. Also, it is worth mentioning that b-metric and F-metric do not have to be continuous functions with two variables as is the case with ordinary metric. In both types of spaces, a convergent sequence is a Cauchy and it has a unique limit. This is what they have in common with ordinary metric spaces. The continuity of mapping in both classes of spaces is sequential, i.e., the same as in ordinary metric spaces. Let us now list the definitions of each of the mentioned types of spaces. For more new details on F-metric spaces and new developments in the metric fixed point theory, one can see some noteworthy papers (Asif et al, 2019), (Aydi et al, 2019), (Derouiche & Ramoul, 2020), (Jahangir et al, 2021), (Kirk & Shazad, 2014), (Mitrovic et al, 2019), (Salem et al, 2020), (Som et al, 2020), (Vujakovic et al, 2020), (Vujakovic & Radenovic, 2020), (Younis et al, 2019a), (Younis et al, 2019b).

Definition 1. ((Bakhtin, 1989), (Czerwik, 1993)) Let X be a nonempty set and s > 1 be a given real number. A function db : X x X ^ [0, is said to be a b-metric with the coefficient s if for all x, y, z e X the following conditions are satisfied:

(db1) db (x, y) = 0 if and only if x = y

(d62) db (x, y) = db (y, x)

(d63) db (x, y) < s [db (x, z) + db (z, y)].

Let F be the set of functions f : (0, ^ (-to, +to) satisfying the following conditions:

Fi) f is non-decreasing,

F2) For every sequence {tn} c (0, +to), we have

lim tn = 0 if and only if lim f (tn) = -to.

n—n—

Definition 2. (Jleli & Samet, 2018) Let X be a (nonempty) set. A function dF : X x X ^ [0, +to) is called a F-metric on X if there exists (f, a) e Fx [0, +to) such that for all x, y e X the following conditions hold:

(dF 1) dF (x, y) = 0 if and only if x = y. (dF2) dF (x, y) = dF (y, x).

(dF3) For every N e N, N > 2 and for every [ui}^=1 c X with (ui,uN) = (x, y), we have

dF (x,y) > 0 yields f (dF (x,y)) < f dF (ui,ui+i)j +

a.

In this case, the pair (X, dF) is called a F-metric space.

Wardowski (Wardowski, 2012) considered a nonlinear function F : (0, +to) ^ +to) with the following characteristics: (F1) F is strictly increasing. (F2) F2) above.

(F3) There exists l e (0,1) such that limt^0+ i'F (t) = 0. Wardowski (Wardowski, 2012) called the mapping T : X ^ X, defined on a metric space (X, d), an F-contraction if there exist t > 0 and F satisfying (F1)-(F3) such that

t + F (d (Tx, Ty)) < F (d (x, y)) whenever d (Tx, Ty) > 0.

The authors in (Asif et al, 2019) take B = {F : (0, +to) ^ +to) : F satisfies F1) and F2)}.

In 2019, A. Asif et al., (Asif et al, 2019) formulated and proved the fixed-point and common fixed-point results for single-valued Reich-type and Kannan-type F-contractions in the setting of F-metric spaces:

Theorem 1. Suppose (f, a) e Bx[0, +to) and (X, dF) is an F-complete F-metric space. Let S, T : X ^ X be self-mappings. Suppose there exist F e F and t > 0 such that

t + F (df (Sx, Ty)) < F (a ■ df (x, y) + b ■ dF (x, Sx) + c ■ df (y, Ty)) (1)

for a, b, c e [0,. + to) such that a + b + c < 1 with

min {df (Sx, Ty), df (x, y), df (x, Sx), df (y, Ty)} > 0,

for all (x, y) e X x X. Then S and T have at most one common fixed point in X.

Corollary 1. Suppose (f, a) eBx[0, and (X, dF ) is an F-complete F -metric space Let S, T : X ^ X be self-mappings. Suppose that k e [0,1), there exist F e F and t > 0 such that

t + F (df (Sx, Ty)) < F (J- (df (x, Sx) + dF (y, Ty))) (2)

with min {dF (Sx, Ty), dF (x, y), dF (x, Sx), dF (y, Ty)} > 0, for all (x, y) e X x X. Then S and T have at most one common fixed point in X.

By replacing S with T, the authors obtained the following result for single mapping.

Corollary 2. Suppose (f, a) e Bx[0, +œ) and (X, dF ) is an F-complete F-metric space Let T : X ^ X be self-mapping. Suppose that for k e [0,1), there exist F e F and t > 0 such that

t + F (df (Tx, Ty)) < F (k (df (x, Tx) + df (y, Ty))) (3)

with min {dF (Tx, Ty), dF (x, Tx), dF (y, Ty)} > 0, for all (x, y) e X x X. Then T have at most one fixed point in X.

Definition 3. Let (X, dF) be an F-complete F-metric space and S, T : X ^ X be self-mappings. Suppose that a + b + c < 1 for a, b, c e [0, +rc>). Then the mapping T is called a Reich-type F-contraction on B (x0, r) ç X if there exist F e F and t > 0 such that for all x, y e B (x0, r)

t + F (df (Sx, Ty)) < F (a ■ df (x, y) + b ■ df (x, Sx) + c ■ df (y, Ty)). (4)

Theorem 2. Suppose (f, a) e Bx[0, +œ) and (X, dF) is an F-complete F-metric space. Let T be a Reich-type F-contraction on B (x0,r) ç X. Suppose that for x0 e X and r > 0, the following conditions are satisfied:

(a) B (x0, r) is F-closed,

(b) df (x0,xi) < (1 - A) r, for xi e X and A = ,

(c) There exist 0 < e < r such that f ( (1 - Afc+1) r) < f (e) - a, where

k e N.

Then S and T have at most one common fixed point in B (x0, r).

Taking S = T in Theorem 2, the authors in ((Asif et al, 2019), Corollary 3.) obtained the following result for single mappings.

Corollary 3. Suppose (f, a) e Bx[0, +ro), (F,t) e Bx (0, , (X, ) is an F-complete F-metric space and T : X ^ X is a self-mapping. Suppose that a + b + c < 1 for a, b, c e [0, +rc>). Suppose that for x0 e X and r > 0, the following conditions are satisfied:

(a) B (x0, r) is F-closed,

(b) t+F (df (Sx, Ty)) < F (a ■ (x, y) + b ■ (x, Sx) + c ■ dr (y, Ty))for all x, y e B (x0, r),

(c) df (xo, xi) < (1 - A) r, for xi e X and A = f+,

(c) There exist 0 < e < r such that f ( (1 - Afc+1) r) < f (e) - a, where

k e N.

Then T has at most one fixed point in B (x0, r).

Corollary 4. Suppose (f, a) e Bx[0, +ro), (F,t) e Bx (0, +ro) , (X, df) is an F-complete F-metric space. Let S, T : X ^ X be a self-mappings and k e [0,1). Suppose that for x0 e X and r > 0, the following conditions are satisfied:

(a) B (x0, r) is F-closed,

(b) t+F (df (Sx, Ty)) < F (a ■ df (x, y) + b ■ df (x, Sx) + c ■ df (y, Ty))for all x, y e B (x0, r),

(c) dF (x0, xi) < (1 - A) r, for xi e X and A = ,

(c) There exist 0 < e < r such that f ( (1 - Afc+1) r) < f (e) - a, where

k e N.

Then S and T have at most one common fixed point in B (x0, r).

Further in the same paper ((Asif et al, 2019), Definitions 6, 8, Theorem 5, Corollary 5.), the authors gave the following:

Definition 4. Let (X, dF) be a metric space. Let CB (X) be the family of all non-empty closed and bounded subsets of X. Let H : CB (X) x CB (X) ^ [0, +œ) be a function defined by

H (A, B) = max < supD (x, B), supD (y, AU , (5)

I xeA y€B I

where D (x, B) = inf {dF (x,y) : y e B}. Then H defines a metric on CB (X) called the Hausdorff-Pompeiu metric induced by dF.

Definition 5. Let (X, dF) be an F-metric space. Suppose F e B and H : CB (X) x CB (X) ^ [0, +rc>) be the Hausdorff-Pompeiu metric function defined in Definition 2. A mapping T : X ^ CB (X) is known as a set-valued Reich-type contraction if there is some t > 0 such that

2t + F (H (Tx, Ty)) < F (a ■ dT (x, y) + b ■ dF (x, Sx) + c ■ dT (y, Ty)) (6)

for (x, y) e X x X and a, b, c e [0, +rc>) such that a + b + c < 1.

Theorem 3. Let (X, dF) be an F-complete F-metric space and (f, a) e Bx[0, +rc>). If the mapping T : X ^ CB (X) is a set-valued Reich-type F-contraction such that F is right continuous, then T has a fixed point in X.

Corollary 5. Suppose (f, a) e Bx[0, +rc>) and (X, dF) is an F-complete F-metric space. Let T : X ^ CB (X) be a Reich-type F-contraction such that F is right continuous. Suppose that for k e [0,1), there exist F e B and t > 0 such that

t + F (H (Tx, Ty)) < F (J (dF (x, Tx) + dF (y, Ty))) (7)

with min {H (Tx, Ty), dF (x, Tx), dF (y, Ty)} > 0 for all (x, y) e X x X. Then T has a fixed point in X.

In the sequel, we will use the following two results:

Lemma 1. ((Mitrovicetal, 2019), Lemma 1.) Let (X,db) (resp. (X,dF) be a

r

b-metric (resp. F-metric) space and {x™}^ the sequence in it such that

db (x„, xn+i) < A ■ db (x„-i, x„) (resp. df (x„, xn+i) < A ■ df (xn-i, xn)),

(8)

for all n e N, where A e [0,1). Then {xn}+=1 is a db-Cauchy sequence in (X, db) (resp.dF-Cauchy sequence in (X, dF)).

Lemma2. Let {x„+l}raeNu{0} = {Tx4raenu{0} = {T™x0}„eNu{0} ,T°x0 =

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

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

df (xn,xn+i) = df (xm,xm+i) < df (xm_i,xm) < ... < df (xn,xn+i),

which is a contradiction. □

Some improved results

Firstly, since F : (0, +to) ^ (-to, +to) it follows that (1) is possible only if dF (Sx, Ty) > 0 where x,y e X. Also, the condition (F1) yields that a ■ dF (x, y) + b ■ dF (x, Sx) + c ■ dF (y, Ty) > 0 for all x, y e X for which dF (Sx, Ty) > 0. This means that at least of a, b, c e [0, +to) must be distinct of 0. Now we can improve the formulation of Theorem 1 and all its corollaries from (Asif et al, 2019) and give new proofs as the following.

Theorem 4. Suppose (f, a) e B x [0, +to) and (X, dF) is an F-complete F-metric space. Let S, T : X ^ X be self mappings. Suppose there exist a strictly increasing function F : (0, +to) ^ (-to, +to) and t > 0 such that

df (Sx, Ty) > 0 yields

t + F (df (Sx, Ty)) < F (a ■ df (x, y) + b ■ df (x, Sx) + c ■ df (y, Ty)), (9)

for a, b, c e [0, +to) such that a2 + b2 + c2 > 0 and a + b + c < 1.

Then S and T have at most one common fixed point in X, if at least one of

the mappings S or T is continuous.

Proof. Already, we first eliminate the function F. Indeed, from (9) if

df (Sx, Ty) > 0 follows

dF (Sx, Ty) < a ■ dF (x, y) + b ■ dF (x, Sx) + c ■ dF (y, Ty), (10)

where a, b, c e [0, +to), a2 + b2 + c2 > 0 and a + b + c < 1. Further, we give the proof in several steps: The step 1.

The point x is a fixed of S if and only if it is a fixed point of T. Let Sx = x and Tx = x. Putting x = y = x in (10) we get

df (x, Tx) = df (Sx, Tx) < a ■ df (x, x) + b ■ df (x, Sx) + c ■ df (x, Tx)

= a ■ 0 + b ■ 0 + c ■ dF (x, Tx),

i.e., (1 - c) dF(x,Tx) < 0. Since, c e [0,1) we obtain the contradiction. Therefore, Tx = x.

Conversely, let x = Tx and Sx = x. In this case, we get

(Sx, x) = (Sx, Tx) < a ■ (x, x) + b ■ (x, Sx) + c ■ (x, x)

= a ■ 0 + b ■ df (x, Sx) + c ■ 0,

i.e., (1 - b) dF (Sx,x) < 0 which is a contradiction, because b e [0,1). Hence, it follows that Sx = x. The step 2.

The uniqueness of a possible common fixed point for S and T. Let x = y be two common fixed points for S and T. Then putting in (10) x = x and y = y we get:

df (Sx, Ty) < a ■ df (x, y) + b ■ df (x, Sx) + c ■ df (y, Ty),

i.e., (1 - a) ■ dF (x, y) < 0. Which is a contradiction with x = y. This means that a possible common fixed point for S and T is unique. The step 3.

In this step, we shall prove the existence of at least one common fixed point of S and T.

Therefore, suppose that x0 is an arbitrary point and define a sequence

|x„} by

x2n+1 = Sx2n and x2n+2 = Tx2n+1,

for n e Nu {0} . It is clear that df («2^+1,^2^+2) > 0 and df (x2n+3,x2n+2 ) > 0 for all n e Nu {0} . Now, by (10) we further get (x = x2n, y = x2n+1)

df (x2n+1,x2n+2) < a-df (x2n,x2n+1)+b-df (®2n, ^n+O+C-df (x2n+1, x2n+2)

i.e., df (x2n+1,x2n+2) < k ■ df (x2n,x2n+1) where k = f+ e [0,1). And similar df (x2n+3,x2n+2) < k ■ df (x2n+2,x2n+1). Hence, for all n e N we have

df (xn, xn+1) < k ■ df (x„_1, x„) < df (x„_1, x„).

According to Lemmas 9 and 10, we have that the sequence {xn} is a dF-Cauchy in an F-complete F-metric space (X, dF) and xn = xm whenever n = m. This further means that there is (unique) x* e X such that

xn ^ x* as n ^

Firstly, let S be continuous. Then x2n+1 = Sx2n ^ Sx* = x* since in each F-metric space the subsequence of each convergent sequence

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converges to the unique limit. Now, we will prove that also Tx* = x*. Indeed, if Tx* = x* then by using (10) with x = y = x* we get

dF (x*, Tx*) = dF (Sx*, Tx*) < a-dF (x*, x*)+Mf (x*, Sx*)+c-dF (x*, Tx*)

= a-dF(x*,x*)+6-dF(x*,Sx*)+c-dF(x*,Tx*) = a-0+6-0+c-dF(x*,Tx*).

Finally, we obtain that (1 - c) ■ dF (x*,Tx*) < 0 which is a contradiction because we suppose dF (x*, Tx*) = 0.

If the mapping T is continuous, the proof is similar. The theorem is completely proved. □

Remark 1. Our Theorem 11 generalizes, improves, complements and unifies the corresponding Theorem 3 from (Asif et al, 2019) in several directions. First of all, it is worth to notice that some parts of the proof for Theorem 3 are doubtful. Namely, the authors in their proof use that F-metric dF is a continuous function with two variables (dF (xn, yn) ^ dF (x, y) if dF (xn,x) ^ 0 and dF (yn,y) ^ 0), which is not case. Also, it is clear that the function F in their Theorem 3 and in both Corollaries 1 and 2 is superfluous.

The next two corollaries follows from our Theorem 11.

Corollary 6. Suppose (f, a) e B x [0, +to) and (X, dF) is an F-complete F-metric space. Let S, T : X ^ X be self mappings. Suppose there exist strictly increasing function F : (0, +to) ^ (-to, +to) and t > 0 such that

dF (Sx, Ty) > 0 yields

for k e [0,1).

By replacing S with T, we get the following result for single mapping:

Corollary 7. Suppose (f, a) e B x [0, +to) and (X, dF) is an F-complete F-metric space. Let T : X ^ X be self mapping. Suppose there exist strictly increasing function F : (0, +to) ^ (-to, +to) and t > 0 such that

(11)

dT (Tx, Ty) > 0 yields

t + F (dF (Tx, Ty)) < F^2(dF (x, Tx) + cdT (y,Ty))^ , (12) for k G [0,1).

Then T has at most one fixed point in X, if it is continuous. Remark 2. Now we give the following Important Notice:

It is useful to note that the other results from (Asif et al, 2019) can be repaired and supplemented in the same or similar way. It should also be said that the results on Hausdorff-Pompeiu metric given in (Asif et al, 2019) are dubious. This will be discussed in another of our papers.

The immediate consequences of our Theorem 11 are the following new contractive conditions that complement the ones given in (Collaco & Silva, 1997), (Rhoades, 1977) for usual metric spaces. For more contractive conditions in the framework of metric spaces see (Ciric, 2003), (Consentino & Vetro, 2014), (Dey et al, 2019), (Karapinar, et al), (Piri & Kumam, 2014), (Salem et al, 2020), (Wardowski & Dung, 2014). In the sequel we will obtain several new contractive conditions in the framework of F-metric spaces.

Corollary 8. Suppose (f, a) gBx [0, +œ) and (X, dF ) is an F-complete F-metric space. Let S,T : X ^ X be self mappings. Suppose there exist ti > 0 such that dF (Sx, Ty) > 0 yields

ti + dF (Sx, Ty) < a ■ dF (x, y) + b ■ dF (x, Sx) + c ■ dF (y, Ty), (13)

for a, b, c g [0, such that a2 + b2 + c2 > 0 and a + b + c < 1.

Then S and T have at most one common fixed point in X, if one of the

mappings S or T is continuous.

Corollary 9. Suppose (f, a) eBx [0, and (X, dF ) is an F-complete F-metric space. Let S,T : X ^ X be self mappings. Suppose there exist t2 > 0 such that dF (Sx, Ty) > 0 yields

T2 + dF (Sx, Ty) < a ■ dF (x,y), (14)

for a G [0,1) .

Then S and T have at most one common fixed point in X, if one of the mappings S or T is continuous.

Corollary 10. Suppose (f, a) eBx [0, +œ) and (X, dF ) is an F-complete F-metric space. Let S, T : X ^ X be self mappings. Suppose there exist t3 > 0 such that dF (Sx, Ty) > 0 yields

T3 + dF (Sx, Ty) < b ■ dF (x, Sx) + c ■ dF (y, Ty), (15)

for b, c g [0, +œ) such that b2 + c2 > 0 and b + c < 1.

Then S and T have at most one common fixed point in X, if one of the

mappings S or T is continuous.

Corollary 11. Suppose (f, a) g B x [0, +œ) and (X, dF) is an F-complete F-metric space. Let S, T : X ^ X be self mappings. Suppose there exist t4 > 0 such that dF (Sx, Ty) > 0 yields

t4--1-<--1-, (16)

dF (Sx, Ty) b ■ dF (x, Sx) + c ■ dF (y, Ty)

for b, c g [0, +œ) such that b2 + c2 > 0 and b + c < 1.

Then S and T have at most one common fixed point in X, if one of the

mappings S or T is continuous.

Corollary 12. Suppose (f, a) g B x [0, +œ) and (X, dF) is an F-complete F-metric space. Let S, T : X ^ X be self mappings. Suppose there exist t5 > 0 such that dF (Sx, Ty) > 0 yields

T5 - , , * , + dF (Sx, Ty) <-- 1

dF (Sx, Ty) ' b ■ dF (x, Sx) + c ■ dF (y, Ty)

+ b ■ dF (x, Sx) + c ■ dF (y, Ty), (17)

for b, c g [0, +œ) such that b2 + c2 > 0 and b + c < 1.

Then S and T have at most one common fixed point in X, if at least one of

the mappings S or T is continuous.

Corollary 13. Suppose (f, a) g B x [0, +œ) and (X, dF) is an F-complete F-metric space. Let S, T : X ^ X be self mappings. Suppose there exist t6 > 0 such that dF (Sx, Ty) > 0 yields

11

T6 + --, , ^ „ < -

1 - exp (df (Sx, Ty)) 1 - exp (b ■ df (x, Sx) + c ■ df (y, Ty)) '

(18)

for b, c e [0, +rc>) such that b2 + c2 > 0 and b + c < 1.

Then S and T have at most one common fixed point in X, if one of the

mappings S or T is continuous.

Proof. As each of the functions F (r) = r, i = 173, F4 (r) = -1, F5 (r) = -1 + r, F6 (r) = 1_exp(r) is strictly increasing on (0, +rc>) , the proof immediately follows by our Theorem 11 and their corollaries. □

Example 1. Finally, we give the following simple example that support our Theorem 11 with S = T. Suppose that X = {2n + 1: n e N}. Define the dF-metric given by the following

, , , f 0 if x = y dF (x,y) = { elx_yl if x = y.

Let F (r) = -e_r and T : X ^ X is defined by

3 if n e {1,2}

T (2n + 1>=' 2n — 1 if n > 3.

It is clear that dF is a F-metric and F is strictly increasing on (0, +rc>). All the conditions of Theorem 11 are satisfied. Indeed, putting in equation (9)

b = c = 0, we get for x = y :

T - e_|Tx_Ty <-o ■ e_|x_y|,

i.e., e_|Tx_Ty| > o ■ e_|x_y|. Taking x = 2n + 1,y = 2m + 1,n = m we further obtain e_|2n_2m| > o ■ e_|2n_2m|. Since n = m this means that there exists o e [0,1) such that (9) holds true, i.e., T has a unique fixed point in X = {2n + 1 : n e N} , which is x = 3. Note that limr^+0 F (r) = -1, then Theorem 3 from (Asif et al, 2019) is not applicable here. This shows that our results are proper generalizations of the ones from (Asif et al, 2019).

Conclusion

In this article, we obtained several new contractive conditions in the framework of F-metric spaces. Our results improve, extend, complement, generalize, and unify various recent developments in the context of F-metric spaces. An example shows that the main results of (Asif et al, 2019) are not applicable in our case. We think that this is a useful contribution in the framework of F-contraction introduced by D. Wardowski.

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References

Asif, A., Nazam, M., Arshad, M. & Kim, S.O. 2019. F-Metric, F-contraction and Common Fixed-Point Theorems with Applications. Mathematics, 7(7), 586. Available at: https://doi.org/10.3390/math7070586.

Aydi, H., Karapinar, E., Mitrovic, Z.D. & Rashid, T. 2019. A remark on "Existence and uniqueness for a neutral differential problem with unbounded delay via fixed point results F-metric spaces". Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (RACSAM), 113, pp.3197-3206. Available at: https://doi.org/10.1007/s13398-019-00690-9.

Bakhtin, I.A. 1989. The contraction mapping principle in quasimetric spaces. Func. An., Gos. Ped. Inst. Unianowsk, 30, pp.26-37.

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.

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.

Consentino, M. & Vetro, P. 2014. Fixed point result for F-contractive mappings of Hardy-Rogers-Type. Filomat, 28(4), pp.715-722. Available at: https://doi.org/10 .2298/FIL1404715C.

Czerwik, S. 1993. Contraction mappings in b-metric spaces. Acta Mathematica et Informatica Universitatis Ostraviensis, 1, pp.5-11 [online]. Available at: https: //dml.cz/handle/10338.dmlcz/120469 [Accessed: 20 November 2020].

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

Derouiche, D. & Ramoul, H. 2020. New fixed point results for F-contractions of Hardy-Rogers type in b-metric spaces with applications. Journal of Fixed Point Theory and Applications, 22(art.number:86). Available at: https://doi.org/10.1007/ s11784-020-00822-4.

Dey, L.K., Kumam, P. & Senapati, T. 2019. Fixed point results concerning a - F - contraction mappings in metric spaces. Applied General Topology, 20(1), pp.81-95. Available at: https://doi.org/10.4995/agt.2019.9949.

Jahangir, F., Haghmaram, P. & Nourouzi, K. 2021. A note on F-metric spaces. Journal of Fixed Point Theory and Applications, 23(art.number:2). Available at: https://doi.org/10.1007/s11784-020-00836-y.

Jleli, M. & Samet, B. 2018. On a new generalization of metric spaces. Journal of Fixed Point Theory and Applications, 20(art.number:128). Available at: https: //doi.org/10.1007/s11784-018-0606-6.

Karapinar, E., Fulga, A. & Agarwal, R. 2020. A survey: F-contractions with related fixed point results. Journal of Fixed Point Theory and Applications, 22(art.number:69). Available at: https://doi.org/10.1007/s11784-020-00803-7.

Kirk, W.A. & Shahzad, N. 2014. Fixed Point Theory in Distance Spaces. Springer International Publishing Switzerland.

Mitrovic, Z.D., Aydi, H., Hussain, H. & Mukheimer, A. 2019. Reich, Jumgck, and Berinde comon fixed point results on F-metric spaces and an application. Mathematics, 7(5), 387. Available at: https://doi.org/10.3390/math7050387.

Piri, H. &Kumam, P. 2014. Some fixed point theorems concerning F-contraction in complete metric spaces. Fixed Point Theory and Applications, art.number:210. Available at: https://doi.org/10.1186/1687-1812-2014-210.

Rhoades, B.E. 1977. A comparison of various definitions of contractive mappings. Trans. Amer. Math. Soc., 226, pp.257-290. Available at: https://doi.org/10.1 090/S0002-9947-1977-0433430-4.

Saleem, N., Iqbal, I., Iqbal, B. & Radenovic, S. 2020. Coincidence and fixed points of multivalued F-contractions in generalized metric space with applications. Journal of Fixed Point Theory and Applications, 22(art.number:81). Available at: https://doi.org/10.1007/s11784-020-00815-3.

Som, S., Bera, A. & Dey, L.K. 2020. Some remarks on the metrizability of F-metric spaces. Journal of Fixed Point Theory and Applications, 22(art.number:17). Available at: https://doi.org/10.1007/s11784-019-0753-4.

Vujakovic, J., Mitrovic, S., Pavlovic, M. & Radenovic, S. 2020. On recent results concerning F-contraction in generalized metric spaces. Mathematics, 8(5), 767. Available at: https://doi.org/10.3390/math8050767.

Vujakovic, J. & Radenovic, S. 2020. On some F-contraction of Piri-Kumam-Dung type mappings in metric spaces. Vojnotehnicki glasnik/Military Technical Courier, 68(4), pp.697-714. Available at: https://doi.org/10.5937/vojtehg68-27385.

Wardowski, D. 2012. Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory and Applications, art.number:94. Available at: https://doi.org/10.1186/1687-1812-2012-94.

Wardowski, D. & Van Dung, N. 2014. Fixed points of F-weak contractions on complete metric spaces. Demonstratio Mathematica, 47(1), pp.146-155. Available at: https://doi.org/10.2478/dema-2014-0012.

Younis, M., Singh, D. & Goyal, A. 2019a. A novel approach of graphical rectangular b-metric spaces with an application to the vibrations of a vertical heavy hanging cable. Journal of Fixed Point Theory and Applications, 21(art.number:33). Available at: https://doi.org/10.1007/s11784-019-0673-3.

Younis, M., Singh, D. & Petrusel, A. 2019b. Applications of Graph Kannan Mappings to the Damped Spring-Mass System and Deformation of an Elastic Beam. Discrete Dynamics in Nature and Society, art.ID:1315387. Available at: https://doi.org/10.1155/2019/1315387.

ПЕРЕСМОТР И УЛУЧШЕНИЕ НЕКОТОРЫХ НОВЫХ РЕЗУЛЬТАТОВ В F-МЕТРИЧЕСКИХ ПРОСТРАНСТВАХ

Зоран Д. Митровича, Мудасир Йоунис6, Милое Д. Раовичв а Университет в г. Баня-Лука, электротехнический факультет, г Баня-Лука, Республика Сербская, Босния и Герцеговина, корреспондент б Университетский технологический институт РГПВ, кафедра прикладной математики, г Бхопал, М.П, Индия

в Крагуевацкий университет, машиностроительный факультет, г Кралево, 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 устанавливается несколько новых сжимающих условий в контексте так называемых F-метрических пространств. Основная цель статьи заключается в обобщении, расширении, улучшении, дополнении, объединении ранее опубликованных результатов в существующей литературе. Мы использовали только свойство (F1) Вардовского, а также одну хорошо известную лемму для доказательства того, что последовательность Пикара тождественна F-Коши в рамках F-метрического пространства.

Методы: В статье применены методы метрической теории неподвижной точки.

Результаты: Сформулированы новые результаты о F-сжатии двух отображений S и Т в контексте F-полных F-метрических пространств.

Выводы: Полученные результаты значительно улучшены по сравнению с некоторыми недавно опубликованными результатами. В заключении приводится пример, доказывающий, что результаты, представленные в данной статье, являются соответствующим обобщением недавних результатов.

Ключевые слова: F-метрическое пространство, F-сжатие, неподвижная точка.

РЕВИЗША И ПОБО^ША^Е НЕКИХ НОВИХ РЕЗУЛТАТА У Т-МЕТРИЧКИМ ПРОСТОРИМА

Зоран Д. Митрови^3, Мудасир иоунисб, Мило]е Д. Рарви^8 а Универзитету Ба^о] Луци, Електротехнички факултет, Ба^а Лука, Република Српска, Босна и Херцеговина, аутор за преписку б Универзитетски институт за технологи]у - РГПВ, Оде^е^е за приме^ену математику, Бопал, М. П, Инди]а

в Универзитету Крагу|евцу, Машински факултет, Кра^ево, Република Срби]а

ОБЛАСТ: математика

ВРСТА ЧЛАНКА: оригинални научни рад

Сажетак:

Увод/цил>: Ова] рад успоставъа неколико нових контрак-тивних услова у контексту такозваних Т-метричких простора. Главни циъ }е генерализаци]а, проширеъе, побоъ-шак>е, допуна и об}едик>ек>е веЬ доби}ених резултата у посто]еПо] литератури. КоришПено ¡е само своство (Е1) Вардовског, као и ¡една добро позната лема за доказ да ¡е Пикаров низ Т-Коши]ев у оквиру Т-метричког простора.

Методе: КоришПене су методе метричке теорбе фиксне тачке.

Резултати: Об}авъени су нови резултати у вези са Т-контракци}ама за два пресликаваъа у оквиру Т-комплетних Т-метричких простора.

Закъучак: Доби}ени резултати представъа}у знача]на по-боъшаъа, као и праву генерализаци]у неких недавно об}а-въених резултата, што показу}е пример наведен на кра}у рада.

Къучне речи: Т-метрички простор, Е-контракци]а, фиксна тачка.

Paper received on / Дата получения работы / Датум приема чланка: 28.11.2020. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 01.03.2021.

Paper accepted for publishing on / Дата окончательного согласования работы / Датум коначног прихвата^а чланка за об]ав^ива^е: 03.03.2021.

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