Ordered b-metric spaces and Geraghty type contractive mappings Текст научной статьи по специальности «Математика»

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Abstract:

The paper shows a new approach to proving the recent fixed point results in ordered b-metric as well as ordered metric spaces, established by several authors, with much shorter and nicer proofs. An example is given to illustrate our results.

Key words: fixed point, b-metric, comparable, well order, Geraghty mapping, b-Cauchy, b-complete.

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ORDERED B-METRIC SPACES AND GERAGHTY TYPE CONTRACTIVE ? MAPPINGS

Sumit C. Chandoka, Mirko S. Jovanovicb, Stojan N. 8

Radenovic0 0

a Thapar University, School of Mathematics, Patiala, India e-mail: sumit.chandok@thapar.edu,

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ORCID iD: http://orcid.org/0000-0003-1928-2952 2

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University of Belgrade, Faculty of Electrical Engineering, Belgrade, CD

Republic of Serbia,

e-mail: msj@sbb rs, l co

ORCID iD: http://orcid.org/0000-0002-7760-1301 S

: University of Belgrade Faculty of Mechanical Engineering, Belgrade, «

Republic of Serbia, e-mail: radens@beotel.rs,

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

https://dx.doi.org/10.5937/vojtehg65-13266

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FIELD: Mathematics, Subject Classification: 47H10, 54H25, 46Nxx ARTICLE TYPE: Original Scientific Paper ARTICLE LANGUAGE: English ^

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Introduction and preliminaries

One of important generalizations of metric spaces are so-called b-metric spaces (type metric spaces by some authors). This concept was introduced by Bakhtin in 1989 and Czerwik in 1993.

Consistent with (Bakhtin, 1989, pp.26-37) and (Czerwik, 1993, pp.511), the following definition and results will be needed in the sequel.

Definition 1.1. (Bakhtin, 1989), (Czerwik, 1993) Let X be a (nonempty) set and s > 1 be a given real number. A function d : X x X ^ [0,+ro) is a b-metric if and only if, for all x, y, z e X, the following conditions are satisfied:

(b 1) d (x, y) = 0 if and only if x = y,

(b 2) d (x, y ) = d (y, x),

(b 3) d (x, z) < s(d (x, y)+d (y, z)).

The pair(X, d) is called a b-etric space.

It should be noted that the class of b-metric spaces is effectively larger than that of metric spaces, since a b-metric is a metric when s = 1. The following example shows that, in general, a b-metric does not necessarily need to be a metric, see also (Aghajani, et al, 2014), (Abbas, et al, 2016, pp.1413-1429), (Ansari, et al, 2016), (Ding, et al, 2016, pp.151-164), (DjukiC, et al, 2011), (Huang, et al, 2015a, pp.808-815), (Huaping, et al, 2015), (Huang, et al, 2015b, pp.800-807), (Hussain, et al, 2012), (Hussain, et al, 2013), (Jleli, et al, 2012, pp.175-192), (Jovanovic, et al, 2010), (Kadelburg, et al, 2015, pp.57-67), (Khamsi, Hussain, 2010, pp.3123-3129), (Parvaneh, et al, 2013), (Roshan, et al, 2015), (Roshan, et al, 2014, pp.229-245), (Zabihi, Razani, 2014).

Example 1.1. Let (X, d)be a metric space, andp(x, y) = (d(x, y))p, p >1 is a real number. Then p is a b-metric with s = 2 p~1, but p is not a metric on X.

Otherwise, for more concepts such as b-convergence, b-completeness, b-Cauchy sequence and b-closed set in b-metric spaces, we refer the reader to (Aghajani, et al, 2014, pp.941-960), (Abbas, et al, 2016, pp.1413-1429), (Ansari, et al, 2016), (Djukic, et al, 2011), (Huang, et al, 2015a, pp.808-815), (Huaping, et al, 2015), (Huang, et al, 2015b, pp.800-807), (Hussain, et al, 2012), (Hussain, et al, 2013), (Jleli, et al, 2012, pp.175-192), (Jovanovic, et al, 2010), (Kadelburg, et al, 2015, pp.57-67), (Khamsi, Hussain, 2010, pp.3123-3129), (Parvaneh, et al,

2013), (Roshan, et al, 2015), (Roshan, et al, 2014, pp.229-245), (Zabihi, Razani, 2014) and the references mentioned therein. Also, for the concepts such as partial order, comparable, well ordered, nondecreasing, increasing, dominated, dominating and other, we refer the reader to (Aghajani, et al, 2014, pp.941-960), (Abbas, et al, 2016, pp.1413-1429), (Ansari, et al, 2016).

The following three lemmas are very significant in the theory of a fixed point in the framework of metric and 6-metric spaces. Also, we use these in the proof of our main results.

Lemma 1.2. (Aghajani, et al, 2014, pp.941-960, Lemma 2.1) Let (X, d) be a b-metric space with s > 1, and suppose that {xn} and {yn} are b-convergent to x, y respectively, then we have

-1d(x, y)< lim„^d(xn, yn) < lim„^d(xn,yn) < s2d(x,y). (1.1)

s

In particular, if x = y, then we have limn^ro d (xn, yn) = 0. Moreover, for each z e X we have

-d(x,z)< limn^rod(xn,z)< limn^d(xn,z)< sd(x,z). (1.2)

s

Lemma 1.3. (Jovanovic, et al, 2010, Lemma 3.1) Let{yn}be a sequence in a b-metric space (X, d) with s > 1, such that

d (n, yn+- )< M (yn_i, yn) (1.3)

for some Me[0,—),and each n = 1,2,... Then {yn}is a b-Cauchy

s

sequence in a b-metric space (X, d).

Lemma 1.4. (Radenovic, et al, 2012, pp.625-645, Lemma 2.1), (Jleli, et al, 2012, pp.175-192, Lemma 2.1) Let (X, d) be a metric space and let {yn} be a sequence in X such that d(yn, yn+1) is nonincreasing and that

lim d (yn, yn+-) =°. (1.4)

If {y2n} is not a Cauchy sequence, then there exist an s>0 and two sequences {mk} and {nk} of positive integers such that the following four sequences tend to s+ when k ^ ro:

d(y2mk , y2nk d(y2mk , y2nk +1 d^2mk-1 , y2nk d^2mk-1 , y2nk +1 ).

Main results

Let ¥ be the family of all nondecreasing functions w:[0,+») ^ [0,+») such that limn^ w" (t) =0 for all t >0. If , then w(() < t for all t >0 and w(0) = 0. Our first result is the following:

Theorem 2.1. Let (X, be a partially ordered set and there exists a b-metric d on X such that (X, d) is a b-complete b-metric space. Suppose s >1 and f: X ^ X is an increasing mapping with respect to ^ such that there exists an element x0 e X with x0 ° fx0. Assume that

s • 1 +1Sd ( y) ■ d(fx, fy) < w(M (x, y)) + L • N(x, y) 1 + ^ d (x, fx)

for all comparable elements x, y e X, where L > 0,

M ( y )=max{d (x, y }

and

N (x, y) = min{d (x, .A), d (x, fy), d (y, fx), d (y, fy)}.

If

(1) f is continuous, or

(2) whenever{xn} is a nondecreasing sequence in X such that

xn ^ u e X, one has xn < u for all n e N,

then f has a fixed point. Moreover, the set of fixed points of f is well ordered if and only if f has one and only one fixed point.

Proof Suppose that xn ^ xn+1 for all n = 0,1,2,..., where xn+1 = fxn = fnx0. In this case, we have xn < xn+1 for all n = 0,1,2,.. Therefore, putting x = xn, y = xn+j in (2.1) we shall prove that

d (xn+1, xn+2 - d (xn , xn+1 ) (2.2)

s

for all n = 0,1,2,.... Indeed, then (2.1) becomes

S 1 + f (,) d (

1 + 2 d (xn ' xn+1) f f W Vl^

<W

max <

d (x x ) d (x"' X"+i)d (x"+i'X"+2)

v

1 + d(x"+1, x"+2 )

+ L rnn f(x^ x"+J,d (x", x"+2),d (x"+i, x"+i),d (x"+i, x"+2)}.

Since, i< 1 + sd(x"»x"+1),d(x",x"+((x"+1- x"+2) < d(x , x+1) and 1 + \d(xn, xn+1) 1 + d((+1, x"+2) " "

d (x«+1, x"+1)=0, we have s •d (x"+1, x"+2 )<¥(d (x", x"+1)) < d (x", x"+1) Hence, (2.2) follows.

Further, using (2.2), we have

d (2 x„, f2 x"+1) <1 d ((x", fx"+1) < ~r d (x"'x"+1)-s s

As -1- e [0,1), therefore by using Lemma 1.4, the sequence

s2 s

{f2xn}0 = {x2,x3,...} is a b-Cauchy sequence. This further implies that the sequence {fxn}0 = {x13x2,...} is a b-Cauchy sequence. Since (X,d) is b-Complete, {x"} b-converges to a point u e X.

(1) First, we suppose that f is continuous. Therefore, we have

u = lim x"+1 = lim fx" = f (lim x") = fu,

that is, u is a fixed point of f.

(2) Further, consider (2) of theorem holds. Using the assumption of (X, d, we have xn < u. Now, we show that fu = u. Firstly, we have

1 d (u, fu ) < d (u, xn+1) + d (fx", fu ). s

Now, using the assumption xn< u and inequality (2.1), we have

1 1 + 2 d (xn, xn+1)

- d fu )< d(u, x"+1)+ ( 2 (-) M(x", u)

s s(1 + sd (xn, u ))

1 + 1 d (( , xn+l )

2 " ""'■■ L ■ N(xn,u)

^(l + sd (xn, u ))

Since M(xn,u)^ 0 and N(xn,u0 as n , the result follows, i.e.,

fu = u.

From Theorem 2.1, we have the following result which is an improvement from the corresponding results (Theorems 2.7 and 2.8) of (Ansari, et al, 2016).

Corollary 2.1. Let (X, <) be a partially ordered set and there exists a b-metric d on X such that (X, d) is a b-complete b-metric space. Suppose s > 1 and f: X ^ X is an increasing mapping with respect to < such that there exists an element x0 e X with x0 < fX0. Assume that

1 +1Sd(x y) ■ d(fx, fy) < /(d (x, y)> M (x, y) + L ■ N(x, y)

(2.3)

1 + 2 d (x, fx)

for all comparable elements x, y e X,

11 where L > 0, /: [0,+») ^ [0, -) with /(tn) ^ - implies tn ^ 0,

s s

M ( y )=max{d (x, y ),%ff}

and

N (x, y) = min{d (x, fx), d (x, fy), d (y, fx), d (y, fy)}.

If

(1) f is continuous, or

(2) whenever {xn} is a nondecreasing sequence in X such that

xn ^ u e X, one has xn < u for all n e N,

then f has a fixed point. Moreover, the set of fixed points of f is well ordered if and only if f has one and only one fixed point.

Proof. Since /(d(x,y))<^, the condition (2.3) implies

J J + sdi^A. d fa fy ) < M (x, y ) + Z,. N (x, y ), 4)

1 + 2 d (x, fx ) (24)

where Z, = s • Z. On the similar lines of Theorem 2.1, we have the result.

On the similar lines of Theorem 2.1, we have the following result. Theorem 2.2. Let (X, be a partially ordered set and suppose that there exists a b-metric d on X such that (X, d) is a b-complete b-metric space (with parameter s > 1 ). Let f : X ^ X be an increasing mapping with respect to < such that there exists an element x0 e X with

x0 ^ fx0. Suppose that

s • d (fx, fy ) < J3(d (x, y ))M (x, y )+Z • N (x, y ), (2.5)

for all comparable elements x, y e X,

where Z > 0, ^ : [0,+œ) ^ [0,1) with f3(tn ) ^1 implies tn ^ 0,

ss

M (x, y )=max{d (x, y idfdSf}

and

N (x, y ) = min{d (x, jx\ d (x, fy ), d (y, fx), d (y, fy )}.

If

(1) f is continuous, or

(2) whenever {xn} is a nondecreasing sequence in X such that

xn ^ u e X, one has xn < u for all n e N,

then f has a fixed point. Moreover, the set of fixed points of f is well ordered if and only if f has one and only one fixed point. Proof The condition (2.5) implies

d (fx, fy ) < \ • M (x, y )+Zi • N (x, y ), (2.6)

s

for all comparable elements x, y e X, where Z, = Z > 0. The rest of the

s

proof is similar to Theorem 2.1.

Remark 2.1. Since the proofs of the main results in (Ansari, et al, 2016), (Zabihi, Razani, 2014) are strongly dependent of Lemma 1.2 of Aghajani et al. (Aghajani, et al, 2014, pp.941-960), it is too complex to deal

with them. Our approach in Theorems 2.1-2.2, as well as in Corollary 2.1 covers all the results of (Aghajani, et al, 2014, pp.941-960) without utilizing the lemma mentioned above. It is clear that our proofs are much shorter and nicer.

Also, it is not hard to see that the main results in (Abbas, et al, 2016, pp.1413-1429) have much shorter proofs by the application of our approach, that is, without using Lemma 1.2 of (Aghajani, et al, 2014, pp.941-960).

In the sequel, we consider all three results in the case where s = 1, that is, (X, d) is a standard metric space. Here we have to use Lemma 1.4 to obtain our results.

Theorem 2.3. Let (X,be a partially ordered set and suppose that there exists a metric d on X such that (X, d) is a complete metric space. Suppose f: X ^ X is an increasing mapping with respect to < such that there exists an element x0 e X with x0 < fX0. Assume that

1 +1d( ^) ■ dfy ) < W(M(x, y)) + L ■ N(x, y) 1 + ^ d (x, fx)

for all comparable elements x, y e X, where L > 0,

M ( y )=max{d (x, y }

N (x, y) = min{d (x, fx), d (x, fy), d (y, fx), d (y, fy)}.

(2.7)

and If

(1) f is continuous, or

(2) wheneve r{xn} is a nondecreasing sequence in X such that xn ^ u e X, one has xn ^ u for all n e N,

then f has a fixed point. Moreover, the set of fixed points of f is well ordered if and only if f has one and only one fixed point.

Proof First we suppose that xn ^ xn+1 for all n = 0,1,2,... Then, by taking x = xn, y = xn+j in (2.7), we get

1 +d(x"' x"+1) • d(xn+1, xn+2) < (xn, xn+1))+ L • N(, xn+1),

1+2 d (x"' x"+1)

where

M(xn,xn+1 )= max{d(xn,x^), d^"j-'*^)"+2)1 = d(xn,x^),

I 1 +d ((+1, xn+2) J

d (xn+1 , xn+2 ) < 1

because ~—tt-r < 1, and

1 + d(xn+1, xn+2 )

N(xn, xn+1 ) = min {d(xn , xn+1d(xn, xn+2 d(xn+1, xn+1 d(xn+1, xn+2 )} = 0.

Since

1 +,d(x"'x"+1) (xn,xn+1 ))<d(,xn+1) and N(xn,x^) (28)

1+2 d (x"' x"+1) '

becomes d x"+ 2 )< d (x" , x"+1 )>

i.e., d(xn,xn+1) is a decreasing sequence. Therefore, there exists r > 0

such that lim^» d(xn> xn+1) = r. Assume that r > 0, from (2.8), we have

1 + r 1

• r < r O — r < 0,

1 + 2 2

which is a contradiction. Hence lim

d ixn, xn+i )= 0.

Now, we suppose that the sequence {xn} is not a Cauchy sequence in a metric space {X, d ). By putting x = xm{k ), y = xn{k ) in (2.7), we obtain

1 + d))Xn(k)) )_)+i5xn(k)+i)< ¥{m(xm{k),xnW))+ L • N(xm{k),xnW), (2.9)

1 + 2 d(Cm{k)) xm{k)+1)

where

M(xm{k)) xn{k))

d{xm{k)) xm{k)+1 d^-{k)) x-

\ U\Xm{k ))Xm {k )+1 )' U\Xn{k ])Xn{k 1+1, = max\ d{)) xm{k)+1 ))-U-^-U-—

1 + d {x

and

\xm{k )+1) xn{k )+1, N{xm{k)) xn{k))

- min{¿/{xm,ka, xm(kv j), d(x,k), xn,k)+ ), d(xn{k), xm(k^ ), d(x,k), xn,k^ )}

m(k ) Xm(k )+1 ^ " \Xm(k ) Xn(k )+1 ^ " VXn(k ) Xm(k )+1 ^ " VXn(k > Xn(k)+1,

Now, letting to the limit in (2.9), as k ^ <», and using Lemma 1.4, we get

1 + s

-s < W(s)+ L ■ 0 < s O

2

s t \ 1

--s < y/(s) + L • 0 < s O — s < 0,

1 2

1 +— s 2

which is a contradiction. Hence the sequence {xn} is a Cauchy sequence. The rest of the proof is the same as in Theorem 2.1.

Corollary 2.2. Let (X,be a partially ordered set and suppose there exists a metricd on X such that (X, d) is a complete metric space. Suppose f: X ^ X is an increasing mapping with respect to ^ such that there exists an element x0 e X with x0 < fx0. Assume that

1 +/(x'^A • d(fx, Jy)<fi(d(x,y))M(x,y) + L • N(x,y) 2 10)

1 + 2 d (x, fx) (210)

for all comparable elements x, y e X,

where L > 0, J:[0,+c») ^ [0,1) with J(tn1" implies tn ^ 0, M (x, y) = max jd (x, y }

and

N (x, y) - min{d (x fx), d (x, fy), d (y, fx), d (y, fy)}

If

(1) f is continuous, or

(2) whenever {xn} is a nondecreasing sequence in X such that

xn ^ u e X, one has xn ^u for all n e N,

then f has a fixed point. Moreover, the set of fixed points of f is well ordered if and only if f has one and only one fixed point.

Proof. Since (x, y))< 1, the condition (2.10) implies that

1 +1d(x>yA • d(fx,fy)<M(x,y) + L• N(x,y).

(2.11)

On the similar lines of Theorem 2.3, we have the result.

1 + ^ d (x, fx)

Remark 2.2. It is not hard to see that both functions y and ¡3 in all results are superfluous. But in ournext result, the function ¡ is not superfluous.

Theorem 2.4. Let (X, <) be a partially ordered set and suppose that there exists a metric d on X such that (X, d) is a complete metric space. Let f: X ^ X be an increasing mapping with respect to < such that the- | re exists an element x0 e X with x0 < fx0. Suppose that

d (fx, fy) < ¡(d (x, y ))M (x, y)+L • N (x, y), (2.12)

for all comparable elements x, y e X, where L > 0, ¡:[0,+w) ^ [0,1) with ¡(tn) ^ 1" implies tn ^ 0,

M (x, y) =max{d (x, y), ff j

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N (x, y) = min{d (x, jx\ d (x, fy), d (y, fx), d (y, fy)}.

If

(1) f is continuous, or

(2) whenever {xn} is a nondecreasing sequence in X such that

xn ^ u e X, one has xn ^ u for all n e N,

then f has a fixed point. Moreover, the set of fixed points of f is well §

ordered if and only if f has one and only one fixed point. is

The following example support our theoretical result given with C

Corollary 2.1. ot

Example 2.3. Let X = {0,1,2} and define the partial order ^ on X -g

by ^ := {(0,0, (1,1) (2,2), (0,1)}. Consider the function f: X ^ X given as | f 0 = f 1 = 1, f 3 = 0 which is nondecreasing with respect to <. Let x0 =0. Hence fx0 = f0 = 1, so x0<fx0. Now, define the b-metric on X

by d(x, y) = (x - y)2 for all x, y e X. Then (X, d) is a b-complete b-metric space with s = 2. It is easy to verify that this example satisfies all the

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conditions of Corollary 2.1 for each ¡:[0,») ^ [0,-2) with tn ^ 0 +

whenever ¡(tn -2 -.

Finally, we formulate the following result (Geraghty fixed point theorem in the framework of a ¿»-complete ¿»-metric space):

Theorem 2.5. Let (X, d) be a b-complete b-metric space and let s >1. Suppose that a mapping f: X ^ X satisfies the condition

d ((x, fy )<3(d (x, y ))d (x, y), for all x, y e X, where 3 :[0,») ^ [0,1) with tn ^ 0 + whenever p(tn) ^ 1 - for each sequence tn e (0, <»).

Question. Prove or disprove Theorem 2.5.

References

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Aghajani, A., Abbas, M., & Roshan, J.R., 2014. Common fixed point of generalized weak contractive mappings in partially ordered-metric spaces. Math. Slovaca, 4, pp.941-960.

Ansari, A.H., Razani, A., & Hussain, N., 2016. Fixed and coincidence points for hybrid rational Geraghty contractive mappings in ordered b-metric spaces, to appear in. Int. J. Nonlinear Anal. Appl. Available at: http://dx.doi.org/10.22075/IJNAA.2016.453.

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Djukic, D., Kadelburg, Z., & Radenovic, S., 2011. Fixed points of Geraghty-type mappings in various generalized metric spaces. Abstr. Appl. Anal. Article ID 561245, 13 pages.

Huang, H., Vujakovic, J., & Radenovic, S., 2015a. A note on common fixed point theorems for isotone increasing mappings in ordered b-metric spaces. J. Nonlinear Sci. Appl., 8, pp.808-815.

Huang, H., Paunovic, Lj., & Radenovic, S., 2015b. On some new fixed point results for rational Geraghty contractive mappings in ordered b-metric spaces. J. Nonlinear Sci. Appl., 8, pp.800-807.

Huaping, H., Radenovic, S., & Vujakovic, J., 2015. On some recent coincidence and immediate consequences in partially ordered b-metric spaces. Fixed Point Theory Appl., 63.

Hui-Sheng, D., Imdad, M., Radenovic, S., & Vujakovic, J. ,2016. On some fixed point results in b-metric, rectangular and b-rectangular metric spaces. Arab J. of Math. Sci, pp.151-164. Available at: http://dx.doi.org/10.1016/j.ajmsc.2015.05.003.

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Jleli, M., Rajic, V.C., Samet, B., & Vetro, C., 2012. Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations. J. Fixed Point Theory Appl, pp.175-192. Available at: http://dx.doi.org/10.1007/s11784-012-0081-4.

Jovanovic, M., Kadelburg, Z., & Radenovic, S., 2010. Common fixed point results in metric-type spaces. Fixed Point Theory Appl. Article ID 978121, 15 pages.

Kadelburg, Z., Paunovic, Lj., & Radenovic, S., 2015. A note on fixed point theorems for weakly T-Kannan and weakly T-Chatterjea contractions in b-metric spacesa. Gulf Journal of Mathematics, 3(3), pp.57-67.

Khamsi, M.A., & Hussain, N., 2010. KKM mappings in metric type spaces. Nonlinear Anal., 73, pp.3123-3129.

Parvaneh, V., Roshan, J.R., & Radenovic, S., 2013. Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations. Fixed Point Theory Appl., 130.

Radenovic, S., Kadelburg, Z., Jandrlic, D., & Jandrlic, A., 2012. Some results on weakly contractive maps. Bulletin of the Iranian Mathematical Society, 38(3), pp.625-645.

Roshan, J.R., Parvaneh, V., Shobkolaei, N., Sedghi, S., & Shatanawi, W., 2013. Common fixed points of almost generalized (y ,P) s - contractive mappings in

ordered b-metric spaces. Fixed Point Theory Appl., 159.

Roshan, J.R., Parvaneh, V., Radenovic, S., & Rajovic, M., 2015. Some coincidence point results for generalized weakly contractions in ordered b-metric spaces. Fixed Point Theory Appl., 68.

Roshan, J.R., Parvaneh, V., & Kadelburg, Z., 2014. Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces. J. Nonlinear Sci. Appl., 7, pp.229-245.

Zabihi, F., & Razani, A., 2014. Fixed point theorems for hybrid rational Geraghty contractive mappings in ordered b-metric spaces. J. Appl. Math., Article ID 929821, 9 pages.

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УПОРЯДОЧЕННЫЕ Б МЕТРИЧЕСКИЕ ПРОСТРАНСТВА И СЖИМАЮЩИЕ ОТОБРАЖЕНИЯ ТИПА СБРАСИТУ

Сумит Чандока, Мирко С. Йованович6, Стоян Н. Раденовичв а Университет Тапар, Факультет математики, Патиала, Индия б Университет в Белграде, Факультет электротехники, г.Белград,

Республика Сербия в Университет в Белграде, Факультет машиностроения, г.Белград, Республика Сербия

ОБЛАСТЬ: математика

ВИД СТАТЬИ: оригинальная научная статья

ЯЗЫК СТАТЬИ: английский

Резюме:

В настоящей статье, благодаря новому методу, представлены новейшие результаты исследований неподвижной точки, проведенные разными авторами. Представление данных результатов подкреплены примерами.

Ключевые слова: неподвижная точка, б-метрика, сравнительный, упорядоченный, СБРАСИТУ-отображение, б-Коши, б-комплет.

УРЕЪЕНИ Б-МЕТРИЧКИ ПРОСТОРИ И КОНТРААКТИВНА ПРЕСЛИКАВА^А ГЕРАГХТШЕВОГ ТИПА

Сумит Чандока, Мирко С. иованови^, Сто^ан Н. Раденови^в а Универзитет у Тапару, Математички факултет, Патиала, Инди]а

б Универзитет у Београду, Електротехнички факултет, Београд,

Република Срби]а в Универзитет у Београду, Машински факултет, Београд, Република Срби]а

ОБЛАСТ: математика

ВРСТА ЧЛАНКА: оригинални научни чланак иЕЗИК ЧЛАНКА: енглески

Сажетак:

КоришЯеъем новог приступа, у раду су представъени недавни резултате фиксне тачке, ко}у }е установило више аутора, на много краЯи и лепши начин. Наведен }е и пример ко\и то илустру}е.

Къучне речи: фиксна тачка, б-метрика, упоредив, добро уре^ен, Герагхщево пресликава^е, б-Коши]ев, б-комплет.

(http://creativecommons.Org/licenses/by/3.0/rs/).

Paper received on / Дата получения работы / Датум приема чланка: 24.01.2017. ^

Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 11. 03. 2017. со

Paper accepted for publishing on / Дата окончательного согласования работы / Датум

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(http://creativecommons.org/licenses/by/3.0/rs/). ^

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Technical Courier» (www.vtg.mod.gov.rs, втг.мо.упр.срб). Данная статья в открытом доступе и '

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