h/called upper class of type II and ˛ˇ-contractive mappings

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Vol. 19 (2018), No. 2, pp. 787–793 DOI: 10.18514/MMN.2018.2077

ON ˛ˇ-CONTRACTIVE AND ADMISSIBLE MAPPINGS

S. CHANDOK, A.H. ANSARI, T. DO ˇSENOVI ´C, AND S. RADENOVI ´C Received 06 September, 2016

Abstract. In this paper, we introduce the concept of a pair.f; h/called upper class of type II and

˛_ˇ-contractive mappings. We obtain that all the corresponding established results of Hussain et al. [7] are immediately consequences of our main result. Our main result generalizes and modifies several existing results in literature. Also, an example is given to support the main result.

2010Mathematics Subject Classification: 47H10; 54H25

Keywords: admissible mappings, contractive mappings, ˛_ˇ-contractive, fixed point, C-class function

1. INTRODUCTION AND PRELIMINARIES

It is well known that Banach contraction principle [2] is one of the most interest- ing and useful result in nonlinear analysis and whole mathematics in general. This famous result has also very large applications in various fields such as engineering, economic, computer sciences, and many others. This theorem has been extended and generalized by various authors (see, e.g., [7–9,11,12]).

In an attempt to generalize this significant principle, many researchers have extended the following result in certain directions.

Theorem 1 ([3–6,8]). Let .X; d / be a complete metric space andT W X !X be a mapping. Assume that there exists a functionˇWŒ0;1/!Œ0; 1such that, for any bounded sequenceftngof positive real numbers,ˇ.tn/!1impliestn!0and d.T x; T y/ˇ.d.x; y//d.x; y/for allx; y2X. ThenT has a unique fixed point.

In this paper we introduce a concept of pair.f; h/which we denote as upper class of type II and˛_ˇ-contractive mappings to show the theorems in [7] are immediately consequences of our main approach.

Now, we introduce some definitions and new notations which will be used in the sequel.

The first author is thankful to Thapar Institute of Engineering & Technology, for SEED grant.

c 2018 Miskolc University Press

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Definition 1. We say thathWŒ0;C1/³!Ris a function of subclass of type II if it is continuous and

x; y1)h.1; 1; ´/h.x; y; ´/

Example1. DefinehWR^CR^CR^C!Rby:

(a) h.x; y; ´/D.´Cl/^xy; l > 1;

(b) h.x; y; ´/D.xyCl/^´; l > 1;

(c) h.x; y; ´/D´;

(d) h.x; y; ´/Dx^myⁿ´^p; m; n; p2N;

(e) h.x; y; ´/D^x^m^C^xⁿ₃^y^p^C^y^q´^k; m; n; p; q; k2N

for allx; y; ´2R^C:Thenhis a function of subclass of type II.

Definition 2. Suppose thatf WŒ0;C1/²!RandhWŒ0;C1/³!R. The pair .f; h/is called upper class of type II iff is a continuous function, ha subclass of type II with

0s1)f .s; t /f .1; t /;

andh.1; 1; ´/f .s; t /)´st:

Example2. DefinehWR^CR^CR^C!RandFWR^CR^C!Rby:

(a) h.x; y; ´/D.´Cl/^xy; l > 1;F.s; t /DstCl;

(b) h.x; y; ´/D.xyCl/^´; l > 1;F.s; t /D.1Cl/^st; (c) h.x; y; ´/D´; F .s; t /Dst;

(d) h.x; y; ´/Dx^myⁿ´^p; m; n; p2N;F.s; t /Ds^pt^p

(e) h.x; y; ´/D^x^m^C^xⁿ₃^y^p^C^y^q´^k; m; n; p; q; k2N;F.s; t /Ds^kt^k for allx; y; ´; s; t2R^C. Then the pair.F; h/is an upper class of type II.

Definition 3. Let .X; d / be a metric space and T W X !X be a mapping. A nonempty subsetF ofX is called invariant underT ifT x2F, for everyx2F:

Definition 4. LetT WX !X be a mapping, F a nonempty subset ofX which is invariant underT and˛WFF !Œ0;C1/. We say thatT is an˛_F-admissible mapping if˛.x; y/1implies˛.T x; T y/1;for allx; y 2 F:

Remark1. A mappingT is called an˛-admissible mapping (see [12]) if we take F DX in Definition4.

Definition 5. A function WŒ0;C1/!Œ0;C1/is called altering distance function if the following properties are satisfied:

(1) is continuous and non-decreasing;

(2) ¹.f0g/D0.

We denote the set of all altering distance functions.

The following result will be used in the sequel.

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Lemma 1([1,9–11]). Suppose.X; d /is a metric space. Letfxngbe a sequence inX such thatd.xn; xnC1/!0asn! 1. Iffxngis not a Cauchy sequence then there exist an" > 0and sequencesfm.k/gandfn.k/gof positive integers such that the following sequences tend to"whenk! 1:

d.x_m.k/; x_n.k//; d.x_{m.k/ 1}; x_n.k/_C₁/; d.x_{m.k/ 1}; x_n.k//;

d.x_{m.k/ 1}; x_{n.k/ 1}/; d.x_m.k/_C₁; x_n.k/_C₁/:

2. MAIN RESULTS

In this section by using the new concept we consider, discuss, improve and generalize the main results from [7]. It is worth mentioning here that in our approach the implicationˇ .tn/!1)tn!0holds also for unbounded sequences.tn/:There- fore, our new results generalize the recent results of [7] in several directions.

Definition 6. Let.X; d /be a metric space,F a nonempty subset ofX,T WX!X and˛WFF!Œ0;C1/. A mappingT is said to be˛_ˇ-contractive mapping if there exists aˇWŒ0;C1/!Œ0; 1/with the property thattn!0wheneverˇ .tn/!1as well as for allx; y2F;the following condition holds:

h .˛ .x; T x/ ; ˛ .y; T y/ ; .d .T x;T y///f .ˇ.d.x; y//; .d.x; y/// ; (2.1) where the pair.f; h/is a upper class of type II and 2 .

Theorem 2. Let.X; d /be a complete metric space andF be a nonempty closed subset ofX. Suppose that T WX !X is an˛F-admissible mapping and F is invariant underT. Further assume thatT is an˛_ˇ-admissible contractive mapping.

Suppose that there existsx02F such that˛.x0; T x0/1and one of the following conditions holds,

(a)T is continuous.

(b) iffxng is a sequence inF such thatxn!´, ˛.xn; xnC1/1, for alln, then

˛.´; T ´/1.

ThenT has a fixed point.

Proof. Letx02F such that˛.x0; T x0/1. Now, we construct a sequencefxng inF byxnDT xn 1, for n1, such that˛.xn; xnC1/D˛.xn; T xn/1. Substituting xDxn 1and yDxnin (2.1), we obtain

h.1; 1; .d.xn; xnC1///h .˛ .xn 1; xn/ ; ˛ .xn; xnC1/ ; .d .xn; xnC1///

f .ˇ.d.xn 1; xn//; .d.xn 1; xn///

which implies that

.d .xn;xnC1//ˇ.d.xn 1; xn// .d.xn 1; xn// (2.2) .d.xn 1; xn//:

As 2 , we have

d .xn;xnC1/d.xn 1; xn/; (2.3)

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for everyn2N. Therefore,fd .xn;xnC1/gis a decreasing sequence, so there exists some r0, such that

nlim!1d .xn;xnC1/Dr: (2.4) Further from (2.2), we have

.d .xn;xnC1//

.d.xn 1; xn// ˇ.d.xn 1; xn//1:

Lettingn! 1in the above inequality, we have lim

n!1ˇ.d.xn 1; xn//D1, and this implies that

nlim!1d .xn;xnC1/D0: (2.5) Now, we will show thatfxngis a Cauchy sequence. Suppose, to the contrary, that fxngis not a Cauchy sequence.

By Lemma 1, there existsı >0 for which we can find subsequencesfxnkgand fxm_kgoffxngwithn_k> m_k>ksuch that

klim!1d.xnk; xmk/D lim

k!1d.xnk 1; xmk 1/Dı: (2.6) SettingxDxm_k 1andyDxn_k 1in (2.1), we obtain

h 1; 1; d.xn_k; xm_k/

h ˛ xm_k 1; xm_k

; ˛ xn_k 1; xn_k

; d.xn_k; xm_k/ f .ˇ d xm_k 1; xn_k 1

; d xm_k 1; xn_k 1 /;

i.e. d.xn_k; xm_k/

ˇ d xm_k 1; xn_k 1

d xm_k 1; xn_k 1

, which implies that

d.xn_k; xm_k/

d xm_k 1; xn_k 1ˇ d xmk 1; xnk 1

1: (2.7)

Lettingk! 1and using (2.6) and (2.7), we obtain

klim!1d.xn_k 1; xm_k 1/D0ı; (2.8) which is a contradiction.

This shows thatfxngis a Cauchy sequence and hence it is convergent in the complete setF. Hencexn!´2F asn! 1.

First, we suppose thatT is continuous. Therefore, we have

´D lim

n!1xnC1D lim

n!1T xnDT lim

n!1xnDT ´:

Next, we suppose that condition (b) holds. Therefore, ˛.´; T ´/D1. Now, by (2.1), we have

h .1; 1; .d .T ´;xnC1/// h .˛ .´; T ´/ ; ˛ .xn; T xn/ ; .d .T ´;xnC1///

f .ˇ .d .´; xn// ; .d .´; xn// /;

which implies that

.d .T ´;xnC1//ˇ .d .´; xn// .d .´; xn// :

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Takingn! 1and using the properties of andˇ, we haved.T ´; ´/D0, that is,

´DT ´.

3. SOME CONSEQUENCES OF THE MAIN RESULT

If h .x; y; ´/D.´Cl/^xy; l > 1, f .x; y/DxyCl, .t /Dt, and F DX in Theorem2, we have Theorem 4 of [7].

Corollary 1. Let .X; d / be a complete metric space and T WX !X be an˛- admissible mapping. Assume that there exists a functionˇWŒ0;1/!Œ0; 1/with the property thattn!0wheneverˇ .tn/!1, such that

.d .T x;T y/Cl/˛.x;T x/˛.y;T y/

ˇ .d .x; y// d .x; y/Cl for allx; y 2X. Suppose that either

(a)T is continuous, or

(b) if fxngis a sequence in X such that xn!´, ˛.xn; xnC1/1 for all n, then

˛.´; T ´/1:

If there existsx02X such that˛.x0; T x0/1, thenT has a fixed point.

If h .x; y; ´/D.xyCl/^´; l >1,f .x; y/D.1Cm/^xy,mD1, .t /Dt, andF D X in Theorem2, we have Theorem 6 of [7].

.˛ .x; T x/ ˛ .y; T y/Cl/d .T x;T y/

2ˇ .d.x;y//d.x;y/

for allx; y 2X. Suppose that either (a)T is continuous, or

(b) iffxngis a sequence in X such thatxn!´, ˛.xn; xnC1/ 1 for all n, then

˛.´; T ´/1

If h . x; y; ´/ Dxy´; f .x; y/Dxy; .t /Dt, and F DX in Theorem 2, we have Theorem 8 of [7].

˛.x; T x/˛.y; T y/d.T x; T y/ˇ.d.x; y//d.x; y/

for allx; y 2X. Suppose that either (a)T is continuous, or

(b) if fxngis a sequence in X such that xn!´, ˛.xn; xnC1/1 for all n, then

˛.´; T ´/1:

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Remark2. Leth.x; y; ´/Dxy´, f .x; y/Dxy, .t /Dt, ˛.x; y/D1,F DX for allx; y; ´2X andt > 0. Then we get Theorem1.

Example3. LetX DŒ1;1/be endowed with a usual metricd.x; y/D jx yjfor allx; y2X andT WX !X be defined by

T .x/D ( _x

C14

8 1x4

x²

4 ; x > 4 Define the function˛; ˇ;and given by

˛.x; y/D

1; x; y2Œ1; 4

0; otherwise ˇ.t /D 1

1Ct; .t /Dt ThenT is˛-admissible and we obtain1yx4

˛ .x; T x/ ˛ .y; T y/ d.T x; T y/Dd

xC14

8 ;yC14 8

D1 8jx yj

ˇ.d.x; y// .d.x; y//

Hence, T satisfies all the assumptions of Theorem 2 with h.x; y; ´/Dxy´ and f .s; t /Dst and thus it has a fixed point (which isxD2).

ACKNOWLEDGEMENTS

The authors are thankful to the learned referee.

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Authors’ addresses

S. Chandok

School of Mathematics, Thapar Institute of Engineering & Technology, Patiala–147004, Punjab, India

E-mail address:sumit.chandok@thapar.edu

A.H. Ansari

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran E-mail address:analsisamirmath2@gmail.com

T. Doˇsenovi´c

Faculty of Technology, University of Novi Sad, Bulevar cara Lazara 1, Serbia E-mail address:tatjanad@tf.uns.ac.rs

S. Radenovi´c

Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia, Department of Mathematics, University of Novi Pazar, Novi Pazar, Serbia

E-mail address:radens@beotel.rs