Cyclic Generalized ϕ-contractions in b-metric Spaces and an Application to Integral Equations

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1 Filomt 8:10 (014) DOI 10.98/FIL N Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: Cyclic Generlized ϕ-contrctions in b-metric Spces nd n Appliction to Integrl Equtions Hemnt Kumr Nshine Zorn Kdelburg b Deprtment of Mthemtics Dish Institute of Mngement nd Technology Sty Vihr Vidhnsbh-Chndrkhuri Mrg Mndir Hsud Ripur-49101(Chhttisgrh) Indi b University of Belgrde Fculty of Mthemtics Studentski trg Beogrd Serbi Abstrct. We introduce the notion of cyclic generlized ϕ-contrctive mppings in b-metric spces nd discuss the existence nd uniqueness of fixed points for such mppings. Our results generlize mny existing fixed point theorems in the literture. Exmples re given to support the usbility of our results. Finlly n ppliction to existence problem for n integrl eqution is presented. 1. Introduction nd Preliminries The Bnch s fixed point theorem (or the contrctions mpping principle) is the most importnt metricl fixed point theorem in solving existence problems in mny brnches of mthemticl nlysis. There is gret number of generliztions of the Bnch contrction principle. The underlying metric spce cn be generlized in mny wys. For exmple new notion of b-metric spce ws introduced by Bkhtin in [4] nd then extensively used by Czerwik in [8]. Since then some reserch works hve delt with fixed point theory for single vlued nd multivlued opertors in b-metric (sometimes lso clled metric type) spces (see [ ] nd the references therein). Definition 1.1. (Czerwik [8 9]) Let X be non-empty set nd s 1 rel number. A function d : X X R + (nonnegtive rel numbers) is sid to be b-metric if for ll x y z X (M1) d(x y) = 0 if nd only if x = y; (M) d(x y) = d(y x); (M3) d(x z) s[d(x y) + d(y z)]. The pir (X d) is clled b-metric spce with prmeter s. Obviously ech metric spce is b-metric spce (for s = 1). However Czerwik [8 9] hs shown tht b-metric on X need not be metric on X. The following simple exmples cn be used to show this. 010 Mthemtics Subject Clssifiction. Primry 47H10; Secondry 54H5 Keywords. b-metric spce cyclic mp fixed point control function Received: 11 September 013; Accepted: 08 Mrch 014 Communicted by V. Rkočević The second uthor is thnkful to the Ministry of Eduction Science nd Technologicl Development of Serbi. Emil ddresses: drhknshine@gmil.com nshine_09@rediffmil.com (Hemnt Kumr Nshine) kdelbur@mtf.bg.c.rs (Zorn Kdelburg)

2 H.K. Nshine Z. Kdelburg / Filomt 8:10 (014) Exmple 1.. Let X = x 1 x x 3 } nd d : X X R + be such tht d(x 1 x ) = x 3 d(x 1 x 3 ) = d(x x 3 ) = 1 d(x n x n ) = 0 d(x n x k ) = d(x k x n ) = 0 for n k = 1 3. Then d(x n x k ) x 3 [d(x n x i ) + d(x i x k )] n k i = 1 3. Hence (X d) is b-metric spce (with s = x/3) nd not metric spce if x > 3. Exmple 1.3. Let (X ρ) be metric spce nd d(x y) = (ρ(x y)) p where p > 1 is rel number. Then d is b-metric with s = p 1. Condition (M3) follows esily from the convexity of function f (x) = x p (x > 0). Let (X d) be b-metric spce. As in the metric cse the b-metric d induces topology. For every r > 0 nd ny x X we set B(x r) = y X : d(x y) < r}. The topology τ(d) on X ssocited with d is given by setting U τ(d) if nd only if for ech x U there exists some r > 0 such tht B(x r) U. The spce X will be equipped with the topology τ(d). In prticulr sequence x n } converges to point x X if lim n d(x n x) = 0. Almost ll the concepts nd results obtined for metric spces cn be extended to the cse of b-metric spces. Lemm 1.4. (see e.g. [1 19]) Let (X d) be b-metric spce with prmeter s nd y n } sequence in X such tht d(y n y n+1 ) qd(y n 1 y n ) n N. Then y n } is Cuchy sequence in X provided tht sq < 1. It is well known tht in stndrd metric spce (X d) the function d is continuous in both vribles in the sense tht if x n } y n } re sequences in X such tht x n x y n y s n then d(x n y n ) d(x y) s n. A b-metric need not posses this property s the following exmple shows. Exmple 1.5. [11 Exmple ] Let X = N } nd let d : X X R be defined by 0 if m = n 1 m d(m n) = 1 n if one of m n is even nd the other is even or 5 if one of m n is odd nd the other is odd (nd m n) or otherwise. Then considering ll possible cses it cn be checked tht for ll m n p X we hve d(m p) 5 (d(m n) + d(n p)). Thus (X d) is b-metric spce (with s = 5/). Let x n = n for ech n N. Then d(n ) = 1 0 s n n tht is x n but d(x n 1) = 5 = d( 1) s n. One of the remrkble generliztions of the Bnch contrction principle ws reported by Kirk Srinivsn nd Veermni [13] vi so-clled cyclic contrctions. A mpping T : A B A B is clled cyclic if T(A) B nd T(B) A where A B re nonempty subsets of metric spce (X d). Moreover T is clled cyclic contrction if there exists k (0 1) such tht d(tx Ty) kd(x y) for ll x A nd y B. Notice tht lthough contrction is continuous cyclic contrctions need not be. This is one of the importnt gins of this pproch. Definition 1.6. ([13 0]) Let (X d) be metric spce. Let p be positive integer A 1 A... A p be nonempty subsets of X nd T : Y Y. Then is sid to be cyclic representtion of Y with respect to T if

3 H.K. Nshine Z. Kdelburg / Filomt 8:10 (014) (i) A i i = 1... p re nonempty closed sets nd (ii) T(A 1 ) A... T(A p 1 ) A p T(A p ) A 1. Following [13] number of fixed point theorems on cyclic representtions of Y with respect to self-mpping T hve ppered (see e.g. [ ]). In this pper we introduce new vrint of cyclic contrctive mppings nmed s cyclic generlized ϕ-contrctions in b-metric spces nd then derive the existence nd uniqueness of fixed points for such mppings. Our min result generlizes nd improves mny existing theorems in the literture. Some exmples re provided to demonstrte the vlidity of our results. As n ppliction in the lst section the existence of solution of n integrl eqution is proved under pproprite conditions.. Min Result All the wy through this pper by R + we designte the set of ll nonnegtive rel numbers while N is the set of ll nturl numbers. We denote by Φ the set of functions ϕ : [0 + ) [0 + ) with ϕ(t) < t for ech t > 0(0) = 0. We introduce the notion of cyclic generlized ϕ-contrction in b-metric spces s follows. Definition.1. Let (X d) be b-metric spce with prmeter s. Let p be positive integer A 1 A... A p be nonempty subsets of X nd. An opertor T : Y Y stisfies cyclic generlized ϕ-contrction for some ϕ Φ if (I) is cyclic representtion of Y with respect to T; (II) for ny (x y) A i A i+1 i = 1... p (with A p+1 = A 1 ) d(tx Ty) M(x y) + L minϕ(d(x Tx))(d(y Ty))(d(x Ty))(d(y Tx))} (1) where L 0 nd d(x Tx) + d(y Ty) d(y Tx) + d(x Ty) } M(x y) = mx ϕ(d(x y))(d(x Tx)). () The min result of this section is s follows: Theorem.. Let (X d) be complete b-metric spce p N A 1 A... A p nonempty closed subsets of X nd. Suppose T : Y Y is cyclic generlized ϕ-contrctive mpping for some ϕ Φ. Then T hs unique fixed point. Moreover the fixed point of T belongs to p. Proof. Let x 0 A 1 (such point exists since A 1 ). Define the sequence x n } in X by x n+1 = Tx n n = We shll prove tht lim d(x n x n+1 ) = 0. n (3) If for some k we hve x k+1 = x k then it is esy to show (using (1)) tht (3) holds. Hence we cn suppose tht d(x n x n+1 ) > 0 for ll n. From condition (I) we observe tht for ll n there exists i = i(n) 1... p} such tht (x n x n+1 ) A i A i+1. Let δ n = d(x n x n+1 ). Now we clim tht for ll n N we hve δ n < δ n 1. (4)

4 Indeed from condition (1) we hve H.K. Nshine Z. Kdelburg / Filomt 8:10 (014) d(x n x n+1 ) = d(tx n 1 Tx n ) M(x n 1 x n ) + L minϕ(d(x n 1 Tx n 1 ))(d(x n Tx n ))(d(x n 1 Tx n ))(d(x n Tx n 1 ))} = M(x n 1 x n ). (5) By () we hve M(x n 1 x n ) d(xn 1 Tx n 1 ) + d(x n Tx n ) = mx ϕ(d(x n 1 x n ))(d(x n 1 Tx n 1 )) } d(xn Tx n 1 ) + d(x n 1 Tx n ) ϕ d(xn 1 x n ) + d(x n x n+1 ) = mx ϕ(d(x n 1 x n ))(d(x n 1 x n )) } d(xn x n ) + d(x n 1 x n+1 ) ϕ } d(xn 1 x n ) + d(x n x n+1 ) d(xn 1 x n+1 ) = mx ϕ(d(x n 1 x n )). Consider the following possibilities. If M(x n 1 x n ) = ϕ(d(x n 1 x n )) by (5) nd using the fct tht ϕ(t) < t for ll t > 0 we hve δ n = d(x n x n+1 ) ϕ(d(x n 1 x n )) < d(x n 1 x n ) If M(x n 1 x n ) = ϕ d(x n 1 x n )+d(x n x n+1 ) we get d(xn 1 x n ) + d(x n x n+1 ) d(x n x n+1 ) ϕ wherefrom δ n = d(x n x n+1 ) < 1 4s 1 d(x n 1 x n ) < δ n 1. If M(x n 1 x n ) = ϕ ( 1 d(x n 1 x n+1 ) ) we get 1 d(x n x n+1 ) ϕ d(x n 1 x n+1 ) = δ n 1. d(x n 1 x n ) + d(x n x n+1 ) 4s < 1 4s d(x n 1 x n+1 ). On the other hnd by the property (M3) we hve d(x n 1 x n+1 ) s[d(x n 1 x n ) + d(x n x n+1 )]. Thus we hve d(x n x n+1 ) < 1 4s d(x n 1 x n ) + 1 4s d(x n x n+1 ) which implies tht δ n = d(x n x n+1 ) < 1 4s 1 d(x n 1 x n ) < δ n 1.

5 H.K. Nshine Z. Kdelburg / Filomt 8:10 (014) Then in ll cses we hve δ n < δ n 1 for ll n N. Therefore we conclude tht (4) holds. Now from (4) it follows tht the sequence δ n stisfies 0 < δ n < δ n 1 < δ n () < δ n 3 () 3 < < δ 0 () n nd pssing to the limit s n we hve lim δ n = lim d(x n x n+1 ) = 0. n n Moreover by Lemm 1.4 (4) implies tht x n } is Cuchy sequence. From the completeness of X there exists z X such tht lim x n = lim Tx n 1 = z. n n (6) We shll prove tht z p A i. i=1 (7) From the condition (I) nd since x 0 A 1 we hve x np } n 0 A 1. Since A 1 is closed from (6) we get tht z A 1. Agin from the condition (I) we hve x np+1 } n 0 A. Since A is closed from (6) we get tht z A. Continuing this process we obtin (7). Now we shll prove tht z is fixed point of T. Indeed from (7) since for ll n there exists i(n) 1... p} such tht x n A i(n) pplying (II) with x = x n nd y = z nd using (M3) we obtin d(z Tz) s[d(z Tx n ) + d(tx n Tz)] sd(z x n+1 ) + sm(x n z) + sl minϕ(d(x n x n+1 ))(d(z Tz))(d(x n Tz))(d(x n+1 z))} = sd(z x n+1 ) } d(xn x n+1 ) + d(z Tz) d(xn Tz) + d(x n+1 z) + s mx ϕ(d(x n z))(d(x n x n+1 )) + sl minϕ(d(x n x n+1 ))(d(z Tz))(d(x n Tz))(d(x n+1 z))} sd(z x n+1 ) + 1 mx d(x n z) d(x n x n+1 ) d(x n x n+1 ) + d(z Tz) d(x } n Tz) + d(x n+1 z) + L mind(x n x n+1 ) d(z Tz) d(x n Tz) d(x n+1 z)}. Now using tht d(x n Tz) s[d(x n z) + d(z Tz)] we get tht d(z Tz) sd(z x n+1 ) + 1 mx d(x n z) d(x n x n+1 ) d(x n x n+1 ) + d(z Tz) d(x n z) + d(z Tz) + d(x } n+1 z) + L mind(x n x n+1 ) d(z Tz) d(x n Tz) d(x n+1 z)}. Pssing to the limit s n we hve d(z Tz) 1 d(z Tz) which is only possible if d(z Tz) = 0 i.e. Tz = z. conclusion.) (Note tht continuity of d ws not needed for this

6 H.K. Nshine Z. Kdelburg / Filomt 8:10 (014) We clim tht z is the unique fixed point of T. Assume to the contrry tht Tu = u nd u z. Then (1) implies tht d(z u) = d(tz Tu) M(z u) + L minϕ(d(z Tz))(d(u Tu))(d(z Tu))(d(u Tz))} where } d(z Tz) + d(u Tu) d(u Tz) + d(z Tu) M(z u) = mx ϕ(d(z u))(d(z Tz)) } d(z u) d(z u) d(z u) < mx = contrdiction. Hence z = u. 3. Consequences In this section we derive some fixed point theorems from our min result given by Theorem.. If we tke p = 1 nd A 1 = X in Theorem. then we get immeditely the following fixed point theorem. Corollry 3.1. Let (X d) be complete b-metric spce nd let T : X X stisfy the following condition: there exists ϕ Φ such tht d(tx Ty) M(x y) + L minϕ(d(x Tx))(d(y Ty))(d(x Ty))(d(y Tx))} where L 0 nd d(x Tx) + d(y Ty) d(y Tx) + d(x Ty) } M(x y) = mx ϕ(d(x y))(d(x Tx)). for ll x y X. Then T hs unique fixed point. Remrk 3.. Corollry 3.1 extends mny existing fixed point theorems in the literture see e.g. [3 5 6] to complete b-metric spce. Corollry 3.3. Let (X d) be complete b-metric spce p N A 1 A... A p nonempty closed subsets of X nd T : Y Y. Suppose tht there exists nondecresing function ϕ Φ such tht () is cyclic representtion of Y with respect to T; (b) for ny (x y) A i A i+1 i = 1... p (with A p+1 = A 1 ) where d(tx Ty) ϕ ( M 1 (x y) ) + L minϕ(d(x Tx))(d(y Ty))(d(x Ty))(d(y Tx))} M 1 (x y) = mx d(x y) d(x Tx) d(x Tx) + d(y Ty) d(y Tx) + d(x Ty) } Then T hs unique fixed point. Moreover the fixed point of T belongs to p Proof. It follows from Theorem. by observing tht if ϕ is nondecresing we hve ϕ ( M 1 (x y) ) d(x Tx) + d(y Ty) d(y Tx) + d(x Ty) } = mx ϕ(d(x y))(d(x Tx)).

7 H.K. Nshine Z. Kdelburg / Filomt 8:10 (014) Corollry 3.4. Let (X d) be complete b-metric spce p N A 1 A... A p nonempty closed subsets of X nd T : Y Y. Suppose tht there exists ϕ Φ such tht () is cyclic representtion of Y with respect to T; (b) for ny (x y) A i A i+1 i = 1... p (with A p+1 = A 1 ) d(tx Ty) ϕ(d(x y)). Then T hs unique fixed point. Moreover the fixed point of T belongs to p Remrk 3.5. Corollry 3.4 is similr to Theorem.1 in [0] but considered in complete b-metric spce. Tking in Corollry 3.4(t) = kt with k (0 1) we obtin Theorem 1.3 in [13] in complete b-metric spce. Corollry 3.6. Let (X d) be complete b-metric spce p N A 1 A... A p nonempty closed subsets of X nd T : Y Y. Suppose tht there exists ϕ Φ such tht () is cyclic representtion of Y with respect to T; (b) for ny (x y) A i A i+1 i = 1... p (with A p+1 = A 1 ) d(x Ty) + d(y Tx) d(tx Ty) ϕ. Then T hs unique fixed point. Moreover the fixed point of T belongs to p Remrk 3.7. Tking in Corollry 3.6(t) = kt with k (0 1) we obtin n nlogue of Theorem 3 from [1] in complete b-metric spce. Corollry 3.8. Let (X d) be complete b-metric spce p N A 1 A... A p nonempty closed subsets of X nd T : Y Y. Suppose tht there exists ϕ Φ such tht () is cyclic representtion of Y with respect to T; (b) for ny (x y) A i A i+1 i = 1... p (with A p+1 = A 1 ) d(tx Ty) mxϕ(d(x Tx))(d(y Ty))}. Then T hs unique fixed point. Moreover the fixed point of T belongs to p Remrk 3.9. Tking in Corollry 3.8(t) = kt with k (0 1) we obtin n nlogue of Theorem 5 from [1] in complete b-metric spce. Corollry Let (X d) be complete b-metric spce p N A 1 A... A p nonempty closed subsets of X nd T : Y Y. Suppose tht there exists ϕ Φ such tht () is cyclic representtion of Y with respect to T; (b) for ny (x y) A i A i+1 i = 1... p (with A p+1 = A 1 ) d(tx Ty) mxϕ(d(x y))(d(x Tx))(d(y Ty))}. Then T hs unique fixed point. Moreover the fixed point of T belongs to p The following result (see Theorem 7 in [1]) extends Reich s fixed point theorem [7] to complete b-metric spces.

8 H.K. Nshine Z. Kdelburg / Filomt 8:10 (014) Corollry Let (X d) be complete b-metric spce p N A 1 A... A p nonempty closed subsets of X nd T : Y Y. Suppose tht there exist three positive constnts b c with + b + c < 1 such tht () is cyclic representtion of Y with respect to T; (b) for ny (x y) A i A i+1 i = 1... p (with A p+1 = A 1 ) d(tx Ty) d(x y) + bd(x Tx) + cd(y Ty). Then T hs unique fixed point. Moreover the fixed point of T belongs to p Proof. It follows from Corollry 3.10 by tking ϕ(t) = ( + b + c)t. Corollry 3.1. Let (X d) be complete b-metric spce p N A 1 A... A p nonempty closed subsets of X nd T : Y Y. Suppose tht there exist four positive constnts with < 1 such tht () is cyclic representtion of Y with respect to T; (b) for ny (x y) A i A i+1 i = 1... p (with A p+1 = A 1 ) d(x Ty) + d(y Tx) d(tx Ty) 1 d(x y) + d(x Tx) + 3 d(y Ty) + 4. Then T hs unique fixed point. Moreover the fixed point of T belongs to p Proof. It follows from Theorem. by tking ϕ(t) = ( )t. Remrk Corollry 3.1 extends nd generlizes the well known fixed point theorem of Hrdy nd Rogers [10] to complete b-metric spces. It lso improves Theorem 3.7 of [1]. 4. Exmples We present some exmples showing how our results cn be used. Exmple 4.1. Consider the set X = R equipped with the function d : X X R + given s d(x y) = (x y). Then d is b-metric with prmeter s = by Exmple 1.3. Let A 1 = [0 + ) nd A = ( 0]. Then A 1 A = X nd A 1 A = 0}. Consider the mpping T : X X given by Tx = 0 if x = 0 x 6 sin 1 x otherwise. Then obviously X = A 1 A is cyclic representtion of X with respect of T. Let x A 1 \ 0} nd y A \ 0} (the other possibility is treted symmetriclly). Then [ d(tx Ty) = x 6 sin 1 x + y 6 sin 1 ] y = 1 [ x 36 sin 1 x + y sin 1 ] y 1 18 (x + y ) ( d(x Tx) = x + x 6 sin 1 x ) x ( d(y Ty) = y + y 6 sin 1 y ) y.

9 H.K. Nshine Z. Kdelburg / Filomt 8:10 (014) Hence d(tx Ty) 1 18 (x + y ) 1 (d(x Tx) + d(y Ty)) mx d(x y) d(x Tx) d(x Tx) + d(y Ty) + L mind(x Tx) d(y Ty) d(x Ty) d(y Tx)} } d(x Ty) + d(y Tx) for rbitrry L > 0. Tke the function ϕ Φ given s ϕ(t) = t 9 (note tht ϕ(t) < t 4 = t for t > 0). Then d(x Tx) + d(y Ty) d(x Ty) + d(y Tx) } d(tx Ty) mx ϕ(d(x y))(d(x Tx)) + L minϕ(d(x Tx))(d(y Ty))(d(x Ty))(d(y Tx))}. The previous inequlity is lso stisfied if one of x y (or both) is equl to 0; hence ll the conditions of Theorem. (s mtter of fct of Corollry 3.8) re stisfied. Obviously T hs unique fixed point 0 belonging to A 1 A. Exmple 4.. Consider the b-metric spce (X d) given in Exmple 1.5 nd the mpping T : X X given s 6n if n N Tn = if n =. If A 1 = n : n N} } nd A = 6n : n N} } then A 1 A is cyclic representtion of X with respect to T. Tke ϕ Φ given s ϕ(t) = t (note tht ϕ(t) < 1 5 t = t ). In order to check the contrctive condition (1) consider the following cses. If x y N then d(tx Ty) = d(6x 6y) = x 1 y x 1 y ( 1 = ϕ x 1 ) y ϕ(d(x y)) M(x y) + L minϕ(d(x Tx))(d(y Ty))(d(x Ty))(d(y Tx))} nd (1) holds. If x = nd y is n even integer then d(tx Ty) = d( 6y) = 1 1 6y ϕ = ϕ(d( y)) y M(x y) + L minϕ(d(x Tx))(d(y Ty))(d(x Ty))(d(y Tx))}. Finlly if x = nd y is n odd integer then d(x y) = 5 nd (1) trivilly holds. Hence ll the conditions of Theorem. (s mtter of fct of Corollry 3.4) re stisfied. Obviously T hs unique fixed point belonging to A 1 A. 5. An Appliction to Integrl Equtions Inspired by [16] we will consider the following integrl eqution for n unknown function u: u(t) = G(t s) f (s u(s)) ds t [ b] (8)

10 H.K. Nshine Z. Kdelburg / Filomt 8:10 (014) where f : [ b] R R nd G : [ b] [0 + ) re given continuous functions. Let X be the set C[ b] of rel continuous functions on [ b] nd let d : X X R + be given by d(u v) = mx t b [u(t) v(t)]. It is esy to see tht d is b-metric with prmeter s = nd tht (X d) is complete b-metric spce. If T : X X is defined by Tu(t) = G(t s) f (s u(s)) ds t [ b] then it is cler tht function u is solution of the given eqution (8) if nd only if it is fixed point of the mpping T. We will prove the existence nd uniqueness of the fixed point of T under the following conditions. (I) There exist functions α β X nd rel numbers α 0 β 0 such tht nd tht α 0 α(t) β(t) β 0 t [ b] Tα(t) β(t) nd Tβ(t) α(t) t [ b]. (II) The function f (s ) is nonincresing i.e. for ech s [ b] x y R x y = f (s x) f (s y). (III) mx G (t s) ds 1 t b b. (IV) There exists nondecresing function ϕ Φ such tht for ech s [ b] nd ll x y R stisfying (x α 0 nd y β 0 ) or (x β 0 nd y α 0 ) the following inequlity holds: [ f (s x) f (s y)] ϕ((x y) ). Note tht (since s = ) hs to stisfy the condition ϕ(t) < t 4 ϕ(t) = t 5 (t) = t 4(1+t) (t) = 1 4 ln(1 + t) nd mny others. for t > 0. Exmples of such function re Theorem 5.1. Under the conditions (I) (IV) the eqution (8) hs unique solution u C = u X : α(t) u(t) β(t) t [ b]}. X nd it belongs to Proof. Consider closed subsets A 1 = u X : u(t) β(t) t [ b]} nd A = u X : u(t) α(t) t [ b]} of the spce (X d). We will prove tht T(A 1 ) A nd T(A ) A 1 i.e. A 1 A = Y is cyclic representtion of Y with respect to T. Indeed let u A 1 i.e. u(s) β(s) for ech s [ b]. Using the condition (II) nd tht G(t s) is nonnegtive we get tht G(t s) f (s u(s)) G(t s) f (s β(s)) t s [ b] which implies tht Tu(t) = G(t s) f (s u(s)) G(t s) f (s β(s)) = Tβ(t) α(t) t [ b] by (I). Hence Tu(t) α(t) t [ b] i.e. Tu A. The inclusion T(A ) A 1 cn be proved in similr wy.

11 H.K. Nshine Z. Kdelburg / Filomt 8:10 (014) Let now (u v) A 1 A i.e. u(t) β(t) β 0 t [ b] nd v(t) α(t) α 0 t [ b]. Then using the conditions (III) (IV) nd the Cuchy-Schwrz inequlity we obtin for t [ b]: [Tu(t) Tv(t)] = 1 b } G(t s)[ f (s u(s)) f (s v(s))] ds G (t s) ds [ f (s u(s)) f (s v(s))] ds ϕ([u(s) v(s)] ) ds 1 (b )ϕ(d(u v)). b Thus d(tu Tv) ϕ(d(u v)) holds. We conclude tht ll the conditions of Corollry 3.3 re fulfilled. Hence the mpping T hs unique fixed point u A 1 A = C i.e. the eqution (8) hs unique solution belonging to this set. References [1] R. P. Agrwl M. A. Alghmdi N. Shhzd Fixed point theory for cyclic generlized contrctions in prtil metric spces Fixed Point Theory Appl. 01:40 doi: / [] M. Akkouchi A common fixed point theorem for expnsive mppings under strict implicit conditions on b-metric spces Act Univ. Plcki. Olomuc. Fc. rer. nt. Mthemtic 50 (011) [3] G. V. R. Bbu M. L. Sndhy M. V. R. Kmesvri A note on fixed point theorem of Berinde on wek contrctions Crpthin J. Mth. 4 1 (008) 8 1. [4] I. A. Bkhtin The contrction mpping principle in qusimetric spces Funct. Anl. Ulinowsk Gos. Ped. Inst. 30 (1989) [5] V. Berinde Generl constructive fixed point theorems for Ćirić-type lmost contrctions in metric spces Crpthin J. Mth. 4 (008) [6] D. W. Boyd J. S. Wong On nonliner contrctions Proc. Amer. Mth. Soc. 0 (1969) [7] S. Reich Some remrks concerning contrction mppings Cnd. Mth. Bull. 14 (1971) [8] S. Czerwik Contrction mppings in b-metric spces Act Mth. Inform. Univ. Ostrviensis 1 (1993) [9] S. Czerwik Nonliner set-vlued contrction mppings in b-metric spces Atti Sem. Mt. Fis. Univ. Moden 46 (1998) [10] G. E. Hrdy R. D. Rogers A generliztion of fixed point theorem of Reich Cnd. Mth. Bull. 16 (1973) [11] N. Hussin V. Prvneh J.R. Roshn Z. Kdelburg Fixed points of cyclic (ψ L A B)-contrctive mppings in ordered b-metric spces with pplictions Fixed Point Theory Appl. 013:56 (013) doi: / [1] M. Jovnović Z. Kdelburg S. Rdenović Common fixed point results in metric-type spces Fixed Point Theory Appl. 010 Article ID pges doi: /010/ [13] W. A. Kirk P. S. Srinivsn P. Veermni Fixed points for mppings stisfying cyclicl contrctive conditions Fixed Point Theory 4 (003) [14] E. Krpınr Fixed point theory for cyclic wek φ-contrction Appl. Mth. Lett. 4 (011) [15] E. Krpınr K. Sdrngni Fixed point theory for cyclic (φ-ψ)-contrctions Fixed Point Theory Appl. 69 (011) doi: / [16] H. K. Nshine Cyclic generlized ψ-wekly contrctive mppings nd fixed point results with pplictions to integrl equtions Nonliner Anl. 75 (01) [17] H. K. Nshine Z. Kdelburg Nonliner generlized cyclic contrctions in G-metric spces nd pplictions to integrl equtions Nonliner Anl. Model. Control. 18 (013) [18] H. K. Nshine Z. Kdelburg S. Rdenović Fixed point theorems vi vrious cyclic contrctive conditions in prtil metric spces Publ. de l Inst. Mth. 93(107) (013) [19] S. L. Singh S. Czerwik K. Król A. Singh Coincidences nd fixed points of hybrid contrctions Tmsui Oxf. J. Mth. Sci. 4 (008) [0] M. Pcurr I. A. Rus Fixed point theory for cyclic ϕ-contrctions Nonliner Anl. 7 (010) [1] M. A. Petric Some results concerning cyclicl contrctive mppings Generl Mth. 18 (010) [] I. A. Rus Cyclic representtions nd fixed points Ann. T. Popoviciu Seminr Funct. Eq. Approx. Convexity 3 (005)