Compared with the previous work, the aim of  this paper is to introduce the more general concept of the generalized $F$-Suzuki type contraction mappings in $b$-metric spaces, and to establish some fixed point theorems in the setting of $b$-metric spaces. Our main results unify, complement and generalize the previous works in the existing literature

Huaping Huang

Stojan Radenović

Sumit Chandok

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Sahand Communications in Mathematical Analysis (SCMA) Vol. 11, No. 1 (2018), 81-89http://scma.maragheh.ac.irDOI: 10.22130/scma.2018.52976.155Some Fixed Point Results for the Generalized F -suzuki TypeContractions in b-metric SpacesSumit Chandok1, Huaping Huang2∗, and Stojan Radenović3Abstract. Compared with the previous work, the aim of this pa-per is to introduce the more general concept of the generalizedF -Suzuki type contraction mappings in b-metric spaces, and to es-tablish some fixed point theorems in the setting of b-metric spaces.Our main results unify, complement and generalize the previousworks in the existing literature.1. Introduction and PreliminariesBefore the start of this article, let us denote the letters N, R and R+as the set of all natural numbers, the set of all real numbers and theset of all nonnegative real numbers, respectively. We denote Ψ as theset of all functions ψ : [0,∞) → [0,∞) such that ψ is continuous andψ−1(0) = 0.Recently, Wardowski [13] introduced a new type of contractive map-pings called F -contraction and proved a fixed point theorem as a gener-alization of the Banach contraction principle as follows:Definition 1.1 ([13]). Let F : R+ → R be a mapping satisfying(F1) F is strictly increasing, i.e. for all α, β ∈ R+, α < β impliesthat F (α) < F (β);(F2) For any sequence {αn} of positive numbers, limn→∞αn = 0 andlimn→∞F (αn) = −∞ are equivalent;(F3) There exists k ∈ (0, 1) such that limα→0+αkF (α) = 0.2010 Mathematics Subject Classification. 47H10, 54H25.Key words and phrases. Fixed point, Generalized F -Suzuki contraction, b-metricspace.Received: 16 November 2016, Accepted: 15 June 2017.∗ Corresponding author.8182 S. CHANDOK, H. HUANG, AND S. RADENOVIĆFor some examples satisfying Definition 1.1, the reader may refer to[14] and [13].Definition 1.2 ([13]). Let (X, d) be a metric space. A map T : X → Xis said to be F -contraction on X if there exist F in Definition 1.1 andτ > 0 such thatx, y ∈ X with d(Tx, Ty) > 0 ⇒ τ + F (d(Tx, Ty)) ≤ F (d(x, y)).Theorem 1.3. Let (X, d) be a complete metric space and T : X → Xan F -contraction. Then T has a unique fixed point x∗ ∈ X and, forevery x0 ∈ X a sequence {Tnx0}n∈N is convergent to x∗.Many researchers have focused on F -contractions and obtained someresults in the field (see [1–3, 5, 8, 10–12] and their references citedtherein). Motivated by [13], throughout this paper, we introduce theconcept of the generalized F -Suzuki contractions and obtain some rele-vant fixed point results in the setting of b-metric spaces.To begin with, we shall give a definition and a lemma which will beused in the sequel.Definition 1.4 ([4, 6]). Let X be a (nonempty) set and s ≥ 1 a givenreal number. A function d : X × X → [0,+∞) is called a b-metric ifand only if, for all x, y, z ∈ X, the following conditions are satisfied:(b1) d (x, y) = 0 if and only if x = y;(b2) d (x, y) = d (y, x);(b3) d (x, z) ≤ s [d (x, y) + d (y, z)].The pair (X, d) is called a b-metric space.It should be noted that, the class of b-metric spaces is effectively largerthan that of metric spaces, since a b-metric is a metric when s = 1. Thefollowing example shows that in general a b-metric need not necessarilybe a metric (see also [7, 9]).Example 1.5. Let (X, ρ) be a metric space, and d (x, y) = (ρ (x, y))p,where p > 1 is a real number. Then d is a b-metric with coefficients = 2p−1, but d is not a metric on X.Otherwise, for more concepts such as b-convergence, b-completenessand b-Cauchy sequence in b-metric spaces, we refer the reader to [7, 9]and the references mentioned therein.Lemma 1.6 ([9], Lemma 3.1). Let {yn} be a sequence in the b-metricspace (X, d) with s ≥ 1, such that(1.1) d (yn+1, yn+2) ≤ λd (yn, yn+1) ,for some λ ∈ [0, 1s ) and each n = 1, 2, . . .. Then {yn} is a b-Cauchysequence in (X, d).SOME FIXED POINT RESULTS FOR THE GENERALIZED · · · 832. Main ResultsIn the following, let us denote F as the collection of all mappingsF : R+ → R satisfying(F1) F is strictly increasing;(F2) F is continuous on (0,∞).Definition 2.1. Let (X, d) be a b-metric space with coefficient s ≥ 1.A map T : X → X is said to be a generalized F -Suzuki contraction ifthere exists F ∈ F such that, for all x, y ∈ X with x ̸= y,(2.1)12sd(x, Tx) < d(x, y) ⇒ F (sεd(Tx, Ty)) ≤ F (MT (x, y))− ψ(MT (x, y)),where ψ ∈ Ψ and ε > 1 is a constant andMT (x, y) = max{d(x, y), d(T 2x, y),d(Tx, y) + d(x, Ty)2s,d(T 2x, x) + d(T 2x, Ty)2s, d(T 2x, Ty) + d(T 2x, Tx),d(T 2x, Ty) + d(Tx, x), d(Tx, y) + d(y, Ty)}.Remark 2.2. Compared with Definition 2.1 of [11], Definition 2.1 hasmore general character, since if ε = 5, then Definition 2.1 is reduced toDefinition 2.1 of [11]. That is to say, Definition 2.1 is a large generaliza-tion of Definition 2.1 of [11].Theorem 2.3. Let (X, d) be a b-complete b-metric space with coefficients > 1 and T : X → X be a generalized F -Suzuki contraction. Then Thas a unique fixed point x∗ ∈ X and, for every x0 ∈ X the iterativesequence {Tnx0}n∈N b-converges to x∗.Proof. Take x0 ∈ X and let xn+1 = Txn for all n ∈ N. If there existsn0 ∈ N such that d(xn0 , Txn0) = 0, then x = xn0 becomes a fixed pointof T , which completes the proof. So, in the rest of the proof, we alwaysassume that 0 < d(xn, Txn), for all n ∈ N. Hence, we have12sd(xn, Txn) < d(xn, Txn) = d(xn, xn+1), for all n ∈ N.Then by (2.1) we haveF (sεd(Txn, Txn+1)) ≤ F (MT (xn, xn+1))− ψ(MT (xn, xn+1)).84 S. CHANDOK, H. HUANG, AND S. RADENOVIĆSincemax {d(xn, xn+1), d(xn+2, xn+1)}≤MT (xn, xn+1)= max{d(xn, xn+1), d(xn+2, xn+1),d(xn, xn+2)2s,d(xn+2, xn)2s,d(xn+2, xn+1), d(xn+1, xn), d(xn+1, xn+2)}≤ max{d(xn, xn+1), d(xn+2, xn+1),s[d(xn, xn+1) + d(xn+1, xn+2)]2s}≤ max {d(xn, xn+1), d(xn+2, xn+1)} ,we getMT (xn, xn+1) = max{d(xn, xn+1), d(xn+2, xn+1)}.ThenF (sεd(xn+1, xn+2)) = F (sεd(Txn, Txn+1))(2.2)≤ F (max{d(xn, xn+1), d(xn+2, xn+1)})− ψ(max{d(xn, xn+1), d(xn+2, xn+1)})≤ F (max{d(xn, xn+1), d(xn+2, xn+1)}) .By the monotonicity of F , it follows immediately from (2.2) thatsεd(xn+1, xn+2) ≤ max{d(xn, xn+1), d(xn+2, xn+1)}.(2.3)Ifd(xn, xn+1) < d(xn+2, xn+1),then (2.3) givessεd(xn+1, xn+2) ≤ d(xn+2, xn+1),which implies that d(xn+1, xn+2) = 0, a contradiction. Ifd(xn, xn+1) ≥ d(xn+2, xn+1),then (2.3) gives thatd(xn+1, xn+2) ≤1sεd(xn, xn+1).Hence, by using Lemma 1.6, {xn} is a b-Cauchy sequence. As (X, d)is b-complete, then {xn} b-converges to some point x∗ ∈ X. Therefore,limn→∞ d(xn, x∗) = 0.SOME FIXED POINT RESULTS FOR THE GENERALIZED · · · 85We claim that, for every n ∈ N,12sd(xn, Txn) < d(xn, x∗),(2.4)or12sd(Txn, T2xn) < d(Txn, x∗).(2.5)Indeed, suppose, on the contrary, that there exists m ∈ N such that12sd(xm, Txm) ≥ d(xm, x∗),12sd(Txm, T2xm) ≥ d(Txm, x∗).(2.6)Then2sd(xm, x∗) ≤ d(xm, Txm) ≤ s [d(xm, x∗) + d(x∗, Txm)] ,which implies that d(xm, x∗) ≤ d(x∗, Txm). As a result, we haved(Txm, T2xm) ≤1sεd(xm, Txm)(2.7)≤ 1sε· [sd(xm, x∗) + sd(x∗, Txm)]≤ 1sε· 2sd(x∗, Txm).It follows from (2.6) and (2.7) thatd(Txm, T2xm) ≤1sεd(Txm, T2xm).This is a contradiction unless d(Txm, T2xm) = 0. Consequently, (2.4)or (2.5) hold.If (2.4) is true, thenF (sεd(xn+1, Tx∗)) = F (sεd(Txn, Tx∗))(2.2)≤ F (MT (xn, x∗))− ψ(MT (xn, x)).86 S. CHANDOK, H. HUANG, AND S. RADENOVIĆSinced(x∗, Tx∗) ≤MT (xn, x∗)= max{d(xn, x∗), d(xn+2, x∗),d(xn+1, x∗) + d(xn, Tx∗)2s,d(xn+2, xn) + d(xn+2, Tx∗)2s, d(xn+2, Tx∗) + d(xn+2, xn+1),d(xn+2, Tx∗) + d(xn+1, xn), d(xn+1, x∗) + d(x∗, Tx∗)}≤ max{d(xn, x∗), d(xn+2, x∗),d(xn+1, x∗) + d(xn, Tx∗)2s,s[d(xn+2, xn+1) + d(xn+1, xn)] + d(xn+2, Tx∗)2s,d(xn+2, Tx∗) + d(xn+2, xn+1), d(xn+2, Tx∗) + d(xn+1, xn),d(xn+1, x∗) + d(x∗, Tx∗)},taking limit from both sides of the above inequality, we getlimn→∞MT (xn, x∗) = d(x∗, Tx∗).Thus by (2.2) and the continuity of ψ, it is easy to see thatF (sεd(x∗, Tx∗)) ≤ F (d(x∗, Tx∗))− ψ(d(x∗, Tx∗)) ≤ F (d(x∗, Tx∗)),which yields that x∗ = Tx∗.If (2.5) is true, using a similar method as the above, we have x∗ = Tx∗.Now we show that T has only one fixed point. Indeed, if y∗ ∈ X isanother fixed point of T , then0 =12sd(x∗, Tx∗) < d(x∗, y∗),and from (2.2), we obtainF (sεd(x∗, y∗)) = F (sεd(Tx∗, T y∗))≤ F (MT (x∗, y∗))− ψ(MT (x∗, y∗))= F (d(y∗, x∗))− ψ(d(y∗, x∗))≤ F (d(y∗, x∗)).This gives d(y∗, x∗) = 0. Hence y∗ = x∗. This completes the proof. □Remark 2.4. Compared with Theorem 2.2 of [11], Theorem 2.3 has asharp superiority. Indeed, because of the arbitrary value of the constantε > 1, it evidently contains the special case ε = 5. Accordingly, ourSOME FIXED POINT RESULTS FOR THE GENERALIZED · · · 87Theorem 2.3 greatly generalizes Theorem 2.2 from [11]. Otherwise, ourproof of Theorem 2.3 is simpler than the one of Theorem 2.2 from [11]since the former avoids the discontinuity problems of b-metric while thelatter deals with it too complicate.The following three theorems can be obtained easily by repeating thesteps in the proof of Theorem 2.3 and therefore we omit their proofs.Theorem 2.5. Let (X, d) be a b-complete b-metric space with coefficients > 1 and T : X → X be a map such that, for every x, y ∈ X,12sd(x, Tx) < d(x, y) ⇒ F (sεd(Tx, Ty)) ≤ F (MT (x, y)),where F ∈ F and ε > 1 is a constant. Then T has a unique fixedpoint x∗ ∈ X, and for every x0 ∈ X the iterative sequence {Tnx0}n∈Nb-converges to x∗.Theorem 2.6. Let (X, d) be a b-complete b-metric space with coefficients > 1 and T : X → X be a map such that, for every x, y ∈ X12sd(x, Tx) < d(x, y) ⇒ F (d(Tx, Ty)) ≤ F (MT (x, y))− ψ(N(x, y)),where F ∈ F, ψ ∈ Ψ andN(x, y) = max{d(x, y), d(x, Tx), d(y, Ty), d(T 2x, y), d(T 2x, Ty),d(T 2x, Tx),d(Tx, y) + d(x, Ty)2,d(T 2x, x) + d(T 2x, Ty)2}.Then T has a unique fixed point x∗ ∈ X, and for every x0 ∈ X theiterative sequence {Tnx0}n∈N b-converges to x∗.Theorem 2.7. Let (X, d) be a b-complete b-metric space with coefficients > 1 and T : X → X be a map such that, for every x, y ∈ X12sd(x, Tx) < d(x, y) ⇒ F (d(Tx, Ty)) ≤ F (MT (x, y))− ψ(d(x, y)),where F ∈ F and ψ ∈ Ψ. Then T has a unique fixed point x∗ ∈ X, andfor every x0 ∈ X, the iterative sequence {Tnx0}n∈N b-converges to x∗.Remark 2.8. Since a b-metric space is a metric space when s = 1,then we easily obtain the corresponding results in the setting of metricspaces.Acknowledgment. The research was partially supported by the Na-tional Natural Science Foundation of China (11271045).88 S. CHANDOK, H. HUANG, AND S. RADENOVIĆReferences1. M. Abbas, M. Berzig, T. Nazir, and E. Karapınar, Iterative ap-proximation of fixed points for presic type F -contraction operators,Uni. Pol. Bucharest Sci. Bul. A-Appl. Math. Phy., 78 (2) (2016),pp. 147-160.2. H.H. Alsulami, E. Karapınar, and H. Piri, Fixed points of gener-alized F-Suzuki type contraction in complete b-metric spaces, Dis.Dyn. Nat. Soc., Volume 2015, Article ID 969726, 8 pages.3. H.H. Alsulami, E. Karapınar, and H. Piri, Fixed points of modifiedF -contractive mappings in complete metric-like spaces, J. Funct.Spaces, Volume 2015, Article ID 270971, 9 pages.4. I.A. Bakhtin, The contraction principle in quasimetric spaces,Funct. Anal., 30 (1989), pp. 26-37.5. L. Ćirić, S. Chandok, and M. Abbas, Invariant approximation re-sults of generalized nonlinear contractive mappings, Filomat, 30(2016), pp. 3875-3883.6. S. Czerwik, Contraction mappings in b-metric spaces, Acta Math.Inform. Univ. Ostrav., 1 (1993), pp. 5-11.7. H. Ding, M. Imdad, S. Radenović, and J. Vujaković, On somefixed point results in b-metric, rectangular and b-rectangular metricspaces, Arab J. Math. Sci., 22 (2016), pp. 151-164.8. N.V. Dung, and V.L. Hang, A fixed point theorem for generalizedF -contractions on complete metric spaces, Vietnam J. Math., 43(2015), pp. 743-753.9. M. Jovanović, Z. Kadelburg, and S. Radenović, Common fixed pointresults in metric-type spaces, Fixed Point Theory Appl., Volume2010, Article ID 978121, 15 pages.10. E. Karapınar, M.A. Kutbi, H. Piri, and D. ÓRegan, Fixed points ofconditionally F -contractions in complete metric-like spaces, FixedPoint Theory Appl., 2015 (2015), Article ID 126, 14 pages.11. H. Piri, and P. Kumam, Fixed point theorems for generalized F -Suzuki-contraction mappings in complete b-metric spaces, FixedPoint Theory Appl., 2016 (2016), Article ID 90, 13 pages.12. S. Shukla, S. Radenović, and Z. Kadelburg, Some fixed point the-orems for ordered F -generalized contractions in 0-f-orbitally com-plete partial metric spaces, Theory Appl. Math. Comput. Sci., 4(2014), pp. 87-98.13. D. Wardowski, Fixed points of a new type of contractive mappingsin complete metric spaces, Fixed Point Theory Appl., 2012 (2012),Article ID 94, 11 pages.SOME FIXED POINT RESULTS FOR THE GENERALIZED · · · 8914. D. Wardowski, and N.V. Dung, Fixed points of F -weak contractionson complete metric spaces, Demonstratio Math., 67 (2014), pp. 146-155.1 School of Mathematics, Thapar University, Patiala-147004, India.E-mail address: sumit.chandok@thapar.edu2 School of Mathematical Sciences, Beijing Normal University, Lab-oratory of Mathematics and Complex Systems, Ministry of Education,Beijing 100875, PR China.E-mail address: mathhhp@163.com3 Faculty of Mechanical Engineering, University of Belgrade, KraljiceMarije 16, 11120, Beograd, Serbia.E-mail address: radens@beotel.net

Some Fixed Point Results for the Generalized $F$-suzuki Type Contractions in $b$-metric Spaces

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Some Fixed Point Results for the Generalized $F$ -suzuki Type Contractions in $b$ -metric Spaces

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Some Fixed Point Results for the Generalized FFF-suzuki Type Contractions in bbb-metric Spaces

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Some Fixed Point Results for the Generalized $F$ -suzuki Type Contractions in $b$ -metric Spaces