Coincidence and common fixed point results for β-quasi contractive mappings on metric spaces endowed with binary relation

Mathematical Sciences, Jul 2016

Coincidence and common fixed point theorems for \(\beta\)-quasi contractive mappings on metric spaces endowed with binary relations and involving suitable comparison functions are presented. Our results generalize, improve, and extend several recent results. As an application, we study the existence of solutions for some class of integral equations.

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Coincidence and common fixed point results for β-quasi contractive mappings on metric spaces endowed with binary relation

Math Sci Coincidence and common fixed point results for b-quasi contractive mappings on metric spaces endowed with binary relation M. I. Ayari 0 1 2 3 M. Berzig 0 1 2 3 I. Ke´dim 0 1 2 3 0 Mathematics Subject Classification 47H10 1 Carthage University, Faculte ́ des Sciences de Bizerte , 7021 Bizerte, Zarzouna , Tunisia 2 Tunis University, E ́ cole Nationale Supe ́rieure d'Inge ́nieurs de Tunis , 5 Avenue Taha Hussein, BP 56, Bab Manara, 1008 Tunis , Tunisia 3 Carthage University, Institu National Des Sciences Applique ́e et de Technologie de Tunis, Centre Urbain Nord , BP 676-1080 Tunis , Tunisia Coincidence and common fixed point theorems for b-quasi contractive mappings on metric spaces endowed with binary relations and involving suitable comparison functions are presented. Our results generalize, improve, and extend several recent results. As an application, we study the existence of solutions for some class of integral equations. Fixed point theory; Ordered metric spaces; Transitive binary relation - 54H25 Introduction The Banach contraction principle [ 1 ] may be considered as one of most powerful tools for establishing existence and uniqueness of solutions for various non linear problems. & dðTx; TyÞ q maxfdðx; yÞ; dðx; TxÞ; dðy; TyÞ; dðx; TyÞ; dðy; TxÞg; for all x; y 2 X. It is worthwhile noting that the quasi-contractive condition subsumes the original contraction and may also be applied to non continuous mappings. Later, Samet and Turinici [ 3 ], established an important fixed point result where the contractive condition is required only on binary related elements of X. Several known results from literature can be derived from their main theorem, for instance we cite those of Kannan [ 4 ], Chatterjea [ 5 ], Hardy and Rogers [ 6 ], Ran and Reurings [ 7 ], Nieto and Lo´ pez [ 8 ], C´ iric´ [ 9 ] and Kirk et al. [ 10 ]. More recently, there have been some attempts to generalize the fixed point results for metric spaces endowed with binary relation. For more details, we refer the reader to [ 11–19 ]. In this paper, some coincidence point results in complete metric spaces endowed with transitive binary relations are first highlighted. Subsequently, the existence and uniqueness of common fixed point theorem for two mappings are established under some suitable conditions. Moreover, some coincidence and common fixed point theorems involving amorphous binary relation are proven. Finally, as an application, existence study of solutions to some integral ð1Þ equation is presented. The paper is divided into five sections. Section introduces the notation used herein, presents a number of definitions and recalls some useful results. Coincidence and common fixed point theorems are stated in Sect. 1, while their proofs are the subject of Sect. 2. Several consequences are subsequently derived in Sect. 3. Finally, the existence of solutions for some class of Urysohn integral equation is shown in Sect. 4. Preliminaries Let us introduce some definitions and recall some basic preliminary results which will be needed in the following sections. Throughout this paper, we denote by N the set of 0 . Let (X, d) be a all positive integer and N0 ¼ N [ f g metric space, R be a binary relation on X and g; T : X ! X be two mappings. Denote by Cðg; TÞ :¼ fx 2 X : gx ¼ Tx g: the set of coincidence points of g and T. Let x0 2 X, and suppose that T ðXÞ gðXÞ. Hence, we can choose x1 2 X satisfying gx1 ¼ Tx0. Again from T ðXÞ gðXÞ; we can choose x2 2 X satisfying gx2 ¼ Tx1: Continuing this process, we construct a sequence fxng satisfying gxnþ1 ¼ Txn for all n 2 N0: ð2Þ Denote by Oðx0Þ the set of all sequences fxng satisfying (2). Define also the g-orbit set, at s ¼ fxkgk2N0 2 Oðx0Þ, from p to q with q [ p [ 0, by Op;qðg; sÞ ¼ fgxp; gxpþ1; . . .; gxqg. Similarly, define the infinite g-orbit set at s from p by Op;1ðg; sÞ ¼ fgxp; gxpþ1; gxpþ2; . . .g. Definition 1.1 We say that (X, d) is g-orbitally complete if and only if for all x0 2 X and s 2 Oðx0Þ: every Cauchy sequence in O0;1ðg; sÞ converges in X. Definition 1.2 ([ 11 ]) A subset D of X is called R-g-directed if for every x; y 2 D, there exists z 2 X such that gxRgz and gyRgz. Definition 1.3 ([ 3 ]) We say that ðX; d; RÞ is regular if for a sequence fxng in X, if we have xnRxnþ1 for all n 2 N0 and limn!1 dðxn; xÞ ¼ 0 for some x 2 X; then there exists a subsequence fxnðkÞg of fxng such that xnðkÞRx for all k 2 N0. Definition 1.4 ([ 11 ]) Let X be a non empty set, g; T : X ! X be two mappings and R be a binary relation. We say that T is g-comparative if x; y 2 X; gxRgy ¼) TxRTy: Definition 1.5 Let b 2 ð0; þ1Þ and u : ½0; þ1Þ ! ½0; þ1Þ be a function satisfying the properties: (P1) u is nondecreasing; (P2) (P3) (P4) limn!1 unbðtÞ ¼ 0 for all t [ 0, where unb denote the n-th iterate of ub and ubðtÞ ¼ uðb tÞ; there exists s 2 ð0; þ1Þ such that Pn1¼1 unbðsÞ \1; ðid ubÞ ubðtÞ ub ðid ubÞðtÞforallt 0; where id : ½0; þ1Þ ! ½0; þ1Þ is the identity func (...truncated)


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M. I. Ayari, M. Berzig, I. Kédim. Coincidence and common fixed point results for β-quasi contractive mappings on metric spaces endowed with binary relation, Mathematical Sciences, 2016, pp. 105-114, Volume 10, Issue 3, DOI: 10.1007/s40096-016-0183-z