Turkish Journal of Analysis and Number Theory, 2018, Vol. 6, No. 2, 43-48 Available online at http://pubs.sciepub.com/tjant/6/2/2 ©Science and Education Publishing DOI:10.12691/tjant-6-2-2 Generalized Dynamic Process for Generalized Multivalued F-contraction of Hardy Rogers Type in b-metric Spaces Abdullah Shoaib 1,* , Awais Asif 2 , Muhammad Arshad 2 , Eskandar Ameer 2 1 Department of Mathematics and Statistics, Riphah International University, Islamabad - 44000, Pakistan 2 Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan *Corresponding author: abdullahshoaib15@yahoo.com Received September 27, 2016; Revised February 07, 2018; Accepted March 28, 2018 Abstract The aim of this paper is to establish common fixed point results for multivalued mappings satisfying generalized F-contractive conditions of Hardy Rogers type with respect to generalized dynamic process in b-metric space. Our results improve and generalize several well known results in the existing literature. Keywords: fixed point, generalized F-contraction, b-metric space, generalized dynamic process, Hausdorff metric Cite This Article: Abdullah Shoaib, Awais Asif, Muhammad Arshad, and Eskandar Ameer, “Generalized Dynamic Process for Generalized Multivalued F-contraction of Hardy Rogers Type in b-metric Spaces.” Turkish Journal of Analysis and Number Theory, vol. 6, no. 2 (2018): 43-48. doi: 10.12691/tjant-6-2-2. 1. Introduction and Preliminaries Let X be a non empty set, : f X X → be a mapping. A point x X ∈ is called a fixed point of f if . x fx = Fixed points results of mappings, which satisfies some specific contractive conditions on some space have been very useful in research activity (see [1-30]). Recently, Wardowski [30] introduced a new concept of contraction named F-contraction and proved a fixed point theorem which generalizes Banach contraction principle. Klim et al. [22] further established fixed point result for F-contractive mapping in dynamic process. Cosentino et al. [16] further generalized this concept as F-Contractive Mappings of Hardy-Rogers-Type. Arshad et al. [4] proved fixed point result in GF α − − contraction of Hardy-Rogers-type. Following this direction of research, in this paper, we will present some fixed point results of Hardy-Rogers-type for multivalued mappings in b-metric space with generalized dynamic process. This paper contain common fixed point results for two mappings. Throughout our paper , R + N and ( ) CB X represent set of real numbers, set of natural numbers and family of non- empty closed bounded subsets X respectively. Definition 1 [10] Let X be a non-empty set and let 1 s ≥ be a given real number. A function : d X X R + × → is called a b-metric provided that, for all , , xyz X ∈ 1) (, ) 0 dxy = iff x y = 2) (, ) (,) dxy dyx = 3) (,) [(, ) ( , )]. dxz sdxy dyz ≤ + The pair ( , ) Xd is called a b-metric space. Definition 2 [15] Let ( , ) Xd be a b-metric space. Then a sequence { } n x in X is called a Cauchy sequence if and only if for all 0 ε > there exist () n N ε ∈ such that for each , () nm n ε > , we have ( , ) . n m dx x ε < Definition 3 [30] Let : F R R + → be a mapping satisfying: (F1) F is strictly increasing. (F2) for each sequence { } n a R + ⊂ of positive numbers lim 0 n n a →∞ = if and only if ( ) lim n n Fa →∞ = −∞ (F3) there exists ( ) 0,1 k ∈ such that 0 lim a + → () 0. k aFa = We denote with  the family of all functions F that satisfy the conditions (F1)-(F3). Let ( , ) Xd be a metric space. A self-mapping T on X is called an F-contraction if there exist R τ + ∈ such that (( , ) ((, )), for all , with ( , ) 0. F d Tx Ty Fdxy xy X d Tx Ty τ + ≤ ∈ > Theorem 4 [30] Let ( , ) Xd be a complete metric space and let : T X X → be an F-contraction. Then T has a unique fixed point x X ∗ ∈ and for every 0 x X ∈ a sequence 0 { } n nN T x ∈ is convergent to . x ∗ Definition 5 [16] Let ( , ) Xd be a metric space. A self- mapping T on X is called a generalized F-contraction of Hardy-Rogers-type if there exist F ∈  and R τ + ∈ such that (( , )) ( (, ) (, ) (, ) (, ) (, )), F d Tx Ty F dxy d x Tx d y Ty d x Ty Ld y Tx τ α β γ δ + ≤ + + + +