Let ((X, d) ) be a complete metric space and (T:X to operatorname{CB}(X) ) be a multivalued mapping, (g:Xto X ) be a continuous self-map and (alpha : [0,infty)to[0,1) ) a mapping satisfying (limsup_{rto t^alpha(r)<1 ), for each (tin[0,infty) ).
Let (T : X to operatorname{CB}(X) ) be a multivalued mapping, (g : X to X ) be a continuous self-map, and (alpha: (0,infty) to[0,1) ) a mapping satisfying (limsup_{r to t^ alpha(r) < 1 ) for every (t in[0,infty) ).
Let ((X,d)) be a metric space, (T:Xto X) be a mapping, and (alpha :Xtimes Xto{{-infty}}cup 0,infty)) be a symmetric function.
Since f is a T-cyclic (( alpha,beta ) )-admissible mapping and (alpha ( Tx_{0} ) geq1) then (beta ( fx_{0} ) =beta ( Tx_{1} ) geq1), which implies (alpha ( Tx_{2} ) =alpha ( fx_{1} ) geq1).
Let (T : X to operatorname{CB}(X) ) be a multivalued mapping, and (alpha: (0,infty) to[0,1) ) a mapping satisfying (limsup_{r to t^ alpha(r) < 1 ) for every (t in [0,infty) ).
Let T be a self-mapping on X and (alpha: X times X to [0,+infty)) be a function. We say that T is an α-admissible mapping if x,yin X, quad alpha x,y geq1 quadLongrightarrow quadalpha (Tx,Ty geq1. [19].
Let T be a self-mapping on X and (alpha: X times X rightarrow [0, +infty)) be a function. We say that T is an α-admissible mapping if x,y in X, quad alpha x,y geq1 quad Rightarrow quad alpha(Tx,Ty geq1. [15].
Let T be a self-mapping on X and (alpha: X times X rightarrow [0, +infty)) be a function. We say that T is a triangular α-admissible mapping if T is α-admissible and x,y,z in X, quad alpha x,z geq1 quadtext{and}quad alpha z,y geq1 quad Rightarrow quad alpha (x,y geq1. Very recently, Popescu [16] improved the notion of α-admissible as follows.
Let T be a self-mapping on X and (alpha, eta: X times X to [0,+infty)) be two functions.
Scale bar = 20 μm (PDF 661 kb) ESM Fig. 7 Two-colour heat map representation of beta and alpha cell enriched genes in INS-GFP+ cell populations derived using different protocols.
