Some results on best proximity points of cyclic alpha-psi contractions in Menger probabilistic metric spaces

Abstract

This paper investigates properties of convergence of distances of p-cyclic a-w-type contractions on the union of the p subsets of a space X defining probabilistic metric spaces and Menger spaces. The paper also investigates the characterization of both Cauchy and G-Cauchy sequences which are convergent, in particular, to best proximity points. On the other hand, the existence and uniqueness of fixed points and best proximity points of pcyclic a-w-type contractions are also investigated. The fixed points of the p-composite self-mappings, which are obtained from the p-cyclic self-mapping restricted to each of the p subsets in the cyclic disposal, are also investigated while a generalization and some illustrative examples are also given.

(C , Ψ * , G ) Class of Contractions and Fixed Points in a Metric Space Endowed with a Graph

“…In [20], Samet et al introduced the concept of α-admissible mappings, and proved fixed point theorems for α-ψ contractive-type mappings, which paved a way to prove new results and generalise existing results in the fixed point theory. For some recent results on fixed point theorems of α-admissible mappings, the reader may refer to [21][22][23][24].…”

On fixed points and convergence results of sequences generated by uniformly convergent and point-wisely convergent sequences of operators in Menger probabilistic metric spaces

“…Fixed point theory is an important tool to investigate the convergence of sequences to limits and unique limits in metric spaces and normed spaces. See, for instance, Pap et al ( 1996 ), Sehgal and Bharucha-Reid ( 1972 ), Schweizer and Sklar ( 1960 ), Eldred and Veeramani ( 2006 ), De la Sen ( 2010a , b ), Choudhury et al ( 2011 , 2012 ), De la Sen and Karapinar ( 2014 , 2015a , b ), Beg et al ( 2001 ), Roldan et al ( 2014 ), Jleli et al ( 2014 ), Roldán-Lopez-de-Hierro et al ( 2015 ), Khan et al ( 1984 ), Choudhury and Das ( 2008 ), Gopal et al ( 2014 ), Takahashi ( 1970 ), Shimizu and Takahashi ( 1996 ), Kaewcharoen and Panyanak ( 2008 ), Karpagam and Agrawal ( 2009 ), Suzuki ( 2006 ), Di Bari et al ( 2008 ), Rezapour et al ( 2011 ), Derafshpour et al ( 2010 ), Al-Thagafi and Shahzad ( 2009 ), Karpagam and Agrawal ( 2009 ), Dutta et al ( 2009 ), Chang et al ( 2001 ), Chen et al ( 2012 ), Chen ( 2012 ), Berinde ( 2007 ), De la Sen et al ( 2015 ) and the wide list of references cited in those papers. In particular, fixed point theory is also a relevant tool to investigate iterative schemes and stability theory of continuous-time and discrete-time dynamic systems, boundedness of the trajectory solutions, stability of equilibrium points, convergence to stable equilibrium points and the existence oscillatory solution trajectories.…”

“…On the other hand, fixed point theory is, in particular, receiving important research attention in the framework of probabilistic metric spaces. See, for instance, Schweizer and Sklar ( 1960 , 1983 ), Pap et al ( 1996 ), Sehgal and Bharucha-Reid ( 1972 ), Choudhury et al ( 2011 , 2012 ), De la Sen and Karapinar ( 2015a ), Beg et al ( 2001 ) and references therein. Also, Menger probabilistic metric spaces are a special class of the wide class of probabilistic metric spaces which are endowed with a triangular norm, (Pap et al 1996 ; Sehgal and Bharucha-Reid 1972 ; Choudhury et al 2011 ; De la Sen and Karapinar 2015a , b ; Choudhury and Das 2008 ; Gopal et al 2014 ) and which are very useful in the context of fixed point theory since the triangular norm plays a close role to that of the norm in normed spaces.…”

“…See, for instance, Schweizer and Sklar ( 1960 , 1983 ), Pap et al ( 1996 ), Sehgal and Bharucha-Reid ( 1972 ), Choudhury et al ( 2011 , 2012 ), De la Sen and Karapinar ( 2015a ), Beg et al ( 2001 ) and references therein. Also, Menger probabilistic metric spaces are a special class of the wide class of probabilistic metric spaces which are endowed with a triangular norm, (Pap et al 1996 ; Sehgal and Bharucha-Reid 1972 ; Choudhury et al 2011 ; De la Sen and Karapinar 2015a , b ; Choudhury and Das 2008 ; Gopal et al 2014 ) and which are very useful in the context of fixed point theory since the triangular norm plays a close role to that of the norm in normed spaces. In probabilistic metric spaces, the deterministic notion of distance is considered to be probabilistic in the sense that, given any two points x and y of a metric space, a measure of the distance between them is a probabilistic metric F x , y ( t ), rather than the deterministic distance d ( x , y ), which is interpreted as the probability of the distance between x and y being less than t ( t > 0), (Sehgal and Bharucha-Reid 1972 ).…”

“…Such a contraction was proved to be useful for multivalued mappings while it generalizes the previous concept of C -contraction. On the other hand, cyclic contractions on subsets of complete Menger spaces were discussed in Choudhury et al ( 2011 , 2012 ), De la Sen and Karapinar ( 2015a ). Also, some types of contractions in complete probabilistic Menger spaces have been studied through the use of the so-called altering distances.…”