Nonlinear Dynamics, Fixed Points and Coupled Fixed Points in Generalized Gauge Spaces with Applications to a System of Integral Equations

Abstract

We will discuss discrete dynamics generated by single-valued and multivalued operators in spaces endowed with a generalized metric structure. More precisely, the behavior of the sequence(fn(x))n∈Nof successive approximations in complete generalized gauge spaces is discussed. In the same setting, the case of multivalued operators is also considered. The coupled fixed points for mappingst1:X1×X2→X1andt2:X1×X2→X2are discussed and an application to a system of nonlinear integral equations is given.

Applications of Graph Kannan Mappings to the Damped Spring-Mass System and Deformation of an Elastic Beam

“…Jachymski [1] gave a unified version of Banach's contraction principle, by considering the graph structure instead of an order structure on a metric space. This result has been extended and generalized by many researchers in different ways (see, e.g., [2][3][4][5][6][7][8][9][10][11]). Recently, Shukla et al in [10] proposed the notion of graphical metric spaces as a graphical version of metric spaces.…”

Coupled fractals in complete metric spaces

“…There are many applications of the coupled fixed point theorems for solving different problems related to systems of integral and differential equations, see [2,4,6,9,22].…”

Generalized Contractions to Coupled Fixed Point Theorems in Partially Ordered Metric Spaces

“…More work on fixed points, common fixed points results in cone metric spaces, partially ordered metric spaces and others spaces can see from [12][13][14][15][16][17][18][19][20][21][22][23][24]. Recently, The existence and uniqueness of coupled fixed points on ordered sets have been investigated by many authors with various conditions on the mappings, readers may refer to [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] and references therein.…”