Discontinuity, Nonlinearity, and Complexity
Interval Metric with Diameter Distance and Its Application to Fixed Point Theory
Discontinuity, Nonlinearity, and Complexity 11(2) (2022) 235241  DOI:10.5890/DNC.2022.06.004
Jeena M. S., Lovelymol Sebastian
Research and Post Graduate Department of Mathematics, St.Thomas College,
Palai, 686574, India
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Abstract
The concept of interval metric is defined in 2010. In the general interval metric, the domain is any set of points and range is one dimensional interval space $ IR $. The tool used here to prove its metric properties is the Moore interval module. In this paper, we extend the concept of interval metric to the function space $ C[a,b] $. We are giving a detailed extension of the concept of metric from a single value to an interval by taking all the possible values from the minimummaximum interval, and the range of all such possible values is considered as the diameter distance. We also prove the completeness property and Banach fixed point theorem for the new metric in $ C[a,b] $.
References

[1] 
Trindade, R.M.P., Bedregal, B.R.C., and DoriaNeto, A.D. An Interval Metric, 4, ISBN 978953307067.


[2] 
Santana, F. and Santiago, R. (2013), Interval metrics, topology and continuous functions, Comp. Appl. Math. 32, 459â€“470.


[3] 
Kreyszing, (1989), Introductory functional analysis with applications, Wiley.


[4] 
Lyra, A., Neto, A.D.D., Bedregal, B.R.C., and Trindade, R.M.P. (2019), A Polar Representation for Complex Interval Numbers, ReCiC, Computer Science Magazine.
% 

[5] 
%Viana, H.O., De Novais, D.P.S., and Trindade, R.M.P. (2010), Mathematical Treatment of Uncertainty in the Speech Recognition Process, Mathematical Models for Engineering Science, 3, 474252
%
%
%
% 

[6] 
% Trindade, R.M.P., Bedregal, B.R.C., and DoriaNeto, A.D. (2010), Basic concepts of interval signal processing, International Journal of Electrical, Computer, and Systems Engineering, 4, 135139.
