Fixed Point Theory in Fractals

Fixed point theory is an exciting area of interest with diversity of applications not only in the different disciplines of mathematics but also in the emerging areas of engineering sciences. The importance of the fixed point theory lies mainly in the fact that the equations arising in the most physical formulations may easily be reduced to fixed point equations or inclusions. We come across the usefulness of this theory in the existence theory of differential equations, integral equations, partial differential equations, dynamic programming, fractal theory, discrete dynamics and theory of chaos, population dynamics, differential inclusions, system analysis and other diverse disciplines. In this project report we have focused on the applicable aspect of fixed point theory in fractals. After giving a necessary introduction of the subject we have presented a brief review of the literature. Some fractal generating algorithms are also discussed and based upon them we have generated the fractals using Matlab programs. An experimental approach to iterated function system (IFS) is studied via function iterations such as Picard, Mann and Ishikawa iterates. Our work on Ishikawa orbits generalizes the recent result of Singh et al [77]. We have studied the nature of the famous logistic maps for different values of the parameter r by using these iterative procedures. The Julia sets for a transcentental function (i.e. λ ez ) are computed.. We used Matlab programs to compute and plot the obtained results. Goldman [41] presented a constructive procedure based on the de Casteljau subdivision algorithm for generating Iterated Function System (IFS) and proved that every Bezier curve is an attractor of IFS. We extended his results to rational Bezier curves and such curves are shown to have fractal nature