MTH604: Fixed Point Theory and Applications (Spring 2021)

This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem for multi-valued mappings are also educated.

Basic concepts: metric spaces, complete metric spaces, Vector spaces (linear spaces) normed spaces, Banach spaces, Banachs contraction principle, non-expansive mappings and related fixed point theorems. Contractive maps, properties of fixed points set and minimal sets. Multi-valued mappings and related fixed point theorems. Best approximation theorems.

  1. Define fixed point.
  2. Find the fixed point of $f(x)=x^2-4x+6$, where $x\in (0,3)$.
  3. Define orbit of $x_0$ under $F$, where $F$ is some real valued function.
  4. Draw the orbit of $2$ under $\exp(x)$, $x\in \mathbb{R}$.
  5. Let $F:E\to \mathbb{R}$ be a function. Then prove that $p$ is fixed point of $F$ iff $p$ is the zero of $F(x)-x$.
  6. Prove that $x^2+1$, where $x\in \mathbb{R}$ does not have fixed point.
  7. Show graphically that $\cos x$ has fixed point for $x\in \mathbb{R}$.
  8. Show graphically that $e^x$, $x\in \mathbb{R}$ doesn't have fixed point.
  9. State and prove intermediate value theorem.
  10. State and prove the fixed point theorem.
  11. Define attracting, repelling and neutral fixed point theorem.
  12. Consider a function $f(x)=x^2-1$. Find its fixed points and also find the nature of fixed point.
  13. Define orbit of the point $x_0$ under function $f$.
  14. Consider $C(x)=\cos (x)$. Draw the cobweb representation of the orbit of $-2$ under $C$.
  15. Consider $C(x)=\cos (x)$. Draw the cobweb representation of the orbit of $0$ under $C$.
  16. Let $L(x)=-1.4x$. Draw the cobweb representation of the orbit of $0.1$ under $L$.
  17. Consider a function $L(x)=mx$, where $m\in \mathbb{R}$. Find the value of $m$ for which $x=0$ is attracting fixed point.
  18. Let $F(x)=2(\sin x)^2$. Find the interval in which $F$ has attracting fixed point.
  19. State the mean value theorem.
  20. State and prove attracting fixed point theorem.
  21. State and prove repelling fixed point theorem.
  22. Find the fixed point of $f(x)=2x-2x^2$ by iteration method by taking initial guess $x_0=0.1$.

Notes and videos can be downloaded from the following university official page:

  1. B. Bollobas: W. Fulton, A. Katok, F. Kirwan and P. Sarnak: Fixed Point Theory and Applications, Cambridge University Press, 2001.
  2. K. Geoble and W.A Kirk: Topics in Metric Fixed Theory, Cambridge university Press, 1990
  3. M.C. Joshi And R.K. Bose: Some Topics in Nonlinear Functional Analysis, John Wiley and Sons, 1985
  4. James Dugundji and A. Granas: Fixed Point Theory, Vol. 1, Polish Scientific Publishers, 1982
  5. W.A. Kirk and B. Sims: Handbook of Metric Fixed Point Theory, Klawer Academic Publishers 2001
  6. M. Aslam Noor, Principles of Variational Inequalities Lapt-Lambert Academic Publishing AG & Co. Saarbrucken, Germany 2009.