Fixed point theory is one of the classical as well as modern (active) areas for investigating new methods of solutions of equations (linear and nonlinear), which arise in a physical phenomenon or in related areas. Fixed point theorems give the conditions under which mappings (single or multivalued) have solutions. In this theory we write equations in the form of operator equations then under certain conditions on these operators we obtain their solutions. Over the last 50 years or so the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. In particular fixed point techniques have been applied in such diverse fields as biology, economics, engineering, game theory, and physical sciences. Banach contraction principle (BCP) is one of the most important results in fixed point theory. The BCP and its several generalizations for single and multivalued mappings in metric, topological, locally convex and uniform spaces have played a significant role in dealing with various problems arising in nonlinear analysis and, in particular, are powerful tools in the study of dynamic processes of, convergence of generalized sequences of random iterations and also in the study of the problem concerning the existence and uniqueness of fixed points, coincidence points, stationary points and invariant sets of mappings.
I began studying this discipline in 2003 and have made contributions in following directions:
1. Fixed point of single and multi-valued operators
2. Stationary points of multi-valued operators.
3. Best proximity points of contractive operators.
4. Stability results of fixed points of single operators over variable domains.
My future research plans are to continue working in the areas described above, and also to begin a study of chaotic dynamical system, fractals and stability analysis.