Applications of Fixed-Point Theory in Differential Equations

Fixed-point theory is the most dynamic area of research, with numerous applications both in pure and applied mathematics, along with physical, chemical and social sciences. This theory is useful as it describes the ways to establish the existence and uniqueness of solutions of algebraic, differential and integral equations. Also, the theory has a number of applications in game theory, control theory, eigenvalue problems, boundary value problems and best approximation problems. The theory has gained a remarkable scope of research in nonlinear analysis also. The origin of fixed-point theory lies in the method of successive approximations used to establish the existence and uniqueness of solutions of differential equations by Liouville and Picard independently. The formal theoretic approach of the fixed-point originated from the work of Picard.