Existence of solutions for fixed point theorem

The development of a mathematical model based on diffusion has received a great dealof attention in recent years, many scientist and mathematician have tried to apply basicknowledge about the differential equation and the boundary condition to explain anapproximate the diffusion and reaction model. The subject of fractional calculus attracted much attentions and is rapidly growing area of research because of itsnumerous applications in engineering and scientific disciplines such as signal processing, nonlinear control theory, viscoelasticity, optimization theory [1], controlled thermonuclear fusion, chemistry, nonlinear biological systems, mechanics,electric networks, fluid dynamics, diffusion, oscillation, relaxation, turbulence, stochastic dynamical system, plasmaphysics, polymer physics, chemical physics, astrophysics, and economics. Therefore, it deserves an independent theoryparallel to the theory of ordinary differential equations (DEs).In the development of non-linear analysis, fixed point theory plays an important role. Also, it has been widely used in different branches of engineering and sciences. Banach fixed point theory is a essential part of mathematical analysis because of its applications in various area such as variational and linear inequalities, improvement and approximation theory. The fixed-point theorem in diffusion equations plays a significant role to construct methods to solve the problems in sciences and mathematics. Although Banach fixed point theory is a vast field of study and is capable of solving diffusion equations. The main motive of the research is solving the diffusion equations by Banach fixed point theorems and Adomian decomposition method. To analysis the drawbacks of the other fixed-point theorems and different solving methods, the related works are reviewed in this paper.