Dear Sachin,

I don't know if I had understood your but the book Chandrasekhar

Chandrasekhar, S., Radiative transfer, Dover Publ., New York, 1960.

Or just look some Hammerstein integral equation, in every book of Ioannis K. Argyros you will found some of them.

I hope I have helped you,

Alberto.

I can recommend to check the book:

Stability by Fixed Point Theory for Functional Differential Equations (Dover Books on Mathematics) (Inglés)  nov 2006

by  T A Burton

It concentrates on dynamic systems with delays but  the ideas can be useful for other kinds of differential equations.

Basically , the basic actions to take  are : a) To write the solution in an integral form, b) to take norms of the formula and may be pass it with an inequality to the norm of the expression under the integral symbol, c) to upper-bound the above expression by real constants, functions etc. ( some upperbounding constant is  supposed to be less than one in the contractive case -or equal to one in the non-expansive, non-contractive  one- ,d) Compare the solutions for two distinct initial conditions by checking conditions of this error to converge asymptotically to zero and the solutions are bounded.

Probably , in that case,  you have a fixed point, which is also an equilibrium point. You have , as a result, either global or local Lyapunov asymptotic stability depending on the region where the iniital conditions and trajectory solution belong to. Fixed Point Theory can be used in this context to prove uniqueness of the solution for each given init.cond. and also  for stability analysis.