There are many many papers studies the sufficient condition of existences of solution of fractional differential equations.
Why we are not find the exact solution direct? and, what are disciplinary which use the theory of existence and uniqueness solutions?
If exact solutions can be found for a class of FDEs, there is no need to discuss the sufficient conditions for existences of solutions. But the fact is, it is so hard to get exact solutions of general FDEs. Then we consider the existence, uniqueness, numerical algorithms, ...
There are no special reasons different with ODEs or PDEs, I think.
The only rational difference equation we can solve in detail is the so called Riccati equation. The problem of finding the forbidden set of rational difference equations is of paramount importance.
Article The Forbidden Set of xn+1=p+xn-1xn
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Hello, Sabri. It depends on your interests. Myself, as a mathematician, existence and/or uniqueness of solutions of functional equations oif any kind is importiant in itself, because if you have amathematical model of some phenomenon and the model does NOT predict that the future behavior is uniquely determined by the current situation and past history, you are in deep theoretical trouble and it might be necessarfy to substitute by a stochastic one or similar (like fuzzy). Besides,. as Zhao pointed out, there are so few fractional DEs which have explicit solutions that building up a good theory of existence and uniqueness is a must.
Dear Sabri, It might be helpful for you to take a look st the paper I have uploaded.
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