I look for mathematical theory of Differential equations like
Picard–Lindelöf theorem and Carathéodory's existence theorem that deal with existence/uniqueness of differential equations. I hope some purely theoretical reference can address such theoretical methods not just applied methods.
Theory of Ordinary Differential equations" by Coddington. If you find the previous one advanced, there is "An Introduction to Ordinary Differential Equations" by Coddington too. The first one is very complete and have a lot of things that I have never seen in other books. The second one is very basic (there is almost no qualitative theory), but proves the validity of some techniques usually used to solve ordinary differential equations. They also gives some explicit formulas for solution in special cases.Carmen Chicone's book "Ordinary Differential Equations with Applications" is good too if you want to study better the qualitative theory.
http://b-ok.org/book/442378/88eab8
http://b-ok.org/book/442443/4cc40c
What type of differential equations you are interested in? ODEs or PDEs?
For ODEs:
"The Theory of Differential Equations: Classical and Qualitative" by Kelley, Walter G., Peterson, Allan C.
"Differential equations: with applications and historical notes" by G.F.Simmons.
"Theory of Ordinary differential equations" by E.A. Coddington.
"Differential quations, dynamical systems and introduction to chaos" by Hirsch, Smale and Devaney.
For PDEs:
"Partial differential equations" by L.C. Evans.
"Introduction to partial differential quations" by G.B. Folland's.
Introduction to partial differential equations by sankara rao is used to study PDE. Theory of ODE by E.A. Coddington
It looks like you look for classical references on existence and uniqueness theorems. In this case, it might be a good idea to have a look at the following texts:
Murray, Francis J.; Miller, Kenneth S. Existence theorems for ordinary differential equations. New York University Press, New York, 1954.
Coddington, Earl A.; Levinson, Norman Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955 (Chapter 2).
Hartman, Philip Ordinary differential equations. John Wiley & Sons, Inc., New York-London-Sydney, 1964 (Chapter 2).
For a modern exposition, you may like to check
O'Regan, Donal Existence theory for nonlinear ordinary differential equations. Mathematics and its Applications, 398. Kluwer Academic Publishers Group, Dordrecht, 1997.
Kelley, Walter G.; Peterson, Allan C. The theory of differential equations. Classical and qualitative. Second edition. Universitext. Springer, New York, 2010.
That is the best: "Differential quations, dynamical systems and introduction to chaos" by Hirsch, Smale and Devaney.
(But if you are lucky enough to read in Hungarian: https://moly.hu/konyvek/toth-janos-simon-l-peter-differencialegyenletek)
:):)
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