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A fixed point index approach to some differential equations. (English) Zbl 0685.55001

Topological fixed point theory and applications, Proc. Conf., Tianjin/China 1988, Lect. Notes Math. 1411, 9-14 (1989).
[For the entire collection see Zbl 0679.00011.]
The ideal situation in the differential equations, ideal both for proving the existence of the solution and for finding this solution numerically - is when the equation is given in the explicit form, i.e. when the higher order derivative is represented as an explicit function of the unknown function and its lower order derivatives, i.e. \(x'(t)=F(x(t),t)\), \(x''(t)=F(x(t),x'(t),t),..\). In some cases, however (e.g. in case of the satellite motion) in the first approximation one has explicit equations, but detailed analysis reveals that one should add additional terms to the right-hand side, depending on the higher order derivative, so we come to equations of the type \(x'(t)=F(x(t),x'(t),t)\) that are implicit in \(x'(t)\). In case for every t and x(t) this implicit equation determines the unique value of \(x'(t)\) we again obtain the above case. But sometimes \(x'\) is not uniquely determined, and in this case we cannot apply already known existence theorems. The author proves new theorems for this case by reducing it to the case of differential inclusions \(x'\in G(x(t),t)\), where G is the set of all solutions of the equation \(z=F(x(t),t,z)\)- i.e. the set of all fixed points of F. So the known results about fixed points allow to obtain a good reduction. The resulting author’s theorems are sufficiently general, they cover higher order equations, partial differential equations and differential equations in Banach spaces.
Reviewer: O.Kosheleva

MSC:

55M20 Fixed points and coincidences in algebraic topology
47H10 Fixed-point theorems

Citations:

Zbl 0679.00011