Abstract
This paper deals with the behavior of solutions of ordinary differential equations in a Hilbert Space. Under certain conditions, we obtain lower estimates or upper estimates (or both) for the norm of solutions of two kinds of equations. We also obtain results about the uniqueness and the quasi‐uniqueness of the Cauchy problems of these equations. A method similar to that of Agmon‐Nirenberg is used to study the uniqueness of the Cauchy problem for the non‐degenerate linear case.
Highlights
This paper deals with the behavior of solutions of ordinary differential equations in a Hilbert Space
In the third part we investigate graphs of lit’(.t)II and the uniqueless of the Cauchy problem for equations (0.1) anG (0.2) and ir part 4 we study several examples of quasi-liwear oralnary and prtial differential equations
For some consta,t C _> 0 for each u(t) from a dense subset DB of the Hilbert space H, for each solution of equation (1.1) we have the following estimate with the same C as in condition (1.5)
Summary
In the first part uf section we study equation (0.I), in the secofd part, equation (0.2). For some constant C, for each u(t) = 0 from a dense subset DB ef the Hilbert space H, for each non-trivial solutior k,’(t) of equation (I.I) we have the following estimate with the same C as in (I.26): for t < tO (1.27). From (1.27) ano (1.29), we have that in the situation of Theorem 1.9 ii) the function (1.22) is strongly monotonic and in the situation of Theorem 1.9 i), the function (1.23) is strongly monotonic; since our estimates (1.27) and (1.29) are true to, for each pair t, in I, with t tO From the montonicity of these functions, we obtain the following statement: THEOREM 1.]0 i) Under the conditions of Theorem 1.9, each non-trivial solution u(t) of equation (1.1), satisfies u(t) 0 for each t O) If u(t m 0 for some to > O, u(t) 0 in
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