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Some properties of uniformly upper-right semicontinuous functions and application to differential equations. (English. Russian original) Zbl 0701.34008
Math. Notes 45, No. 1-2, 104-108 (1989); translation from Mat. Zametki 45, No. 2, 22-29 (1989).
The paper proves two theorems concerning a class \(\Pi\) (\({\mathcal T})\) of vector functions \(\phi\) defined on a bounded time interval \({\mathcal T}\). The class \(\Pi\) (\({\mathcal T})\) is sufficiently general and includes the class of absolutely upper semicontinuous functions and also sums of uniformly continuous and nondecreasing functions. Theorem 1 shows that if \(\{\phi_ k\}\) is a sequence of upper right-equisemicontinuous functions that are uniformly bounded on the closed time interval T and the sequence converges pointwise to the limit function \(\phi\), then the latter is a member of the class \(\Pi\) (\({\mathcal T}).\)
Theorem 2 shows that if g is the maximum solution on the time interval \({\mathcal T}\) for the Cauchy problem \(dy/dt=f(t,y),\) \(y_ 0=g(t_ 0)\) where f is a measurable function on a certain domain A which is an open connected set in \({\mathcal T}\times R^ n\), then g is upper semicontinuous on \({\mathcal T}\) with respect to the initial values and the term on the right-hand side of the differential equation.
Reviewer: S.K.Lakshmana Rao
MSC:
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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