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.

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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\textit{R. I. Kozlov}, Math. Notes 45, No. 1--2, 104--108 (1989; Zbl 0701.34008); translation from Mat. Zametki 45, No. 2, 22--29 (1989)

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##### References:

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