On Nemytskij operator in the space of absolutely continuous set-valued functions
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Jakub J. Ludew
Abstract
We consider the Nemytskij operator, defined by (Nφ)(x) ≔ G(x, φ(x)), where G is a given set-valued function. It is shown that if N maps AC(I, C), the space of all absolutely continuous functions on the interval I ≔ [0, 1] with values in a cone C in a reflexive Banach space, into AC(I, 𝒦), the space of all absolutely continuous set-valued functions on I with values in the set 𝒦, consisting of all compact intervals (including degenerate ones) on the real line ℝ, and N is uniformly continuous, then the generator G is of the form
G(x, y) = A(x)(y) + B(x),
where the function A(x) is additive and uniformly continuous for every x ∈ I and, moreover, the functions x ↦ A(x)(y) and B are absolutely continuous. Moreover, a condition, under which the Nemytskij operator maps the space AC(I, C) into AC(I, 𝒦) and is Lipschitzian, is given.
© de Gruyter 2011
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- Oscillation criteria for a class of third-order nonlinear neutral differential equations
- Ellipses numbers and geometric measure representations
- Duals of homogeneous weighted sequence Besov spaces and applications
- On positive definite and stationary sequences with respect to polynomial hypergroups
- Relative functional entropy in convex analysis
- Damped wave equations with dynamic boundary conditions
- On Nemytskij operator in the space of absolutely continuous set-valued functions
Artikel in diesem Heft
- Oscillation criteria for a class of third-order nonlinear neutral differential equations
- Ellipses numbers and geometric measure representations
- Duals of homogeneous weighted sequence Besov spaces and applications
- On positive definite and stationary sequences with respect to polynomial hypergroups
- Relative functional entropy in convex analysis
- Damped wave equations with dynamic boundary conditions
- On Nemytskij operator in the space of absolutely continuous set-valued functions
