On differential and integral equations in locally convex spaces. (English) Zbl 0855.34071

In the paper there is proved a Kneser-type theorem for the continuous solution set of the Volterra equation \[ x(t) = g(t) + \int_{A(t)} f \bigl( t,s,x(s) \bigr) ds, \quad t \in A. \] Further an existence theorem for continuous solutions of the Urysohn equation \[ x(t) = g(t) + \lambda \int_A f \bigl( t,s,x(s) \bigr) ds, \quad t \in A,\;\lambda \in \mathbb{R}^1, \] is proved. The solutions are considered in the sequentially complete locally convex space containing a compact barel. Nearby \(A = \langle 0, a_1 \rangle \times \langle 0, a_2 \rangle \times \cdots \times \langle 0, a_n \rangle\), \(a_i > 0\) and \(A(t) = \{s \in \mathbb{R}^n\mid 0 \leq s_i \leq t_i,\;i = \overline {1,n}\}\).


34G20 Nonlinear differential equations in abstract spaces
45D05 Volterra integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
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