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On the solvability of a complete second-order differential equation in Banach space. (English. Russian original) Zbl 0814.34048

Ukr. Math. J. 45, No. 10, 1629-1635 (1993); translation from Ukr. Mat. Zh. 45, No. 10, 1449-1454 (1993).
Summary: For the complete second-order differential equation with unbounded operator coefficients \(u''+ A(t) u'+ B(t)u= f\), \(u(0)= u_ 0\), \(u'(0)= u_ 1\), the Cauchy problem is studied. By using the “commutant method”, we construct the coercive solution of this problem in a Hölder space in the case where the operator \(B\) has the same “strength” as the operator \(A^ 2\).

MSC:

34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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