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On a variational statement of a nonlocal boundary value problem for a fourth-order ordinary differential equation. (English. Russian original) Zbl 1183.34020
Differ. Equ. 45, No. 3, 335-343 (2009); translation from Differ. Uravn. 45, No. 3, 325-333 (2009).
The paper deals with the existence of solutions to a differential equation of the form \[ (k_1(x)u''(x))''-(k_2(x)u'(x))'+k_3(x)u(x)=f(x) \]
satisfying the boundary conditions
\[ u(-a)=u'(-a)=u'(0)=0\text{ and }\int_{-\xi}^0k_2(x)u'(x)dx-k_1(0)u''(0))+k_1(-\xi)u''(-\xi)=0. \]
The authors formulate a minimizing functional defined on an appropriate Hilbert space and then they show that the minimum of the functional is a solution of the original boundary value problem.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
58E30 Variational principles in infinite-dimensional spaces
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