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Nonlinear boundary value problem for a system of nonlinear ordinary differential equations. (English) Zbl 0569.34015
We investigate a system of nonlinear differential equations with a special type of stable nonlinear boundary conditions. We suppose that by means of this system we can define a monotone operator T with a potential J. We seek a weak solution in the set M, a subset of a Sobolev space V. This set M is not a subspace because we consider nonlinear boundary conditions. The variational formulation of a weak solution is obtained in terms of the derivative of the functional J along the curves passing through the point of its minimum, and the test functions are taken from the set \(M_ u\), which is the manifold tangent to the set M at the point u. The proof of existence is not too difficult, but to the author’s knowledge no similar theorem is known for a monotone operator with nonlinear boundary conditions. For this formulation it seems to be essential that the corresponding Sobolev space V be an algebra and that the identical imbedding V to C be completely continuous, which implies that every bounded closed part of the set M is weakly compact. This is the reason why we restrict ourselves to ordinary differential equations. The main result is Theorem 4.1, where we give a sufficient condition for ”local” uniqueness of the weak solution. This condition is a relation between the monotonicity of the operator T and the ”curvature” of the set M expressed in terms of the second derivatives of the functions involved in the boundary conditions. As a limit case, for linear boundary conditions, the ”curvature” is equal to zero and we get the global uniqueness without any additional condition.
34B15 Nonlinear boundary value problems for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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