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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.
MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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