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On existence and uniqueness of solutions for ordinary differential equations with nonlinear boundary conditions
| Content Provider | Semantic Scholar |
|---|---|
| Author | Calamai, Alessandro |
| Copyright Year | 2005 |
| Abstract | – We prove an existence and uniqueness theorem for a nonlinear functional boundary value problem, that is, an ordinary differential equation with a nonlinear boundary condition. The proof is based on a Global Inversion Theorem of Ambrosetti and Prodi, which is applied to the boundary operator restricted to the manifold of the global solutions to the equation. Our result is a generalization of an analogous existence and uniqueness theorem of G. Vidossich, as it is shown with some examples. 1. – Introduction and preliminaries We consider a functional boundary value problem (in short, BVP) of the form (1) x′ = f(t, x), L(x) = r, where f : [a, b] × R → R is a continuous map with continuous partial derivative with respect to the second variable, and L is a nonlinear map from the space C([a, b],R) into R of class C, i.e. Fréchetdifferentiable with continuous derivative L′ : C([a, b],R) → L(C([a, b],R),R). Our aim is to prove an existence and uniqueness theorem for the functional BVP (1), generalizing an analogous result of Vidossich [4, Theorem 3]. For this purpose, we will use the following Global Inversion Theorem of Ambrosetti and Prodi [1, Theorem 1.8, p. 47]. We recall that if X and Y are metric spaces and F : X → Y is a continuous map, F is said to be proper if F−1(K) is compact for any compact set K ⊂ Y ; furthermore, F is said to be locally invertible at a point x ∈ X if there exist neighborhoods U of x in X and V of y = F (x) in Y |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://dipmat.univpm.it/~calamai/Pub/Ca04_BUMI.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
