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Topological characterization of the solution set of Fredholm equations with \(f\)-compactly contractive perturbations and its applications. (English, Russian) Zbl 1014.47034
Russ. Math. 45, No. 1, 33-45 (2001); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2001, No. 1, 36-48 (2001).
This article deals with the degree theory for completely continuous and \(f\)-compactly restricting perturbations \(f-g\) (in particular, \(f\)-condensing) of \(C^r\)-smooth mappings \(f\) which are defined on open subsets \(X\) of a real Banach space \(E\), Fredholm on the zero set of \(f-g\), and take values in a real Banach space \(F\). The authors introduce and study the index \(\text{ind}_2 (f-g,X,0)\) of the zero set of \(f-g\) on \(X\) with values in the ring of Rholin-Thom nonoriented bordisms and show that this index has the usual “degree” properties: the non-triviality of the zero set, if this index is nonzero, the invariance under homotopies from a natural class of deformations, and so on. As applications, the authors prove the solvability in the space \(C^2([0,1],\mathbb R^n)\) of the following nonlinear second-order boundary value problem \[ a_0\ddot x^m(t)+ a_1\ddot x^{m-1}(t)+\cdots+ a_m=G\bigl(t, x(t),\dot x(t),\ddot x(t)\bigr), \quad x(0)=c_0,\;x(1)=c_1. \] The main difference of the theory presented in this article from the previous is that the authors assume the Fredholm property of \(f\) only at points of some set containing zeros of \(f-g\).

47H11 Degree theory for nonlinear operators
58B15 Fredholm structures on infinite-dimensional manifolds
47J05 Equations involving nonlinear operators (general)
34B15 Nonlinear boundary value problems for ordinary differential equations
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