Fredholm mappings and the generalized boundary value problem (Q1343051)

The author proves several assertions concerning various properties (e.g. generic properties, surjectivity, bifurcation problem) of the mapping \(F: X\to Y\) (\(X\) and \(Y\) are Banach spaces) which is the sum of a linear bounded Fredholm mapping of index zero and of a completely continuous mapping. These results are applied to boundary value problems of the form \(A(x)+ f(t, x, x',\dots, x^{(m)})= q(t)\), \(a\leq t\leq b\), \(\ell_ j(x)= 0\), \(j= 1,\dots, n\), where \(-\infty< a< b< \infty\), \(0\leq m\leq n- 1\), \(f: [a,b]\times \mathbb{R}^{m+ 1}\to \mathbb{R}\) is continuous, \(q\in C([a, b], \mathbb{R})\), \(A\) is an \(n\)th order linear differential expression with continuous coefficients and \(\ell_ j\) are linear continuous functionals on \(C^{n- 1}([a, b],\mathbb{R})\). In particular, the author proves that if a certain a priori estimate is satisfied then the given problem possesses bifurcation points. Furthermore, he proves that the existence of at most finitely many solutions is generic for the given type of boundary value problems. In the cases when the corresponding linearized homogeneous problem is positive or self-adjoint some more results are obtained, as well.