On an inverse eigenvalue problem for a semilinear Sturm–Liouville operator

The following problem is considered: −u″+f(u)=λu,x∈(0,1),u=u(x),u(0)=1,u′(0)=u(1)=0, where λλ is a spectral parameter. The inverse problem is studied: a subsequence λn→+∞λn→+∞ of the sequence of eigenvalues is given and odd ff is the unknown quantity. A description of the whole class of solutions of this problem is obtained. In addition, it is proved that there exists at most one function ff such that an auxiliary function is nondecreasing.