Spectrum of nonlinear integral equations and the widths of function classes (Q1326081)

Consider the class, \(W^ K_ p\), of functions on the interval \([0,1]\), defined by a kernel \(K(s,t)\) with some specific properties: \(W^ K_ p = \{x \mid x (s) = \int K(s,t) u(t)dt,\;u \in L_ p,\;\| u \|_ p \leq 1\}\). The author investigates the isoperimetric problem: \(\| x \|_ q \to \sup\), \(x \in W^ K_ p\), which is reduced to a system of nonlinear integral equations. Along with this problem, the Kolmogorov width \(d_ n (W^ K_ p, L_ q)\) and the Bernshtein width \(b_ n (W^ K_ p, L_ q)\) are computed precisely.