Nonlinear singular integral equations in Lebesgue spaces
Using the theory of monotone operators, existence and uniqueness for singular integral equations of the type \[ \lambda_1a(x)u(x)+{\lambda_2\over\pi}\int_a^b {\bigl(w(x)+w(s)\bigr){\mathcal K}(x,s)u(s)\over s-x}ds+ \lambda_3F\bigl(x,u(x)\bigr)=f(x), \] of singular Hammerstein integral equations of the type \[ u(x)+{\lambda\over\pi}\int_a^b {\bigl(w(x)+w(s)\bigr){\mathcal K}(x,y)F\bigl(s,u(s)\bigr)\over s-x}ds=f(x), \] and of \[ u(x)+\lambda F\Bigl(x,{1\over\pi}\int_a^b {\bigl(w(x)+w(s)\bigr){\mathcal K}(x,s)u(s)\over s-x}ds\Bigr)=f(x) \] are established in a weighted \(L_p\) space \(L_p(\rho)\). The main hypotheses are that \(\lambda_1\lambda_3\geq0\), that \({\mathcal K}\) is Hölder continuous (and in some cases symmetric), that the weight function \(\rho\) and the functions \(w,a\geq0\) satisfy certain integral estimates, and that the nonlinearity \(F(x,t)\) is monotone with respect to \(t\) and satisfies some growth estimates such that, roughly speaking, it grows like \(\rho(x)| t|^{p}/t\) or, for the last equation, like \((| t|/\rho(x))^{1/(p-1)}\,\text{sgn}\,t\). For the first equation, also an analogue for complex-valued function is obtained. Besides standard theory of monotone operators, the proof for the first equation makes use of a remarkable integral identity involving Riesz type integral operators by which the term involving \(\lambda_2\) can be eliminated for monotonicity considerations.
scientific article; zbMATH DE number 5924759
scientific article