An Application of B. N. Sadovskij’s Fixed Point Principle to Nonlinear Singular Equations

Abstract

This paper is concerned with the applicability of Sadovskij’s fixed point principle to the solvability of nonlinear singular integral equations of the form $x=\lambda SFx$, with $F$ being a non-linear superposition operator, and $S$ a singular linear integral operator. If the operator $F$ acts in the "little" Hölder space $H_{\alpha }^0$, or in some space $J_{\alpha ,\beta }$ of functions which satisfy an integral-type Hölder condition, $F$ turns out to be $k$-condensing under fairly general hypotheses such that the above equation has it solution at least for small $\lambda$. On the other hand, under these general hypotheses the classical fixed point principles of Schauder does not apply immediately, and that of Banach–Caccioppoli not at all; in fact, it is shown that the Bunach-Caccioppoli principle applies only if the above equation is linear. Moreover, in this paper it is shown that the above equation may be studied as well by means of the topological degree theory for vector fields involving condensing operators.