Existence of integrals and the solution of integral equations

Abstract:

Functions are from R to N or $R \times R$ to N, where R denotes the real numbers and N denotes a normed complete ring. If S, T and G are functions from $R \times R$ to N, each of $S({p^ - },p),S({p^ - },{p^ - }),T({p^ - },p)$ and $T({p^ - },{p^ - })$ exists for $a < p \leqslant b$, each of $S(p,{p^ + }),S({p^ + },{p^ + }),T(p,{p^ + })$ and $T({p^ + },{p^ + })$ exists for $a \leqslant p < b$, G has bounded variation on [a, b] and $\smallint _a^bG$ exists, then each of \[ \int _a^b S \left [ {G - \int G } \right ]T\quad {\text {and}}\quad \int _a^b {S\left [ {1 + G - \prod {(1 + G)} } \right ]} \;T\] exists and is zero. These results can be used to solve integral equations without the existence of integrals of the form \[ \int _a^b {\left | {G - \int G } \right | = 0} \quad {\text {and}}\quad \int _a^b {\left | {1 + G - \prod {(1 + G)} } \right |} = 0.\] This is demonstrated by solving the linear integral equation \[ f(x) = h(x) + (LR)\int _a^x {(fG + fH)} \] and the Riccati integral equations \[ f(x) = w(x) + (LRLR)\int _a^x {(fH + Gf + fKf)} \] without the existence of the previously mentioned integrals.