Nonlinear operations and the solution of integral equations

Abstract:

The letters S, G and H denote a linearly ordered set, a normed complete Abelian group with zero element 0, and the set of functions from G to G that map 0 into 0, respectively. In addition, if $V \in H$ and there exists an additive function $\alpha$ from $S \times S$ to the nonnegative numbers such that $\left \| {V(x,y)P - V(x,y)Q} \right \| \leqslant \alpha (x,y)\left \| {P - Q} \right \|$ for each $\{ x,y,P,Q\}$ in $S \times S \times G \times G$, then $V \in \mathcal {O}\mathcal {S}$ only if $\smallint _x^yVP$ exists for each $\{ x,y,P\}$ in $S \times S \times G$, and $V \in \mathcal {O}\mathcal {P}$ only if $_x{\Pi ^y}(1 + V)P$ exists for each $\{ x,y,P\}$ in $S \times S \times G$. It is established that $V \in \mathcal {O}\mathcal {S}$ if, and only if, $V \in \mathcal {O}\mathcal {P}$. Then, this relationship is used in the solution of integral equations of the form $f(x) = h(x) + \smallint _c^x[U(u,v)f(u) + V(u,v)f(v)]$, where U and V are in $\mathcal {O}\mathcal {S}$. This research extends known results in that requirements pertaining to the additivity of U and V are weakened.