A unification theory of Krasnoselskii for differential equations (Q395389)

Ted Burton has been an active contributor to research in delay differential equations and related matters for probably around fifty years. This nice paper illustrates his continuing research activity. Of concern is the following 1958 result of Mark Krasnoselskii. Let \(M\) be a closed convex set in a Banach space \(X\): Let NEWLINE\[NEWLINE A; B : M \rightarrow X NEWLINE\]NEWLINE satisfy NEWLINE\[NEWLINE A(M ) + B(M ) \subset M; A(M )\, \text{is precompact}; B\, \text{is a strict contraction}.NEWLINE\]NEWLINE Then \(A(x) + B(x) = x\) has a solution \(x\) in \(M\): The authors show how this result applies in diverse settings, including fractional differential equations, neutral differential equations, and an old problem of Volterra in integral and integrodifferential equations. This is a well written, interesting paper.