Equivalence of differential, fractional differential, and integral equations: Fixed points by open mappings

  • T. A. Burton
  • I. K. Purnaras

Abstract

It is an elementary fact that if $xf(t,x) \geq 0$ then the solution of the initial value problem \[ x'(t) =-f(t,x(t)), \quad x(0) \in \Re\] is bounded by $|x(0)|$.  Under the same conditions on $f$, can we say the same about the solution of the fractional differential equation of $Caputo$ type \[ ^cD^q x(t) =-f(t,x(t)),\quad  x(0) \in \Re,\quad  0<q<1? \] We can and we can also say the same for some rather general integral equations \[ x(t) =x(0) -\int^t_0 A(t-s)f(s,x(s))ds \] with great ease.  This rests on a transformation we developed in 2011 and have used extensively.  In this note we study these properties and use another idea, apparently new.  By that same transformation it is not difficult to find a closed convex set which is mapped into its interior and this gives rise to some interesting fixed point results. In the fourth section we discuss an annotated bibliography of examples we have considered since 2011 using the transformation.  We point out that our new result of mapping into the interior of our set extends all of those results.  The classical heat equation is treated in detail using the open mapping.  In the fifth section we refine $xf(t,x) \geq 0$ and obtain asymptotic results showing that not only are solutions bounded by $|x(0)|$, but they may tend to zero.  Section 6 extends all of this by introducing large persistent perturbations in a very simple form.  Section 7 then offers a generalization of the work in Section 6.

Published
2017-08-27