Fixed points and stability of a nonconvolution equation
T. A. Burton · Proceedings of the American Mathematical Society · 2004
In this note we consider an equation of the form \[ x ′ ( t ) = − ∫ t − r t a ( t , s ) g ( x ( s ) ) d s x’(t)=-\int ^{t}_{t-r} a(t,s)g(x(s))ds \] and give conditions on a a and g g to ensure that the zero solution is asymptotically stable. When applied to the classical case of a ( t , s ) = a ( t − s ) a(t,s)=a(t-s) , these conditions do not require that a ( r ) = 0 a(r)=0 , nor do they involve the sign of a ( t ) a(t) or the sign of any derivative of a ( t ) a(t) .