Scalar nonlinear integral equations

Rok, strany: 2007, 41 - 56

In a series of papers [T. A. Burton: extit{Integral equations, Volterra equations, and the remarkable resolvent: contractions}, Electron. J. Qual. Theory Differ. Equ. extbf{2} (2006), 1–17], [T. A. Burton: extit{Integral equations, $L^p$-forcing, remarkable resolvent: Liapunov functionals}, Nonlinear Anal. extbf{68} (2008), 35–46], [T. A. Burton: extit{Integral equations, large and small forcing functions: periodicity}, Math. Comput. Modelling extbf{45} (2007), 1363–1375.] we have studied scalar linear integral equations and the role of the resolvent in duplicating the forcing function so that large forcing functions frequently have little effect on the solution, while small forcing functions can exert enormous control over the behavior of solutions. Much of that work depends entirely on linearity. The purpose of this paper is to see what can be proved in the nonlinear case and just how much techniques must be changed. The following question is forever before us. What is the qualitative difference between the solutions of [ x(t)=(t+1)^{((1) / (2))} sin (t+1)^{((1) / (3))} + sin t -intlimits^t₀ C(t,s)x(s) ds ] and [ x(t)=sin t -intlimits^t₀ C(t,s) x(s) ds ] under ``reasonable" conditions on $C(t,s)$? There would be no story to tell unless it were true that there is essentially no difference at all. The large function $(t+1)<sup>(</sup>(1) / (2))sin (t+1)<sup>(</sup>(1) / (3))$ is simply swallowed up, while the tiny function $sin t$ exerts enormous control over everything. The example presents us with three problems. Replace $x(s)$ in the integral by $gl(x(s)r)$ where $g(x)$ has the sign of $x$. First, determine exactly which properties on $C(t,s)$ constitute ``reasonable" conditions. Then determine a vector space of big functions, like $a(t)=t$, which have little effect on solutions. Finally, determine a vector space of small functions, $a(t)$, which ``rule the solution with an iron hand."

ISO 690:

Burton, T. 2007. Scalar nonlinear integral equations. In Tatra Mountains Mathematical Publications, vol. 38, no.4, pp. 41-56. 1210-3195.

APA:

Burton, T. (2007). Scalar nonlinear integral equations. Tatra Mountains Mathematical Publications, 38(4), 41-56. 1210-3195.