A fixed point theorem, a perturbed differential equation, and a multivariable Volterra integral equation

Abstract:

A fixed point theorem is obtained for an equation of the form $u = T[p,f + G[u]]$. This theorem is then applied to a functionally perturbed ordinary differential equation of the form $u’(t) = f(t) + A(t,u(t)) + G[u](t);u(0) = p$, and, as a consequence of this, Fredholm integrodifferential equations of the form $z’(t) = f(t) + \phi (t,z(t)) + \int _0^\infty {\alpha (t,s)\omega (z(s))ds,z(0) = p}$. Applications are also made to a multivariable Volterra integral system \[ \begin {array}{*{20}{c}} {{u_1}(s,t) = {g_1}(s,t) + \int _0^s {\int _0^t {F\left [ {{u_1}(x,y),{u_2}(x,y),{u_3}(x,y)} \right ]dy\;dx} } } \hfill \\ {{u_2}(s,t) = {g_2}(s,t) + \int _0^t {F\left [ {{u_1}(s,y),{u_2}(s,y),{u_3}(s,y)} \right ]} \;dy} \hfill \\ {{u_3}(s,t) = {g_3}(s,t) + \int _0^s {F\left [ {{u_1}(x,t),{u_2}(x,t),{u_3}(x,t)} \right ]} \;dx,} \hfill \\ \end {array} \] and, as a corollary to this, a differential equation of the form \[ \begin {array}{*{20}{c}} \hfill {\frac {{{\partial ^2}}}{{\partial s\partial t}}u(s,t) = f(s,t) + F\left [ {u(s,t),\frac {\partial }{{\partial s}}u(s,t),\frac {\partial }{{\partial t}}u(s,t)} \right ],} \\ \hfill {u(s,0) = \phi (s),u(0,t) = \psi (t).} \\ \end {array} \] These last two equations are set in a Banach space so as to allow applications to integrodifferential equations such as \[ \begin {array}{*{20}{c}} {\frac {{{\partial ^2}}}{{\partial s\partial t}}u(s,t,z) = f(s,t,z) + H\left ( {z,u(s,t,z),\frac {\partial }{{\partial s}}u(s,t,z),\frac {\partial }{{\partial t}}u(s,t,z)} \right )} \\ { + \int _0^1 {K\left ( {z,r,u(s,t,r),\frac {\partial }{{\partial s}}u(s,t,r),\frac {\partial }{{\partial t}}u(s,t,r)} \right )} \;dr,} \\ {u(s,0,z) = \sigma (s,z),u(0,t,z) = \tau (t,z).} \\ \end {array} \]