Stability of Nonlinear Computing Schemes

Stability of Ritz procedure for nonlinear two-point boundary value problem

“…Thus by a well known result of Vi~ik-Browder-Petryshyn ( [6] for example) from the relation (t4) it follows that the Eq. (t2) has always one solution.…”

“…Thus, the Ritz equation is: J(") (a C")) = o (t 6) where Jr (a (")) is the vector: (Jx r (at")), 1,(") (at")) ..... J,(")(ar Definition I [6]. A n is said to lie in ano n = (a (hI, rn, bn) neighborhood of jInl if:…”

On the Stability of the Ritz-Galerkin Method for Hammerstein Equations

“…u^xvAy)) = Giul,-• •, um) = G(u), then it has been shown in [1] that there exists a positive constant C. such that We will denote the system (2.9) by (2-11) Tmum=0. Definition 1 [7]. An operator Am is said to lie in an Í2m = (um, rm,bm)…”

“…Definition 2 [7]. The computing scheme (2.11) is stable at {um} if for each rm there exist neighborhoods Vmi0, r\m), numbers pm and constants s and t such that, if formly bounded, then we have from (2.7), lim,,,^^ xp(w)= +°°, which contradicts the combination of (2.16) and (2.14).…”

“…The computing scheme (2.11) is stable at {um} if for each rm there exist neighborhoods Vmi0, r\m), numbers pm and constants s and t such that, if formly bounded, then we have from (2.7), lim,,,^^ xp(w)= +°°, which contradicts the combination of (2.16) and (2.14). Tucker has proved [7] that the uniform boundedness of {H"mHOT}, together with (2.15), ensures that the computing scheme (2.11) is stable. Q.E.D.…”