By Robert E. 1941- Gaines
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Extra resources for Coincidence Degree, and Nonlinear Differential Equations
The quest is for an existence theory for such problems which a) Employs sufficient conditions : for existence of the greatest pos- sible generality. b) Employs hypotheses which may be readily verified for a given problem. c) Provides an accompanying mechanism for approximating the solutions. In part V we present an existence theory which partially meets objectives and b). Consistent with the pursuit of a) we concentrate on methods a) and results which apply even when the nonlinear function f does not display such special structure as : monotonicity, tiability, theoretical or "Lipschitzicity".
0)Iker L, ~ i] ker L, 0] # 0. I, to problems having a coincidence degree equal to one. 8) Brouwer degree occuring in the corresponding (3) has absolute value strictly greater that one. 9) L'-compact, any homotopy of the following form, (x,1) = Nx, L'x = I[(L'-L)x+N (x,l)], i s [0,I] (IV. 10), fd[L'>L' -~+N),~]I = td[(L,~),e]l. ( I V . l, the left-hand member of (IV. t1) is one and the right hand member strictly greater than one, a contradiction. 5. I is a global one. 2 an useful local existence theorem.
Remark. We used only xf(t,x,y) > - Ixl ¢(IYl), 0 < Ixl < R in the proof rather than the full force of condition b). S~. SMppose a) There exist a(t), 8(t) E C2[a,b] such that ~(a) < o < S(a), a(b) < o < S(b) a(t) < ~(t) , t ~ [a,b] a"(t) > f(t,~(t),B(t)), t @ [a,b]. 8. 12) has at least one solution satisfying ~(t) ~ x(t) ~ ~(t). Proof. 8. 5. 13) ~(a) ~(b) < 0 < ~(b) To construct ~(t),for example,we suppose f(t,x,x') > - ~(x,x') and consider the solution ~(t) to x" = - ¢(x,x') x(a) = M x'(a) = o.
Coincidence Degree, and Nonlinear Differential Equations by Robert E. 1941- Gaines