STABILITY OF SOLUTIONS BIFURCATING FROM MULTIPLE-EIGENVALUES
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Let X and Y be real Banach spaces and G:X R be a twice continuously differentiate function which is not necessarily linear. Suppose G(u0, 0) = 0 and the dimension of the null space of Gu(u0, 0) is m, where 1 m < . Usually, S = {(u, ):G(u, ) = 0}, in a neighborhood of (u0, 0), consists of a finite number of curves emanating from (u0, 0). We will determine the stability of points, (u, ), in S (i.e., the maximum of the real parts of the spectrum of Gu(u, ) for each (u, ) S) using a general perturbation theorem of Kato. Our results contain as a special case the stability theorems of Crandall and Rabinowitz for the case m = 1. We will also tie our stability theorems together with some bifurcation results of Decker and Keller. Finally we apply our results to systems of reaction diffusion equations. 1981.