nonlinear problems for systems of differential equations

h’onlineor Analysis. 7’heory, Methods & Appkotionr. Vol 4, No 6. p. 1213 0 Pergamon Press Ltd. 1980 Printed in Great Britain 0362.546X/SO’llOl-1213 $02 00/O ADDENDUM AND CORRIGENDUM NONLINEAR PROBLEMS FOR SYSTEMS OF DIFFERENTIAL EQUATIONS* GIUSEPPE ANICHINI Istituto Matematico ‘U. Dini’, V. le Morgagni 67/A, 50134 Firenze, Italy Received 21 November 1979 Key words and phrases: Boundary value problems, differential equations, nonlinear operators, fixed point theorems. THEOREM 3.1 is true and the method of proof is correctly based upon showing the existence of a fixed point of the multivalued transformation U, provided that the following hypothesis is assumed : (iv)’ There exists a linear operator AU: Y + Ker DU which is continuous with respect to U, for all u E X, such that (2.6) holds. It is easy to see that, since the nonlinearity of the original boundary operator T is shifted on the operator H, this additional property does not destroy the goodness of the results. Finally, when ,!,ff has a linear continuous right inverse and, in particular when LE has the inverse, this hypothesis is trivially satisfied, which is how it is seen almost always in the applications. * This journal, Vol. 1, No. 6, p. 691 (1977). 1213

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Page 1: Nonlinear problems for systems of differential equations

h’onlineor Analysis. 7’heory, Methods & Appkotionr. Vol 4, No 6. p. 1213

0 Pergamon Press Ltd. 1980 Printed in Great Britain

0362.546X/SO’llOl-1213 $02 00/O

ADDENDUM AND CORRIGENDUM

NONLINEAR PROBLEMS FOR SYSTEMS OF DIFFERENTIAL EQUATIONS*

GIUSEPPE ANICHINI

Istituto Matematico ‘U. Dini’, V. le Morgagni 67/A, 50134 Firenze, Italy

Received 21 November 1979

Key words and phrases: Boundary value problems, differential equations, nonlinear operators, fixed point theorems.

THEOREM 3.1 is true and the method of proof is correctly based upon showing the existence of a fixed point of the multivalued transformation U, provided that the following hypothesis is assumed :

(iv)’ There exists a linear operator AU: Y + Ker DU which is continuous with respect to U, for all u E X, such that (2.6) holds.

It is easy to see that, since the nonlinearity of the original boundary operator T is shifted on the operator H, this additional property does not destroy the goodness of the results.

Finally, when ,!,ff has a linear continuous right inverse and, in particular when LE has the inverse, this hypothesis is trivially satisfied, which is how it is seen almost always in the applications.

* This journal, Vol. 1, No. 6, p. 691 (1977).

1213

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