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By H.A. Antosiewicz

ISBN-10: 0120596504

ISBN-13: 9780120596508

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VQI- I dt ! - Pi Let y and P denote the following Banach s p a c e s : y = r n L i=i q (G) (G) ; = n L i i=i 27 p i q i , P. > 1 . x L. D. BERKOVITZ The norm of an element z = (z , . . , z11) in p is given by IWI = i S IU l ll*p ! 1/2 . i=i i A s i m i l a r formula gives the norm of an element y = (y , . . , y ) in ty . Let 7 element be a Banach space and let the norm of an 0 in 7 be denoted by mapping from 7 to y . ,y) (G). 0 0 in J in 7 in V » where each i M0 = z = (z , . • • , z ) in is an element 2 , where each z * not be linear, is in L (G).

In the next section we shall give s e v e r a l lower closure theorems that cover m o s t c a s e s of interest. Theorem 4. 2 below is the s i m p l e s t , is the e a s i e s t to apply, and covers many special c a s e s of interest. 1 . I (0» » u i ) } A sequence of a d m i s s i b l e pairs is said to be weakly lower closed whenever the following holds. 3) 4. l i m inf J ( 0 k > u k ) k -* oo > J(0,u). Lower Closure T h e o r e m s . We first list a set of assumptions that will be in force for all of our t h e o r e m s .

Sc. INTERNATIONAL CONFERENCE ON DIFFERENTIAL EQUATIONS Paris Ser A. 274(1972), 62-65. T4] L. Cesari, Existence theorems for abstract multidimensional control problems, J. Optimization Theory Appl. 6(1970), 210-236. T5] , Closure theorems for orientor fields, Bull. Amer. Math. Soc. 79(1973), 684-689. Purdue University, West Lafayette, Indiana 39 A SINGULAR CAUCHY PROBLEM AND GENERALIZED TRANSLATIONS B. L. J. Braaksma 0, Introduction. Generalized translations have been introduced by Delsarte [ 2 ] by means of a generalized Taylor formula.

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International Conference on Differential Equations by H.A. Antosiewicz

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