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Proposition 1 . 3 - Moreover, if F is monotone increasing, then the solution of ( 1 . 5 9 ) is unique. Remark 1 . 61) Of course, in Proposition 1 . 1 , we must have: s(x,O) 5 so(x) 5 g(x,O). Remark 1 . 62) IF(s1) - F(s2)I 5 I,IS1 - 521 for every s l , s Z e R. Without l o s s of generality we can assume that L conveniently. Remark 1 . 4 - = 1, by choosing u F is said to be monotone increasing if: The way of applying Proposition 1 1 t o problem ( 1 . S O ) is quite apparent. The Lipschitz continuity of F is a consequence of I F ' ( 5 ) 1 5 1, Similarly Propositions 1 .

An enzyme electrode for acetylocholine, Biochimica et Biophysica Acta, 527 (1978) 277-281. D. , Difference Methods for Initial Value Problems (Wiley-Interscience, New York, 1957). N. , Electron microscopic and kinetic studies dealing with an artificial enzyme membrane, Application to a cytochemical model with the horseradish peroxidase-3,3l-diaminobenzidine system, J. Hystochem. Cytochem. Vol. 2 2 , N " l 1 (1974) 1048-1059. , Electron microscopy as a tool for studying artificial enzyme membranes, in: Thomas, D.

8). For example, in Thomas' experiments, glucose concentration is observed in the receptor compartment during several hours. 6), ( 2 . 8). The numerical simulations give the results shown in Fig. 5 and end with steady state . prol'iles as hep'icieh 'in P'ig. b an> wYic\, are represenieh \by equa- l ,s,p) = 0, = 0, p'(0) = p'(1) = 0. 12) is satisfied there is no more pumping action, since s ' ( 1 ) = 0 To summarize the above discussion, the modeling of the glucose pump system results in evolution problems governed by equations ( 2 .

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