By Ivar Ekeland, Roger Témam

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2) are used to prove that they are the same. 45). 45) be the fixed points of a certain completely continuous operator, and then, to apply the Schauder fixed point theorem [12]. 55) R ! x/j; 8 y 2 X x2Œ0;L x2Œ0;L we can define the operator T W X ! X/ is bounded. 45). X/ is not bounded, there would exist a sequence fyn g X such that kuyn kX ! 1: Moreover, from the hypotheses of the theorem, the sequence of functions fb. ; yn . 0; L/ and, passing to a subsequence if necessary, we may assume that fb.

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0; 1/ of positive measure. 2 on Œ0; 1 and Conditions of this type are referred to as nonuniform nonresonance conditions with respect to the first positive eigenvalue of the associated linear homogeneous problem. 45). 1). 45). 2). , none of these hypotheses implies the other). 9) and studies the limits of kˇkp for p ! 1C and p ! 1 (see [3] for further details). 4. 45) where the following requirements are fulfilled: 1. f and fu are continuous on Œ0; L R: 2. x; u/ in Œ0; L R. e. x; 0/ dx D 0: 0 3.