For the blank in (1)′ we may substitute various conditions that undermine J.
Remark 3.2 Compared with [16], we substitute the condition that F is positively homogeneous by the condition (iii), when F u,x,.) is C-explicitly quasiconvex.
Substituting the conditions as in Eqs.
The characteristic equation analysis is achieved by substituting input conditions into variables UI, R and Otx in Eq. (1).
Substituting these conditions in the Langevin equation, equation (18), we obtain physical insight into the shortcomings of the conventional heuristic method.
Now, we consider a new set of real functions, say Θ. Precisely, we modify the set Ψ by substituting the condition (psi_{3}) by another condition.
Substituting the condition (iv) in Theorem 19 by the condition that X is ordered complete, we can get the following result.
Step 7 Substitute the complementarily conditions with their equivalent linear constraints to calculate the upper bound of outer level objective function.
(30) In the above summation, we substitute the supplementary conditions (27) and approximate it with the Euler wavelet, we can get y(x approxbigl(Y^{mathrm{T}}P^{alpha}+ tilde{Y}^{mathrm{ T}}bigr Psix), (31) where Ỹ is an m̂-vector.
The classical Leray-Schauder principle refers to the maps from a Banach space into itself that verifies the condition (i) above and the following substitute of the condition (ii): There exists a number such that implies for all .
We remark that the condition (I-C) by itself cannot be a substitute for the condition (L-S), and an additional assumption on the domain of needs to be made in order to guarantee the existence of a fixed point for.
