(3.7) Notice that solutions (x_{h}) of problem (3.1), (3.5) and fixed points of (F h,cdot)) are connected in the following way: if (x_{h}) is a solution of (3.1), (3.5), then (x^{h} = Q_{h}x_{h}) is a fixed point of (F h,cdot)), and, conversely, if (x_{h}) is a fixed point of (F h,cdot)), then (P_{h}x^{h}) is a solution of problem (3.1), (3.5).
We have determined the magnetic ground state of the Fe-Cr solid solutions, and noticed that calculations without the correct magnetic ground state would lead to incorrect results of the lattice and elastic constants.
Notice that, are the solutions of, and,, respectively.
Notice that uniqueness of solutions cannot be guaranteed when we include variable coefficients p and q and functional boundary conditions.
Notice that is a solution of (A.17), passing through, and is by Lemma A.1.
Notice that is an optimal solution of (P3) implies is an optimal solution of (P4) which implies is an optimal solution of (P5).
end{aligned} (3.17) Notice that problem (3.13 - 3.14) has solutions if and only if the operator (mathcal{H}) has fixed points.
It can be observed from the results that the sphericity was not clearly influenced by increasing the concentration of the hardening solution, but it was noticed that hardening solutions with lower molarity of CuCl2 (0.5 and 1.0%) yielded microspheres with higher sphericity values.
We notice that is a distributional solution to (3.5) if and only if is a distributional solution to (C.16).
It may be noticed that numerical solutions are very close to exact solutions.
