For example, in the system both equations are satisfied by the solution x = 2, y = 3.
Since the first objective function is already determined by the solution (x, y, z) of the first-stage problem, the recourse function only captures the value of the second objective function, i.e., of the expected uncovered demand.
By Remark 2.1, the solution x ( t ) can be defined for all t ∈ [ 0, + ∞ ).
By Corollary 3.2, the solution x ( t ) of Eq. (39) is mean square stable.
By definition of the solution, X ˜ ( t ) = Y ˜ ( t ) if t ∈ [ t 0 − σ, t 0 ].
Therefore, it is impossible to improve the solution x by rotating to v. From analysis of both cases, we can see that solution x = α g i + β g j,α > 0,β > 0 is the optimal solution to Eq. (43).
Obviously, by the method of steps, once the solution x ¯ ( t ) of the system (2.12) on [ 0, τ ] is known, continuing the above process, we can easily obtain the solution x ¯ ( t ) of the system (2.12) on [ τ, 2 τ ], [ 2 τ, 3 τ ], … .
By Figure 3, the solutions x and (x_{1}) are almost the same on ([0,frac{pi}{2}]).
Then by the continuity of the solutions X and Z, there exists (tau >0) such that, for all (tin 0,tau)), we have (0< X t)< Z(t)).
By the recursive method, the solutions ( x n, y n ) ∈ D 1 of system (1.1) satisfy the conditions of theorem and 0 < x n < x n + 1 < x ∗, 0 < y n < y n + 1 < y ∗ ( n = 1, 2, … ).
This condition is satisfied for example by the principal solution x ^ , for which x ^ 0 = 0, x ^ 1 = 1 / r 0 for all λ ∈ ℝ. (2.4).
