Sentence examples for be the minimal solution from inspiring English sources

Let (u) be the minimal solution to (2.1) in (L^2(Omega,e^{psi -varphi })).

Assume that the hypotheses of Theorem 3.1 hold for f, h and f ˆ, h ˆ with the same fixed functions y, z, c y, c z and p. Let x ∗ be the minimal solution of PBVP (1.1) in [ y, z ], x ˆ ∗ the minimal solution of the PBVP D x = f ˆ ( t ) + h ˆ ( t, x ), x ( 0 ) = x ( T ) (3.25). in [ y, z ].

Further, let w ̃ and w ̄ be the minimal solutions of the corresponding Riccati equations (4) and (6), respectively.

and is the minimal solution of this equation.

Thus, there exists a fixed point x ¯ of the map F such that | x ¯ | < 1 / 2 (it is easy to check that x ¯ is the minimal solution of the equation e x − 2 = x ).

Since r ̃ k ≤ 0, c ̃ k ≥ 0, it follows from Lemma 3 that w k ≥ w k h for large k, where w k h = r k Φ ( Δ h k / h k ) is the minimal solution of (4).

Therefore (v) is the minimal solution to (bar{partial }v=beta :=e^psi (alpha +u,bar{partial }psi )) in (L^2(Omega,e^{-varphi })) and by Hörmander's estimate begin{aligned} int _Omega |v|^2e^{-varphi }dlambda le int _Omega |beta |^2_{ipartial bar{partial }varphi }e^{-varphi }dlambda.

An element is said to be the minimal norm solution of SFP (1.1) if.

Definition 2.2 An element w ˜ = ( x ˜, y ˜ ) ∈ Γ is said to be the minimal norm solution of SEP (1.1) if ∥ w ˜ ∥ = inf w ∈ Γ ∥ w ∥.

Definition 2.2 An element x ˜ ∈ Γ is said to be the minimal norm solution of SEP (1.1) if ∥ x ˜ ∥ = inf x ∈ Γ ∥ x ∥.

Therefore, for an appropriate constant (tau>1), equation (3.5) is always solvable, and if solution of (3.5) is not unique, the (xi_{max}) will be understood as the minimal solution of the equation.