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Furthermore, if the conditions (E0), (E1), (E2) are satisfied, problem (1.1) has at least one solution z 1 ∈ E. Remark 1.2 For the hypotheses ∇ F ( x, 0, 0 ) ≡ 0 and F ( x, 0, 0 ) ≡ 0, problem (1.1) admits the trivial solution ( u, v ) = 0.

Since all the conditions of Theorem 1 are satisfied, problem (1) has at least three nonnegative solutions x 1, x 2, x 3 such that ∥ x i ′ ∥ ≤ d for i = 1, 2, 3, and b ≤ min [ 0, η ] x 1 ( t ), a < ∥ x 2 ∥ 1 with  min [ 0, η ] x 2 ( t ) < b, ∥ x 3 ∥ 1 < a.

Thus all the assumptions of Lemma 2.1 are satisfied and problem (1.1) has at least one solution on J. □.

If conditions of (10) and (30)–(60) of Theorem 4.2 are satisfied, then problem (4.43) with (1.2), (1.3), and (1.4) has at least one solution when the parameter is small enough.

Then one can verify easily that the function x satisfied boundary value problem (1.1).

If, zhowever, nmin exceeds nmax, the two constraints cannot both be satisfied and the problem becomes unsolvable.

If the condition (7) or equivalent condition (detmathbf {A}=0) is satisfied, then the discrete problem (9) does not have the unique solution [21].

Thus in the case ϕ ≤ θ (i.e., if condition C is not satisfied), the solution of problem (61) obtained by using bracketing method[30, 31] lies at most away from the optimal solution.

Also, Clearly, f ω − ρ 1, ρ 1 < r 1 and f ( ρ 2, ρ 2 / c ) ω > r 2, so condition (S2) in the previous theorem is satisfied and, therefore, the problem (5.1 - 5.2 5.1 - 5.2east one solution.

Clearly | f ( t, x ) | ≤ | t 1 3 e − 2 t / ( 1 + t x 2 ) | ≤ 1 = M. Hence, all the conditions of Theorem 3.2 are satisfied and consequently the problem (4.1 - 4.2 4.1 - 4.2east one solution.

Using the viewpoint of Triplet Structure Model, we analyze the satisfied constraints of posed problems.