Sentence examples for a solution of the following problem from inspiring English sources

Hence ψ x ˜ ( y ) + 1 2 ∥ y ∥ 2 − 1 2 ∥ x ˜ ∥ 2 ≥ 0, ∀ y ∈ C. It then follows from Proposition 3.3 that x ˜ is a solution of the following problem: f ( x ˜, y ) + φ ( y ) + 〈 y − x ˜, J x ˜ − J z 〉 ≥ φ ( x ˜ ), ∀ y ∈ C. The existence result relaxes the results of Ceng and Yao [1] and Cholamjiak and Suantai [2], because of the ξ-monotonicity of f.

Denote, then is a solution of the following problem: (3.17).

where is a solution of the following problem: (3.28).

ψ M blows up at s = S, moreover, it is a solution of the following problem: (3.1).

For this purpose, we notice that u is a solution of problem (1.1 - 1.2 1.1 - 1.2nly if u is andonlyifn of the following problem: L v = λ g ( t ) f ( u ), 0 < t < 1, (2.2).

(u(t)) is a solution of the following problem: left { textstylebegin{array}{l} (^{mathrm{C}}D_{0^^{beta}u)(t)=f t,varphi_{q} u(t))),quad tin[0,1], u(0)=u(1), end{array}displaystyle right.

For each, we let be the set of all solutions of the following problem.

The WD can be obtained as a solution of the following optimization problem: (8).

Let, and be a classical solution of the following problem.

With the similar method, by setting, we obtain a classical solution of the following problem.

Hence is an -approximate solution of the following problem: (3.16).