Sentence examples for multiple solution problem from inspiring English sources

Constructing approximating minimizing sequences of functions twice plays a key role in the proof of looking for optimal controls, which enable us to deal with the multiple solution problem of feasible pairs.

It is apparent that the range of δ ^ given by (11) is ± 0.5 or half the subcarrier spacing, and it has no multiple solution problem, requires no channel knowledge and only takes a one step computation process without adaptive iterations.

In general, because of an insufficient observed dataset, the multiple solutions problem becomes a serious issue for the 3D gravity inversion.

In [5], Wei established the existence of multiple solutions for problem (1.1) by means of bifurcation theory.

In this article, we consider multiple solutions of problem (1) in the non-resonance by using the mountain pass theorem and Morse theory.

In this article, under the guidance of [8], we consider multiple solutions of problem (1) with the asymmetric nonlinearity by using variational methods and Morse theory.

In this paper we consider multiple solutions of problem (1.1) in the cases of resonance and non-resonance by using the mountain pass theorem and Morse theory.

For and, the existence of a nontrivial solution as well as multiple solutions for problem is established in [5] and [7] by using critical point theory and invariant sets of descent flow.

We will use the Ekeland variational principle [15] and a version of the mountain pass theorem in [1] to study the existence of multiple solutions of problem (1.1) in ({mathbb {R}}^{N}).

In this paper, influenced by [1] and [2], we consider the existence of multiple solutions for problem (1) near resonance at the higher eigenvalue, where (F x,s,t)) satisfies the superquadratic condition (2).

In [7], we have studied the existence of multiple solutions for problem (1) by using the variant fountain theorem without the condition (c