Sentence examples for method the existence from inspiring English sources

By using the variational method, the existence and multiplicity of positive solutions are obtained.

By mixed monotone method, the existence and uniqueness are established for singular higher-order continuous and discrete boundary value problems.

Applying the Banach contraction theorem and a modified Picard method, the existence and uniqueness of a classical solution is given.

In particular, in [6], by critical point method, the existence of periodic and subharmonic solutions of equation (1.5).

On the basis of fixed point theorems and successive iteration method, the existence of positive solutions was obtained.

Compared with traditional hot air drying method, the existence of optimal material layer thickness is a significant phenomenon in microwave thin layer drying process.

With the help of the variational methods, the existence of a non-trivial solution is obtained.

By using variational methods, the existence of at least two positive solutions was obtained.

By virtue of variational methods, the existence criteria of infinitely many nontrivial high or small energy solutions are established.

With the aids of the variational methods, the existence and multiplicity of homoclinic orbits for (1.2) have been extensively investigated in many recent papers; see [2 24].

In this paper, by using the variational methods, the existence of nontrivial solutions and the concentration phenomena of the solutions to equation (1.1) were established.