Sentence examples for solutions were established from inspiring English sources

Two nontrivial solutions were established by the mountain pass theorem and Ekeland's variational principle.

By utilizing the monotone iterative method, the sufficient conditions which guarantee the existence of solutions were established.

have been studied in [1, 3, 8], respectively, and existence criteria of periodic solutions were established for these equations.

In the scheme of these definitions, many interesting existence results for mild solutions were established by various fixed-point theorems.

Using a fixed point theorem for operators on a cone, sufficient conditions for the existence of multiple positive solutions were established.

was first introduced in [25], in which the global existence of dissipative solutions were established, where α, β, θ and γ are constant parameters.

Some existence results of sign-changing solutions are established.

The continuation of solutions is established and sufficient conditions for continuous and Lipschitzian parametric dependence of solutions are also provided.

Firstly, four SUS solutions are established to satisfy demands and detailed descriptions are introduced.

Limiting cases are examined and fair agreements with well-known solutions are established.

A correlation between the current theoretical and associated ASTM solutions was established.