Sentence examples for minimal solution and from inspiring English sources

Under the hypotheses (H1)–(H4), the IVP (3.4) has the minimal solution and maximal solution in.

Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability).

Furthermore, (alpha^) and (beta^) are a minimal solution and a maximal solution of (1.1) in Ω, respectively.

Therefore, we are obliged to distinguish between the concepts of minimal solution and least solution (or maximal and greatest solutions), unfortunately often identified in the literature on lower and upper solutions.

After that, the individuals are sorted by the energy values from minimum to maximum and at the same time, the minimal solution and the minimal energy are saved as hmin and Emin respectively.

It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions.

A pair ( x 0, y 0 ) ∈ gr F is called a minimal solution of F on A if ( F ( A ) − y 0 ) ∩ − D = { 0 Y } ; A pair ( x 0, y 0 ) ∈ gr F is called an ideal minimal solution of F on A if ( F ( A ) − y 0 ) ⊆ D. The sets of all minimal solutions and ideal minimal solutions of (SOP) are denoted by Min ( F, A ) and IMin ( F, A ), respectively.

The first result ensures the existence of maximal and minimal solutions, and the second one establishes the existence of the greatest and the least solutions in a particular case.

After establishing a comparison result of the nonlinear Riemann-Liouville fractional differential equation of order ({pin 2, 3]}), we obtain the existence of maximal and minimal solutions, and the uniqueness result for fractional differential equations.

Thus is a weakly minimal solution of and the proof is complete.

Therefore, by Theorem 3.1, problem (1.1) has a minimal solution (underline{u}) and a maximal solution (overline{u}) in ([y_{0},x_{0}]).