Sentence examples for upper functions and from inspiring English sources

Suppose that there exist lower and upper functions and of problems (1.1) and (1.2), respectively.

Moreover, it is pointed out that lower and upper functions, and the correspondent first derivatives, are not necessarily ordered.

By integration, from (iii) and (2.1), we obtain (2.2). that is, lower and upper functions, and their first derivatives are also well ordered.

By using these statements, in particular, the method of lower and upper functions and monotone iterative techniques can be developed to derive solvability results for hyperbolic equations subjected to various initial conditions (Darboux, Cauchy, Goursat, etc).

By making a series of a priori estimates and applying lower and upper functions techniques and Leray-Schauder degree theory, the authors obtain the existence and location result of solutions to the problem.

We apply lower and upper functions technique and topological degree method to prove the existence of solutions by making a priori estimates for the third derivative of all solutions of problems (1.1) and (1.2).

In the presence of an ordered pair of lower and upper functions, the existence and location results for problems (1.1) and (1.2) can be obtained.

are, respectively, lower and upper functions of (4.1) and (4.2).

Since problem (1.4a - 1.4b 1.4a - 1.4br, the existence resingular [3] are proved by a combinathen of thexistenceof loweresultspper functinns with regularization and sequential techniques.

In Section 2, we give the definition of lower and upper functions to problems (1.1) and (1.2) and obtain some a priori estimates.

define a pair of lower and upper functions of problems (1.1) and (1.2) if the following conditions are satisfied: (i),, (ii),, (iii).