Sentence examples for imposed on the function from inspiring English sources

Suppose the conditions imposed on the function (psi(I)) are satisfied.

In these fixed point theorems, very simple conditions are imposed on the function ϕ.

Under certain assumptions imposed on the function q, we obtain necessary conditions for the existence of nontrivial solutions to (1.1).

The following assumptions are imposed on the function f: (i) f ′ ( N t ) < 0 for N t ∈ [ 0, ∞ ) ; i.e., as the density increases, f decreases continuously.

In this paper, inspired by Sedghi et al. and Hu's work mentioned above, we prove some common fixed point theorems for ϕ-contractive mappings in fuzzy metric spaces, in which a very simple condition is imposed on the function ϕ.

M. Rassias [4] extended the problem to ∥ f ( x + y ) − f ( x ) − f ( y ) ∥ ≤ ε ( ∥ x ∥ p + ∥ y ∥ p ) x, y ∈ E 1, for some ε ≥ 0 and some 0 ≤ p < 1. Subsequently, in 1994, P. Gavruta [5] generalized the problem to ∥ f ( x + y ) − f ( x ) − f ( y ) ∥ ≤ ϕ ( x, y ) with certain conditions imposed on the function ϕ.

Under the assumptions imposed on the functions r ( x, s ) and ξ ( x, y ) in Section 1.2, from Lemma B.1 (in Appendix 2) we obtain the following statement.

The restrictions imposed on the functions γ, Q, and the right-hand side of Eq. (1) guarantees that, by virtue of (11) and (15), the kernel k 1 ( x, t ) is a kernel with weak singularity.

Usually, these problems are posed by appropriate differential equations and boundary conditions to be imposed on the unknown function or functions.

Comparison of results from the proposed objective function and conventional methods indicates that the new changes imposed on the objective function has caused the algorithm output to be sensitive to the variations of grade, domain's boundaries and the thickness of mineralization domain.

The problem is complex and a time consuming process due to mixed design variables and inequality constraints imposed on the objective function.