Sentence examples for the dominating function from inspiring English sources

Based on this, we also assume the dominating function λ that in (1.1) satisfies (1.2) in this paper.

Based on this, in this paper, we always assume that the dominating function λ also satisfies it.

This leads to the counterparts of the explicit representation in Proposition 23 and to the dominating function in Corollary 24 by using the area formula (e.g., [35, § 3.3, Theorem 2]) rather than the change of variables formula for Lebesgue integrals.

On the other hand, a Radon measure μ as in (1.2) on R d is also an upper doubling measure by taking the dominating function λ ( x, r ) = C 0 r d.

Remark 1 If we take the dominating function λ ( x, r ) to be μ ( Q ( x, r ) ), then the measure doubling is a special case of upper doubling.

As studied in Theorem KST, it is more practical to put an assumption on the dominated function ψ than to put one on the mapping T itself.

Indeed, the author [25] considered some suitable generalized distances without assuming that the dominated functions possess the lower semicontinuity property.

It should be mentioned that in our results, the dominated functions need not possess the lower semicontinuity property.

Finally, applying Theorem 2.1, we can establish a new Caristi-type fixed point theorem without assuming that the dominated functions possess the lower semicontinuity property.

Furthermore, we also discuss the results with the lower semicontinuities of the dominated functions in place of the original orbital continuities of the mappings.

It should be mentioned that in this paper we remove the lower semicontinuity property of the dominated functions in Caristi-type cyclic maps.