Sentence examples for proved corresponding from inspiring English sources

Precisely, Sehgal and Bharucha-Reid [9] introduced probabilistic q-contractions and proved corresponding unique fixed point results by giving a generalization of the classical Banach fixed point principle.

The proof presented here uses a substantial simplification of the argument due to Vodev [265] who proved corresponding upper bounds in even dimension [266, 267], following an earlier contribution by Intissar [142].

We prove corresponding maximal inequalities in non-commutative Lq-spaces over a semifinite von Neumann algebra.

We also prove corresponding existence results using bifurcation theory and the continuation method.

However, at the level of individual proteins, we were unable to prove corresponding gene regulation patterns for specific genes/proteins.

Influenced from Bloch's principle ([1] or [4]), that is, there is a normal criterion corresponding to every Liouville-Picard type theorem, Fang and Yuan [5] proved a corresponding normality criterion for inequality (1.2).

Meanwhile, they proved the corresponding fixed point theorems.

As their special cases, they proved the corresponding inequalities involving Riemann-Liouville and Erdélyi-Kober type fractional integral operators.

As a characterization of H p n p ( R n ), Edmunds and Triebel [3] proved the corresponding Hardy-type inequality with a logarithmic correction as follows.

Alber and Guerre-Delabrere [1] introduced the concept of weak contractions in Hilbert spaces and proved the corresponding fixed point result.

Specially, Li and Liu [4] stated and proved the corresponding boundedness and compactness characterizations for C φ from H ∞ (U n ) to B log α ( U n ).