Sentence examples for under suitable hypotheses from inspiring English sources

Under suitable hypotheses we prove the convergence.

under suitable hypotheses on and under the additional hypothesis (1.6).

under suitable hypotheses on and the additional hypothesis: (1.6).

In particular, we establish Ekeland type and Caristi type results under suitable hypotheses for mappings and cyclic mappings.

They proved that under suitable hypotheses S r is single-valued and firmly nonexpansive with F ( S r ) = EP ( G ).

We also prove that under suitable hypotheses on the external random forces the solution turns out to be unique.

We consider a Banach spaceVof functions from some manifoldMtoRn′and a functionf:M×Rn′→Rn: under suitable hypothesis, there is a homeomorphism ofVtakingZ={v∈V∣∫M f(m, v(m)) dm=0} to a closed subspace of codimensionn.

and obtained the existence of many solutions depending on the value of the parameter λ which lie in some different intervals under some suitable hypotheses.

It should be noted that inequality for assumed in Theorem 4.2 holds in a natural way under some suitable hypotheses on each complete CMC spacelike surface that lies between two spacelike slices [5, Section ].

Under a locally nonnegative assumption on the nonlinearity f and some other suitable hypotheses, positive solutions are sought by considering the corresponding truncated problem, constructing the variational framework and combining the sub-supersolution method with the mountain pass lemma.

We discuss the Witten index in terms of Krein's spectral shift function, and prove invariance of the Witten index under suitable relative trace class hypotheses.