Sentence examples for author proved that from inspiring English sources

In [10], the author proved that problem (1.3) has a positive solution for (q<0).

For example, in [3], the author proved that system (1.1) is globally stable while (theta _{i}geq 1) and (theta _{i}<1) ((i=1,2)).

Recently, in [9], the author proved that problem (1.3) has a positive ground state solution for (q<0) and (f x,u =a(x vert uvert ^{p-1}u), (3< p<5).

By means of Ljusternik-Schnirelmann theory the author proved that problem (1.3) has at least (operatorname{cat}_{overline{Omega}} overline{Omega })+1) solutions for p near the critical Sobolev exponent 6, where cat denotes the Ljusternik-Schnirelmann category.

Under the assumption that 0 < α η 2 < 2, Tariboon and the author proved that problem (1.1 - 1.3) has at least one positive solution in the super-linear case or in the sub-linear one.

In [1], the author proved that the functions G α and g α are pointwise comparable.

Then the authors proved that converges strongly to, where.

Matrix equation (2) was studied in [15], and based on some conditions, the authors proved that matrix equation (2) has positive definite solutions.

By applying the Manásevich-Mawhin continuation theorem, the authors proved that equation (1.2) has at least one positive T-periodic solution.

In this study, the authors proved that inhibition of caspase-3 reduced GluA1 dephosphorylation and removal from the PSD, and also reversed hippocampal function deficits at the onset of AD.

In the first, the author proves that to apply physics' principles of pro bability to events is to say that it is impossible that anything will happen, or has happened, or else the world of living things does not exist.