Sentence examples for extended the existence from inspiring English sources

In [5], Zhang investigated a kind of singular Liénard equation, and by applications of his theory, they extended the existence result of (1.1) to (ain 0,0.1532)).

Recently, Dhage [9] extended the existence of PPF dependent fixed points to PPF common dependent fixed points for mappings satisfying the weaker contractive conditions.

After that Dhage [11] extended the existence of PPF dependent fixed points to PPF common dependent fixed points for mappings satisfying the weaker contractive conditions.

By introducing the Itô stochastic calculus with G-Brownian motion, Revathi and Sakthivel extended the existence and uniqueness results to a class of nonautonomous stochastic neutral differential equations with infinite delay in real separable Hilbert spaces [19, 20].

For the class of P-homeomorphisms, the classical result of Denjoy can be easily extended, the existence of the conjugating map for this class was proved by Herman in [5].

For example, Bao et al. [1] extended the existence and uniqueness of mild solutions to a class of more general stochastic neutral partial functional differential equations under non-Lipschitz conditions.

Especially, our Theorem 3.1 extends the existence of solution in the literature to the case of generalized extended nonlinear variational inequality (2.1).

Daniel E. Lieberman, a human evolutionary biologist at Harvard who was not involved in the research, wrote in a commentary for the journal that the hominin foot "is a valuable addition to the fossil record as it extends the existence of Ardipithecus-like feet by a million years".

In this section, we extend the existence results presented in Section 3 to the global situation.

Our result gives both the existence and approximation of solutions to Caputo fractional order hybrid differential equations and also extends the existence results for ordinary hybrid differential equations.

We try to extend the existence of solutions for semilinear Dirichlet problems on the Heisenberg balls in [9] to general Carnot domains.