Sentence examples for obtained a unique from inspiring English sources

In 2012, Chen [17] introduced the below notion of cyclic orbital stronger Meir-Keeler contraction, and obtained a unique fixed-point theorem for such class of mappings.

In 2005, Nieto and Rodríguez-López [6] extended Ran and Reurings's theorems for nondecreasing mappings and obtained a unique solution for a first-order ordinary differential equation with periodic boundary conditions.

Subsequently, Nieto and Rodríguez-López [12] extended the results in [11] for nondecreasing mappings and obtained a unique solution for a first order ordinary differential equation with periodic boundary conditions (see also, [13 19]).

The point x is called coincidence point of a pair (( f,T ) ). Abbas et al. [34] introduced a generalization of 'condition (B)' for a pair of self-maps and obtained a unique point of coincidence.

Subsequently, Nieto and Rodríguez-López [12] extended the results in [11] for non-decreasing mappings and obtained a unique solution for a first-order ordinary differential equation with periodic boundary conditions (see also [13 19]).

A mapping f : A ∪ B → A ∪ B is called a cyclic map if f ( A ) ⊆ B and f ( B ) ⊆ A. In 2010, Karpagam and Agrawal [2] introduced the notion of cyclic orbital contraction, and obtained a unique fixed point theorem for such a map.

To obtain a unique solution, two additional equations are required.

The clump types were individually calibrated to obtain a unique set of parameter values for each.

At least four neighboring points with heights are required to obtain a unique solution.

In this paper, we will obtain a unique strong solutions for (1.1) under (4< p<+infty).

Additional conditions are given to obtain a unique solution to the Robin problem.