Sentence examples for under the contractive condition from inspiring English sources

The following two results hold under the contractive condition (2.12).

Later on, Sehgal and Bharucha-Raid [32] proved a fixed point theorem for a mapping under the contractive condition in a complete Menger PM-space.

Under the contractive condition (2.1) assumed in the above proposition, the assumption of PD-operators and the existence of a unique common fixed point are equivalent conditions.

It is demonstrated that, under the contractive condition, the weakened linear growth condition and the non-Lipschitz condition, a neutral stochastic functional differential equation in the G-frame has at most one solution.

Theorem 3.1 is reformulated under weaker conditions if the contractive condition holds for consecutive members of the subsequences generated by the iterative scheme.

So, we satisfy the contractive condition.

(3) The contractive condition is replaced by the new contractive condition defined by (2.42).  .

The contractive condition is replaced by the new contractive condition defined by (2.42).

Most recently, Kutbi et al. [13] established fixed point results for multi-valued operator under the generalized contractive condition via the generalized Hausdorff distance operator of Cho and Bae [12].

Many new fixed point theorems have been proved under the nonlinear contractive condition by using the theorem of cone and monotone iterative technique.

Remark 2.1 We have proved Theorem 2.2 under the simplest contractive condition d ( A ( x ), A ( y ) ) < d ( S ( x ), S ( y ) ). Analogous results for commuting or compatible mappings require a Banach-type or d ( A ( x ), A ( y ) ) ≤ ϕ ( d ( S ( x ), S ( y ) ) ) or Meir-Keeler-type contractive condition [19].