Sentence examples for using the contraction principle from inspiring English sources

First, by using the contraction principle, the existence and uniqueness of mild solutions are given.

Remark 3.3 The existence and uniqueness results for solutions of problem (3.1) can be obtained by using the contraction principle.

Recently, Klimek investigated existence and uniqueness of the solution of sequential fractional differential equations with Hadamard derivative by using the contraction principle in [16].

Sukavanam and Kumar [47] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order by using the contraction principle and Schauder's fixed-point theorem.

In [21]n th-order fuzzy differential equations (NFDEs) under generalized differentiability were discussed and the existence and uniqueness theorem of solution of NFDE was proved by using the contraction principle in a Banach space.

Surendra Kumar and Sukavanam [33] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order using the contraction principle and the Schauder fixed-point theorem.

For the local existence and uniqueness, we use the method of successive approximations under the Lipschitz condition, and for global existence and uniqueness, we use the contraction principle under suitable conditions.

Uniqueness criteria of the steady state in porous catalysts are derived by using the contraction mapping principle with different norms.

Then the proof now can be finished by using the contraction mapping principle.

In section 4, we prove Theorem 1.2 by using the contraction mapping principle.

Then (mathcal{F}) is a contraction mapping, the proof now can be finished by using the contraction mapping principle.