Sentence examples for by using some fixed from inspiring English sources
In step 7, we obtain the smoothed mean μ ̂ k, t and standard deviation σ ̂ k, t by using some fixed smoothing parameter α where 0<α<1.
They obtained the existence and multiplicity results of positive solutions by using some fixed point theorems.
By using some fixed point theorems, they obtained the existence and multiplicity result of positive solution to this problem.
By using some fixed point index theorems on a cone for differentiable operators, the authors obtain the existence of positive solutions to the above boundary value problem.
By using some fixed point theorems, the existence theorems of mild solutions are obtained, our results are also more general than some well-known results.
By using some fixed point theorems, they obtained the existence, uniqueness and multiplicity of positive solutions to singular positone and semipositone problems.
By using some fixed-point theorems and some properties of the Green function, the existence of positive solutions was obtained.
In 1969, Reinermann investigated some problems by using approximate fixed point property ([1]).
Next we will show by using Banach fixed point theorem that has a unique fixed point.
Some new existence results are obtained by using standard fixed point theorems when the right-hand side of the inclusion has convex as well as non-convex values.
