Sentence examples for been proved for a from inspiring English sources

Following the Banach contraction principle, most of the fixed point results have been proved for a self-mapping defined on an abstract space.

Their results have been proved for a selfmapping T of X in the case when Y is a normal Banach space.

Quite recently, approximate controllability has been proved for a large number of abstract control systems involved with fractional derivatives; see [35] and the references therein.

Furthermore, after the solvability has been proved for a general variational inequality, a new problem has been raised: does this general variational inequality have maximum and minimum solutions (with respect to the partial order)?

After that, the rapid approximate-controllability has been proved for a lot of systems governed by (linear/semilinear) heat equations in the mathematical control community; see [7] and the references therein.

As an application of the first theorem, a coupled coincidence and coupled common fixed point theorem has been proved for a hybrid pair of multivalued and single-valued mappings.

Existence is proved for A-optimal, D-optimal and Q-optimal minimax designs.

Its potential to learn complex action sequences has been proven for a variety of problems, such as navigation tasks.

If, in addition, ( X, d ) is complete then z j ∈ X, ∀ j ∈ p ¯. Property (iii) has been proven for a sequence S = { n k }.

Such decomposition has been proven for a large class of Liénard systems [41] [44].

Uniqueness is proved for a regular solution.