The key ingredient is a precised concentration-compactness lemma related to V-dependent limiting problems.
We found geometry-dependent limiting values of the stimulation current, above which electric tissue damage may occur.
The theorem on the Gronwall's type integral inequalities with two dependent limits is established.
The present study involves some Gronwall's type integral inequalities with two dependent limits.
In paper [9] the following useful inequality with two dependent limits was established.
Section 2 includes some new integral inequalities with two dependent limits and relevant proofs.
The theory of integral inequalities with several dependent limits and its applications to differential equations has been investigated in [10 14].
Corrections are derived for constant and time dependent limits of integration the latter is used to approximate dissipation rates in advective diffusive systems.
Vivekanandan and Fixen [64] have reported that large-scale manure applications to a silty clay loam results in a linear increase in available P (Bray P1), up to a (presumably) soil dependent limit.
It is well known that various initial-boundary value problems for the integral-differential equation of the hyperbolic type with two dependent limits can be reduced to the initial value problem (1) in a Hilbert space H; see [1 3].
Using (14 - 17 14 - 17e theorem anduthehe discretheoremog of aboutnwall thee integral inequality with two discretet limits [4, 6], we obtanalogx − N + 1 ≤ k ≤ N ∥ A ( u k + u k − 1 ) 2 ∥ H ≤ M ˜ [ ∥ A u 0 ∥ H + ∥ A 1 / 2 u 0 ′ ∥ H + ∥ f 0 ∥ H + ∑ k = − N + 1 N ∥ f k − f k − 1 ∥ H ]. (18).
