Using the technique of measures of noncompactness we prove that under certain assumptions this equation possesses solutions that are convex of order for each, with being a given integer.
With this relationship, the biaxial load bearing capacity can be roughly predicted for metallic foams under biaxial loading; hence this equation can be used as the strength-design criterion for metallic foams under biaxial loads.
Generalizing this equation under the network model and the cluster model, and it becomes (8).
Under the condition (q^=p), this equation can be simplified as iq_{t}+q_{xx}+ibetabigl vert qvert ^{2}q bigr) _{x}+ialpha q_{x}+vert qvert ^{2}q+rq=0.
Let us consider the following ordinary differential equation: x^{prime prime }(t)+Ax^{prime }(t)+Bx t)=0,quad tin [ 0,+ infty), (2.1) with constant positive coefficients A and B. We demonstrate that the exponential stability of this equation, under corresponding conditions on coefficients and delays, implies the exponential stability of delay differential equation (1.3).
where the function ψ 0 ( x ) is a solution of equation (23), moreover the number of independent solutions of this equation under these conditions is equal to 1. Proof Since v ( x ) is a harmonic function, a solution of problem (17 - 18) can be found in the form of the Poisson integral v ( x ) = ∫ ∂ Ω P ( x, y ) ν ( x ) d s x where ν ( x ) is an unknown function.
It should also be noted that we obtain this equation under the assumption that the total fluence rate, Φ, can be approximated as the incident field, Φ i, since Φ i ≫ Φ s [ 2].
In order to simulate evolution under this master equation, it is convenient to eliminate fast-rotating terms by expressing the master equation in a frame rotating with (H_{Q}).
We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.
Under a non-cohesive condition, this equation would be valid for the entire wedge except at the boundary along the basement (e.g., Dahlen [1984]).
