On the other hand, obviously, one branch of the set W u ( O ) ∖ { O } lies in B ( A 1 ), and the other in B ( A 2 ).
In this paper, we are concerned with the existence of an unbounded branch of the set of solutions for nonlinear elliptic equations of p ( x ) -Laplacian type subject to the Dirichlet boundary condition { − div ( ϕ ( x, | ∇ u | ) ∇ u ) = μ | u | p ( x ) − 2 u + f ( λ, x, u, ∇ u ) in Ω, u = 0 on ∂ Ω, (B).
When p ( x ) is a constant function, the existence of an unbounded branch of the set of solutions for equations of p-Laplacian type operator is obtained in [11] (for generalizations to unbounded domains with weighted functions, see also [16, 17]).
In the present paper, we are concerned with the existence of an unbounded branch of the set of solutions for the p ( x ) -Laplacian problem with degeneracy subject to the Neumann boundary condition { − div ( w ( x ) | ∇ u | p ( x ) − 2 ∇ u ) + | u | p ( x ) − 2 u = μ g ( x ) | u | p ( x ) − 2 u + f ( λ, x, u, ∇ u ) in Ω, ∂ u ∂ n = 0 on ∂ Ω, (B).
If the set ⋃ t ∈ [ 0, 1 ] { u ∈ X : J ( u ) + G ( u ) = t F ( 0, u ) }. is bounded, then the set { ( λ, u ) ∈ R × X : J ( u ) + G ( u ) = F ( λ, u ) }. has an unbounded connected set C ⊆ ( R ∖ { 0 } ) × X such that C ¯ intersects { 0 } × X. Finally we establish the existence of an unbounded branch of the set of solutions for Neumann problem (B) thereby using Lemma 4.2.
Positive selection on the branch was considered as detected, if Γ i - Var(Γ i ) ≥ 0.45 under the condition that Γ i - Var(Γ i ) = Γmax - Var Γmax), where Var(Γ i ) was the variance of Γ i distance at the branch i, Var Γmax) was the variance of the Γmax distance that corresponded to the longest tree branch of the set of tree branches R ∈ [ a, b, c, d].
Lemma 5 If the equilibrium curves E 1 and E 2 are irreducible conics, then all branches of the sets E 1 = { ( x, y ) ∈ R 2 : B 1 x 2 + C 1 x y + ( A 1 − β 1 ) x − γ 1 y − α 1 = 0 }, E 2 = { ( x, y ) ∈ R 2 : C 2 y 2 + B 2 x y + ( A 2 − γ 2 ) y − β 2 x − α 2 = 0 }. are the graphs of monotone functions of one variable on an invariant attracting set B : = [ m 1, M 1 ] × [ m 2, M 2 ] for system (1).
In La Bête humaine (1890; The Human Beast) he analyzes the hereditary urge to kill that haunts the Lantier branch of the family, set against the background of the French railway system, with its powerful machinery and rapid movement.
The Night Wolves were instrumental in the illegal annexation of Crimea where the Ukrainian branch of the bikers set up and manned checkpoints during Russia's covert military invasion of the southern peninsula.
TiVo knows that, which is why the company is trying to branch out of the set-top box business and into building software that it can license to much larger media companies.
For further evaluation the interrelationships of Annelida sensu lato, we performed additional phylogenetic analyses with a smaller taxon set comprising 30 species (all species from the lophotrochozoan branch of the larger taxon set).
