id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-198	Mavinga, Nsoki; Morris, Quinn A.; Robinson, Stephen B.	Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions	2023	22	.pdf	application/pdf	9836	573	88	First, we establish that the functional J is uniformly Lipschitz in α, β, and x. Note that |J(α2, β2, x)− J(α1, β1, x)| = 1 2 ∣∣∣(α2 − α1)‖x+‖2(m,ρ) + (β2 − β1)‖x−‖2(m,ρ) ∣∣∣ ≤ 1 2µ1 ‖x‖2(c,σ) (|α2 − α1|+ |β2 − β1|) ≤ 1 2µ1 K (|α2 − α1|+ |β2 − β1|) . = ‖x2 − x1‖2(c,σ) − 〈α2x2 − α1x1, x2 − x1〉(m,ρ) + 〈s2(x2 + y2)− − s1(x1 + y1)−, x2 − x1〉(m,ρ) = ‖x2 − x1‖2(c,σ) − α2‖x2 − x1‖2(m,ρ) − (α2 − α1)〈x1, x2 − x1〉(m,ρ) + s2〈(x2 + y2)− − (x1 + y1)−, x2 − x1〉(m,ρ) + (s2 − s1)〈(x1 + y1)−, x2 − x1〉(m,ρ) (2.4)	cache/ejde-198.pdf	txt/ejde-198.txt