Topics in Nonlinear Functional Analysis

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Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference (2015, background): “By virtue of the homotopy invariance of the Leray–Schauder degree (Nirenberg, 2001), we have deg(h(·; 0), Σ, 0) = deg(h(·; 1), Σ, 0).”
On stationary patterns of a reaction–diffusion model with autocatalysis and saturation law (2008, positive): “In addition, lemma 3.1 claims that (1, 1) is non-degenerate (namely, zero is not the eigenvalue of the linearized problem of (3.7 )a t(1, 1)) and is also linearly stable as the unique constant steadystate positive solution of (3.7) with µ = 0. Hence, by the well-known Leray–Schauder degree formula (”
Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus (2018, neutral): “[46,56]) for bifurcations with a one-dimensional kernel.”
Stability of discrete solitons in nonlinear Schrödinger lattices (2005, background): “Proof. The statement follows from the Implicit Function Theorem (see Theorem 2.7.2 in [ 25 ]), since the Jacobian matrix for the system (2.8) is non-singular at n = (0) n , n 2 Z, while the right-hand side of the system (2.8) is analytic in ".”
Non-constant positive steady states of the Sel'kov model ☆ (2003, background): “Then hðu; tÞa0 for all uA@S and tA½0; 1� : Bythe homotopyinvariance of the Leray– Schauder degree [ 31 ],”