Periodic solutions to parameter-dependent equations with a $$\phi $$-Laplacian type operator

Abstract

We study the periodic boundary value problem associated with the φ-Laplacian equation of the form (φ(u ′ )) ′ +f (u)u ′ +g(t, u) = s, where s is a real parameter, f and g are continuous functions, and g is T -periodic in the variable t. The interest is in Ambrosetti-Prodi type alternatives which provide the existence of zero, one or two solutions depending on the choice of the parameter s. We investigate this problem for a broad family of nonlinearities, under non-uniform type conditions on g(t, u) as u → ±∞. We generalize, in a unified framework, various classical and recent results on parameter-dependent nonlinear equations.Mathematics Subject Classification (2010). 34B08, 34B15, 34C25.

On the periodic Ambrosetti–Prodi problem for a class of ODEs with nonlinearities indefinite in sign

“…A real outbreak of papers devoted to the study of nonlinear problems which are indefinite in sign dates back to the eighties of the last century both in the PDEs and the ODEs settings, together with a parallel renewed interest towards ecological models (see, e.g., the monograph [2]). First relevant progresses in relaxing the uniform coercivity assumption (1.4) were achieved in the recent papers [7,8,3]; precisely, the following result for equation (1.3) was obtained in [7].…”

Ambrosetti–Prodi Alternative for Coupled and Independent Systems of Second-Order Differential Equations

“…[4,5] for Neumann boundary conditions; ref. [6][7][8][9][10] for periodic problems; ref. [11] for parametric problems with (p, q)-Laplacian equations; ref.…”

Bifurcation Results for Periodic Third-Order Ambrosetti-Prodi-Type Problems

“…The study of the periodic orbits of differential equations is an important line of research, namely: to obtain sufficient conditions for the non-existence and multiplicity for strongly nonlinear differential equations [6]; the existence of periodic orbits as limit cycles [7], or as solutions of the φ-Laplacian generalized Liénard equations [8]; solvability of higher-order periodic problems with fully differential equations [9], and singular third order problems via cones theory [10]; equations with asymptotically sign-changed nonlinearities [11], or with anti-periodic boundary conditions [12]; oscillations of nonlinear even order differential equations [13].…”