Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity

Abstract

TWe prove the existence of T −periodic solutions for the second order non-linear equation u √ 1 − u 2 = h(t)g(u), where the non-linear term g has two singularities and the weight function h changes sign. We find a relation between the degeneracy of the zeroes of the weight function and the order of one of the singularities of the non-linear term. The proof is based on the classical Leray-Schauder continuation theorem. Some applications to important mathematical models are presented.

Existence and multiplicity of periodic solutions to an indefinite singular equation with two singularities. The degenerate case

“…Moreover, there exists β * ≥ β * such that the equation (3) has no T −periodic solution for every β > β * . The conditions of Theorem 2 are the ones expected according to the literature available on the subject, see e.g., [1,21,22,29,30]. Hence, at least for this case, the conditions [R]-[F] seem to be connected more with the multiplicity of the periodic problem than with its mere solvability.…”

“…where h ∈ L(R/T Z), β > 0 is a parameter, and g : (A, B) → (0, +∞) is a continuous function with −∞ ≤ A < B ≤ +∞. See, e.g., [1,3,5,6,7,8,9,10,11,15,16,17,19,29,30]. An elementary observation is that any solution of (1) satisfying the above-mentioned boundary conditions verifies T 0 h(t)g(u)dt = 0, and, consequently the function h (called the weight function) has to change its sign.…”

“…An elementary observation is that any solution of (1) satisfying the above-mentioned boundary conditions verifies T 0 h(t)g(u)dt = 0, and, consequently the function h (called the weight function) has to change its sign. As a result, the equations of type (1) are so-called indefinite equations. The terminology indefinite was introduced by Hess and Kato in [23] under a rather different context.…”

“…Finally, to see results dealing with λ ∈ (0, 1) we refer to [17,18]. In a recent series of papers [1,20,22], the authors have dealt with indefinite singular equations with two singularities, i.e., they considered the equation (1) with A, B ∈ R. The appearance of another singularity substantially changes the attitude to the problem, mainly due to the lack of monotony of the nonlinear term. This aspect carries serious difficulties to apply directly the arguments used above and, consequently, new strategies have to be proposed.…”