Positive subharmonic solutions to superlinear ODEs with indefinite weight

Abstract

We study the positive subharmonic solutions to the second order nonlinear ordinary differential equationwhere g(u) has superlinear growth both at zero and at infinity, and q(t) is a T -periodic sign-changing weight. Under the sharp mean value condition T 0 q(t) dt < 0, combining Mawhin's coincidence degree theory with the Poincaré-Birkhoff fixed point theorem, we prove that there exist positive subharmonic solutions of order k for any large integer k. Moreover, when the negative part of q(t) is sufficiently large, using a topological approach still based on coincidence degree theory, we obtain the existence of positive subharmonics of order k for any integer k ≥ 2. * Work partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Progetto di Ricerca 2016: "Problemi differenziali non lineari: esistenza, molteplicità e proprietà qualitative delle soluzioni". It is also partially supported by the project "Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations" (T.1110.14) of the Fonds de la Recherche Fondamentale Collective, Belgium.

Subharmonic Solutions In Reversible Non-Autonomous Differential Equations

“…# characters appearing in E chi := Irr ( G ) [3]+ Irr ( G ) [5]+ Irr ( G ) [9]+ Irr ( G ) [10] + Irr ( G ) [14]+ Irr ( G ) [18]; # find orbit types in E orbtyps := ShallowCopy ( OrbitTypes ( chi ) ); Remove ( orbtyps ); # find maximal orbit types in H -0 max_orbtyps := MaximalElem e nt s ( orbtyps ); Print ( List ( max_orbtyps , IdCCS ) );…”

“…We also use the list of all irreducible G-representations generated by GAP. Using this list, the corresponding basic G-degrees are easily computed by the GAP program, so the exact # characters appearing in E chi := Irr ( G ) [2]+ Irr ( G ) [3]+ Irr ( G ) [4]+ Irr ( G ) [5] + Irr ( G ) [6]+ Irr ( G ) [7]+ Irr ( G ) [8]+ Irr ( G ) [9] + Irr ( G ) [17]+ Irr ( G ) [18]+ Irr ( G ) [19]+ Irr ( G ) [20] + Irr ( G ) [25]+ Irr ( G ) [26]+ Irr ( G ) [30]; # find orbit types in E orbtyps := ShallowCopy ( OrbitTypes ( chi ) ); Remove ( orbtyps ); # find maximal orbit types in H -0 max_orbtyps := MaximalElem e nt s ( orbtyps ); Print ( List ( max_orbtyps , IdCCS ) );…”

“…Regarding the degree theory (cf. [13,16]), it has been successfully applied to non-Hamiltonian systems in [4,9]) (see also [5,14,28,30]).…”

High multiplicity and chaos for an indefinite problem arising from genetic models

“…where µ(•) is the Möbius function, defined on N \ {0} by µ(1) = 1, µ(l) = (−1) q if l is the product of q distinct primes and µ(l) = 0 otherwise. We refer to [18,Remark 4.1] for an interesting discussion on this formula.…”

Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach

“…A simple comparison of the two theorems shows that when m grows, the number of different subhamonics found by Theorem 5.2 largely exceeds the number of those obtained by Theorem 5.1. The number of m-order subharmonics can be precisely by a combinatorial formula coming from the study of aperiodic necklaces [27].…”