We study the periodic boundary value problem associated with the second order nonlinear equation u''+(λa+(t)−μa−(t))g(u)=0, where g(u) has superlinear growth at zero and sublinear growth at infinity. For λ,μ positive and large, we prove the existence of 3^m−1 positive T-periodic solutions when the weight function a(t) has m positive humps separated by m negative ones (in a T-periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.

PB - American Mathematical Society UR - http://urania.sissa.it/xmlui/handle/1963/35264 N1 - AMS Subject Classification: 34B15, 34B18, 34C25, 34C28, 47H11. U1 - 35568 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Positive subharmonic solutions to nonlinear ODEs with indefinite weight JF - Communications in Contemporary Mathematics Y1 - 2018 A1 - Alberto Boscaggin A1 - Guglielmo Feltrin AB -We prove that the superlinear indefinite equation u″ + a(t)up = 0, where p > 1 and a(t) is a T-periodic sign-changing function satisfying the (sharp) mean value condition ∫0Ta(t)dt < 0, has positive subharmonic solutions of order k for any large integer k, thus providing a further contribution to a problem raised by Butler in its pioneering paper [Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations 22 (1976) 467–477]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincaré–Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it).

VL - 20 UR - https://doi.org/10.1142/S0219199717500213 ER - TY - JOUR T1 - An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators JF - Topol. Methods Nonlinear Anal. Y1 - 2017 A1 - Guglielmo Feltrin A1 - Fabio Zanolin PB - Nicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies VL - 50 UR - https://doi.org/10.12775/TMNA.2017.038 ER - TY - JOUR T1 - Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities JF - Communications on Pure & Applied Analysis Y1 - 2017 A1 - Guglielmo Feltrin KW - Leray-Schauder topological degree; KW - positive solutions KW - Sturm-Liouville boundary conditions KW - Superlinear indefinite problems AB -We study the second order nonlinear differential equation

\begindocument $ u'' + \sum\limits_i = 1^m α_ia_i(x)g_i(u) - \sum\limits_j = 1^m + 1 β_jb_j(x)k_j(u) = 0,\rm $ \enddocument

where $\alpha_i, \beta_j>0$, $a_i(x), b_j(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen[0, L\mathclose]$, and the nonlinearities $g_i(s), k_j(s)$ are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation $u"+a(x)u.p = 0$, with $p>1$.When the positive parameters $\beta_j$ are sufficiently large, we prove the existence of at least $2.m-1$positive solutions for the Sturm-Liouville boundary value problems associated with the equation.The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets.Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.

VL - 16 UR - http://aimsciences.org//article/id/1163b042-0c64-4597-b25c-3494b268e5a1 ER - TY - JOUR T1 - Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree JF - Journal of Differential Equations Y1 - 2017 A1 - Guglielmo Feltrin A1 - Fabio Zanolin KW - Coincidence degree KW - Multiplicity results KW - Neumann boundary value problems KW - Positive periodic solutions KW - subharmonic solutions KW - Superlinear indefinite problems AB -We study the periodic boundary value problem associated with the second order nonlinear differential equationu″+cu′+(a+(t)−μa−(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill's equation. We prove the existence of 2m−1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G.J. Butler in 1976. The proof is based on Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.

VL - 262 UR - http://www.sciencedirect.com/science/article/pii/S0022039617300219 ER - TY - JOUR T1 - A note on a fixed point theorem on topological cylinders JF - Ann. Mat. Pura Appl. Y1 - 2017 A1 - Guglielmo Feltrin AB -We present a fixed point theorem on topological cylinders in normed linear spaces for maps satisfying a property of stretching a space along paths. This result is a generalization of a similar theorem obtained by D. Papini and F. Zanolin. In view of the main result, we discuss the existence of fixed points for maps defined on different types of domains and we propose alternative proofs for classical fixed point theorems, as Brouwer, Schauder and Krasnosel’skii ones.

PB - Springer Verlag UR - http://urania.sissa.it/xmlui/handle/1963/35263 N1 - AMS Subject Classification: 47H10, 37C25, 47H11, 54H25. U1 - 35567 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case JF - Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 449–474. Y1 - 2016 A1 - Alberto Boscaggin A1 - Guglielmo Feltrin A1 - Fabio Zanolin AB -We study the periodic and Neumann boundary value problems associated with the second order nonlinear differential equation u''+cu'+λa(t)g(u)=0, where g:[0,+∞[→[0,+∞[ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when ∫a(t)dt 0 is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.

PB - Cambridge University Press UR - http://urania.sissa.it/xmlui/handle/1963/35262 N1 - AMS Subject Classification: Primary 34B18; 34C25; Secondary 34B15; 47H11; U1 - 35566 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - THES T1 - Positive solutions to indefinite problems: a topological approach Y1 - 2016 A1 - Guglielmo Feltrin KW - positive solutions AB - The present Ph.D. thesis is devoted to the study of positive solutions to indefinite problems. In particular, we deal with the second order nonlinear differential equation u'' + a(t) g(u) = 0, where g : [0,+∞[→[0,+∞[ is a continuous nonlinearity and a : [0,T]→R is a Lebesgue integrable sign-changing weight. We analyze the Dirichlet, Neumann and periodic boundary value problems on [0,T] associated with the equation and we provide existence, nonexistence and multiplicity results for positive solutions. In the first part of the manuscript, we investigate nonlinearities g(u) with a superlinear growth at zero and at infinity (including the classical superlinear case g(u)=u^p, with p>1). In particular, we prove that there exist 2^m-1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large. Then, for the Dirichlet problem, we solve a conjecture by Gómez‐Reñasco and López‐Gómez (JDE, 2000) and, for the periodic problem, we give a complete answer to a question raised by Butler (JDE, 1976). In the second part, we study the super-sublinear case (i.e. g(u) is superlinear at zero and sublinear at infinity). If a(t) has m positive humps separated by negative ones, we obtain the existence of 3^m-1 positive solutions of the boundary value problems associated with the parameter-dependent equation u'' + λ a(t) g(u) = 0, when both λ>0 and the negative part of a(t) are sufficiently large. We propose a new approach based on topological degree theory for locally compact operators on open possibly unbounded sets, which applies for Dirichlet, Neumann and periodic boundary conditions. As a byproduct of our method, we obtain infinitely many subharmonic solutions and globally defined positive solutions with complex behavior, and we deal with chaotic dynamics. Moreover, we study positive radially symmetric solutions to the Dirichlet and Neumann problems associated with elliptic PDEs on annular domains. Furthermore, this innovative technique has the potential and the generality needed to deal with indefinite problems with more general differential operators. Indeed, our approach apply also for the non-Hamiltonian equation u'' + cu' + a(t) g(u) = 0. Meanwhile, more general operators in the one-dimensional case and problems involving PDEs will be subjects of future investigations. PB - SISSA N1 - The research work described in this Ph.D. thesis has produced 10 papers. U1 - 35528 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems JF - Adv. Differential Equations 20 (2015), 937–982. Y1 - 2015 A1 - Guglielmo Feltrin A1 - Fabio Zanolin AB -We prove the existence of positive periodic solutions for the second order nonlinear equation u'' + a(x) g(u) = 0, where g(u) has superlinear growth at zero and at infinity. The weight function a(x) is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.

PB - Khayyam Publishing UR - http://projecteuclid.org/euclid.ade/1435064518 N1 - AMS Subject Classification: 34B18, 34B15, 34C25, 47H11. U1 - 35388 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Existence of positive solutions of a superlinear boundary value problem with indefinite weight JF - Conference Publications Y1 - 2015 A1 - Guglielmo Feltrin KW - boundary value problem KW - indefinite weight KW - Positive solution; existence result. KW - superlinear equation AB -We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change sign. We assume that the function $g\colon\mathopen[0,+∞\mathclose[\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, including the superlinear case $g(s)=s^p$, with $p>1$. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.

VL - 2015 UR - http://aimsciences.org//article/id/b3c1c765-e8f5-416e-8130-05cc48478026 ER - TY - JOUR T1 - Multiple positive solutions for a superlinear problem: a topological approach JF - J. Differential Equations 259 (2015), 925–963. Y1 - 2015 A1 - Guglielmo Feltrin A1 - Fabio Zanolin AB -We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation u''+f(x,u)=0. We allow x ↦ f(x,s) to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that f(x,s)/s is below λ_1 as s→0^+ and above λ_1 as s→+∞. In particular, we can deal with the situation in which f(x,s) has a superlinear growth at zero and at infinity. We propose a new approach based on the topological degree which provides the multiplicity of solutions. Applications are given for u'' + a(x) g(u) = 0, where we prove the existence of 2^n-1 positive solutions when a(x) has n positive humps and a^-(x) is sufficiently large.

PB - Elsevier UR - http://urania.sissa.it/xmlui/handle/1963/35147 N1 - Work presented at the "Special Session 21" of the "10th AIMS Conference on Dynamical Systems, Differential Equations and Applications" (Madrid, July 7-11, 2014). U1 - 35387 U2 - Mathematics U4 - 1 U5 - MAT/05 ER -