Landesman-Lazer Condition Research Articles

Dynamic bifurcation of nonautonomous evolution equations under Landesman–Lazer condition with cohomology methods

In this article we study the dynamic bifurcation of nonautonomous evolution equations by using cohomology methods. First, we construct a homotopy equivalence relation between the nonautonomous system and a product flow. Then, we slightly extend some continuation theorems on bifurcations for autonomous equations, and prove some new cohomology consequences on the reduced singular groups. Based on this homotopy equivalence relation and these conclusions, we establish some typical results on the dynamic bifurcation from infinity of the abstract nonautonomous evolution equation. Finally, we consider the parabolic equation ut−Δu=λu+f(x,u)+g(x,t) associated with the Dirichlet boundary condition, where f(x,u) satisfies the appropriate Landesman–Lazer type condition. Some new results on the dynamical behaviors of this equation near resonance of the equation are derived.

An extension of the Poincaré–Birkhoff Theorem to systems involving Landesman–Lazer conditions

AbstractWe provide multiplicity results for the periodic problem associated with Hamiltonian systems coupling a system having a Poincaré–Birkhoff twist-type structure with a system presenting some asymmetric nonlinearities, with possible one-sided superlinear growth. We investigate nonresonance, simple resonance and double resonance situations, by implementing some kind of Landesman–Lazer conditions.

Resonance with Landesman-Lazer conditions for parameter-dependent equations: a multiplicity result via the Poincaré-Birkhoff theorem

<p>We investigate the resonance problem and prove the existence of multiple periodic solutions to a second order parameter-dependent equation $ x''+f(t, x) = sp(t) $. We weaken the usual requirement on the sublinearity of perturbations when $ |x| $ becomes large; and develop a more general method to investigate the rotational characterizations of the Landesman-Lazer conditions. Furthermore, $ f $ does not satisfy the common sign condition, and even the global existence of the solution is not guaranteed.</p>

Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions

We consider the boundary value problem $$\displaylines{ - \Delta u + c(x) u = \alpha m(x) u^+ - \beta m(x) u^- +f(x,u), \quad x \in \Omega, \cr \frac{\partial u}{\partial \eta} + \sigma (x) u =\alpha \rho (x) u^+- \beta \rho (x) u^- +g(x,u), \quad x \in \partial \Omega, }$$ where \((\alpha, \beta) \in \mathbb{R}^2\), \(c, m \in L^\infty (\Omega)\), \(\sigma, \rho \in L^\infty (\partial\Omega)\), and the nonlinearities f and g are bounded continuous functions. We study the asymmetric (Fucik) spectrum with weights, and prove existence theorems for nonlinear perturbations of this spectrum for both the resonance and non-resonance cases. For the resonance case, we provide a sufficient condition, the so-called generalized Landesman-Lazer condition, for the solvability. The proofs are based on variational methods and rely strongly on the variational characterization of the spectrum. See also https://ejde.math.txstate.edu/special/02/m2/abstr.html

Spectral problem for the Laplacian and a selfadjoint nonlinear elliptic boundary value problem

In this paper, we present some connections between the spectral problem,
 −Δu(x) = λ1u(x) in Ω,u(x) = 0 on ∂Ω
 and selfadjoint boundary value problem,
 Δu(x) − λ1u(x) + g(x, u(x)) = h(x) in Ω,u(x) = 0 on ∂Ω,
 where λ1 is the smallest eigenvalue of −∆, Ω ⊆ Rn is a bounded domain, h ∈ L2(Ω) and the nonlinear function g is a Caratheodory function satisfying a growth condition. We initially investigate the existence of solutions for the spectral problem by considering the selfadjoint boundary value problem. The selfadjoint boundary value problem is then considered for both existence and estimation results. We use degree argument in order to show that the selfadjoint boundary value problem has a solution instead of the Landesman-Lazer condition or the monotonocity assumption on the second argument of the function g.
 In this paper, we present some connections between the spectral problem,
 and selfadjoint boundary value problem,
 where λ1 is the smallest eigenvalue of −∆, Ω ⊆ Rn is a bounded domain, h ∈ L2(Ω) and the nonlinear function g is a Caratheodory function satisfying a growth condition. We initially investigate the existence of solutions for the spectral problem by considering the selfadjoint boundary value problem. The selfadjoint boundary value problem is then considered for both existence and estimation results. We use degree argument in order to show that the selfadjoint boundary value problem has a solution instead of the Landesman-Lazer condition or the monotonocity assumption on the second argument of the function g.

Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth

<p style='text-indent:20px;'>We prove the existence and multiplicity of subharmonic solutions for bounded coupled Hamiltonian systems. The nonlinearities are assumed to satisfy Landesman-Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is based on phase plane analysis and a higher dimensional version of the Poincaré-Birkhoff twist theorem by Fonda and Ureña. The results obtained generalize the previous works for scalar second-order differential equations or relativistic equations to higher dimensional systems.</p>

Non-resonance and Double Resonance for a Planar System via Rotation Numbers

We consider a planar system $$z'=f(t,z)$$ under non-resonance or double resonance conditions and obtain the existence of $$2\uppi $$ -periodic solutions by combining a rotation number approach together with Poincare-Bohl theorem. Firstly, we allow that the angular velocity of solutions of $$z'=f(t,z)$$ is controlled by the angular velocity of solutions of two positively homogeneous system $$z'=L_i(t,z),i=1,2$$ , whose rotation numbers satisfy $$\rho (L_1)>n$$ and $$\rho (L_2)<n+1$$ , namely, nonresonance occurs in the sense of the rotation number. Secondly, we prove the existence of $$2\uppi $$ -periodic solutions when the nonlinearity is allowed to interact with two positively homogeneous system $$z'=L_i(t,z),i=1,2$$ , with $$\rho (L_1)\ge n$$ and $$\rho (L_2)\le n+1$$ , which gives rise to double resonance, and some kind of Landesman–Lazer conditions are assumed at both sides.

On Landesman-Lazer conditions and the fundamental theorem of algebra

We give an elementary proof of a Landesman-Lazer type result for systems by means of a shooting argument and explore its connection with the fundamental theorem of algebra.

A continuation principle for Fredholm maps II: application to homoclinic solutions

AbstractWhen studying the behavior of autonomous ordinary differential equations under time‐dependent perturbations vanishing for , their equilibria generically persist locally as homoclinic solutions. Using an abstract and flexible continuation theorem, we find even global branches of such homoclinic solutions for parametrized nonautonomous ordinary differential equations. Our approach is based on degree‐theoretical arguments. In particular, Landesman–Lazer conditions are proposed to obtain the existence of homoclinic solutions by means of a nonzero degree.

Some existence theorems for semilinear Neumann problems with Landesman-Lazer condition revisited

In this paper, existence theorems are established for Neumann problems for semilinear elliptic equations at resonance together with Landesman-Lazer condition revisited. Our existence results follow as an application of the Saddle point Theorem together with a standard eigenspace decomposition.

Quasilinear problems under local Landesman–Lazer condition

This paper presents results on the existence and multiplicity of solutions for quasilinear problems in bounded domains involving the p-Laplacian operator under local versions of the Landesman–Lazer condition. The main results do not require any growth restriction at infinity on the nonlinear term which may change sign. The existence of solutions is established by combining variational methods, truncation arguments and approximation techniques based on a compactness result for the inverse of the p-Laplacian operator. These results also establish the intervals of the projection of the solution on the direction of the first eigenfunction of the p-Laplacian operator. This fact is used to provide the existence of multiple solutions when the local Landesman–Lazer condition is satisfied on disjoint intervals.

A Landesman–Lazer Local Condition for Semilinear Elliptic Problems

The objective of this paper is to study the existence, multiplicity and non existence of solutions for semilinear elliptic problems under a local Landesman–Lazer condition. There is no growth restriction at infinity on the nonlinear term and it may change sign. In order to establish the existence of solution we combine the Lyapunov–Schmidt reduction method with truncation and approximation arguments via bootstrap methods. In our applications we also consider the existence of a bifurcation point which may have multiple positive solutions for a fixed value of the parameters.

Nonautonomous nonlinear ODEs: Nonresonance conditions and rotation numbers

We deal with the T -periodic problem associated with a nonlinear scalar differential equation like (1) x ″ + f ( t , x ) = 0 , where, for x → 0 and | x | → + ∞ , the nonlinearity f is assumed behave linearly, with a time-dependent coefficient. We prove that the Landesman–Lazer conditions at zero and at infinity possess a rotational effect on “small” and “large” (in the phase-plane) solutions of (1) . As a consequence, we are able to generalize previous multiplicity results in the resonant case - i.e., when the linearizations at zero and/or at infinity are resonant, through the use of the Poincaré–Birkhoff fixed point theorem.

On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay

For eachx0∈[0,2π)andk∈N, we obtain some existence theorems of periodic solutions to the two-point boundary value problemu′′(x)+k2u(x-x0)+g(x,u(x-x0))=h(x)in(0,2π)withu(0)-u(2π)=u′(0)-u′(2π)=0wheng:(0,2π)×R→Ris a Caratheodory function which grows linearly inuasu→∞, andh∈L1(0,2π)may satisfy a generalized Landesman-Lazer condition(1+sign(β))∫02πh(x)v(x)dx&lt;∫v(x)&gt;0gβ+(x)vx1-βdx+∫v(x)&lt;0gβ-(x)vx1-βdxfor allv∈N(L)\{0}. HereN(L)denotes the subspace ofL1(0,2π)spanned bysin⁡kxandcos⁡kx,-1&lt;β≤0,gβ+(x)=lim infu→∞(gx,uu/u1-β), andgβ-(x)=lim infu→-∞(gx,uu/u1-β).

Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions

In this paper, we discuss a new class of fractional order differential switched systems with coupled nonlocal initial and impulsive conditions in Rn. We firstly derive a solution formula for this system. Secondly, we utilize three well-known fixed point methods to present the existence results. Moreover, we use Schauder topological degree theory to show a new existence result for resonant case: Landesman–Lazer conditions. Finally, we introduce the concepts of Ulam's type stability and present new stability results in the space of fractional version piecewise continuous functions.

Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

AbstractWe prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations ofNplanar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman–Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincaré–Birkhoff Theorem. Different situations, including Lotka–Volterra systems, or systems with singularities, are also illustrated.

Landesman–Lazer conditions for difference equations involving sublinear perturbations

We study the existence and uniqueness for discrete Neumann and periodic problems. We consider both ordinary and partial difference equations involving sublinear perturbations. All the proofs are based on reformulating these discrete problems as a general singular algebraic system. Firstly, we use variational techniques (specifically, the Saddle Point Theorem) and prove the existence result based on a type of Landesman–Lazer condition. Then we show that for a certain class of bounded nonlinearities this condition is even necessary and therefore, we specify also the cases in which there does not exist any solution. Finally, the uniqueness is discussed.

Landesman–Lazer conditions for resonant p-Laplacian problems with jumping nonlinearities

We study the existence of solutions of the p-Laplacian Dirichlet problem(1)−ϕp(u′)′=λϕp(u)+h(x,u)+f(x,u,u′),x∈(0,1),(2)u(0)=u(1)=0, where λ∈R, p>1, ϕs(ξ):=|ξ|s−1sgnξ for s⩾1, ξ∈R, the function h:[0,1]×R→R has the formh(x,ξ)=a+∞(x)ϕp(ξ+)−a−∞(x)ϕp(ξ−),(x,ξ)∈[0,1]×R, with ξ±:=max⁡{±ξ,0}, and a±∞∈L1(0,1), and the function f:[0,1]×R2→R is continuous, and satisfies|f(x,ξ,η)|⩽K(x)(1+|ξ|q−1),(x,ξ,η)∈[0,1]×R2, for some q∈[1,p) and K∈L1(0,1).The dominant asymptotic behaviour of equation (1) as u→±∞ is determined by the coefficients a±∞, and we allow a−≠a+, in which case the problem is said to be jumping. If the positively homogeneous problem obtained from (1)–(2) by setting f≡0 has a non-trivial solution then the problem is said to be resonant, and λ is said to be a half-eigenvalue. Assuming that the problem (1)–(2) is both jumping and resonant, we will obtain a solution under certain ‘Landesman–Lazer’ conditions on f.

On the second order periodic problem at resonance with impulses

In this paper we consider the periodic problem for the second order equation at resonance with impulses in the derivative. The impulses are considered at fixed times and depend on the actual value of the solution in a nonlinear way. We formulate rather general sufficient condition in terms of the asymptotic properties of both nonlinear restoring force and nonlinear impulses which generalizes classical Landesman–Lazer condition. Moreover, our condition implies existence results for some open problems with vanishing and oscillating nonlinearities.

Landesman-Lazer condition revisited: the influence of vanishing and oscillating nonlinearities

In this paper we deal with semilinear problems at resonance. We present a sufficient condition for the existence of a weak solution in terms of the asymptotic properties of nonlinearity. Our condition generalizes the classical Landesman-Lazer condition but it also covers the cases of vanishing and oscillating nonlinearities.