Nonmonotone Nonlinearity Research Articles

Semilinear wave equation with non-monotone nonlinearity

We prove the existence of weak solutions to a semilinear wave with non-monotone asymptotically linear nonlinearity when the forcing is dominated by a trigonometric polynomial. See also https://ejde.math.txstate.edu/special/02/c1/abstr.html

Optimal regularity results for the one-dimensional prescribed curvature equation via the strong maximum principle

A refined version of the strong maximum principle is proven for a class of second order ordinary differential equations with possibly discontinuous non-monotone nonlinearities. Then, exploiting this tool, some optimal regularity results, recently established by López-Gómez and Omari in [15–17] for the bounded variation solutions of non-autonomous quasilinear equations driven by the one-dimensional curvature operator, are substantially improved by admitting general prescribed curvatures and by incorporating general boundary conditions. The novel approach developed here yields a new, deeper, interpretation of the assumptions introduced in our previous papers, simultaneously clarifying their meaning and making fully transparent their connection with the strong maximum principle.

Hemivariational inequalities on graphs

In this paper, a new class of hemivariational inequalities is introduced. It concerns Laplace operator on locally finite graphs together with multivalued nonmonotone nonlinearities expressed in terms of Clarke’s subdifferential. First of all, we state and prove some results on the subdifferentiability of nonconvex functionals defined on graphs. Thereafter, an elliptic hemivariational inequality on locally finite graphs is considered and the existence and uniqueness of its weak solutions are proved by means of the well-known surjectivity result for pseudomonotone mappings. In the end of this paper, we tackle the problem of hemivariational inequalities of parabolic type on locally finite graphs and we prove the existence of its weak solutions.

Fast reaction limit and forward-backward diffusion: A Radon–Nikodym approach

We consider two singular limits: a fast reaction limit with a non-monotone nonlinearity and a regularization of the forward-backward diffusion equation. We derive pointwise identities satisfied by the Young measure generated by these problems. As a result, we obtain an explicit formula for the Young measure even without the non-degeneracy assumption used in the previous works. The main new idea is an application of the Radon–Nikodym theorem to decompose the Young measure.

Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary

<abstract><p>This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.</p></abstract>

A semilinear wave equation with non-monotone nonlinearity and forcing flat on characteristics

A semilinear wave equation with non-monotone nonlinearity and forcing flat on characteristics

A-posteriori reduced basis error-estimates for a semi-discrete in space quasilinear parabolic PDE

We prove a-posteriori error-estimates for reduced-order modeling of quasilinear parabolic PDEs with non-monotone nonlinearity. We consider the solution of a semi-discrete in space equation as reference, and therefore incorporate reduced basis-, empirical interpolation-, and time-discretization-errors in our consideration. Numerical experiments illustrate our results.

H∞ Switched-Gain Based Observer vs Nonlinear Transformation Based Observer for a Vehicle Tracking Model

The main objective of this paper is to provide a comparison between two nonlinear observer design methods for a vehicle tracking problem based on a kinematic model. This challenging tracking issue is difficult to solve because of the nature of the nonlinearities in the dynamics process, which are all non-monotonic. First, we will extend the switched-gain based approach, established recently in the literature, to systems with disturbances in both the state dynamics and in the output measurements, and investigate its H∞ performance. Then we will introduce a new way to avoid non-monotonic nonlinearities by using a specific nonlinear transformation to bring the system into a suitable form for which available techniques in the literature can be applied. Finally, numerical comparisons between the two design methods are provided, where the H∞ based switched gain observer has better performance with respect to measurement noise.

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Nonmonotonic Nonlinearities

Corner Boundary Layer in Boundary Value Problems for Singularly Perturbed Parabolic Equations with Nonmonotonic Nonlinearities

Uniqueness of large solutions for non-monotone nonlinearities

In this paper we construct a general class of non-monotone positive nonlinearities, f(u), for which the differential equation u′′=f(u) has a unique large solution in (−T,T). In most of the available literature it is required f(u) to be strictly increasing in order to establish the uniqueness of the large solution.

Travelling Waves for a Nonlocal Delay Differential Equation

A class of nonlocal delay differential equation with nonmonotone nonlinearity that arises from population dynamics is considered. The existence of travelling waves is established by constructing two auxiliary monotone dynamical systems combined with a global coupling condition. Our results require neither the compactness nor the reflection invariance of the solution semiflow.

New frequency-domain ℓ2-stability criteria for nonlinear MIMO discrete-time systems with constant and varying time delays

New frequency-domain -stability criteria are derived for nonlinear discrete-time MIMO systems, having a linear time-invariant block with the transfer function , in negative feedback with an aperiodic matrix gain , and a linear combination of a vector of certain classes of (generalised) first-and-third-quadrant non-monotone nonlinearities , having arguments with constant and time-varying delays, but without restrictions on their slopes. The framework does not employ Lyapunov–Krasovskii functionals involving linear matrix inequalities (LMIs) or their equivalent. The new stability criteria seem to be the most general for nonlinear and time variying time-delay systems, and have the following structure: (1) positive definiteness of the real part (as evaluated on ) of the product of and an algebraic sum of general causal and anticausal matrix multiplier functions of z. (2) An upper bound on the -norm of the inverse Fourier transform of the multiplier function, the -norm being weighted by certain novel, quantitative characteristic parameters (CPs) of the nonlinearities without the asumptions of monotonicity, slope restrictions and the like. And (3) constraints on certain global averages of the generalised eigenvalues of , that are expressed in terms of (i) CPs of the nonlinearities, (ii) their coefficients, and, in general, (iii) time-delays in their arguments, a trade-off among all the three being possible. These global averages imply a restriction on the rate of variation of in a new sense. The literature results turn out to be special cases of the results of the present paper. Examples illustrate the stability theorems.

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

TWe prove the existence of $T$-periodic solutions for the second order non-linear equation $ {\left( {\frac{{u'}}{{\sqrt {1 - {{u'}^2}} }}} \right)^\prime } = 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.

Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity

We prove bifurcation at infinity for a semilinear wave equation depending on a parameter \begin{document}$λ$\end{document} and subject to Dirichlet-periodic boundary conditions. We assume the nonlinear term to be asymptotically linear and not necessarily monotone. We prove the existence of L∞ solutions tending to \begin{document}$+∞$\end{document} when the bifurcation parameter approaches eigenvalues of finite multiplicity of the wave operator. Further details are presented in cases of simple eigenvalues and odd multiplicity eigenvalues.

Existence of infinitely many solutions of a beam equation with non-monotone nonlinearity

In this paper, we prove a new critical point theorem without the (PS)∗ condition or the (WPS)∗ condition. Using this theorem, we prove the existence of infinitely many solutions of the following sublinear beam equation {utt+uxxxx+f(x,t,u)=0for(x,t)∈Q=(0,π)×(0,2π)u(0,t)=u(π,t)=uxx(0,t)=uxx(π,t)fort∈Ru(x,t)=u(t+2π,x)fort∈(0,π)×R. Unlike the previous results on this equation, we do not impose any monotone conditions on the variable ξ in the nonlinearity f(x,t,ξ).

Periodic Navier Solutions for the Plate Equation with Non-Monotone Nonlinearities: the Multidimensional Case

AbstractWe discuss the solvability of the periodic Navier problem for the plate equation with forced vibrationsxtt(t, y)+Δ2x(t, y)+l(t, y, x(t, y)) = 0 in higher dimensions with side lengths being irrational numbers and the nonlinearity being superlinear. We also derive a new dual variational method.

A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities

Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X⁎. Let T:X⊇DT→2X⁎ be maximal monotone of type Γdϕ (i.e., there exist d≥0 and a nondecreasing function ϕ:0,∞→0,∞ with ϕ(0)=0 such that 〈v⁎,x-y〉≥-dx-ϕy for all x∈DT, v⁎∈Tx, and y∈X),L:X⊃D(L)→X⁎ be linear, surjective, and closed such that L-1:X⁎→X is compact, and C:X→X⁎ be a bounded demicontinuous operator. A new degree theory is developed for operators of the type L+T+C. The surjectivity of L can be omitted provided that RL is closed, L is densely defined and self-adjoint, and X=H, a real Hilbert space. The theory improves the degree theory of Berkovits and Mustonen for L+C, where C is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when L is monotone, a maximality result is included for L and L+T. The theory is applied to prove existence of weak solutions in X=L20,T;H01Ω of the nonlinear equation given by ∂u/∂t-∑i=1N(∂/∂xi)Aix,u,∇u+Hλx,u,∇u=fx,t, x,t∈QT; ux,t=0, x,t∈∂QT; and ux,0=ux,T, x∈Ω, where λ>0, QT=Ω×0,T, ∂QT=∂Ω×0,T, Aix,u,∇u=∂/∂xiρx,u,∇u+aix,u,∇u (i=1,2,…,N), Hλx,u,∇u=-λΔu+gx,u,∇u, Ω is a nonempty, bounded, and open subset of RN with smooth boundary, and ρ,ai,g:Ω¯×R×RN→R satisfy suitable growth conditions. In addition, a new existence result is given concerning existence of weak solutions for nonlinear wave equation with nonmonotone nonlinearity.

Solutions ofp-Laplace Equations with Infinite Boundary Values: The case of Non-Autonomous and Non-Monotone Nonlinearities

AbstractFor a non-negative and non-trivial real-valued continuous functionhΩ× [0, ∞) such thath(x, 0) = 0 for allx∈Ω, we study the boundary-value problemwhereΩ⊆ ℝN,N⩾ 2, is a bounded smooth domain and Δp:= div(|Du|p–2DDu) is thep-Laplacian. This work investigates growth conditions onh(x, t) that would lead to the existence or non-existence of distributional solutions to (BVP). In a major departure from past works on similar problems, in this paper we do not impose any special structure on the inhomogeneous termh(x, t), nor do we require any monotonicity condition onhin the second variable. Furthermore,h(x, t) is allowed to vanish in either of the variables.

Asymptotic Behavior, Spreading Speeds, and Traveling Waves of Nonmonotone Dynamical Systems

In this paper, we study the asymptotic behavior, the spreading speed, and the existence/nonexistence of traveling waves of a class of nonmonotone discrete-time dynamical system. As a byproduct, we also obtain some results on the global attractivity of a nontrivial constant fixed point and on the existence of a nonconstant fixed point. We then apply the main results to three model systems: (i) a spatially nonlocal integro-difference equation; (ii) a reaction-diffusion equation with spatial nonlocality and time delay in the reaction term; and (iii) an equation with nonlocal diffusion and delayed nonmonotone nonlinearity in the reaction term. The obtained results for these three equations improve some existing ones by removing the symmetry of the kernel functions and relaxing the conditions on the nonlinear terms.

On improving Popov's criterion for nonlinear feedback system stability

For the L2-stability of a nonlinear single-input–single-output (SISO) feedback system, described by an integral equation and with the forward block transfer function and a first- and third-quadrant non-monotone nonlinearity in the feedback path, we derive an interesting generalization of the celebrated criterion of Popov [(1962). Absolute stability of nonlinear systems of automatic control. Automation and Remote Control, 22(8), 857–875]: , where is a constant. The generalization entails the addition of a general causal+anticausal O'Shea–Zames–Falb multiplier function whose time-domain -norm is constrained by certain characteristic parameters (CPs) of the nonlinearity obtained from certain novel algebraic inequalities. If the nonlinearity is monotone or belongs to any prescribed subclass of , its CPs are reduced, thereby relaxing the time-domain constraint on the multiplier. An important special feature of the new stability results is a partial bridging of the significant gap between the Popov criterion and the stability results that appeared post-Popov in the form of considering monotone and other subclasses of nonlinearities in exchange for weakening the restrictions on the phase angle behaviour of . Extensions to time-varying nonlinearities more general than those in the literature are also presented. Numerical examples are given to illustrate the theorems and to demonstrate their superiority over the existing literature.