Unbounded Nonlinearities Research Articles

О СОГЛАСОВАННЫХ ДВУСТОРОННИХ ОЦЕНКАХ РЕШЕНИЙ КВАЗИЛИНЕЙНЫХ ПАРАБОЛИЧЕСКИХ УРАВНЕНИЙ И ИХ АППРОКСИМАЦИЙ

In this article, for the linearized difference scheme that approximates the Dirichlet problem for the homogeneous multidimensional quasi-linear parabolic equation with unbounded nonlinearity, two-sided point-wise estimates of the solution are established which are fully consistent with the same estimates for the differential problem. It is interesting to note that the proved two-sided estimates do not depend on diffusion coefficient. The direct application of such estimates is the proof of the convergence of the considered difference scheme in the grid norm L 2 . An example of the calculation by the Crank–Nicolson difference scheme is given, showing that the violation of the consistency conditions of differential and difference estimates leads to non-monotonic numerical solutions.

A new coercivity condition applied to semipositone integral equations with nonpositive, unbounded nonlinearities and applications to nonlocal BVPs

We consider the perturbed Hammerstein integral equation $$\begin{aligned} y(t)=\gamma _1(t)H_1(\varphi _1(y))+\gamma _2(t)H_2(\varphi _2(y))+\lambda \int _0^1G(t,s)f(s,y(s)) \mathrm{d}s \end{aligned}$$ and demonstrate the existence of at least one positive solution in the case where this problem is semipostione—i.e., f is allowed to be negative on its domain. As applications of this abstract result, we demonstrate existence of at least one positive solution to a variety of boundary value problems in the ordinary differential equations setting as well as the setting of elliptic partial differential equations on annuli. Finally, two novel aspects of this study are that, first, our results allow for f to be strictly negative on its entire domain, and second, it can actually be the case that \(\lim _{y\rightarrow +\infty }f(t,y)=-\infty \), uniformly for \(t\in [0,1]\). In addition, these results can hold even if \(H_1\) and \(H_2\) are piecewise linear on their domains.

Existence of Renormalized Solutions for p(x)-Parabolic Equation with three unbounded nonlinearities

In this article, we study the existence of a renormalized solution for the nonlinear $p(x)$-parabolic problem associated to the equation: $$\frac{\partial b(x,u)}{\partial t} - \mbox{div} (a(x,t,u,\nabla u)) + H(x,t,u,\nabla u) = f - \mbox{div}F \;\mbox{in }\;Q= \Omega\times(0,T)$$with $ f $ $ \in L^{1} (Q),$\; $b(x,u_{0}) \in L^{1} (\Omega)$ and $ F \in (L^{P'(.)}(Q))^{N}. $The main contribution of our work is to prove the existence of a renormalized solution in the Sobolev space with variable exponent. The critical growth condition on $ H(x,t,u,\nabla u)$\; is with respect to$ \nabla u$, no growth with respect to $u$ and no sign condition or the coercivity condition.

Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities

In this paper we prove the existence and uniqueness of a renormalized solution for nonlinear parabolic equations whose model is \begin{eqnarray} \frac{\partial b(u)}{\partial t} - div\big(a(x,t,u,\nabla u)\big)=f+ div (g), \end{eqnarray} where the right side belongs to $L^{1}(Q)+L^{p'}(0,T;W^{-1,p'}(\Omega))$, where $b(u)$ is a real function of $u$ and where $-div(a(x,t,u,\nabla u))$ is a Leray-Lions type operator with growth $|\nabla u|^{p-1}$ in $\nabla u$, but without any growth assumption on $u$.

Existence theory for nonlinear Sturm-Liouville problems with unbounded nonlinearities

Existence theory for nonlinear Sturm-Liouville problems with unbounded nonlinearities

Existence of a solution for a class of parabolic equations with three unbounded nonlinearities, natural growth terms and L1 data

We give an existence result of a renormalized solution for a class of nonlinear parabolic equations ∂b(x,u)∂t-div(a(x,t,u,∇u))+g(u)|∇u|p=f, where the right side belongs to L1(Ω×(0,T)), b(x,u) is an unbounded function of u and −div(a(x,t,u,∇u)) is a Leray–Lions type operator with growth ∣∇u∣p−1 in ∇u, but without any growth assumption on u. The function g is just assumed to be continuous on R and satisfying a sign condition.

Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces

Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces

Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities

Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities

MORE VECTORIAL BOOLEAN FUNCTIONS WITH UNBOUNDED NONLINEARITY PROFILE

The nonlinearity profile of Boolean functions is a generalization of the most important cryptographic criterion, called the (first order) nonlinearity. It is defined as the sequence of the minimum Hamming distances nlr(f) between a given Boolean function f and all Boolean functions in the same number of variables and of degrees at most r, for r ≥ 1. This parameter, which has a close relationship with the Gowers norm, quantifies the resistance to cryptanalyses by low degree approximations of stream ciphers using the Boolean function f as combiner or as filter. The nonlinearity profile can also be defined for vectorial functions: it is the sequence of the minimum Hamming distances between the component functions of the vectorial function and all Boolean functions of degrees at most r, for r ≥ 1. The nonlinearity profile of the multiplicative inverse functions has been lower bounded in a previous paper by the same author. No other example of an infinite class of functions with unbounded nonlinearity profile has been exhibited since then. In this paper, we lower bound the whole nonlinearity profile of the (simplest) Dillon bent function (x,y) ↦ xy2n/2-2, x, y ∈ 𝔽2n/2 and we exhibit another class of functions, for which bounding the whole profile of each of them comes down to bounding the first order nonlinearities of all functions.

Periodic solutions to second order nonautonomous differential systems with unbounded nonlinearities

We investigate the existence of periodic solutions of differential systems with a skew-symmetric matrix term and unbounded nonlinearities. Under new generalized Ahmad-Lazer-Paul conditions, some existence results are obtained and some results in the literature are improved.

Existence of solutions to resonant elliptic boundary value problems with unbounded discontinuous nonlinearities

We consider the situation when a resonant elliptic boundary value problem with an unbounded discontinuous nonlinearity is an idealization of a distributed system with continuous nonlinearities with respect to the phase variable and possessing narrow regions in the phase variable domain where the changes of nonlinear parameters are intractable. We study the existence of solutions to problems of this type.

On abstract equations with unbounded nonlinearities and their applications

On abstract equations with unbounded nonlinearities and their applications

A Smooth Center Manifold Theorem which Applies to Some Ill-Posed Partial Differential Equations with Unbounded Nonlinearities

We prove the existence of a smooth center manifold for several partial differential equations, including ill posed equations with unbounded nonlinearities. We also prove smooth dependence on parameters with respect to some perturbations, including unbounded ones. More concretely, we prove an abstract theorem and present applications to several concrete equations: ill posed Boussinesq, equation and system and nonlinear Laplace equations in cylindrical domains. We also consider the effect of some geometric structures.

Existence and location results for hinged beam equations with unbounded nonlinearities

This work presents some existence, non-existence and location results for the problem composed by the fourth-order fully nonlinear equation u ( 4 ) ( x ) + f ( x , u ( x ) , u ′ ( x ) , u ″ ( x ) , u ‴ ( x ) ) = s p ( x ) for x ∈ [ 0 , 1 ] , where f : [ 0 , 1 ] × R 4 → R and p : [ 0 , 1 ] → R + are continuous functions and s is a real parameter, with the Lidstone boundary conditions u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 . This problem models several phenomena, such as, the bending of an elastic beam simply supported at the endpoints. The arguments used apply a lower and upper solutions technique, a priori estimations and topological degree theory. In this paper we replace the usual bilateral Nagumo condition by some one-sided conditions, which enables us to consider unbounded nonlinearities.

Periodic solutions to differential systems with unbounded or periodic nonlinearities

We investigate the existence of (multiple) solutions of differential systems of Josephson-type with unbounded or periodic nonlinearities. Some new results are obtained and some results in the literature are improved.

Existence of critical point for abstract resonant problems with unbounded nonlinearities and applications to differential equations

In this paper, using well-known variational methods, we prove a general abstract theorem on the existence of a critical point for resonant problems with unbounded nonlinearities and then apply it to ordinary and partial differential equations with generalized Ahmad–Lazer–Paul conditions. The abstract result contains several concrete results in the literature and can also be used to deal with some new cases for resonant differential equations.

Solvability of some third-order boundary value problems with asymmetric unbounded nonlinearities

In this paper, we present existence and location results for the third-order separated boundary value problems u ‴ ( t ) = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , with the boundary conditions u ( a ) = A , u ″ ( a ) = B , u ″ ( b ) = C or u ( a ) = A , c 1 u ′ ( a ) - c 2 u ″ ( a ) = B , c 3 u ′ ( b ) + c 4 u ″ ( b ) = C , with c 1 , c 2 , c 3 , c 4 ∈ R + and A , B , C ∈ R . We assume f : [ a , b ] × R 3 → R is a continuous function satisfying one-sided Nagumo-type condition which allows an asymmetric unbounded behaviour. The arguments used concern Leray–Schauder degree and lower and upper solution techniques.

Multiple solutions of a class of Neumann problem for semilinear elliptic equations

The multiplicity results are obtained for solutions of the Neumann problem for nonlinear elliptic equations with unbounded nonlinearity by the Implicit Function Theorem and the Morse theory.

Pairs of solutions of constant sign for nonlinear periodic equations with unbounded nonlinearity

We consider periodic problems driven by the ordinary scalar $p$-Laplacian with a Caratheodory nonlinearity. Using variational techniques, coupled with the method of upper and lower solutions, we obtain two nontrivial solutions, with one positive and the other negative.

A switching controller for systems with hard uncertainties

A novel control scheme is proposed to solve the problem of stabilizing nonlinear uncertain systems with an unknown sign of the control gain and potentially unbounded nonlinearities. If the drift term f(x) has a growth rate such that the overall system has a finite escape time t/sup */, only a guess of a lowerbound, different from zero, of t/sup */ is necessary to design the control law. In spite of the hard uncertainties which affect the system model, the proposed switching controller is capable of asymptotically stabilizing the origin of the state space. The properties of the proposed control algorithm are compared with those of a well-known adaptive-control procedure when applied to the same class of systems.