Multiple periodic solutions of a superlinear forced wave equation

We study the existence of forced vibrations of nonlinear wave equation: (*) $$\begin{array}{*{20}c} {u_{tt} - u_{xx} + g(u) = f(x,t),} & {(x,t) \in (0,\pi ) \times R,} \\ {\begin{array}{*{20}c} {u(0,t) = u(\pi ,t) = 0,} \\ {u(x,t + 2\pi ) = u(x,t),} \\ \end{array} } & {\begin{array}{*{20}c} {t \in R,} \\ {(x,t) \in (0,\pi ) \times R,} \\ \end{array} } \\ \end{array}$$ whereg(ξ)∈C(R,R)is a function with superlinear growth and f(x, t) is a function which is 2π-periodic in t. Under the suitable growth condition on g(ξ), we prove the existence of infinitely many solution of (*) for any given f(x, t).     

On a semilinear volterra integrodifferential equation

The Volterra integrodifferential equation $$\begin{array}{*{20}c} {u_t (t,x) + \smallint '_0 a(t - s)( - \Delta u(s,x) + f(x,u(s,x)))ds = h(t,x),,} \\ {t > 0,x \in \Omega \subset R^N ,} \\ \end{array} $$ together with boundary and initial conditions is considered. The existence of global solutions (in time) is established under weak assumptions onf. An application in heat flow is also indicated.     

The problem of the solvability of nonlinear two-point boundary-value problems

We establish sufficient conditions for the solvability of boundary-value problems of the form $$\begin{gathered} u'' = f(t,u,u'); \hfill \\ \begin{array}{*{20}c} {(u(0),} & {u'(0)) \in S_0 ,} & {(u(1),} & {u'(1)) \in S_1 .} \\ \end{array} \hfill \\ \end{gathered} $$      

Bang-bang property for Bolza problems in two dimensions

Consider the following Bolza problem: $$\begin{gathered} \min \int {h(x,u) dt,} \hfill \\ \dot x = F(x) + uG(x), \hfill \\ \left| u \right| \leqslant 1, x \in \Omega \subset \mathbb{R}^2 , \hfill \\ x(0) = x_0 , x(1) = x_1 . \hfill \\ \end{gathered} $$ We show that, under suitable assumptions onF, G, h, all optimal trajectories are bang-bang. The proof relies on a geometrical approach that works for every smooth two-dimensional manifold. As a corollary, we obtain existence results for nonconvex optimization problems.     

Multiplicity and concentration of positive solutions for the Schr?dinger�CPoisson equations

This paper is concerned with the multiplicity and concentration of positive solutions for the nonlinear Schr?dinger?CPoisson equations $$ \left\{ \begin{array}{l@{\quad}l} -\varepsilon^2\triangle u+V(x)u+\phi(x) u=f(u)& {\rm in}\,{\mathbb R}^3, \\ -\varepsilon^2\triangle \phi=u^2 & {\rm in}\,{\mathbb R}^3, \\ u\in H^1({\mathbb R}^3), u(x) > 0,& \forall x\in{\mathbb R}^3, \\ \end{array} \right. $$ where ???>?0 is a parameter, ${V: {\mathbb R}^3\rightarrow{\mathbb R}}$ is a continuous function and ${f: {\mathbb R}\rightarrow {\mathbb R}}$ is a C ¹ function having subcritical growth. The proof of the main result is based on minimax theorems and the Ljusternik?CSchnirelmann theory.     

On a class of Hammerstein integral equations

By a monotone representation of the nonlinearity we derive sufficient (and partly necessary) conditions for the unique existence of positive solutions of the Hammerstein integral equation $$\begin{array}{*{20}c} {u(x) = \int\limits_D {f(y,u(y)) k(x,y) dy ,} } & {x \in D,} \\ \end{array} $$ and for the convergence of successive approximations towards the solution. Further we study the corresponding nonlinear eigenvalue problem. Essentially we assume that the integral kernel k satisfies appropriate positivity conditions and that, for the nonlinearity f and any y ∈ D, rf(y,r) strictly monotone increases and f(y,r)/r strictly monotone decreases as r>0 increases.     

Metodi variazionali nello studio di problemi al contorno con parte non lineare discontinua

The main purpose of this paper is to use variational methods in the study of problems of following type: $$\left\{ {\begin{array}{*{20}c} {Lu = \lambda f(x,u) in \Omega } \\ {u = 0 on \partial \Omega } \\ \end{array} } \right.$$ Here Ω is supposed to be a bounded domain with smooth boundary ? Ω,L an elliptic operator, λ∈IR andf(x, t) a real function defined on \(\tilde \Omega \times IR\) having one or several simple discontinuities ont. Mainly we are interested in solutions which satisfy (.) a. e., which are most meaningful in physical problems, and we prove various existence theorems for several choices ofL, f and λ. The main difficulty consists in the fact that the functionals related to (.) are not Fréchet differentiable in every point, sincef is discontinuous.     

Dirichlet problems for the quasilinear second order subelliptic equations

In this paper, we study the Dirichlet problems for the following quasilinear second order sub-elliptic equation, $$\left\{ {\begin{array}{*{20}c} {\sum\limits_{i,j = 1}^m {X_i^* (A_{i,j} (x,u)X_j u) + \sum\limits_{j = 1}^m {B_j (x,u)X_j u + C(x,u) = 0in\Omega ,} } } \\ {u = \varphi on\partial \Omega ,} \\ \end{array} } \right.$$ whereX={X ₁, ...,X _m } is a system of real smooth vector fields which satisfies the Hörmander's condition,A _i,j ,B _j ,C ∈C ^∞( $\bar \Omega$ ×R) and (A _i,j (x,z)) is a positive definite matrix. We have proved the existence and the maximal regularity of solutions in the “non-isotropic” Hölder space associated with the system of vector fieldsX.     

On the invariants of Möbius groupsM(R n )

Suppose that $$\operatorname{Re} (a + d^ * ) \in \left\{ {\begin{array}{*{20}c} {( - 2,2),if g(x) is f.p.f. or elliptic,} \\ {\left[ { - 2,2} \right], if g(x) is parabolic,} \\ {( - \infty ,\infty ), if g(x) is loxodromic.} \\ \end{array} } \right.$$ is a Clifford matrix of dimensionn, g(x)=(ax+b)(cx+d) ^?1. We study the invariant balls and the more careful classifications of the loxodromic and parabolic elements inM(R ⁿ ), prove that the loxodromic elements inM(R ^2k+1 ) certainly have an invariant ball, expound the geometric meaning of Ahlfors' hyperbolic elements, and introduce the uniformly hyperbolic and parabolic elements and give their identifications. We prove that $$\operatorname{Re} (a + d^ * ) \in \left\{ {\begin{array}{*{20}c} {( - 2,2),if g(x) is f.p.f. or elliptic,} \\ {\left[ { - 2,2} \right], if g(x) is parabolic,} \\ {( - \infty ,\infty ), if g(x) is loxodromic.} \\ \end{array} } \right.$$ These results are fundamental in the higher dimensional Möbius groups, especially in Fuchs groups.     

Weak solutions for elliptic systems with variable growth in Clifford analysis

In this paper we consider the following Dirichlet problem for elliptic systems: $$\begin{array}{*{20}c} {\overline {DA\left( {x,u\left( x \right),Du\left( x \right)} \right)} = B\left( {x,u\left( x \right),Du\left( x \right)} \right), x \in \Omega ,} \\ {u\left( x \right) = 0, x\partial \Omega } \\ \end{array}$$ where D is a Dirac operator in Euclidean space, u(x) is defined in a bounded Lipschitz domain Ω in ? ⁿ and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned systems in the space W ₀ ^1,p(x) (Ω,C? _n ) under appropriate assumptions.     

Comportement asymptotique des solutions d'un système elliptique conservatif non linéaire

Let M be a compact riemannian manifold; in a previous article we show that every non- negative solution of u_tt + Δ_g u=f(u) on M ×R ⁺, satisfying Dirichlet or Neumann boundary conditions, converges to a (stationary) solution Φ of Δ_g Φ=f(Φ) with exponential decay of ∥u - Φ∥c²(M), if we assume that f behaves like r? r^p - λr. We extend this result to a system in the following form $$\left\{ {\begin{array}{*{20}c} {u_{tt} + \Delta _g u + \alpha u - G_x (u,\upsilon ) = 0,} \\ {u_{tt} + \Delta _g \upsilon + \beta \upsilon - G_x (u,\upsilon ) = 0,} \\ \end{array} } \right.$$ . where G satisfies some growth and convexity properties.     

The buffer phenomenon in one-dimensional piecewise linear mapping in radiophysics

We study the problem of attractors of the two-dimensional mapping $$(u,v) \to (v, - (1 - \mu )u - F(v)),F(v) = \left\{ {\begin{array}{*{20}c} {q_1 forv > 0,} \\ {0forv > 0,} \\ { - q_2 forv > 0,} \\ \end{array} } \right.$$ where 0 < µ ? 1 and q ₁, q ₂ > 0. This mapping is the mathematical model of a self-excited oscillator with relay amplifier and a part of the long transmission line without distortions in the feedback circuit. We prove that, in the system under study, there coexist stable cycles with arbitrarily large periods as the parameter µ decreases properly. We also show that the total number of these cycles increases without bound as µ → 0, i.e., the buffer phenomenon is realized.     

Existence and Multiplicity of Positive Solutions to a Quasilinear Elliptic Equation with Strong Allee Effect Growth Rate

In this paper we consider a p-Laplacian equation with strong Allee effect growth rate and Dirichlet boundary condition $$\left\{\begin{array}{ll} {\rm div} (|\nabla u|^{p-2} \nabla u) + \lambda f(x,u)=0, &\quad x \in \Omega, \\ u=0, &\quad x \in \partial \Omega, \qquad \qquad ^ {(P_\lambda)} \end{array}\right.$$ where Ω is a bounded smooth domain in ${\mathbb{R}^N}$ for ${N \ge 1, p > 1}$ , and λ is a positive parameter. By using variational methods and a suitable truncation technique, we prove that problem (P _λ) has at least two positive solutions for large parameter and it has no positive solutions for small parameter. In addition, a nonexistence result is investigated.     

Über verallgemeinerte Momente additiver Funktionen

In this paper we give characterizations of additive functionsf, for which $$\mathop {\lim \sup }\limits_{x \to \infty } x^{ - 1} \sum\limits_{n \leqslant x} {\varphi (|f(n)|)}$$ is bounded, where φ: ?⁺ → ?⁺ is monotone and or $$\begin{array}{*{20}c} {\varphi (x) = c^x } & {(x \in \mathbb{R}).} \\ \end{array}$$ A typical example is φ (x)=x ^a (a>0) forx≥0.     

Matrix semigroups—generators and relations

Some presentations for the semigroups of all 2×2 matrices and all 2×2 matrices of determinant 0 or 1 over the field GF(p) (p prime) are given. In particular, if <a, b, c‖ R> is any (semigroup) presentation for the general linear group in terms of generators $$A = \left( {\begin{array}{*{20}c} 1 & 0 \\ 1 & 1 \\ \end{array} } \right),B = \left( {\begin{array}{*{20}c} 1 & 1 \\ 0 & 1 \\ \end{array} } \right),C = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & \xi \\ \end{array} } \right),$$ where ζ is a primitive root of 1 modulop, then the presentation $$\langle a,b,c,t|R,t^2 = ct = tc = t,tba^{p - 1} t = 0,b^{\xi - 1} atb = a^{\xi - 1} tb^\xi a^{1 - \xi - 1} \rangle $$ defines the semigroup of all 2×2 matrices over GF (2,p) in terms of generatorsA, B, C and $$T = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 0 \\ \end{array} } \right).$$ Generating sets and ranks of various matrix semigroups are also found.     

Eigenvalues of the Dirac operator in the continuous spectrum

Functionsp(x) andq(x) for which the Dirac operator $$Dy = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } & {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \\ \end{array} } \right)\frac{{dy}}{{dx}} + \left( {\begin{array}{*{20}c} {p(x) q(x)} \\ {q(x) - p(x)} \\ \end{array} } \right)y = \lambda y, y = \left( {\begin{array}{*{20}c} {y_1 } \\ {y_2 } \\ \end{array} } \right), y_1 (0) = 0,$$ has a countable number of eigenvalues in the continuous spectrum are constructed.     

A remark on global existence of solutions of shadow systems

Let ?? be an open, bounded domain in ${\mathbb{R}^n\;(n \in \mathbb{N})}$ with smooth boundary ???. Let p, q, r, d ₁, ?? be positive real numbers and s be a non-negative number which satisfies ${0 < \frac{p-1}{r} < \frac{q}{s+1}}$ . We consider the shadow system of the well-known Gierer?CMeinhardt system: $$ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right. $$ We prove that solutions of this system exist globally in time under some conditions on the coefficients. Our results are based on a priori estimates of the solutions and improve the global existence results of Li and Ni in [4].     

Existence and multiplicity results for Pucci’s operators involving nonlinearities with zeros

We study the problem $$ \left\{\begin{array}{ll} {-\varepsilon^{2}\mathcal{M}^+_{\lambda,\Lambda}(D^{2}u) = f (x, u)} \quad\; {\rm in} \; \Omega,\\ {u = 0} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {\rm on} \; \partial{\Omega}, \end{array} \right.$$ where Ω is a smooth bounded domain in ${\mathbb{R}^{N},N > 2,}$ and show it possesses nontrivial solutions for small values of ε provided f is a nonnegative continuous function which has a positive zero. The multiplicity result is based on degree theory together with a new Liouville type theorem for ${-{M}^+_{\lambda,\Lambda}(D^{2}u) = f(u)}$ in ${\mathbb{R}^{N}}$ for nonnegative nonlinearities with zeros.     

Asymptotic integration of symmetric second-order quasidifferential equations

This paper presents conditions on the coefficients of the equations $$\begin{array}{*{20}c} { - (p(f' - rf))' - \bar rp(f' - rf) + qf = 0,} \\ { - (P(f' - Rf))' - \bar RP(f' - Rf) + Qf = 0,} \\ \end{array}$$ where 1/p, 1/P, q, Q, r, R ∈ ? _loc ¹ (?₊), p, P, q, and Q are real-valued functions, while r and R are complex-valued functions, as well as on the fundamental system of solutions of the second equation, which ensure the asymptotic proximity of the solutions of these equations. The results obtained are applied to the study of the spectral properties of the differential operator generated by the expression $$ - y'' + \sum\limits_{k = 0}^{ + \infty } {h_k \delta (x - x_k )y, x_k \in \mathbb{R}_ + ,} h_k \in R,$$ , in the space ?²(?₊). In particular, we obtain conditions on h _k , x _k under which the limit-disk case is realized for this operator.