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We present a general formulation of the averaging method in the setting of a semilinear equation L x = e N ( x , e ) , being L a linear Fredholm mapping of index zero. Our general approach provides new results even in the classical periodic framework. Among the applications we obtained there are: a partial answer to an open problem related to the Liebau phenomenon, the multiplicity of periodic solutions for a planar system with delay and the existence of solution for a nonlocal chemical reactor.

We study the existence of homoclinic solutions for a class of Lagrangian systems $\begin{array}{} \frac{d}{dt} \end{array} $(∇Φ(u̇(t))) + ∇uV(t, u(t)) = 0, where t ∈ ℝ, Φ : ℝ2 → [0, ∞) is a G-function in the sense of Trudinger, V : ℝ × (ℝ2 ∖ {ξ}) → ℝ is a C1-smooth potential with a single well of infinite depth at a point ξ ∈ ℝ2 ∖ {0} and a unique strict global maximum 0 at the origin. Under a strong force condition around the singular point ξ, via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions u± : ℝ → ℝ2 ∖ {ξ}.

This paper is concerned with the second order differential equation with not necessarily linear nonlocal boundary condition. The existence of solutions is obtained using the properties of the Leray–Schauder degree. The results generalize and improve some known results with linear nonlocal boundary conditions.

In this paper, we discuss the existence and multiplicity of periodic solutions for a class of parametric nonlocal equations with critical and supercritical growth. It is well known that these equations can be realized as local degenerate elliptic problems in a half-cylinder of $$\mathbb {R}^{N+1}_{+}$$ together with a nonlinear Neumann boundary condition, through the extension technique in periodic setting. Exploiting this fact, and by combining the Moser iteration scheme in the nonlocal framework with an abstract multiplicity result valid for differentiable functionals due to Ricceri, we show that the problem under consideration admits at least three periodic solutions with the property that their Sobolev norms are bounded by a suitable constant. Finally, we provide a concrete estimate of the range of these parameters by using some properties of the fractional calculus on a specific family of test functions. This estimate turns out to be deeply related to the geometry of the domain.

We consider the system of difference equations Δ ( Δ ⁢ u n - 1 1 - | Δ ⁢ u n - 1 | 2 ) = ∇ V n ( u n ) + h n , u n = u n + T ( n ∈ ℤ ) , $\Delta\bigg{(}\frac{\Delta u_{n-1}}{\sqrt{1-|\Delta u_{n-1}|^{2}}}\bigg{)}=% \nabla V_{n}(u_{n})+h_{n},\quad u_{n}=u_{n+T}\quad(n\in\mathbb{Z}),$ with Δ ⁢ u n = u n + 1 - u n ∈ ℝ N ${\Delta u_{n}=u_{n+1}-u_{n}\in{\mathbb{R}}^{N}}$ , V n = V n ⁢ ( x ) ∈ C 2 ⁢ ( ℝ N , ℝ ) ${V_{n}=V_{n}(x)\in C^{2}({\mathbb{R}}^{N},\mathbb{R})}$ , V n + T = V n ${V_{n+T}=V_{n}}$ , h n + T = h n ${h_{n+T}=h_{n}}$ for all n ∈ ℤ ${n\in\mathbb{Z}}$ and some positive integer T, V n ⁢ ( x ) ${V_{n}(x)}$ is ω i ${\omega_{i}}$ -periodic ( ω i > 0 ${\omega_{i}>0}$ ) with respect to each x i ${x_{i}}$ ( i = 1 , … , N ${i=1,\ldots,N}$ ) and ∑ j = 1 T h j = 0 ${\sum_{j=1}^{T}h_{j}=0}$ . Applying a modification argument to the corresponding problem with a left-hand member of p-Laplacian type, and using Morse theory, we prove that if all its solutions are non-degenerate, then the difference system above has at least 2 N ${2^{N}}$ geometrically distinct T-periodic solutions.

We identify a family of Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetimes such that the radially symmetric prescribed curvature problem with Neumann boundary condition is solvable on a ball of small radius. Such family contains some examples of interest in Cosmology.

A couple ($\sigma ,\tau $) of lower and upper slopes for the resonant second order boundary value problem $$ x'' = f(t,x,x'), \quad x(0) = 0,\quad x'(1) = \int _0^1 x'(s) {\rm d}g(s), $$ with $g$ increasing on $[0,1]$ such that $\int _0^1 dg = 1$, is a couple of functions $\sigma , \tau \in C^1([0,1])$ such that $\sigma (t) \leq \tau (t)$ for all $t \in [0,1]$, \begin {gather} \sigma '(t) \geq f(t,x,\sigma (t)), \quad \sigma (1) \leq \int _0^1 \sigma (s) {\rm d}g(s),\nonumber \\ \tau '(t) \leq f(t,x,\tau (t)), \quad \tau (1) \geq \int _0^1 \tau (s) {\rm d}g(s),\nonumber \end {gather} in the stripe $\int _0^t\sigma (s) {\rm d}s \leq x \leq \int _0^t \tau (s) {\rm d}s$ and $t \in [0,1]$. It is proved that the existence of such a couple $(\sigma ,\tau )$ implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.

Using Brouwer degree and convex analysis, we obtain geometric conditions for the existence of solutions of multipoint boundary value problems for n-dimensional difference systems of the formΔu(k)=f(k,u(k))(k=0,1,…,T-1),u(T)=∑k=0T-1g(k)u(k),when T is a positive integer, the vector fields f(k, u) point outside a bounded convex set of Rn on its boundary, and the diagonal matrix function g(k) satisfies suitable conditions, which cover the periodic boundary conditions.

In this paper we study the existence and multiplicity of periodic solutions for discontinuous perturbations of the operator u ↦ ( u ′ 1 − | u ′ | 2 ) ′ . The results are obtained by reduction to an equivalent non-singular problem and using the non-smooth critical point theory. Some illustrative examples concerning Filippov type solutions are also provided.

We show that the periodically perturbed N-dimensional relativistic pendulum equation has at least N + 1 geometrically distinct periodic solutions Also we obtain the existence of infinitely many solutions for systems with oscillating potential Both results are obtained by reduction to an equivalent non-singular problem using classical critical point theory.

In this paper we consider the Dirichlet problem with mean curvature operator in Minkowski space : where Ω ⊂ ℝN is a smooth open bounded set and f : Ω × ℝ → ℝ is a Carathéodory function. We show that u is a solution of the above problem iff it is a critical point of the corresponding non-smooth action functional. Applications concerning nontrivial solutions for a class of such Dirichet problems depending on a parameter are provided.

The paper surveys recent results obtained for the existence and multiplicity of radial solutions of Dirichlet problems of the type $$ \nabla \cdot \bigg (\frac {\nabla v}{\sqrt {1 - |\nabla v|^2}}\bigg ) = f(|x|,v) \quad \text {in} \ B_R,\quad u = 0 \quad \text {on} \ \partial B_R , $$ where $B_R$ is the open ball of center $0$ and radius $R$ in $\mathbb R^n$, and $f$ is continuous. Comparison is made with similar results for the Laplacian. Topological and variational methods are used and the case of positive solutions is emphasized. The paper ends with the case of a general domain.

Using continuation theorems of Leray-Schauder degree theory, we obtain existence results for the first order quasilinear boundary value problem (Φ(u))=f (t,u), u(T)=bu(0), where Φ : R → (-a,a) is an homeomorphism such that Φ(0)= 0 and f :[0.T] × R → R is a continuous function, a and T being positive real numbers and b some non zero real number.

In the framework of variational methods, we use a two non-zero critical points theorem to obtain the existence of two positive solutions to Dirichlet boundary value problems for difference equations involving the discrete p -Laplacian operator.

T-periodic solutions of systems of difference equations of the form Δφ[Δu(n-1)]= ∇u F[n,u(n)]+h(n)(n∈Z) where φ=∇Φ, Φ strictly convex, is a homeomorphism of RN onto the ball Ba ⊂ RN , or of Ba onto RN , are considered under various conditions upon F and h. The approach is variational. © 2011 Elsevier Ltd. All rights reserved.

We show that if \({{\mathcal A} \subset \mathbb{R}^N}\) is an annulus or a ball centered at zero, the homogeneous Neumann problem on \({{\mathcal A}}\) for the equation with continuous data $$\nabla \cdot \left(\frac{\nabla v}{\sqrt{1 - |\nabla v|^2}} \right) = g(|x|,v) + h(|x|)$$ has at least one radial solution when g(|x|,·) has a periodic indefinite integral and \({\int_{\mathcal A} h(|x|)\,{\rm{d}}x = 0.}\) The proof is based upon the direct method of the calculus of variations, variational inequalities and degree theory.

In this paper we survey some recent results on the existence and multiplicity of radial solutions for Neumann problems in a ball and in an annular domain, associated to pendulum-like perturbations of the mean extrinsic curvature operator in Minkowski space. Our approach relies on the Leray-Schauder degree, upper and lower solutions, and critical point theory.

We establish existence results for solutions to functional boundary value problems for φ- Laplacian ordinary differential equations assuming there are lower and upper solutions and Lipschitz bounding surfaces for the derivative which we adapt to our problem. Our results apply to some problems which do not satisfy Nagumo growth bounds. Moreover they contain as special cases many results for the p- and ɸ-Laplacians as well as many results where the boundary conditions depend on n-points or even functionals. Our boundary conditions generalize those of Fabry and Habets, Cabada and Pouso, Cabada, O’Regan and Pouso, and many others.

In 1963–64, Palais and Smale have introduced a compactness condition, namely condition (C), on real functions of class C 1 defined on a Riemannian manifold modeled upon a Hilbert space, in order to extend Morse theory to this frame and study nonlinear partial differential equations. This condition and some of its variants have been essential in the development of critical point theory on Banach spaces or Banach manifolds, and are referred as Palais–Smale-type conditions. The paper describes their evolution.

In this paper we study the existence and multiplicity of periodic solutions of pendulum-like perturbations of bounded or singular \({\phi}\)-Laplacians. Our approach relies on the Leray-Schauder degree and the upper and lower solutions method.

In this paper we study the existence of radial solutions for Neumann problems in a ball and in an annular domain, associated to mean curvature operators in Euclidean and Mmkowski spaces. Our approach relies on the Leray-Schauder degree together with some fixed point reformulations of our nonlinear Neumann boundary value problems (C) 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim

In this paper we study the existence and multiplicity of radial solutions for Neumann problems in a ball and in an annular domain, associated to pendulum-like perturbations of mean curvature operators in Euclidean and Minkowski spaces and of the p-Laplacian operator. Our approach relies on the Leray-Schauder degree and the upper and lower solutions method.

Using Leray-Schauder degree theory we obtain various existence results for the quasilinear equation problems (phi(u'))' = f(t, u, u') submitted to nonhomogeneous Dirichlet or nonlinear Neumann-Steklov boundary conditions on [0, T], when phi:]-a, a[ -> R is an increasing homeomorphism, phi(0) = 0. We compare the results with the ones proved earlier in the homogeneous case. (C) 2008 Elsevier Inc. All rights reserved.

A class of linear operators L + lambda I between suitable function spaces is considered, when 0 is an eigenvalue of L with constant eigenfunctions. It is proved that L + lambda I satisfies a strong maximum principle when lambda belongs to a suitable pointed left-neighborhood of 0, and satisfies a strong uniform anti-maximum principle when lambda belongs to a suitable pointed right-neighborhood of 0. Applications are given to various types of ordinary or partial differential operators with periodic or Neumann boundary conditions.

Systems of differential equations of the form $$(\phi(u^\prime))^\prime = f(t, u, u^\prime)$$ with \(\phi\) a homeomorphism of the ball \(B_a \subset {\mathbb{R}^{n}} \rm\,\,{onto}\,\, {\mathbb{R}^{n}}\) are considered, under various boundary conditions on a compact interval [0, T]. For non-homogeneous Cauchy, terminal and some Sturm–Liouville boundary conditions including in particular the Dirichlet–Neumann and Neumann–Dirichlet conditions, existence of a solution is proved for arbitrary continuous right-hand sides f. For Neumann boundary conditions, some restrictions upon f are required, although, for Dirichlet boundary conditions, the restrictions are only upon \(\phi\) and the boundary values. For periodic boundary conditions, both \(\phi\) and f have to be suitably restricted. All the boundary value problems considered are reduced to finding a fixed point for a suitable operator in a space of functions, and the Schauder fixed point theorem or Leray–Schauder degree are used. Applications are given to the relativistic motion of a charged particle in some exterior electromagnetic field.

In this paper, using the Schauder fixed point theorem, we prove existence results of radial solutions for Dirichlet problems in the unit ball and in an annular domain, associated to mean curvature operators in Euclidean and Minkowski spaces.

Using Leray-Schauder degree theory we obtain various existence and multiplicity results for nonlinear periodic boundary-value problems of the form (phi(u'))' = f(t, u, u'), u(0) - u(T) = 0 = u'(0) - u'(T), where phi : R -->] -a, a [ (0 R is a continuous function. (C) 2007 Elsevier Ltd. All rights reserved.

We give some existence results for bounded solutions of linear and nonlinear second order difference equations. In particular, a method of lower and upper solutions for bounded solutions of some nonlinear second order difference equations is obtained, and applied to Duffing-type difference equations.

Наведено огляд останніх результатів щодо нерезонансних і резонансних періодично збуджуваних нелінійних осциляторів: існування періодичних необмежених або обмежених розв'язків для обмежених нелінійних збурень лінійних та кусково-лінійних осциляторів, а також деяких класів плоских гамільтонових систем. This paper surveys recent results about nonresonant and resonant periodically forced nonlinear oscillators. This includes the existence of periodic, unbounded or bounded solutions for bounded nonlinear perturbations of linear and of piecewise linear oscillators, as well as of some classes of planar Hamiltonian systems.

We use Brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second-order and first-order difference equations. We obtain, in particular upper and lower solutions theorems, Ambrosetti-Prodi type results and sharp existence conditions for nonlinearities which are bounded from below or above.

We use Brouwer degree to prove existence and multiplicity results for the solutions of some nonlinear second order difference equations with Dirichlet boundary conditions. We obtain in particular upper and lower solutions theorems, Ambrosetti-Prodi type results, and sharp existence conditions for nonlinearities which are bounded from below or from above.

This paper uses topological degree methods to prove the existence of periodic solutions of some quaternionic-valued ordinary differential equations.

A maximum principle is proved for the weak solutions u is an element of L-infinity (R x T-3) of the telegraph equation in space dimension three u(tt) - Delta(x)u + cu(t) + lambdau = f (t, x), when c > 0, lambda is an element of (0, c(2)/4) and f is an element of L-infinity (R x T-3) (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation u(tt) - Delta(x)u + cu(t) = F(t, x, u). Also, they can be employed in the study of almost periodic solutions of the forced sine-Gordon equation. A counterexample for the maximum principle in dimension four is given. (C) 2004 Elsevier Inc. All rights reserved.

We establish maximum principles for second order difference equations and apply them to obtain uniqueness for solutions of some boundary value problems.

In this paper we extend the guiding function approach to show that there are periodic or bounded solutions for first order systems of ordinary differential equations of the form x1 =f(t,x), a.e. epsilon[a,b], where f satisfies the Caratheodory conditions. Our results generalize recent ones of Mawhin and Ward.

Using some recent extensions of upper and lower solutions techniques and continuation theorems to the periodic solutions of quasilinear equations of p-Laplacian type, we prove the existence of positive periodic solutions of equations of the form (|xʹ|p-2xʹ)ʹ + f(x)xʹ + g(x) = h(t) with p > 1, f arbitrary and g singular at 0. This extends results of Lazer and Solimini for the undamped ordinary differential case.

A maximum principle is proved for the weak solutions u is an element of L-infinity(R x T-3) of the telegraph equation u(tt) - Delta(x)u + cu(t) + lambda(u) = f(t, x), in space dimension three, when c > 0, lambda is an element of (0, c(2)/4] and f is an element of L-infinity (R x T-3) (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation u(tt) - Delta(x)u + cut = F(t, x, it). (C) 2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.

We state and prove some extensions of the fundamental theorem of the method of guiding functions for periodic and for bounded solutions of ordinary differential systems. Those results unify and generalize previous results of Krasnosel'skii, Perov, Mawhin, Walter and Gossez.

The aim of this paper is to consider the radially-symmetric periodic-Dirichlet problem on [0,T] x B-n [a] for the equation u(tt)-Deltau = f(t, x , u), where Delta is the classical Laplacian operator, and B-n[a] denotes the open ball of center 0 and radius a in R-n : When alpha = a/T is a sufficiently large irrational with bounded partial quotients, we combine some number theory techniques with the asymptotic properties of the Bessel functions to show that 0 is not an accumulation point of the spectrum of the linear part. This result is used to obtain existence conditions for the nonlinear problem.

In this paper, we consider the 2@p-periodic problem for the equation x'' + n^2x = f(x) - b(t), where n is a positive integer, b(t) is continuous and 2@p-periodic, and f(x) is bounded and continuous. We give a new formulation for the Lazer-Leach conditions for the existence of 2@p-periodic solutions, and new sufficient conditions for the existence of unbounded sequences of such solutions. (C) 2000 Elsevier Science Ltd. All rights reserved.

This paper is devoted to the computation of the index at infinity for some asymptotically linear completely continuous vector fields a:-T(a:), when the principal linear part x - Ax is degenerate (1 is an eigenvalue of A), and the sublinear part is not asymptotically homogeneous (in particular do not satisfy Landesman-Lazer conditions). In this work we consider only the case of a one-dimensional degeneration of the linear part, i.e.s 1 is a simple eigenvalue of A. For this case we formulate an abstract theorem and give some general examples for vector fields of Hammerstein type and for a two point boundary value problem.

We prove Landesman-Lazer type existence conditions for the solutions bounded on the real line, together with their first derivatives, for some second order nonlinear differential equations of the form x " + cx' + f(t, x) = 0. The proofs use upper and lower solutions techniques together with some limiting arguments.

This paper deals with the existence of solutions to the periodic and the Sturm-Liouville boundary value problems for second order ordinary differential equations with singular nonlinearity. The results are obtained by use of some continuation theorems in the absence of a priori bounds of Capietto-Mawhin-Zanolin.

We use the critical point theory for convex, lower semicontinuous perturbations of C1-functionals to establish existence of multiple radial solutions near resonance for some Neumann problems involving the mean extrinsic curvature operator.

has been considered by Srzednicki [11 13], using a combination of generalized Conley's isolating blocks, the index of singular points of autonomous systems and the Lefschetz fixed point theorem. Recently, it has been shown that slight extensions of Srzednicki's results could be obtained in a simpler and more elementary way using continuation theorems of the Leray Schauder type (see [6, 9]). The aim of the present paper is to show that the same methodology can be adapted to prove the existence of periodic solutions for more general classes of equations and systems. article no. 0054