In this paper we introduce the two possible formulations of the -functional calculus that are based on the Fueter–Sce mapping theorem in integral form and we introduce the pseudo--resolvent equation. In the case of dimension 3 we prove the -resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the -functional calculus.

We study the existence of Liouvillian first integrals for the generalized Liénard polynomial differential systems of the form xʹ = y, yʹ = –g(x) – f(x)y, where f(x) = 3Q(x)Qʹ(x)P(x) + Q(x)2 Pʹ(x) and g(x) = Q(x)Qʹ(x)(Q(x)2 P(x)2 – 1) with P,Q ∈ ℂ[x]. This class of generalized Liénard polynomial differential systems has the invariant algebraic curve (y + Q(x)P(x))2 – Q(x)2 = 0 of hyperelliptic type.

We study the following coupled nonlinear Schr¨odinger system in ℝ3:

where μ1 > 0, μ2 > 0 and β ∈ ℝ is a coupling constant. Irrespective of whether the system is repulsive or attractive, we prove that it has nodal semi-classical segregated or synchronized bound states with clustered spikes for sufficiently small ε under some additional conditions on P(x), Q(x) and β. Moreover, the number of this type of solutions will go to infinity as ε → 0+.

In this paper a non-convex vector optimization problem among infinite-dimensional spaces is presented. In particular, a generalized Lagrange multiplier rule is formulated as a necessary and sufficient optimality condition for weakly minimal solutions of a constrained vector optimization problem, without requiring that the ordering cone that defines the inequality constraints has non-empty interior. This paper extends the result of Donato (J. Funct. Analysis261 (2011), 2083–2093) to the general setting of vector optimization by introducing a constraint qualification assumption that involves the Fréchet differentiability of the maps and the tangent cone to the image set. Moreover, the constraint qualification is a necessary and sufficient condition for the Lagrange multiplier rule to hold.

We prove the existence of solitary wave solutions to the quasilinear Benney system

where , –1 < p < +∞ and a, γ > 0. We establish, in particular, the existence of travelling waves with speed arbitrarily large if p < 0 and arbitrarily close to 0 if . We also show the existence of standing waves in the case , with compact support if – 1 < p < 0. Finally, we obtain, under certain conditions, the linearized stability of such solutions.

We consider the zero-resistivity limit for Hasegawa–Wakatani equations in a cylindrical domain when the initial data are Stepanov almost-periodic in the axial direction. First, we prove the existence of a solution to Hasegawa–Wakatani equations with zero resistivity; second, we obtain uniform a priori estimates with respect to resistivity. Such estimates can be obtained in the same way as for our previous results; therefore, the most important contribution of this paper is the proof of the existence of a local-in-time solution to Hasegawa–Wakatani equations with zero resistivity. We apply the theory of Bohr–Fourier series of Stepanov almost-periodic functions to such a proof.

We show the existence of connecting orbits for a class of singular second-order Hamiltonian systems

where, as opposed to most of the existing literature, we assume that the potential V has not one but any finite number of maxima of equal value. We use variational methods under the assumption that V (t, u) satisfies the so-called ‘strong-force’ condition at the singularity.

By giving some new treatments we can improve a classical result of Walfisz (1963) on the asymptotic formula of Euler's function.

We study the asymptotic behaviour of solutions of the fast diffusion equation near extinction. For a class of initial data, the asymptotic behaviour is described by a singular Barenblatt profile. We complete previous results on rates of convergence to the singular Barenblatt profile by describing a new phenomenon concerning the difference between the rates in time and space.

We consider the following Fife–Greenlee problem:

where Ω is a smooth and bounded domain in ℝn, ν is the outer unit normal to ∂Ω and a is a smooth function satisfying a(x) ∈ (–1, 1) in . Let K, Ω– and Ω+ be the zero-level sets of a, {a < 0} and {a < 0}, respectively. We assume ∇a ≠ 0 on K. Fife and Greenlee constructed stable layer solutions, while del Pino et al. proved the existence of one unstable layer solution provided that some gap condition is satisfied. In this paper, for each odd integer m ≥ 3, we prove the existence of a sequence ε = εj → 0, and a solution with m-transition layers near K. The distance of any two layers is O(ε log(1/ε)). Furthermore, converges uniformly to ±1 on the compact sets of Ω± as j → +∞

We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.

The aim of this work is to represent the elements of the homotopy classes of maps from S2 to S2 ∨ S1 by maps with some minimality root properties.

We consider the following equation:

where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up (p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up . Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x 1 only, and W satisfies

Let D(A) be the domain of an m-accretive operator A on a Banach space E. We provide sufficient conditions for the closure of D(A) to be convex and for D(A) to coincide with E itself. Several related results and pertinent examples are also included.

We show that every modular form on Γ0(2n) (n ⩾ 2) can be expressed as a sum of eta-quotients, which is a partial answer to Ono's problem. Furthermore, we construct a primitive generator of the ring class field of the order of conductor 4N (N ⩾ 1) in an imaginary quadratic field in terms of the special value of a certain eta-quotient.

We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stochastic differential equations.

We prove new results on the existence, non-existence, localization and multiplicity of non-trivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed-point index. Some of the criteria involve a comparison with the spectral radius of some related linear operators. We apply our results to some boundary-value problems with local and non-local boundary conditions of Neumann type. We illustrate in some examples the methodologies used.

We prove that a class of A-stable symplectic Runge–Kutta time semi-discretizations (including the Gauss–Legendre methods) applied to a class of semilinear Hamiltonian partial differential equations (PDEs) that are well posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. Consequently, such time semi-discretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(hp )-close to the original energy, where p is the order of the method and h is the time-step size. Examples of such systems are the semilinear wave equation, and the nonlinear Schrödinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace in which the operators occurring in the evolution equation are bounded, and by coupling the number of excited modes and the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form O(exp(–c/h 1/(1+q))) with c > 0 and q ⩾ 0; for the semilinear wave equation, q = 1, and for the nonlinear Schrödinger equation, q = 2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.

We consider both a problem over ℝn and a boundary problem over an exterior subregion for a Douglis–Nirenberg system of differential operators under limited smoothness assumptions as well as under the assumption of parameter ellipticity in a closed sector Ꮭ in the complex plane with vertex at the origin. We pose each problem on an Lp Sobolev space setting, 1 < p < ∞, and denote by the operator induced in this setting by the problem over ℝn and by Ap the operator induced in this setting by the boundary problem under null boundary conditions. We then derive results pertaining to the Fredholm theory for each of these operators for values of the spectral parameter λ lying in Ꮭ as well as results for these values of λ pertaining to the invariance of their Fredholm domains with p.

We obtain all the solutions of types u(x, y) = f(x) + g(y) and u(x, y) = f(x)g(y) for three known mean-curvature-prescribed equations, namely, the capillary equation, the translating soliton equation and the two-dimensional analogue of the catenary.