Functional Coupled Systems with Generalized Impulsive Conditions and Application to a SIRS-Type Model

In this paper, we consider a first-order coupled impulsive system of equations with functional boundary conditions, subject to the generalized impulsive effects. It is pointed out that this problem generalizes the classical boundary assumptions, allowing two-point or multipoint conditions, nonlocal and integrodifferential ones, or global arguments, as maxima or minima, among others. Our method is based on lower and upper solution technique together with the fixed point theory. The main theorem is applied to a SIRS model where to the best of our knowledge, for the first time, it includes impulsive effects combined with global, local, and the asymptotic behavior of the unknown functions.

Ratio of flavour non-singlet and singlet scalar density renormalisation parameters in $$N_\mathrm {f}=3$$ QCD with Wilson quarks

We determine non-perturbatively the normalisation factor $$r_{\mathrm{m}}\equiv Z_{\mathrm{S}}/Z_{\mathrm{S}}^{0}$$ r m ≡ Z S / Z S 0 , where $$Z_{\mathrm{S}}$$ Z S and $$Z_{\mathrm{S}}^{0}$$ Z S 0 are the renormalisation parameters of the flavour non-singlet and singlet scalar densities, respectively. This quantity is required in the computation of quark masses with Wilson fermions and for instance the renormalisation of nucleon matrix elements of scalar densities. Our calculation involves simulations of finite-volume lattice QCD with the tree-level Symanzik-improved gauge action, $$N_{\mathrm{f}}= 3$$ N f = 3 mass-degenerate $${\mathrm{O}}(a)$$ O ( a ) improved Wilson fermions and Schrödinger functional boundary conditions. The slope of the current quark mass, as a function of the subtracted Wilson quark mass is extracted both in a unitary setup (where nearly chiral valence and sea quark masses are degenerate) and in a non-unitary setup (where all valence flavours are chiral and the sea quark masses are small). These slopes are then combined with $$Z \equiv Z_{\mathrm{P}}/(Z_{\mathrm{S}}Z_{\mathrm{A}})$$ Z ≡ Z P / ( Z S Z A ) in order to obtain $$r_{\mathrm{m}}$$ r m . A novel chiral Ward identity is employed for the calculation of the normalisation factor Z. Our results cover the range of gauge couplings corresponding to lattice spacings below $$0.1\,$$ 0.1 fm, for which $$N_{\mathrm{f}}= 2+1$$ N f = 2 + 1 QCD simulations in large volumes with the same lattice action are typically performed.

The Difference in Facial Movement Between the Medial and the Lateral Midface: A 3D Skin Surface Vector Analysis

Background Our understanding of the functional anatomy of the face is constantly improving. To date, it is unclear whether the anatomic location of the line of ligaments has any functional importance during normal facial movements such as smiling. Objectives It is the objective of the present study to identify differences in facial movements between the medial and lateral midface by means of skin vector displacement analyses derived from 3D imaging and to further ascertain whether the line of ligaments has both a structural and functional significance in these movements. Methods The study sample consisted of 21 healthy volunteers (9 females & 12 males) of Caucasian ethnic background with a mean age of 30.6 (8.3) years and a mean BMI of 22.57 (2.5) kg/m 2. 3D images of the volunteers’ faces in repose and during smiling (Duchenne type) were taken. 3D imaging-based skin vector displacement analyses were conducted. Results The mean horizontal skin displacement was 0.08 (2.0) mm in the medial midface (lateral movement) and was -0.08 (1.96) mm in the lateral midface (medial movement) (p = 0.711). The mean vertical skin displacement (cranial movement of skin toward the forehead/temple) was 6.68 (2.4) mm in the medial midface whereas it was 5.20 (2.07) mm in the lateral midface (p = 0.003). Conclusions The results of this study provide objective evidence for an antagonistic skin movement between the medial and the lateral midface. The functional boundary identified by 3D imaging corresponds to the anatomic location of the line of ligaments.

Higher-Order Functional Discontinuous Boundary Value Problems on the Half-Line

In this paper, we consider a discontinuous, fully nonlinear, higher-order equation on the half-line, together with functional boundary conditions, given by general continuous functions with dependence on the several derivatives and asymptotic information on the (n−1)th derivative of the unknown function. These functional conditions generalize the usual boundary data and allow other types of global assumptions on the unknown function and its derivatives, such as nonlocal, integro-differential, infinite multipoint, with maximum or minimum arguments, among others. Considering the half-line as the domain carries on a lack of compactness, which is overcome with the definition of a space of weighted functions and norms, and the equiconvergence at ∞. In the last section, an example illustrates the applicability of our main result.

Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms

We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The results are applicable to systems of nonlocal second order ordinary differential equations subject to functional boundary conditions, this is illustrated in an example. Our approach is based on the classical fixed point index.

Positive solutions of systems of perturbed Hammerstein integral equations with arbitrary order dependence

Motivated by the study of systems of higher-order boundary value problems with functional boundary conditions, we discuss, by topological methods, the solvability of a fairly general class of systems of perturbed Hammerstein integral equations, where the nonlinearities and the functionals involved depend on some derivatives. We improve and complement earlier results in the literature. We also provide some examples in order to illustrate the applicability of the theoretical results. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

Nonzero Positive Solutions of Elliptic Systems with Gradient Dependence and Functional BCs

We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of nonnegative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.

Existence of solutions for functional boundary value problems at resonance on the half-line

By defining the Banach spaces endowed with the appropriate norm, constructing a suitable projection scheme, and using the coincidence degree theory due to Mawhin, we study the existence of solutions for functional boundary value problems at resonance on the half-line with $\operatorname{dim}\operatorname{Ker}L = 1$ dim Ker L = 1 . And an example is given to show that our result here is valid.