Constant sign and nodal solutions for parametric (p, 2)-equations

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Multiple solutions for Robin ( p , q )-equations plus an indefinite potential and a reaction concave near the origin

AbstractWe consider a Robin problem driven by the (p, q)-Laplacian plus an indefinite potential term. The reaction is either resonant with respect to the principal eigenvalue or$$(p-1)$$(p-1)-superlinear but without satisfying the Ambrosetti-Rabinowitz condition. For both cases we show that the problem has at least five nontrivial smooth solutions ordered and with sign information. When$$q=2$$q=2(a (p, 2)-equation), we show that we can slightly improve the conclusions of the two multiplicity theorems

Energy expenditure rate in level and uphill treadmill walking determined from empirical models and foot inertial sensing data

An empirical model is used for predicting the energy expenditure rate of treadmill walking from walking speed and incline, which are measured by a foot-mounted inertial sensor. The difference between values of the energy expenditure rate obtained by entering measured and true values of these variables in the model equation is less than the errors that are reported to affect model based assessments of the metabolic response to locomotion in humans

Assessment of walking features from foot inertial sensing

An ambulatory monitoring system is developed for the estimation of spatio-temporal gait parameters. The inertial measurement unit embedded in the system is composed of one biaxial accelerometer and one rate gyroscope, and it reconstructs the sagittal trajectory of a sensed point on the instep of the foot. A gait phase segmentation procedure is devised to determine temporal gait parameters, including stride time and relative stance; the procedure allows to define the time intervals needed for carrying an efficient implementation of the strapdown integration, which allows to estimate stride length, walking speed, and incline. The measurement accuracy of walking speed and inclines assessments is evaluated by experiments carried on adult healthy subjects walking on a motorized treadmill. Root-mean-square errors less than 0.18 km/h (speed) and 1.52% (incline) are obtained for tested speeds and inclines varying in the intervals [3, 6] km/h and [ 5, +15]%, respectively. Based on the results of these experiments, it is concluded that foot inertial sensing is a promising tool for the reliable identification of subsequent gait cycles and the accurate assessment of walking speed and incline

Analytical and numerical convexity results for discrete fractional sequential differences with negative lower bound

We investigate relationships between the sign of the discrete fractional sequential difference (Δv 1+a-μ Δμaf)(t) and the convexity of the function t→f(t). In particular, we consider the case in which the bound (Δv 1+a-μ Δμaf)(t) ≥εf(a), for some ε \u3e 0 and where f(a) \u3c 0 is satisfied. Thus, we allow for the case in which the sequential difference may be negative, and we show that even though the fractional difference can be negative, the convexity of the function f can be implied by the above inequality nonetheless. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. We use a combination of both hard analysis and numerical simulation

Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces

AbstractAim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form$$\begin{aligned} \mathscr {L}u{:}{=}\sum _{i,j=1}^{n}\left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}}=\nabla \cdot f,\qquad \hbox {for almost all }x\in \Omega \end{aligned}$$Lu:=∑i,j=1naij(x)uxixj=∇·f,for almost allx∈Ωwhere the coefficients$$a_{ij}$$aijbelong to the Central (that is, Local) Sarason class CVMO andfis assumed to be in some Modified Local Generalized Morrey Spaces$$\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }$$LM~{x0}p,φ. Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results

Nonlinear resonant problems with an indefinite potential and concave boundary condition

We consider a nonlinear elliptic problem driven by the p-Laplacian plus and indefinite potential term. The reaction is (p − 1)-linear and resonant and the boundary term is concave. The problem is nonparametric. Using variational tools, together with truncation and perturbation techniques and critical groups, we show that the problem has at least three nontrivial smooth solutions

Positive solutions for (p, 2)-equations with superlinear reaction and a concave boundary term

We consider a nonlinear boundary value problem driven by the (p, 2)- Laplacian, with a (p − 1)-superlinear reaction and a parametric concave boundary term (a “concave-convex” problem). Using variational tools (critical point theory) together with truncation and comparison techniques, we prove a bifurcation type theorem describing the changes in the set of positive solutions as the parameter λ > 0 varies. We also show that for every admissible parameter λ > 0, the problem has a minimal positive solution uλ and determine the monotonicity and continuity properties of the map λ 7→ uλ