In this paper, we introduce a new differential-difference operator Tξ (ξ ∈ R ) by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between Tξ and the directional derivative ∂ξ. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this o...

We study the following p-Laplacian equation with nonlinear boundary conditions: -Δ(p)u + μ(x)|u|(p-2)u = f(x,u) + g(x,u),x ∈ Ω, | ∇u|(p-2)∂u/∂n = η|u|(p-2)u and x ∈ ∂Ω, where Ω is a bounded domain in ℝ(N) with smooth boundary ∂Ω. We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) and f, g do not need to satisfy the (P.S) or (P.S...

We study the positive solutions to boundary value problems of the form −Δu λf u ; Ω, α x, u ∂u/∂η 1 − α x, u u 0; ∂Ω, where Ω is a bounded domain in R with n ≥ 1, Δ is the Laplace operator, λ is a positive parameter, f : 0,∞ → 0,∞ is a continuous function which is sublinear at ∞, ∂u/∂η is the outward normal derivative, and α x, u : Ω×R → 0, 1 is a smooth function nondecreasing in u. In particul...

In this paper we prove Hopf’s boundary point lemma for the fractional Laplacian. With respect to the classical formulation, in the non-local framework the normal derivative of the involved function u at z ∈ ∂Ω is replaced with the limit of the ratio u(x)/(δR(x)) , where δR(x) = dist(x, ∂BR) and BR ⊂ Ω is a ball such that z ∈ ∂BR. More precisely, we show that lim inf B∋x→z u(x) (δR(x)) > 0 . Als...

Pairs of compact convex sets naturally arise in quasidiierential calculus as the sub-and superdiierentials of the directional derivative of a quasidiierentiable function (see 1]). Since the sub-and superdiierential in a given point are not uniquely determined, minimal representations are of special importance. For the 2-dimensional case, equivalent minimal pairs of compact convex sets are uniqu...

In this paper, we introduce three new types of generalized convex functions, called semi E-pseudoinvex, strictly semi Epseudoinvex and semi E-quasiinvex functions. Then some of their basic properties are studied, We used directional derivative and obtain the new results in this class of functions. As applications of our results, we obtain optimal solutions for multiobjective programming. AMS Su...

R Upper-triangular matrix from QR decomposition v Direction unit vector: vj ; j ∈ [1, N ] Vinf Freestream velocity W Combined weight matrix Wb Derivative observation weight matrix: Wb i,j ≡ wi δi,j Wf Function observation weight matrix: Wf i,j ≡ wi δi,j x Vector of input quantities to f : xj ; j ∈ [1, N ] θ, φ Spherical polar coordinates {d} Vector of Taylor series derivatives: dk ; k ∈ [1, t] ...

We deene and study a directional derivative for two functions of the spectrum of an analytic matrix valued function. These are the maximum real part and the maximum modulus of the spectrum. Results are rst obtained for the roots of polynomials with analytic coeecients by way of Puiseux-Newton series. In this regard, the primary analytic tool is the so called Puiseux-Newton diagram. These result...

An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimates of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to ...

We derive a change of variable formula for non-anticipative functionals defined on the space of Rd -valued right-continuous paths with left limits. The functionals are only required to possess certain directional derivatives, which may be computed pathwise. Our results lead to functional extensions of the Itô formula for a large class of stochastic processes, including semimartingales and Diric...