Related papers: Coordinate-wise Armijo's condition: General case

It is proved that a differentiable with respect to each variable function $f:\mathbb R^2\to\mathbb R$ is a solution of the equation $ \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}=0$ if and only if there exists a function…

We consider the degenerate equation $$\partial\_t f(t,x) - \partial\_x \left( x^{\alpha} \partial\_x f \right)(t,x) =0,$$ on the unit interval $x\in(0,1)$, in the strongly degenerate case $\alpha \in [1,2)$ with adapted boundary conditions…

We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class $\Lambda_{f,g}(a, b)$ of mean values where $f, g$ are continuously differentiable convex functions satisfying the relation…

In this work we study the existence of positive solution to the fractional quasilinear problem, $$ \left\{ \begin{array}{rcll} (-\Delta )^s u &=&\lambda \dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ \mu f &\inn \Omega,\\ u&>&0 & \inn\Omega,\\ u&=&0…

In this paper, we study the quasilinear inequality $ \Delta_m u+f(u)\leq 0$ on a complete Riemannian manifold, where \begin{align*} m>1,\alpha>m-1 \quad and \quad f(t)> 0,\alpha f(t)-tf^{'}(t)\geq 0, \forall t>0. \end{align*} If for some…

We define in the space of n by m matrices of rank n, n less or equal than m, the condition Riemannian structure as follows: For a given matrix A the tangent space of A is equipped with the Hermitian inner product obtained by multiplying the…

Suppose $v(x,y):\mathbb C\rightarrow \mathbb R$ is an entire harmonic polynomial with no critical points in the right half plane. Let $z_1, z_2\in\mathbb C$ lie on a level set of $v$ , and assume ${\rm Re}(z_2)>{\rm Re}(z_1)\geq0$. We give…

For $0 < a \le 1/2$, we define the quadrilateral zeta function $Q(s,a)$ using the Hurwitz and periodic zeta functions and show that $Q(s,a)$ satisfies Riemann's functional equation studied by Hamburger, Heck and Knopp. Moreover, we prove…

We consider systems of the differential inequalities $$\left\{ \begin{aligned} & \sum_{|\alpha| = m_1} \partial^\alpha a_\alpha (x, u_1) \ge f_1 (u_2) & \mbox{in } {\mathbb R}^n, & \sum_{|\alpha| = m_2} \partial^\alpha b_\alpha (x, u_2) \ge…

Let $\mathcal{L}$ be a subspace lattice on a Banach space $X$ and let $\delta:\mathrm{Alg}\mathcal{L}\rightarrow B(X)$ be a linear mapping. If $\vee\{L\in \mathcal{L}: L_-\nsupseteq L\}=X$ or $\wedge\{L_-:L\in \mathcal{L}, L_-\nsupseteq…

In this paper several inequalities of the right-hand side of Hermite-Hadamard inequality are obtained for the class of functions whose derivatives in absolutely value at certain powers are ({\alpha},m)-convex.Some applications to special…

Let $\alpha,\beta,\gamma\in\mathbb{N}$. We prove that given an $r$-colouring of $\mathbb{F}_p$ with $p$ prime, there are more than $c_{r,\alpha,\beta,\gamma} p^2$ solutions to the equation $x^\alpha+y^\beta=z^\gamma$ with all of $x,y,z$ of…

Given a complete, cocomplete category $\mathcal C$, we investigate the problem of describing those small categories $I$ such that the diagonal functor $\Delta:\mathcal C\to {\rm Functors}(I,\mathcal C)$ is a Frobenius functor. This…

By considering Schwarz's map for the hypergeometric differential equation with parameters $(a,b,c)=(1/6,1/2,1)$ or $(1/12,5/12,1)$, we give some analogies of Jacobi's formula $\vartheta_{00}(\tau)^2= F(1/2,1/2,1;\lambda(\tau))$, where…

In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta(g(|\nabla u|_2^2) u + u^r) = a u + b u^p& \mbox{in}~\Omega, u>0& \mbox{in}~\Omega, u= 0& \mbox{on}~\partial\Omega, \end{array}…

We investigate nonnegative solutions $u(x,t)$ and $v(x,t)$ of the nonlinear system of inequalities \[0\leq(\partial_t -\Delta)^\alpha u\leq v^\lambda\] \[ 0\leq (\partial_t -\Delta)^\beta v\leq u^\sigma\] in $\mathbb{R}^n \times\mathbb{R}$,…

Using the properties of geometric mean, we shall show for any $0\le \alpha ,\beta \le 1$, \[f\left( A{{\nabla }_{\alpha }}B \right)\le f\left( \left( A{{\nabla }_{\alpha }}B \right){{\nabla }_{\beta }}A \right){{\sharp}_{\alpha }}f\left(…

Let $(M,g)$ be a Riemannian manifold with Laplace-Beltrami operator $-\Delta$ and let $E\to M$ be a Hermitian vector bundle with a Hermitian covariant derivative $\nabla$. Furthermore, let H(0) denote the Friedrichs realization of…

Let $$ F(x, y) = \prod\limits_{k = 0}^{n - 1}(\delta_kx - \gamma_ky) $$ be a binary form of degree $n \geq 1$, with complex coefficients, written as a product of $n$ linear forms in $\mathbb C[x, y]$. Let $$ h_F = \prod\limits_{k = 0}^{n -…

We study pointwise convergence of the solutions to Schr\"odinger equations with initial datum $f\in H^s(\mathbb R^n)$. The conjecture is that the solution $e^{it\Delta}f$ converges to $f$ almost everywhere for all $f\in H^s(\mathbb R^n)$ if…