Singular $\phi $-Laplacian third-order BVPs with derivative dependance

Smaïl Djebali; Ouiza Saifi

Displaying similar documents to “Singular $φ$ -Laplacian third-order BVPs with derivative dependance”

On a singular multi-point third-order boundary value problem on the half-line

Zakia Benbaziz, Smail Djebali (2020)

Mathematica Bohemica

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We establish not only sufficient but also necessary conditions for existence of solutions to a singular multi-point third-order boundary value problem posed on the half-line. Our existence results are based on the Krasnosel’skii fixed point theorem on cone compression and expansion. Nonexistence results are proved under suitable a priori estimates. The nonlinearity $f = f (t, x, y)$ which satisfies upper and lower-homogeneity conditions in the space variables $x, y$ may be also singular at time $t = 0$ . Two examples...

A singular initial value problem for the equation $u^{(n)} (x) = g (u (x))$

Wojciech Mydlarczyk (1998)

Annales Polonici Mathematici

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We consider the problem of the existence of positive solutions u to the problem $u^{(n)} (x) = g (u (x))$ , $u (0) = u^{'} (0) = . . . = u^{(n - 1)} (0) = 0$ (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition $\int ₀^{δ} 1 / s {[s / g (s)]}^{1 / n} d s < \infty$ is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.

On the optimality and sharpness of Laguerre's lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Yusaku Yamamoto (2017)

Applications of Mathematics

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Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix $A \in ℝ^{m \times m}$ play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on $Tr (A^{- 1})$ and $Tr (A^{- 2})$ have attracted attention recently, because they can be computed in $O (m)$ operations when $A$ is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that...

Commutators on ${(\sum ℓ_{q})}_{p}$

Dongyang Chen, William B. Johnson, Bentuo Zheng (2011)

Studia Mathematica

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Let T be a bounded linear operator on $X = {(\sum ℓ_{q})}_{p}$ with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.

Behaviour of $L^{r}$ -Dini singular integrals in weighted $L^{1}$ spaces

Osvaldo Capri, Carlos Segovia (1989)

Studia Mathematica

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Common zero sets of equivalent singular inner functions

Keiji Izuchi (2004)

Studia Mathematica

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Let μ and λ be bounded positive singular measures on the unit circle such that μ ⊥ λ. It is proved that there exist positive measures μ₀ and λ₀ such that μ₀ ∼ μ, λ₀ ∼ λ, and $| ψ_{μ ₀} | < 1 \cap | ψ_{λ ₀} | < 1 = \emptyset$ , where $ψ_{μ}$ is the associated singular inner function of μ. Let $(μ) = ⋂_{ν; ν \sim μ} Z (ψ_{ν})$ be the common zeros of equivalent singular inner functions of $ψ_{μ}$ . Then (μ) ≠ ∅ and (μ) ∩ (λ) = ∅. It follows that μ ≪ λ if and only if (μ) ⊂ (λ). Hence (μ) is the set in the maximal ideal space of $H^{\infty}$ which relates naturally to the set of measures equivalent...

Fractional Laplacian with singular drift

Tomasz Jakubowski (2011)

Studia Mathematica

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For α ∈ (1,2) we consider the equation $\partial_{t} u = Δ^{α / 2} u + b \cdot \nabla u$ , where b is a time-independent, divergence-free singular vector field of the Morrey class $M ₁^{1 - α}$ . We show that if the Morrey norm ${| | b | |}_{M ₁^{1 - α}}$ is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.

Asymptotic integration of differential equations with singular $p$ -Laplacian

Milan Medveď, Eva Pekárková (2016)

Archivum Mathematicum

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In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p -$ Laplacian, where $1 < p < 2$ . We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t \to \infty$ .

$L^{p} (ℝ ⁿ)$ boundedness for the commutator of a homogeneous singular integral operator

Guoen Hu (2003)

Studia Mathematica

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The commutator of a singular integral operator with homogeneous kernel Ω(x)/|x|ⁿ is studied, where Ω is homogeneous of degree zero and has mean value zero on the unit sphere. It is proved that $Ω \in L {(l o g L)}^{k + 1} (S^{n - 1})$ is a sufficient condition for the kth order commutator to be bounded on $L^{p} (ℝ ⁿ)$ for all 1 < p < ∞. The corresponding maximal operator is also considered.

Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

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We study the existence of positive solutions to ⎧ ${d i v (| \nabla u |}^{p - 2} \nabla u) + q (x) u^{- γ} = 0$ on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is $ℝ^{N}$ or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

Zonghu Xiu, Caisheng Chen (2013)

Annales Polonici Mathematici

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We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ $- {d i v (| x |}^{- a p} {| \nabla u |}^{p - 2} {\nabla u) + h (x) | u |}^{p - 2} u = g (x) {| u |}^{r - 2} u$ , x ∈ Ω, ⎨ ⎩ ${| x |}^{- a p} {| \nabla u |}^{p - 2} \partial u / \partial ν = λ f (x) {| u |}^{q - 2} u$ , x ∈ ∂Ω, where Ω is an exterior domain in $ℝ^{N}$ , that is, $Ω = ℝ^{N} ∖ D$ , where D is a bounded domain in $ℝ^{N}$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function...

Singular non polynomial perturbations of $-\Delta+|x|^{2}$

Franco Nardini (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si studia la perturbazione dello spettro deiroperatore $-\Delta+|x|^{2}$ dovuta all'introduzione di un potenziale singolare non polinomiale e si prova che la serie perturbativa del primo autovalore di tale operatore è sommabile secondo Borel.

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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We study the chemotaxis system with singular sensitivity and logistic-type source: $u_{t} = Δ u - χ \nabla \cdot (u \nabla v / v) + r u - μ u^{k}$ , $0 = Δ v - v + u$ under the non-flux boundary conditions in a smooth bounded domain $Ω \subset ℝ^{n}$ , $χ, r, μ > 0$ , $k > 1$ and $n \geq 1$ . It is shown with $k \in (1, 2)$ that the system possesses a global generalized solution for $n \geq 2$ which is bounded when $χ > 0$ is suitably small related to $r > 0$ and the initial datum is properly small, and a global bounded classical solution for $n = 1$ .

Corrigendum to “Commutators on ${(\sum ℓ_{q})}_{p}$ ” (Studia Math. 206 (2011), 175-190)

Dongyang Chen, William B. Johnson, Bentuo Zheng (2014)

Studia Mathematica

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We give a corrected proof of Theorem 2.10 in our paper “Commutators on ${(\sum ℓ_{q})}_{p}$ ” [Studia Math. 206 (2011), 175-190] for the case 1 < q < p < ∞. The case when 1 = q < p < ∞ remains open. As a consequence, the Main Theorem and Corollary 2.17 in that paper are only valid for 1 < p,q < ∞.

Lower semicontinuous envelopes in $W^{1, 1} \times L^{p}$

Ana Margarida Ribeiro, Elvira Zappale (2014)

Banach Center Publications

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The lower semicontinuity of functionals of the type $\int_{Ω} f (x, u, v, \nabla u) d x$ with respect to the $(W^{1, 1} \times L^{p})$ -weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in $W^{1, 1} \times L^{p}$ for the lower semicontinuous envelope is also provided.

Displaying similar documents to “Singular φ -Laplacian third-order BVPs with derivative dependance”

Displaying similar documents to “Singular $φ$ -Laplacian third-order BVPs with derivative dependance”