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Register a new userOn a lower a priori estimate of minimal eigenvalue of one Sturm-Liouville problem with second-type boundary conditions
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It is proved that for class $A_gamma={qin L_1[0,1]: qgeq 0, int_0^1 q^gamma,dx=1}$, where $gammain (0,1)$, there exists a potential $q_*in A_gamma$ such that minimal eigenvalue $lambda_1(q_*)$ of boundary problem $$ -y+q_*y=lambda y, y(0)=y(1)=0 $$ is equal to $m_gamma=inf_{qin A_gamma}lambda_1(q)$. The equality $m_gamma=1$ for $gammaleq 1-2pi^{-2}$ and the inequality $m_gamma<1$ for $gamma>1-2pi^{-2}$ are also obtained.
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This article considers the eigenvalue problem for the Sturm-Liouville problem including $p$-Laplacian begin{align*} begin{cases} left(vert uvert^{p-2}uright)+left(lambda+r(x)right)vert uvert ^{p-2}u=0,,, xin (0,pi_{p}), u(0)=u(pi_{p})=0, end{cases} end{align*} where $1<p<infty$, $pi_{p}$ is the generalized $pi$ given by $pi_{p}=2pi/left(psin(pi/p)right)$, $rin C[0,pi_{p}]$ and $lambda<p-1$. Sharp Lyapunov-type inequalities, which are necessary conditions for the existence of nontrivial solutions of the above problem are presented. Results are obtained through the analysis of variational problem related to a sharp Sobolev embedding and generalized trigonometric and hyperbolic functions.
We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann Liouville type. We then solve a Dirichlet type Sturm-Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann-Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann-Liouville integrals at those end-points. For each $1/2<alpha<1$ it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as $alpha to 1^-$, and that the fractional operator converges to an ordinary two term Sturm-Liouville operator as $alpha to 1^-$ with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of $alpha$.
We continue the study of a non self-adjoint fractional three-term Sturm-Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral under {it Dirichlet type} boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter $alpha$, $0<alpha<1$, there is a finite set of real eigenvalues and that, for $alpha$ near $1/2$, there may be none at all. As $alpha to 1^-$ we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm-Liouville problem with the composition of the operators becoming the operator of second order differentiation.
On the basis of the theory of Sturm--Liouville problem with distribution coefficients we get the infima and suprema of the first eigenvalue of the problem $-y + (q-lambda) y=0, y(0) -k_0^2 y(0) = y(1) + k_1^2 y(1) = 0$, where $q$ belongs to the set of constant-sign summable functions on $[0,1]$ such that $int_0^1 q dx=pm 1$.
Considering singular Sturm--Liouville differential expressions of the type [ tau_{alpha} = -(d/dx)x^{alpha}(d/dx) + q(x), quad x in (0,b), ; alpha in mathbb{R}, ] we employ some Sturm comparison-type results in the spirit of Kurss to derive criteria for $tau_{alpha}$ to be in the limit point and limit circle case at $x=0$. More precisely, if $alpha in mathbb{R}$ and for $0 < x$ sufficiently small, [ q(x) geq [(3/4)-(alpha/2)]x^{alpha-2}, ] or, if $alphain (-infty,2)$ and there exist $Ninmathbb{N}$, and $varepsilon>0$ such that for $0<x$ sufficiently small, begin{align*} &q(x)geq[(3/4)-(alpha/2)]x^{alpha-2} - (1/2) (2 - alpha) x^{alpha-2} sum_{j=1}^{N}prod_{ell=1}^{j}[ln_{ell}(x)]^{-1} &quadquadquad +[(3/4)+varepsilon] x^{alpha-2}[ln_{1}(x)]^{-2}. end{align*} then $tau_{alpha}$ is nonoscillatory and in the limit point case at $x=0$. Here iterated logarithms for $0 < x$ sufficiently small are of the form, [ ln_1(x) = |ln(x)| = ln(1/x), quad ln_{j+1}(x) = ln(ln_j(x)), quad j in mathbb{N}. ] Analogous results are derived for $tau_{alpha}$ to be in the limit circle case at $x=0$. We also discuss a multi-dimensional application to partial differential expressions of the type [ - Div |x|^{alpha} abla + q(|x|), quad alpha in mathbb{R}, ; x in B_n(0;R) backslash{0}, ] with $B_n(0;R)$ the open ball in $mathbb{R}^n$, $nin mathbb{N}$, $n geq 2$, centered at $x=0$ of radius $R in (0, infty)$.
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