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تسجيل مستخدم جديدThe Sturm--Liouville problem with singular potential and the extrema of the first eigenvalue
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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 $int_0^1 q dx=pm 1$.
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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
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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
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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 $$ -y+q_*y=lambda y, y(0)=y(1)=0 $$ i
s 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.
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} e
nd{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.
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