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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.
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
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 o
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
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} bound
The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different