Abstract

The aim of this paper is to study the nonlinear eigenvalue problem \begin{align*} (P)\quad \begin{cases} \Delta (|\Delta u|^{p(x)-2}\Delta u)-\lambda \zeta(x)|u|^{\alpha(x)-2} u= \mu \xi(x) |u|^{\beta(x)-2}u, \quad x\in\Omega, \\ u\in W^{2,p(\cdot)}(\Omega)\cap W_0^{1,p(\cdot)}(\Omega), \end{cases} \end{align*} where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, with smooth boundary $\partial\Omega$, $N\geq 1$, $\lambda, \mu$ are real parameters, $\zeta$ and $\xi$ are nonnegative functions, $p, \alpha$, and $\beta$ are continuous functions on $ \overline{\Omega}$ such that $1< \alpha(x)< \beta(x)< p(x)<\frac{N}{2}.$ We show that the $p(\cdot)$-biharmonic operator possesses infinitely many eigengraph sequences and also prove that the principal eigengraph exists. Our analysis mainly relies on variational method and we prove Ljusternik-Schnirelemann theory on $C^1$-manifold.

Keywords: $p(\cdot)$-biharmonic operator; nonlinear eigenvalue problems; variational methods; Ljusternik-Schnirelmann theory.

MSC: 35A15, 35J35, 46E35

Abstract

Let $R$ be a commutative Noetherian ring and let $I$ be a semidualizing ideal of $R$. In this paper, it is shown that the $G_{I}$-projective, $G_{I}$-injective, and $G_{I}$-flat dimensions agree with $Gpd _{R\bowtie I}(-)$, $Gid _{R\bowtie I}(-)$, and $Gfd _{R\bowtie I}(-)$, respectively. Also, it is proved that for a non-negative integer $n$ if $\sup \{\mathcal{GP}_{I}-pd _{R}(M) \mid M\in \mathcal{M}(R) \}\leq n$ (or $\sup \{\mathcal{GI}_{I}-id _{R}(M) \mid M\in \mathcal{M}(R) \}\leq n)$, then for every projective $(R\bowtie I)$-module $P$ we have $id _{R\bowtie I}(P)\leq n$, and for every injective $(R\bowtie I)$-module $E$ we have $pd_{R\bowtie I}(E)\leq n$.

Keywords: Amalgamated duplication; semidualizing; $G_{C}$-projective dimension; $G_{C}$-injective dimension; $G_{C}$-flat dimension.

MSC: 13D05, 13H10

Abstract

Let $(X,\|.\|)$ be a normed linear space. Let $K$ be a nonempty closed convex subset of $X$. Let $T:K\to K$ be a contractive-like operator with a nonempty fixed point set $F(T)$. We prove the strong convergence and $T$-stability of Picard S-iteration procedure with respect to the contractive-like operator $T$ which are independent for any arbitrary choices of the sequences $\{\alpha_n\}_{n=0}^\infty$ and $\{\beta_n\}_{n=0}^\infty$ in $[0,1]$.

Keywords: Fixed point; contractive-like operator; Picard S-iteration procedure; $T$-stability.

MSC: 47H10, 54H25

Abstract

This paper develops an Implicit-Explicit method (IMEX) of third order for solving stiff system of ordinary differential equations (ODEs). The method is $L$-stable with respect to the implicit part and allows the use of an arbitrary approximation of the Jacobian matrix. Order and stability conditions are derived and then solved analytically. Automatic stepsize selection based on local error estimation and stability control is made. The estimations for local error and stability control are obtained without significant additional computational cost. The results of numerical experiments confirm the reliability and efficiency of the implemented integration algorithm.

Keywords: Stiff systems of ODEs; implicit-explicit methods; L-stability, local error estimation and stability control; embedded methods.

MSC: 65L04, 65L05, 65L20

Abstract

In this paper, we analyze the Dirichlet problem for an elliptic system on a conical domain. We essentially apply the results from the theory of sums of operators to achieve our main results. We work in the setting of little Hölder spaces.

Keywords: Elliptic systems of second order; linear operators; sums of operators theory; little Hölder spaces.

MSC: 47D06, 47D99

Abstract

In this article, we establish several infinite families of Ramanujan-type congruences modulo 16, 32 and 64 for $\overline{p}_o(n),$ the number of overpartitions of $n$ in which only odd parts are used.

Keywords: Partition; overpartition; congruence.

MSC: 05A17, 11P83