Under a robust Bayesian framework, we study the problem of power estimation in binomial hypothesis testing problems. For testing the binomial mean, the observed powers are maximum likelihood estimates of power functions for one-sided and two-sided problems. With respect to these functions, we show that the observed powers are too large when the data is significant for some beta families of priors. A practical implication about the problem of sample size calculation is discussed.

Some results on the unique solvability and the regularity of solution of the second initial boundary value problem for strongly Schrödinger systems in finite and infinite cylinders with the bases containing conical points are given.

For a class of second order nonlinear differential equations with damping, efficient oscillation criteria are derived by refining the standard integral averaging technique. Examples are provided to illustrate the relevance of new theorems.

In this paper, we shall establish some inequalities related to the functions which are studied in $\left[ 2,5,7,13\right] $ and give several applications.

In the present paper, we give some applications of first order differential subordination and superordinations to obtain sufficient conditions for normalized analytic functions defined by certain linear operators to be subordinated and superordinated to a given univalent function.

We study the global periodicity of difference equations of the form $% x_{n+1}=f(x_n)$, $x_{n+1}=f_n(x_n)$ and $x_{n+2}=f(x_n,x_{n+1})$. We characterize the $n$-cycles %%%(all the solutions of the equation is periodic with period at most $n$) in the case of first order equations and give some partial results for the second order equation. In particular, we find some examples of $3$-cycles which are different from the equation $x_{n+2}=\frac c{x_n x_{n+1}}$, solving a question of [2] and [4].

In this paper, we prove the Hyers-Ulam-Rassias stability of homomorphisms in $C^*$-ternary rings and of derivations on $C^*$-ternary rings for the following generalized Cauchy-Jensen additive mapping: \begin{eqnarray*} 2f\left(\displaystyle\frac{\displaystyle\sum_{j=1}^{p}x_{j}+\sum_{j=1}^{q}y_{j}}{2}+\sum_{j=1}^{d}z_{j}\right) =\sum_{j=1}^{p}f(x_{j})+\sum_{j=1}^{q}f(y_{j})+2\sum_{j=1}^{d}f(z_{j}) . \end{eqnarray*} This is applied to investigate isomorphisms in $C^*$-ternary rings.

We use a property of the Bernardi operator in the theory of the Briot– Bouquet differential subordinations to prove several theorems for some classes of analytic functions defined by using the Dziok-Srivastava operator. Some of these results we obtain applying the convolution property due to Rusheweyh. We take advantage of the Miller–Mocanu differential subordinations.

Prime submodules and weakly prime submodules of Artinian modules are characterized. Furthermore, some previous results on prime modules are generalized.

This paper presents the existence of minimizers and $\Gamma$-convergence for the energey functionals \begin{eqnarray*} E_\epsilon(u) = \int_\Omega \left\{ W(u(x))+\epsilon{|\nabla u(x)|^p}\right\} dx, \mbox{ for all }\epsilon>0,\quad p>1 \end{eqnarray*} with Neumann boundary condition and the constraint \begin{eqnarray*} \int_\Omega u(x) dx = m|\Omega|, \mbox{ where }0 \lt m \lt 1. \end{eqnarray*} The energy functionals discussed in this paper are associated with the Euler-Lagrange $p$-Laplacian equation. We employ the direct method in the calculus of variations to show the existence of minimizers. The $\Gamma$-convergence is achieved with the help of coarea formula and Young's inequality.

Some necessary conditions are established for the nonoscillation of the following PDE with $p-$Laplacian \begin{equation*} \mbox{div}(\|\nabla y\|^{p-2}\nabla y)+c(x)|y|^{p-2}y=0. \end{equation*} Using these results, we obtain some oscillation criteria for the above equation.

Perodic solutions of discrete Rayleigh equations with variable delays can be related to steady state solutions of discrete time neural networks. By means of Mawhin’s continuation theorem, and sharp a priori estimates, existence of periodic solutions is established.

The initial boundary value problem for a Kirchhoff type plate equation in a bounded domain is considered. We prove the existence of global solutions by the similar arguments as in [11]. We derive the blow-up properties of solutions by energy method. Moreover, the estimates of the lifespan of solutions are also given.

Sufficient conditions for exact null controllability of the semilinear integrodifferential stochastic evolution equations in Hilbert space are obtained. It is shown that the exact null controllability of the corresponding linear system with additive term implies the exact null controllability of the semilinear functional integrodifferential stochastic evolution equations. An application to stochastic partial integrodifferential equations is given.