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This paper is devoted to studying a *q*-analogue of Sturm-Liouville operators. We formulate a dissipative *q*-difference operator in a Hilbert space. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which is based on the method of Pavlov and define its characteristic function. Finally, we prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative *q*-Sturm-Liouville difference operator.

Two-weight norm estimates for sublinear integral operators involving Hardy-Littlewood maximal, Calderón-Zygmund and fractional integral operators in variable exponent Lebesgue spaces are derived. Operators and the space are defined on a quasi-metric measure space with doubling condition. The derived conditions are written in terms of *L*
^{p(·)} norms and are simultaneously necessary and sufficient for appropriate inequalities for maximal and fractional integral operators mainly in the case when weights are of radial type.

For a large class of arithmetic functions *f*, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities *f*(*n* + 1) < *f*(*n* + 2) < … < *f*(*n* + κ) holds for in-finitely many positive integers *n*. For other arithmetic functions *f*, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of *n* satisfying σ_{2}(*n* − 1) < σ_{2} < σ_{2}(*n* + 1), where σ_{2}(*n*) = ∑_{d|n}
*d*
^{2}. On the other hand, we prove that for the function *f*(*n*) := ∑_{p|n}
*p*
^{2}, we do have *f*(*n* − 1) < *f*(*n*) < *f*(*n* + 1) in finitely often.

In this paper, a fourth-order nonlinear difference equation is considered. By making use of the critical point theory, we establish various sets of sufficient conditions for the existence and nonexistence of solutions for Neumann boundary value problem and give some new results. Results obtained generalize and complement the existing ones.

^{N}with smooth boundary, λ,

*μ*> 0,

*p*,

*q*,

*r*1,

*r*2,

*α*and

*β*are continuous functions on

*μ*> 0, there exists λ

_{*}sufficiently small, and λ* large enough such that for any λ ∈ (0; λ

_{*}) ∪ (λ*, ∞), the above system has a nontrivial weak solution. The proof relies on some variational arguments based on the Ekeland’s variational principle and some adequate variational methods.

In order to give an excellent description of income distributions, although a large number of functional forms have been proposed, but the four-parameter generalized beta model of the second kind (GB2), introduced by J. B. McDonald [18], is now widely acknowledged which is including many other models as special or limiting cases.One of the fundamentals of statistical inference is the estimation problem of a function of unknown parameter in a probability distribution and computing the variance of the estimator or approximating it by lower bounds.In this paper, we consider two famous lower bounds for the variance of any unbiased estimator, which are Bhattacharyya and Kshirsagar bounds. We obtain the general forms of the Bhattacharyya and Kshirsagar matrices in the GB2 distribution. In addition, we compare different Bhattacharyya and Kshirsagar bounds for the variance of any unbiased estimator of some parametric functions such as mode, mean, skewness and kurtosis in GB2 distribution and conclude that in each case, which bound is better to use. The results of this paper can be useful for researchers trying to find the accuracy of the estimators.

More than two centuries ago Malfatti (see [9]) raised and solved the following problem (the so-called Malfatti’s construction problem): Construct three circles into a triangle so that each of them touches the two others from outside moreover touches two sides of the triangle too. It is an interesting fact that nobody investigated this problem on the hyperbolic plane, while the case of the sphere was solved simultaneously with the Euclidean case. In order to compensate this shortage we solve the following exercise: *Determine three cycles of the hyperbolic plane so that each of them touches the two others moreover touches two of three given cycles of the hyperbolic plane*.

Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, *J. Pure Appl. Algebra*, **168** (2002), 45–52] studied relations between the set of annihilators in a ring *R* and the set of annihilators in a polynomial extension *R*[*x*] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring *R* and a monoid *M* such that the monoid ring *R*[*M*] is quasi-Armendariz. As a consequence we show that if *R* is a right APP-ring, then *R*[*x*]=(*x*
^{n}) and hence the trivial extension *T*(*R*,*R*) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.

We introduce Kurosh elements in division rings based on the idea of a conjecture of Kurosh. Using this, we generalize a result of Faith in {xc[3]} and of Herstein in {xc[6]}.

This paper concerns the existence of mild solutions for some fractional Cauchy problem with nonlocal conditions in the α-norm. The linear part of the equations is assumed to generate an analytic compact bounded semigroup, and the nonlinear part satisfies some Lipschitz conditions with respect to the fractional power norm of the linear part. By using a fixed point theorem of Sadovskii, we establish some existence results which generalize ones in the case of fractional order derivative.