Volume 27, Issue 2

In this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation xn+1=a⁢xn⁢xn-kb⁢xn-c⁢xn-k-1,n=0,1,…,x_{n+1}=\frac{ax_{n}x_{n-k}}{bx_{n}-cx_{n-k-1}},\quad n=0,1,\ldots, where a,b,c{a,b,c} are positive real numbers and the initial conditions x-k-1,x-k,…,x-1,x0{x_{-k-1},x_{-k},\ldots,x_{-1},x_{0}} are real numbers. We show that when a=b=c{a=b=c}, the behavior of the solutions depends on whether k is even or odd.

The paper establishes the basic oscillational properties of differential equations of fractional order. It is also shown that the so-called fractional oscillational equation does not have the basic oscillational properties.

In this work, we give a unification and generalization of Laguerre and Hermite polynomials for which the orthogonal property is replaced by d -orthogonality. We state some properties of these new polynomials.

The main purpose of this paper is to introduce the concept of strongly ideal lacunary quasi-Cauchyness of sequences of real numbers. Strongly ideal lacunary ward continuity is also investigated. Interesting results are obtained.

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

The purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain Ωα⊂ℝ2{\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces ℍps⁢(Ωα){\mathbb{H}^{s}_{p}(\Omega_{\alpha})}, s>1p{s>\frac{1}{p}}, 1<p<∞{1<p<\infty}. The investigation is carried out using the potential method by reducing the problems to an equivalent boundary integral equation (BIE) in the Sobolev–Slobodečkii space on a semi-infinite axis 𝕎ps-1/p⁢(ℝ+){\mathbb{W}^{{s-1/p}}_{p}(\mathbb{R}^{+})}, which is of Mellin convolution type. Applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731], explicit conditions of the unique solvability of this BIE in the Sobolev–Slobodečkii 𝕎pr⁢(ℝ+){\mathbb{W}^{r}_{p}(\mathbb{R}^{+})} and Bessel potential ℍpr⁢(ℝ+){\mathbb{H}^{r}_{p}(\mathbb{R}^{+})} spaces for arbitrary r are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the non-classical setting. The same problem was investigated in a previous paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20 2013, 3, 439–467], but there were made fatal errors. In the present paper, we correct these results.

In this work, we investigate the spectrum denoted by Λ for the p⁢(x){p(x)}-biharmonic operator involving the Hardy term. We prove the existence of at least one non-decreasing sequence of positive eigenvalues of this problem such that sup⁡Λ=+∞{\sup\Lambda=+\infty}. Moreover, we prove that inf⁡Λ>0{\inf\Lambda>0} if and only if the domain Ω satisfies the p⁢(x){p(x)}-Hardy inequality.

The aim of this paper is to study a quasistatic frictional contact problem for viscoelastic materials with long-term memory. The contact boundary conditions are governed by Tresca’s law, involving a slip dependent coefficient of friction. We focus our attention on the weak solvability of the problem within the framework of variational inequalities. The existence of a solution is obtained under a smallness assumption on a normal stress prescribed on the contact surface and on the coefficient of friction. The proof is based on a time discretization method, compactness and lower semicontinuity arguments.

It is shown that any function acting from the real line ℝ{\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function x→exp⁡(x2){x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

The weighted boundedness properties of multilinear operators associated to singular integral operators with non-smooth kernels for extreme cases are obtained.

In this paper, we consider a new class of convex functions, called relative h-preivex functions. Seven new inequalities of Hermite–Hadamard type for relative h-preinvex functions are established using different approaches.

We present the conditions for a block matrix of a ring to have the image-kernel (p,q){(p,q)}-inverse in the generalized Banachiewicz–Schur form. We give representations for the image-kernel inverses of the sum and the product of two block matrices. Some characterizations of the image-kernel (p,q){(p,q)}-inverse in a ring with involution are investigated too.

The aim of this paper is to introduce generalized Suzuki type F-contraction fuzzy mappings and to prove the existence of fixed fuzzy points for such mappings in the setup of complete ordered metric spaces. An example is provided to show the validity of our results, followed by couple of remarks about the comparison of obtained results with the existing results in the literature. An application of our result to the domain of words is also presented.

In this paper, we investigate the absolute convergence of Fourier series of functions in several variables for an odd-dimensional space when these functions have continuous partial derivatives. It should be noted that similar properties for an even-dimensional space were given in [L. D. Gogoladze and V. S. Tsagareishvili, On absolute convergence of multiple Fourier series (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 2015, 9, 12–21; translation in Russian Math. (Iz. VUZ) 59 (2015), no. 9, 9–17]. The obtained results are the best possible in a certain sense.