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For 0 < p < ∞ , we completely characterize the Schatten p -class embedding and fractional embedding of the Bergman-type space A α 2 on the unit ball of C m. The main results of this paper are twofold: (1) For α , β ∈ R , we show that the necessary and sufficient condition for the embedding operator I : A α 2 → A β 2 is in the Schatten p -class is that p (β − α) − 2 m > 0. (2) For a positive Borel measure μ on B m and a real number τ , we prove that the fractional embedding operator I α + τ τ : A α 2 → L 2 (B m , d μ) is in the Schatten p -class if and only if the Berezin transform of μ , denoted by μ ˜ α α + τ , is in L p 2 (B m , d λ) , where d λ (z) = (1 − | z | 2) − m − 1 d v (z) is the Möbius invariant volume measure and d v is the Lebesgue measure on B m , or equivalently, the averaging function of μ , denoted by μ ˆ r α , is in L p 2 (B m , d λ). • For p > 0 , we completely characterize the Schatten p -class embedding and fractional embedding of the Bergman-type space on the unit ball. • The range 0 < p < 2 is an open problem left by H. T. Kaptanoğlu (see [5]). • We prove that the fractional embedding operator is in Schatten p -class iff the Berezin transform of the positive Borel measure is in L − space. [ABSTRACT FROM AUTHOR]

We study limit points of the spectral radii of Laplacian matrices of graphs. We adapted the method used by J. B. Shearer in 1989, devised to prove the density of adjacency limit points of caterpillars, to Laplacian limit points. We show that this fails, in the sense that there is an interval for which the method produces no limit points. Then we generalize the method to Laplacian limit points of linear trees and prove that it generates a larger set of limit points. The results of this manuscript may provide important tools for proving the density of Laplacian limit points in [ 4.38 + , ∞). • We investigate the Hoffman concept of limit points for the Laplacian matrix of a graph. The problem is to determine which real numbers are limit points of the spectral radius of these matrices. • We make progress towards the proof that any real number larger than 4.38+ is such a limit point. • We adapt Shearer's method, used to solve the problem for the adjacency matrix, extending it to linear trees. • We determine larger sets of limit points and provide technical tools to prove the density of Laplacian limit points in [ 4.38 + , ∞). [ABSTRACT FROM AUTHOR]

• Obtain a sharp upper bound for the maximum signless Laplacian spectral radius of a linear-forest-free graphs. • Characterize all extremal graphs which have attained the upper bound for for the maximum signless Laplacian spectral radius. • The main results extends and strengthes several known results and techniques. Let A (G) and D (G) be the adjacency and degree matrices of a simple graph G on n vertices, respectively. The A α -spectral radius of G is the largest eigenvalue of A α (G) = α D (G) + (1 − α) A (G) for a real number α ∈ [ 0 , 1 ]. In this paper, for α ∈ (0 , 1) , we obtain a sharp upper bound for the A α -spectral radius of graphs on n vertices without a subgraph isomorphic to a linear forest for n large enough and characterize all graphs which attain the upper bound. As a result, we completely dertermine the maximum signless Laplacian spectral radius of graphs on n vertices without a subgraph isomorphic to a linear forest for n large enough. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a molecular graph, in which the vertices of V represent atoms and the edges of E the bonds between pairs of atoms. One of the earliest and most widely studied degree-based molecular descriptor, the Randić index of G , is defined as R ( G ) = ∑ u v ∈ E ( d u d v ) − 1 2 , where d u denotes the degree of u  ∈  V . Bollobás and Erdős (1998) generalized this index by replacing − 1 2 with any fixed real number. To facilitate the enumeration of these indices, motivated by earlier work of Dvořák et al. (2011) and Knor et al. (2015) introduced two other indices, that provide lower and upper bounds, respectively: R α ′ ( G ) = ∑ u v ∈ E min { d u α , d v α } and R α ″ ( G ) = ∑ u v ∈ E max { d u α , d v α } . In this paper, we give expressions for computing R α ′ and R α ″ of benzenoid systems and phenylenes, as well as a relation between R α ′ and R α ″ of a phenylene and its corresponding hexagonal squeeze. We also determine the extremal values of R α ′ and R α ″ in benzenoid systems with h hexagons for different intervals for the value of α . [ABSTRACT FROM AUTHOR]

The present paper deals with Hyers–Ulam stability of the first-order linear difference equation Δ h x ( t ) − a x ( t ) = f ( t ) on h Z , where Δ h x ( t ) = ( x ( t + h ) − x ( t ) ) / h and h Z = { h k | k ∈ Z } for the constant stepsize h  > 0; a is a real number; f ( t ) is a real-valued function on h Z . The main purpose of this paper is to find the best HUS constant on h Z . Several relationships between solutions of two different perturbed difference equations are also given. [ABSTRACT FROM AUTHOR]

• Given an explicit expression for s α (G) in terms of resistance distances Ω i j , i , j ∈ V. • Generalize the following well-known equality n s − 1 (G) = Kf (G) to any integer k ≥ − 1 , where Kf (G) = Σ i < j Ω i j is the Kirchhoff index of G. • Express the first Zagreb index and the Laplacian Estrada index in terms of resistance distances. Let G = (V , E) be a simple connected graph with vertex set V = { 1 , 2 , ... , n }. For any real number α , the topological index s α (G) of G is defined as s α (G) = ∑ i = 1 n − 1 μ i α , where μ 1 ≥ μ 2 ≥ ... μ n − 1 > μ n = 0 are the Laplacian eigenvalues of G. In this paper, we first express s α (G) explicitly in terms of resistance distances Ω i j , i , j ∈ V. Then we generalize the following well-known equality n s − 1 (G) = Kf (G) to any integer k ≥ − 1 , where Kf (G) = ∑ i < j Ω i j is the Kirchhoff index of G. As by-products, we get the expressions for the first Zagreb index and the Laplacian Estrada index in terms of the resistance distances. [ABSTRACT FROM AUTHOR]

We investigate inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type whose spectrum consists of a finite set of eigenvalues. For given two finite sets of interlacing real numbers, there exists a class of Sturm–Liouville equations such that the two sets of numbers are exactly the eigenvalues of their associated Sturm–Liouville problems with two different separated boundary conditions. The main approach is to give an equivalent relation between Sturm–Liouville problems of Atkinson type and matrix eigenvalue problems, and the theory of inverse matrix eigenvalue problems. [ABSTRACT FROM AUTHOR]

A k -labeling of a graph G is an injective function ϕ from E ( G ) to m + k real numbers, where m = | E ( G ) | . Let μ G ( v ) = ∑ u v ∈ E ( G ) ϕ ( u v ) . A graph is called antimagic if G admits a 0-labeling with labels in { 1 , 2 , … , | E ( G ) | } such that μ G ( u ) ≠  μ G ( v ) for any pair u, v  ∈  V ( G ). A well-known conjecture of Hartsfield and Ringel states that every connected graph other than K 2 admits an antimagic labeling. Recently, two sets of authors Arumugam, Premalatha, Băca, Semanĭcová-Fen̆ov̆cíková, and Bensmail, Senhaji, Lyngsie independently introduced the weaker notion of a local antimagic labeling, which only distinguishes adjacent vertices by sum with labels in { 1 , 2 , … , | E ( G ) | } . Both sets of authors conjecture that any connected graph other than K 2 admits a local antimagic labeling. In this paper, we prove that every subcubic graph without isolated edges admits a local antimagic labeling with | E ( G )| positive real labels. We also prove that each graph G without isolated edges admits a local antimagic k -labeling, where k = min { Δ ( G ) + 1 , 3 c o l ( G ) + 3 2 } , and col ( G ) is the coloring number of G . Actually, the latter result holds for the list version. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the bi-periodic Lucas matrix sequence and present some fundamental properties of this generalized matrix sequence. Moreover, we investigate the important relationships between the bi-periodic Fibonacci and Lucas matrix sequences. We express that some behaviors of bi-periodic Lucas numbers also can be obtained by considering properties of this new matrix sequence. Finally, we say that the matrix sequences as Lucas, k -Lucas and Pell–Lucas are special cases of this generalized matrix sequence. [ABSTRACT FROM AUTHOR]

The energy of a graph is defined as the sum of the absolute values of its eigenvalues. In 1940 Coulson obtained an important integral formula which makes it possible to calculate the energy of a graph without knowing its spectrum. Recently several Coulson-type integral formulas have been obtained for various energies and some other invariants of graphs based on eigenvalues. For a complex polynomial ϕ ( z ) = ∑ k = 0 n a k z n − k = a 0 ∏ k = 1 n ( z − z k ) of degree n and a real number α , the general energy of ϕ ( z ), denoted by E α ( ϕ ), is defined as ∑ z k ≠ 0 | z k | α when there exists k 0 ∈ { 1 , 2 , … , n } such that z k 0 ≠ 0 , and 0 when z 1 = ⋯ = z n = 0 . In this paper we give Coulson-type integral formulas for the general energy of polynomials whose roots are all real numbers in the case that α ∈ Q . As a consequence of this result, we obtain an integral formula for the 2 l -th spectral moment of a graph. Furthermore, we show that our formulas hold when α is an irrational number with 0 < | α | < 2 and do not hold with | α | > 2. [ABSTRACT FROM AUTHOR]

The irregularity of a simple graph G = ( V , E ) is defined as irr ( G ) = ∑ u v ∈ E ( G ) | d G ( u ) − d G ( v ) | , where d G ( u ) denotes the degree of a vertex u  ∈  V ( G ). This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. Recently, it also gains interest in Chemical Graph Theory, where it is named the third Zagreb index. In this paper, by means of the Laplacian eigenvalues and the normalized Laplacian eigenvalues of G , we establish some new spectral upper bounds for irr( G ). We then compare these new bounds with a known bound by Goldberg, and it turns out that our bounds are better than the Goldberg bound in most cases. We also present two spectral lower bounds on irr( G ). [ABSTRACT FROM AUTHOR]

For a real number β , the general sum-connectivity index χ β (G) of a graph G is defined as ∑ x y ∈ E (G) (d (x) + d (y)) β , where d (x) denote the degree of a vertex x in G. In this paper, we show that for every graph G ≇ P n , if β ≥ 0 , then χ β (L (G)) ≥ { χ β (G) f o r δ (G) ≤ 2 , 2 χ β (G) f o r δ (G) ≥ 3. In addition, for β < 0 , we present the lower bound for χ β (L (G)) in terms of χ β (G). Finally, we establish sharp bounds for χ β (G) + χ β (L (G)) , and characterize the extremal graphs attaining the bounds. [ABSTRACT FROM AUTHOR]

The variable sum exdeg index of a graph G is defined as S E I a ( G ) = ∑ u v ∈ E ( G ) [ a d e g G ( u ) + a d e g G ( v ) ] , where a ≠ 1 is a positive real number. The aim of this paper is applying the majorization technique to obtain the maximum and minimum values of variable sum exdeg index of trees, unicyclic, bicyclic and tricyclic graphs. [ABSTRACT FROM AUTHOR]

Let R and H m × n stand, respectively, for the real number field and the set of all m × n matrices over the real quaternion algebra H = { a 0 + a 1 i + a 2 j + a 3 k | i 2 = j 2 = k 2 = ijk = − 1 , a 0 , a 1 , a 2 , a 3 ∈ R } . For η ∈ { i, j, k }, a real quaternion matrix A ∈ H n × n is said to be η -Hermitian if A η * = A where A η * = − η A * η , and A * stands for the conjugate transpose of A , arising in widely linear modeling. We present a simultaneous decomposition for a set of nine real quaternion matrices involving η -Hermicity with compatible sizes: A i ∈ H p i × t i , B i ∈ H p i × t i + 1 , and C i ∈ H p i × p i , where C i are η -Hermitian matrices, ( i = 1 , 2 , 3 ) . As applications of the simultaneous decomposition, we give necessary and sufficient conditions for the existence of an η -Hermitian solution to the system of coupled real quaternion matrix equations A i X i A i η * + B i X i + 1 B i η * = C i , ( i = 1 , 2 , 3 ) , and provide an expression of the general η -Hermitian solutions to this system. Moreover, we derive the rank bounds of the general η -Hermitian solutions to the above-mentioned system using ranks of the given matrices A i , B i , and C i as well as the block matrices formed by them. Finally some numerical examples are given to illustrate the results of this paper. [ABSTRACT FROM AUTHOR]

A general inequality for non-negative real numbers is proven. Based on it, upper bounds for (ordinary) graph energy, minimum dominating energy, minimum covering energy, Laplacian-energy-like invariant, Laplacian energy, Randić energy, and incidence energy are obtained. [ABSTRACT FROM AUTHOR]

• Different from Podlubny [1], the power-like properties of fractional derivatives at zero are studied in a more general case. • Different from Ross [2], two kinds of estimators are designed in this work for higher-order systems. Besides, the corresponding estimation process can be simplified according to the power-like properties. • Different from Monje et al. [3], Soczkiewicz [4], the estimators designed in this work are suitable for not only the position and velocity but also for the fractional derivatives and integrals of the position. • Different from Hilfer [5], Torvik and Bagley [6], the acceleration is considered as the available output. In addition, the orders of fractional differentiation of the considered systems can be arbitrary real numbers rather than the rational and commensurate orders considered in [5,6]. In this paper, non-asymptotic and robust estimation for fractional integrals of noisy acceleration is considered for a class of fractional linear systems based on the study of the behaviour of fractional derivatives at zero. Particularly, the position and velocity can be estimated from the noisy acceleration via the designed estimators. Such estimates can be of value for health monitoring, reliability assessment, and vibration control of mechanical systems. Based on the existing numerical approaches addressing the proper fractional integrals, our attention is primarily restricted to estimating the unknown initial values. For completing the estimation, two methods based on classical modulating functions and generalized modulating functions are proposed, respectively. Via the results on fractional derivatives at zero, the initial values of the considered systems are discussed, and the corresponding estimator simplifications are performed. In addition, constructions of the required modulating functions are discussed. Finally, some numerical simulations are provided, which demonstrate the validity of the theoretical results and the efficiency of the proposed estimators. [ABSTRACT FROM AUTHOR]

In extremal graph theory studies, it is not yet finished to determine the exact value of ex (n ; C 4) in general case. Erdős and Simonovits [10] conjectured that an n -vertex graph G with ex (n ; C 4) + 1 edges may contain at least n + o (n) 4-cycles. In this paper we investigate the number of 4-cycles in a graph. We show that the number of 4-cycles in an n -vertex graph G is at least (ϵ 4 n − 3) / 4 + ϵ 2 / (2 n) provided that | E (G) | = n (1 + 4 n − 3) / 4 + ϵ , where ϵ is an arbitrary positive real number. This result is an approach solving to this open problem. Furthermore, we prove that if n is large enough, then there are at least ((4 ϵ − 1) 4 n − 3 + 5) / 16 + o (1) 4-cycles in an n -vertex graph with e = (n − 1) (1 + 4 n − 3) / 4 + ϵ edges which has k vertices whose degrees are mutually different, where k ≥ 24 n 3 + 12 n − (12 n) 2 − 1 / 27 3 and ϵ > 0 is a real number. In addition, we get that a triangle-free graph with n vertices and e = n n − 1 / 2 + ϵ edges has at least (2 ϵ (n − 1 − 2) + 1) / 4 + o (1) 4-cycles, where ϵ > 0 is an any real number and n is large enough. This shows that the weak version of Erdős and Simonovits' conjecture [10] holds for such graphs when n is sufficiently large. Finally, we obtain a bipartite version of this result. [ABSTRACT FROM AUTHOR]

In this study, we establish generalized Grüss type inequality and some generalized Čebyšev type inequalities for local fractional integrals on fractal sets R α (0 < α ≤ 1) of real line numbers. [ABSTRACT FROM AUTHOR]

The general connectivity index R α ( G ) of a graph G = ( V , E ) is defined as R α ( G ) = ∑ ( u , v ) ∈ E ( G ) ( d u d v ) α , where ( u, v ) is an edge of G, d u and d v denote the degrees of the vertices u and v , respectively, and α ≠ 0 is an arbitrary real number. In this paper, we give the expressions for computing the general connectivity indices of fluoranthene-type benzenoid systems, and we determine the extremal values of R α ( G ) in f-benzenoid systems with h hexagons for some real number α . Especially, the extremal values of R α ( G ) in cata-catacondensed fluoranthene-type benzenoid systems with h hexagons were studied. [ABSTRACT FROM AUTHOR]

We show that the following system of difference equations x n = x n − 1 y n − 2 a y n − 2 + b y n − 1 , y n = y n − 1 x n − 2 c x n − 2 + d x n − 1 , n ∈ N 0 , where parameters a, b, c, d , as well as initial values x − 2 , x − 1 , y − 2 , y − 1 , are real numbers, is solvable in closed form and by using obtained formulas we give some results on the long term behavior of their solutions. We also give natural explanations and considerably extend some recent results in the literature. [ABSTRACT FROM AUTHOR]

• We determine an upper bound for the quantity of real numbers in a well spaced sequence making the Diophantine approximation over two squares, two cubes and two biquadrates of primes unsolvable. • We prove that under some specific conditions, the values of real linear combinations of two squares, two cubes and two biquadrates of primes can approximate almost all real numbers in a well spaced sequence. • Our results extend some recent publications. Assume non-zero real numbers κ 1 , ... , κ 6 are not all negative. Assume also κ 1 κ 2 is algebraic and not rational, the sequence G is well-spaced and ϑ > 0. In this work, we showed that the quantity of γ ∈ G satisfying 1 ≤ γ ≤ X and making the Diophantine inequality | κ 1 p 1 2 + κ 2 p 2 2 + κ 3 p 3 3 + κ 4 p 4 3 + κ 5 p 5 4 + κ 6 p 6 4 − γ | < γ − ϑ unsolvable in primes p 1 , ... , p 6 is not more than O (X 6 7 + 2 ϑ + ϵ) for any ϵ > 0. [ABSTRACT FROM AUTHOR]

• The Markov jumps QVNNs is split into four RVNNs, which can be analysed by the analysis method in real number field. • The event-triggered mechanism introduced into QVNNs, the stochastic stabilization of Markovian jump QVNNs is explored for the first time. • In order that reducing the conservatism, by constructing a novel augmented LKF containing more hybrid features with respect to the practical sampling pattern with combining some inequality techniques and convex combination optimal skill, a new stabilization condition is derived in the form of LMIs. This makes it easier to verify the results with MATLAB. This paper investigates, through designing an event-triggered sampled-data mechanism, the stochastic stabilization of Markovian jump quaternion-valued neural networks (QVNNs) with partially unknown transition probabilities. First, since the multiplication of quaternions is not commutative, the Markovian jump QVNNs are decomposed into four real-valued neural networks. Then, by constructing a novel augmented Lyapunov–Krasovskii functional (LKF), it is containing more hybrid features, in which concerning the practical sampling pattern, and using the convex combinatorial optimization and inequality technique, the stabilization condition is proposed in the form of linear matrix inequalities (LMIs). Furthermore, the desired event-triggered sampled-data controller gains are obtained for the closed-loop Markovian jump QVNNs with partially unknown transition probability. Finally, the viability and validness of the obtained results are confirmed via three numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the authors establish some new estimates for the remainder term of the midpoint, trapezoid, and Simpson formula using functions whose derivatives in absolute value at certain power are s -convex. Some applications to special means of real numbers are provided as well. [ABSTRACT FROM AUTHOR]

The paper discusses necessary and sufficient conditions for the asymptotic stability of the zero solution of the linear delay difference equation y (n+1)=αy(n)+βy(n-k), where α,β are complex numbers and k is a positive integer. Compared to the case when α,β are real numbers, the stability behavior of this equation turns out to be much richer. In particular, if ∣α∣+∣β∣>1 then, as k monotonously increases, the equation may switch finite times from asymptotic stability to instability and vice versa. We describe an interesting structure of the set of these stability switches, their explicit values and apply the obtained results to some important delay difference equations and their systems. [ABSTRACT FROM AUTHOR]

Let G be a graph with n vertices. A vertex-degree-based topological index is defined from a set of real numbers {ψij} as TI (G) = Σ1⩽i⩽j⩽n-1mijψij where mij is the number of edges of G connecting a vertex of degree i with a vertex of degree j . Many of the well-known topological indices are particular cases of this expression, including all Randic-type connectivity indices. In this work we determine extremal values for TI over the set HSn of hexagonal systems with a fixed number of vertices. [ABSTRACT FROM AUTHOR]

The solution to the third-kind Volterra integral equation (VIE3) usually has unbounded derivatives near the original point t = 0 , which brings difficulties to numerical computation. In this paper, we analyze two kinds of modified multistep collocation methods for VIE3: collocation boundary value method with the fractional interpolation (FCBVM) and that with Lagrange interpolation (CBVMG). The former is developed based on the non-polynomial interpolation which is particularly feasible for approximating functions in the form of t η with the real number η ≥ 0. The latter is devised by using classical polynomial interpolation. The application of the boundary value technique enables both approaches to efficiently solve long-time integration problems. Moreover, we investigate the convergence properties of these two kinds of algorithms by Grönwall's inequality. [ABSTRACT FROM AUTHOR]

Abstract: We give some sufficient conditions under which the zero solution of the next functional-differential equation where are real numbers, is a continuous function such that , and , is asymptotically stable. [Copyright &y& Elsevier]

Abstract: We investigate in this paper the perturbed Lyness difference equation , where are arbitrary positive real numbers and and the initial values , which is a generalization of the Lyness difference equation extensively studied. It is known that for the Lyness difference equation, i.e., the perturbed Lyness difference equation with , all its solutions are periodic or strictly oscillatory. However, one here finds that this perturbed Lyness difference equation possesses the following dichotomy: for , all of its solutions are globally asymptotically stable; for , all the sequences generated by it converge to . We hence find that there exists the essential difference for the properties of solutions between the unperturbed Lyness difference equation and the perturbed Lyness difference equation. [Copyright &y& Elsevier]

Abstract: In this paper, wedefine a new generalized difference sequence space and consider it equipped with the Luxemburg norm under which it is a Banach space and we show that the space possess Banach Saks property of type p, uniform opial property and property , where is a bounded sequence of positive real numbers with for all . Also, we give some results about the fixed point theory for the spaces and . [Copyright &y& Elsevier]

Abstract: Let k be a positive integer and be non-negative real numbers with . We show that if then the rational Riccati difference equation of order k has a unique positive equilibrium point that is stable and attracts all solutions with initial points outside a set of zero Lebesgue measure. This holds in particular if . The case is studied in detail. [Copyright &y& Elsevier]

Abstract: This paper investigates some connections between individual values, such as the Shapley value and the Banzhaf value, and coalition values, such as blockability value, viability value and profitability value, for games in characteristic function form. It turns out that the Shapley value, the Banzhaf value, blockability value, viability value and profitability value assign the same real number to each coalition where the characteristic function is superadditive and inessential. It is also clarified that there is equalization of the Banzhaf value and profitability value of each player in constant-sum game. A new class of games in characteristic function form, which is called equally average coalition solidarity, is proposed in this paper. A proposition which shows that every player’s real number evaluated by the Shapley value corresponds with the real number evaluated by the Banzhaf value of the player in the proposed class of games in characteristic function form is provided. [Copyright &y& Elsevier]

Abstract: The asymptotic behavior of the solutions to a neutral difference equation of the form where is a general retarded argument, is a general deviated argument, and are sequences of real numbers, denotes the forward difference operator , and denotes the jth forward difference operator for , is studied. Examples illustrating the results are also given. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the boundedness character, the periodic character and the global behavior of positive solutions of the difference equation where and is a two periodic sequence of nonnegative real numbers and the initial conditions are arbitrary positive real numbers. [Copyright &y& Elsevier]

Abstract: The paper concerns the differential subordination with the expression combined by arithmetic and geometric means: where and α are real numbers such that , , . For , , we also study the differential subordination Several applications of the studied subordination in the theory of analytic functions are given. [Copyright &y& Elsevier]

Abstract: Relationship between the boundedness and the existence of a finite limit at the infinity of solutions of the linear functional equationwhere is a continuous increasing function satisfying the condition , and f is a continuous function such that there is a finite limit , is studied here. By using obtained results we study behaviour of bounded solutions of the functional equationwhere are real numbers such that , functions and f satisfy above mentioned conditions and . [Copyright &y& Elsevier]

Abstract: Let u and v denote respectively the principal and nonprincipal solutions of the second-order linear equation defined on some half-line of the form . It is shown that for any given real numbers a and b the nonlinear equationhas a solution asymptotic to as under some reasonable conditions on the function f. Examples are given to illustrate the results. [Copyright &y& Elsevier]

Abstract: The system of difference equations , where the sequences , and initial values are real numbers, such that , is solved in closed form, and based on the obtained formulae is studied the asymptotic behaviour of well-defined solutions of the system. [Copyright &y& Elsevier]

Abstract: We show that the system of difference equationswhere the sequences , , , , , , , and initial values , , are real numbers, such that , , , can be solved in closed form, and for the case when all the sequences , , , , and are constant we describe the asymptotic behavior of well-defined solutions of the system. [Copyright &y& Elsevier]

Abstract: We show that the system of difference equationswhere , the sequences , , and initial values are real numbers can be solved in closed form. By using obtained formulae we investigate asymptotic behavior of well-defined solutions of the system for the case when all the sequences , and are constant. [Copyright &y& Elsevier]

Abstract: Using a three-critical-point theorem due to Arcoya and Carmona, we show that, under suitable conditions and for sufficiently large real parameters , the following nonlinear elliptic equation defined on the Sierpinski gasket and with zero Dirichlet boundary conditionshas at least two nonzero weak solutions. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to determine the spectrum and the fine spectrum of the rth difference operator on sequence space . The operator over is defined by with for , where and is either constant or strictly decreasing sequence of positive real numbers satisfying certain conditions. [Copyright &y& Elsevier]

Abstract: Given a sequence of real numbers, we find necessary and sufficient conditions for the existence and uniqueness of a distribution function on , such thatHere are the Chebyshev polynomials of the first kind. [Copyright &y& Elsevier]

Abstract: The continued fraction expansion of a real number may be studied by considering a suitable discrete dynamical system of dimension two. In the special case where the number to be expanded is a quadratic irrational, that is a positive irrational root of a polynomial of degree two, more insight may be gained by considering a new dynamical system of dimension three, where the state vector stores the coefficients of the quadratic polynomials resulting from the expansion process. We show that a number of constants of motions can be derived and exploited to explore the attracting set of the solutions. Links with the solution to Pell’s equations are also investigated. [Copyright &y& Elsevier]

Abstract: In the early years of set theory, Du Bois Reymond introduced a vague notion of infinitary pantachie meant to symbolize an infinity bigger than the infinity of real numbers. Hausdorff reformulated this concept rigorously as a maximal chain (a linearly ordered subset) in a partially ordered set of certain type, for instance, the set under eventual domination. Hausdorff proved the existence of a pantachy in any partially ordered set, using the axiom of choice AC. We show in this note that the pantachy existence theorem fails in the absense of AC, and moreover, even if AC is assumed, hence pantachies do exist, one may not be able to come up with an individual, effectively defined example of a pantachy. [Copyright &y& Elsevier]

Abstract: We show that the system of difference equationswhere the parameters , and initial values , are real numbers, can be solved, developing further the results in the literature. [Copyright &y& Elsevier]

Abstract: We describe the behavior of well-defined solutions of the difference equationwhere , parameters b, c and the initial values x −k ,…, x −1 are real numbers. These results, among others, improve and extend our results in [S. Stević, More on a rational recurrence relation, Appl. Math. E-Notes 4 (2004) 80–85]. [Copyright &y& Elsevier]

Abstract: We prove that the difference equationwhere and are sequences periodic with period two and the initial values x −2, x −1 are real numbers, can be solved explicitly. Some applications of obtained formulae are given. [Copyright &y& Elsevier]

Abstract: We show that the system of difference equationswhere the sequences a n , b n , c n , α n , β n , γ n are two-periodic and initial values x −1, x 0, y −1, y 0 are real numbers, can be solved. [Copyright &y& Elsevier]

Abstract: We show that the system of difference equationswhere the parameters a, b, c, α, β, γ and initial values x −1, x 0, y −1, y 0 are real numbers, can be solved, considerably improving the results in the literature. [Copyright &y& Elsevier]

Abstract: This paper presents a new class of functions f(z) analytic and normalized in the open unit disc (which is related to a shell-like curve and associated with Fibonacci numbers) satisfying the condition thatwhere α is a real number andThe class being closely related to the classes of starlike and convex functions, we apply some basic techniques to investigate certain interesting properties (given below) for this class of functions. Some important observations of the main results are also mentioned. [Copyright &y& Elsevier]