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In this paper, some sufficient conditions for the oscillation of all solutions of second order dynamic equations with a negative sub-linear neutral term are established. The obtained results provide a unified platform that adequately covers both discrete and continuous equations. Furthermore, it covers a wide range of equations by utilizing different time scales. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]

In this research paper, we investigate some new identifies for Sarıkaya fractional integrals which introduced by Sarıkaya and Ertugral in [20]. The fractional integral operators also have been applied to Hermite-Hadamard type integral inequalities to provide their generalized properties. Furthermore, as special cases of our main results, we present several known inequalities such as Simpson, Bullen, trapezoid for convex functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new function class called n-fractional polynomial convex function and their some algebric properties. We obtain some refinements of the right-hand side of Hermite-Hadamard inequality for the class of functions whose derivatives in absolutely value at certain powers are n-fractional polynomial convex. Also, we compare the results obtained with both Hölder, Hölder-İşcan inequalities and power-mean, improved-power-mean integral inequalities and show that the result obtained with Hölder-İşcan and improved power-mean inequalities give better approach than the others. Some applications to special means of real numbers are also given. [ABSTRACT FROM AUTHOR]

In this paper, we find all repdigits which can be expressed as the sum of three Half- companion Pell numbers. To prove our main result, we use the combined approach of lower bounds for linear forms in logarithms of algebraic numbers and a version of the Baker Davenport reduction method. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of entropy solutions for some generalized elliptic p(u)-Laplacian problem when p(u) is a local quantity. We get the results by assuming the right-hand side function f to be an integrable function, and by using the regularization approach combined with the theory of Sobolev spaces with variable exponents. [ABSTRACT FROM AUTHOR]

Let R be a prime ring, I be a nonzero ideal of R, Q be its maximal right ring of quotients and C be its extended centroid. The aim of this paper is to show that if R admits a nonzero b-generalized derivation F such that [F (xm)xn + xnF (xm), xr]k = 0 for all x ∈ I, where m, n, r, k are fixed positive integers, then there exists λ ∈ C such that F (x) = λx unless R ∼= M2(GF(2)), the 2 × 2 matrix ring over the Galois field GF(2) of two elements. [ABSTRACT FROM AUTHOR]

This paper deal with the initial boundary value problem for a higher-order heat equation with logarithmic source term ut + (−∆)mu − ∆ut = uk−2u ln |u| . We obtain blow-up of weak solutions in the finite time, by employing potential well technique and concave technique. In addition, the upper bound of blow-up time is considered. This improves and extends some previous studies. [ABSTRACT FROM AUTHOR]

The present paper aims to derive summation and transformation formulae for the generalized very well-poised hypergeometric functions 2q+5F2q+4(1) and 2q+7F2q+6(1) having arbitrary integral parameter differences. These results are derived with the help of Bailey's transform and the extension of Saalschutz summation theorem for the series r+3Fr+2(1), where r pairs of parameters differ by positive integers. The particularizations of these generalized identities give classical summation theorems due to Dougall, transformation formula due to Whipple and, other related results. Furthermore, the application of 2q+5F2q+4(1) summation to the limiting case, when q→1, of one of the Andrews' q-identities gives a Srivastava-Daoust type multiple hypergeometric series. [ABSTRACT FROM AUTHOR]

In this paper, we give some identities involving derangement numbers. In addition, we derive nonlinear differential equation from the generating function of r-derangement numbers and obtain some identities including generalized harmonic numbers and derangement numbers by using this differential equation. For example, for any positive integers N, n and r, ... [ABSTRACT FROM AUTHOR]

Assume that Tn is the nth term of Tribonacci sequence and Rm is the mth term of Perrin sequence. In this paper, we solve the equation Tn = Rm completely. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a perturbed first order non-convex sweeping process for a class of subsmooth moving sets depending on the time and the state. In the first result we study the existence of solution and the compactness of the attainable set, the perturbation considered here is an upper semi-continuous set-valued mapping with nonempty closed convex values unnecessarily bounded. In the second result, we prove the existence of solution for the autonomous problem under assumptions that do not require the convexity of the values of the perturbation and that weaken the assumption on the upper semi-continuity. Then, we deduce a solution of the time optimality problem and we describe the attainable set. [ABSTRACT FROM AUTHOR]

In this paper we deal with some Diophantine equations and present infinitely many positive integer solutions for each one of them. [ABSTRACT FROM AUTHOR]

In this paper we discuss some convergence and divergence properties of subsequences of logarithmic means of Walsh-Fourier series. We give necessary and sufficient conditions for the convergence regarding logarithmic variation of numbers. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Fibonacci conditional sequence {fn} and the Lucas conditional sequence flng. We derive some properties of Fibonacci and Lucas conditional sequences by using the matrix method. Our results are elegant as the results for ordinary Fibonacci and Lucas sequences. [ABSTRACT FROM AUTHOR]

The main object of this paper is to study an old problem concerning the hyper-sums of powers of integers. First, we establish some important properties of this problem (generating function, explicit formula, congruence). Finally, an explicit formula for the hyper-sums of powers of integers involving the generalized Bernoulli polynomials are also given. [ABSTRACT FROM AUTHOR]

In this paper we introduce r-parameter Srivastava polynomials in r-variable by inserting new indices. These polynomials include the Lagrange polynomials in several variables, which are also known as Chan-Chyan-Srivastava polynomials (W.-C.C. Chan, C.-J. Chyan and H.M. Srivastava, The Lagrange polynomials in several variables, Integral Transforms and Special Functions, 12 (2001) 139-148). We prove several two sided linear generating relations between r-variable and (r-1)-variable Chan-Chyan-Srivastava polynomials. Some special cases of the main results are also presented. [ABSTRACT FROM AUTHOR]

Let k≥3 be an odd integer. In this paper, we investigate all positive integer solutions of the equations x4-kx2y+y2=∓A, x4-kx2y+y2=∓A(k2-4), x4-(k2-4)y2=∓4A, and x2-(k2-4)y4=∓4A with A=∓(k∓2). We show that if k≡1(mod8) and k2-4 be a squarefree integer, then the equation x4-kx2y+y2=(k-2)(k2-4) has no positive integer solutions. Moreover, if k2-4 be a squarefree integer, then the equation x4-kx2y+y2=-(k+2)(k2-4) has no positive integer solutions. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring and L is a Lie ideal of R such that LZ(R). A map F:R⋺R is called a multiplicative generalized derivation if there exists a map d:R⋺R such that F(xy)=F(x)y+xd(y), for all x,y∈R. In the present paper, we shall prove that d is a commuting map on L if any one of the following holds: i) F(uv)=±uv, ii) F(uv)=±vu, iii) F(u)F(v)=±uv, iv) F(u)F(v)=±vu, v) F(u)F(v)±uv∈Z, vi) F(u)F(v)±vu∈Z, vii) [F(u),v]±[u,G(v)]=0, for all u,v∈L. [ABSTRACT FROM AUTHOR]

In this paper we introduce a class of quasipolar rings which is a generalization of J -quasipolar rings. Let R be a ring with identity. An element a ϵ 2 R is called 1-quasipolar if there exists p2 = p ϵ comm2(a) such that a + p is contained in δ(R), and the ring R is called δ-quasipolar if every element of R is δ-quasipolar. We use 1-quasipolar rings to extend some results of J-quasipolar rings. Then some of the main results of J-quasipolar rings are special cases of our results for this general setting. We give many characterizations and investigate general properties of 1-quasipolar rings. [ABSTRACT FROM AUTHOR]

In this paper, some algebraic properties of the joint k-numerical radius, joint k-radius and the joint k-norm of matrices are investigated. Moreover, using the joint higher numerical ranges of diagonal matrices which are convex polyhedrons, a description for the shape of the higher numerical ranges of matrix polynomials is given. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian regular local ring containing a field. Let I be an ideal of R and let l ≥ 0 be an integer. In this paper it is shown that for every finitely generated R-module M and each integer i ≥l the Bass numbers of the R-module HIi(M) are finite, whenever, dimSupp(HIi(R)) ≤ 1, for all i ≥ l. [ABSTRACT FROM AUTHOR]

The paper is about calculating multiple fractional part integrals of the form ...where {x} denotes the fractional part of x and k,m and p are nonnegative integers. We show that these integrals can be expressed as series involving products of Riemann zeta function values and some binomial coefficients. We obtain, as particular cases of our results, new integral representations of Euler's constant as double and triple symmetric fractional part integrals. [ABSTRACT FROM AUTHOR]

In this paper we establish some Tauberian-like conditions in terms of the weighted general control modulo of integer order to retrieve subsequential convergence of a sequence from its boundedness. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the growth and oscillation of solutions and their derivatives of equations of the type f''+ A(z) f' + B(z) f = F(z), where A(z), B(z) (≢0) and F(z) (≢0) are meromorphic functions of finite order. [ABSTRACT FROM AUTHOR]

In this paper, the convergence of Mann iterative process with errors for generalized Lipschitz ϕ-quasi-accretive operators is proved in uniformly smooth Banach spaces. Our results improve the corresponding results of Chidume et al.[2]. [ABSTRACT FROM AUTHOR]

In this paper, the Diophantine equation B1k + B2k + ... Bn+1k = Bn+1l + Bn+2l + ... + Bn+rlfor the positive integer unknowns n ⩾ 2, k, l and r is studied in certain cases, where Bn denotes the nth term of the balancing sequence. [ABSTRACT FROM AUTHOR]

By the work of Abu Muriefah, Al-Rashed, Dujella and the author, the problem of the existence of D(z)-quadruples in the ring ℤ[√-2] has been solved, except for the cases z = 24a + 2 + (12b + 6) √-2,z = 24a + 5 + (12b + 6) √-2, z = 48a = 44 + (24b + 12) √-2. In this paper, we present some new formulas for D(z)-quadruples in these remaining cases, involving some congruence conditions modulo 11 on integers a and b. We show the existence of D(z)-quadruple for significant proportion of the remaining three cases. [ABSTRACT FROM AUTHOR]

Let n×n complex matrices P and Q be nontrivial generalized reflection matrices, i.e., P* = P = P-1 ≠ In, Q* = Q = Q-1 ≠ In. A complex matrix A with order n is said to be a (P, Q) generalized anti-reflexive matrix, if PAQ = -A. We in this paper mainly investigate the (P, Q) generalized anti-reflexive maximal and minimal rank solutions to the system of matrix equation AX = B. We present necessary and sufficient conditions for the existence of the maximal and minimal rank solutions, with (P, Q) generalized anti-reflexive, of the system. Expressions of such solutions to this system are also given when the solvability conditions are satisfied. In addition, in correspondence with the minimal rank solution set to the system, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been provided. [ABSTRACT FROM AUTHOR]

In this paper, we give a recurrence relation for the Tribonacci numbers with indices in aritmetics progression, {Trn+s} for 0 ⩽ s < n. We find sums of {Trn} for arbitrary integer r via matrix methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of the Stancu- Schurer operators based on q-integers and get a Bohman-Korovkin type approximation theorem of these operators. We also compute the rate of convergence by using the first modulus of smoothness and give some numerical result of operators based on Matlab algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we give all the solutions of the Diophantine equation x2 + 7α·11β= yn for the nonnegative integers α, β, x, y, n ≥3, where x and y coprime, except when α.x is odd and β is even. [ABSTRACT FROM AUTHOR]

In this paper, we determine when the equation in the title has an infinite number of positive integer solutions x and y when 0 ≤ n ≤ 10. Moreover, we give all the positive integer solutions of the same equation for 0 ≤ n ≤ 10. [ABSTRACT FROM AUTHOR]

In this paper, we study the statistical approximation properties of real and complex Post-Widder operators based on q-integers. We also obtain a Voronovskaya-type formula in statistical sense for these operators. [ABSTRACT FROM AUTHOR]