Band 52, Heft 1

We use a theorem of Lam and Leung to prove that a submatrix of a Fourier matrix cannot be Hadamard for particular cases when the dimension of the submatrix does not divide the dimension of the Fourier matrix. We also make some observations on the trace-spectrum relationship of dephased Hadamard matrices of low dimension.

In this paper we study in detail a variation of the orthonormal bases (ONB) of L2[0, 1] introduced in [Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2), 1128-1139] by means of representations of the Cuntz algebra ON on L2[0, 1]. For N = 2 one obtains the classic Walsh system which serves as a discrete analog of the Fourier system. We prove that the generalized Walsh system does not always display periodicity, or invertibility, with respect to function multiplication. After characterizing these two properties we also show that the transform implementing the generalized Walsh system is continuous with respect to filter variation. We consider such transforms in the case when the orthogonality conditions in Cuntz relations are removed. We show that these transforms which still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compression, similar to the discrete wavelet transform.

We prove the “hot spots” conjecture on the Vicsek set. Specifically, we will show that every eigenfunction of the second smallest eigenvalue of the Neumann Laplacian on the Vicsek set attains its maximum and minimum on the boundary.

We generalize the Zygmund inequality for the conjugate function to the Morrey type spaces defined on the unit circle T. We obtain this extended Zygmund inequality by introducing the Morrey-Zygmund space on T.

Based on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μ α is the unique Borel probability measure on [0, 1] satisfying μα(E)=∑n=0N-1αnμα(ϕn-1(E)){\mu ^\alpha }(E) = \sum\nolimits_{n = 0}^{N - 1} {{\alpha _n}{\mu ^\alpha }(\varphi _n^{ - 1}(E))} where ϕ n : x ↦ ( x + n )/ N . In Sections 1 and 2 we examine several general properties of the measure μ α and the associated Legendre polynomials in Lμα2L_{{\mu ^\alpha }}^2 [0, 1]. In Section 3, we (1) compute the Laplacian and moment generating function of μ α , (2) characterize precisely when the moments I m = ∫ [0,1] x m dμ α exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first m moments within uniform error ε in O ((log log(1/ ε )) · m log m ). We also state analogous results in the natural case where α is palindromic for the measure ν α attained by shifting μ α to [−1/2, 1/2].

In this paper we investigate the restriction problem. More precisely, we give sufficient conditions for the failure of a set E in ℝ n to have the p -restriction property. We also extend the concept of spectral synthesis to L p (ℝ n ) for sets of p -restriction when p > 1. We use our results to show that there are p -values for which the unit sphere is a set of p -spectral synthesis in ℝ n when n ⩾ 3.

We prove a strong maximum principle for Schrödinger operators defined on a class of postcritically finite fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular we permit both the fractal Laplacian and the potential to be Radon measures on the fractal. As a consequence of our results, we establish a Harnack inequality for solutions of these operators.

Reproducing Kernel Hilbert Spaces (RKHS) and their kernel are important tools which have been found to be incredibly useful in many areas like machine learning, complex analysis, probability theory, group representation theory and the theory of integral operator. In the present paper, the space of Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is demonstrated to be an RKHS and its associated kernel is derived. This extends the possibility of using this new kernel function, which is partly self-affine and partly non-self-affine, in diverse fields wherein the structure is not always self-affine.

Let C(r) denote the curve which is image of the circle |z| = r < 1 under the mapping f . Let L(r) be the length of C(r) and A(r) the area enclosed by the curve C(r). Furthermore M(r) = max|z|=r |f (z)|.We present some relations between these notions for Bazilevič functions.

We give negative answer to the question of Bordulyak and Sheremeta for more general classes of entire functions than in the original formulation: Does index boundedness in joint variables for an entire function F imply index boundedness in the variable zj for the function F? This question is addressed for entire functions of bounded L-index in joint variables and entire functions of bounded L-index in direction. We also present a class of analytic functions in the unit ball which has bounded L-index in joint variables and has unbounded l-index in the variables z1 and z2 for any positive continuous function l : B2 → C.

In this paper, we adopt direct method to prove the Hyers-Ulam-Rassias stability of an additivequadratic-cubic-quartic functional equation f(x+2y)+f(x−2y)=4f(x+y)+4f(x−y)−6f(x)+f(2y)+f(−2y)−4f(y)−4f(−y)$$f(x + 2y) + f(x - 2y) = 4f(x + y) + 4f(x - y) - 6f(x) + f(2y) + f( - 2y) - 4f(y) - 4f( - y)$$ in non-Archimedean ( n , β )-normed spaces.

In this paper, existence and uniqueness of solution for a coupled impulsive Hilfer–Hadamard type fractional differential system are obtained by using Kransnoselskii’s fixed point theorem. Different types of Hyers–Ulam stability are also discussed.We provide an example demonstrating consistency to the theoretical findings.

In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers–Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.

In this manuscript, we introduce a new hybrid contraction that unify several nonlinear and linear contractions in the set-up of a complete metric space. We present an example to indicate the genuine of the proved result. In addition, we consider Ulam type stability and well-posedness for this new hybrid contraction.

In this paper, we establish the existence and uniqueness of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with Riemann-Liouville fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle and Schaefer’s fixed point theorem. An example is included to show the applicability of our results.

The goal of this paper is to give an Ulam-Hyers stability result for a parabolic partial differential equation. Here we present two types of Ulam stability: Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability. Some examples are given, one of them being the Black-Scholes equation.

In this paper, we obtain Hyers-Ulam stability of the functional equations f ( x + y , z + w ) + f ( x − y , z − w ) = 2 f ( x , z ) + 2 f ( y , w ), f ( x + y , z − w ) + f ( x − y , z + w ) = 2 f ( x , z ) + 2 f ( y , w ) and f ( x + y , z − w ) + f ( x − y , z + w ) = 2 f ( x , z ) − 2 f ( y , w ) in 2-Banach spaces. The quadratic forms ax 2 + bxy + cy 2 , ax 2 + by 2 and axy are solutions of the above functional equations, respectively.

Let ( X , ⊥) be an orthogonality module in the sense of Rätz over a unital Banach algebra A and Y be a real Banach module over A . In this paper, we apply the alternative fixed point theorem for proving the Hyers-Ulam stability of the orthogonally generalized k -quadratic functional equation of the form af(kx+y)+af(kx-y)=f(ax+ay)+f(ax-ay)+(2k2-2)f(ax)af(kx + y) + af(kx - y) = f(ax + ay) + f(ax - ay) + \left( {2{k^2} - 2} \right)f(ax) for some |k| > 1, for all a ɛ A 1 := { u ɛ A||u|| = 1} and for all x , y ɛ X with x ⊥ y , where f maps from X to Y .

In this study we introduced and tested retarded conformable fractional integral inequalities utilizing non-integer order derivatives and integrals. In line with this purpose, we used the Katugampola type conformable fractional calculus which has several practical properties. These inequalities generalize some famous integral inequalities which provide explicit bounds on unknown functions. The results provided here had been implemented to the global existence of solutions to the conformable fractional differential equations with time delay.

We consider axial-symmetric stationary flows of the ideal incompressible fluid as an important case of potential solenoid fields. We use an integral expression of the Stokes flow function via the corresponding complex analytic function for solving a boundary value problem with respect to a steady streamline of the ideal incompressible fluid along an axial-symmetric body. We describe the solvability of the problem in terms of the singularities of the mentioned complex analytic function. The obtained results are illustrated by concrete examples of modelling of steady axial-symmetric flows.

A quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric in general. Quasi-metrics are a subject of comprehensive investigation both in pure and applied mathematics in areas such as in functional analysis, topology and computer science. The main purpose of this paper is to extend the convergence and Cauchy conditions in a quasi-metric space by using the notion of asymptotic density. Furthermore, some results obtained are related to completeness, compactness and precompactness in this setting using statistically Cauchy sequences.

We consider special elliptic operators in functional spaces on manifolds with a boundary which has some singular points. Such an operator can be represented by a sum of operators, and for a Fredholm property of an initial operator one needs a Fredholm property for each operator from this sum.

In this paper, we introduce a new kind of q-Balázs-Szabados-Kantorovich operators called q-BSK operators. We give a weighted statistical approximation theorem and the rate of convergence of the q-BSK operators. Also, we investigate the local approximation results. Further, we give some comparisons associated with the convergence of q-BSK operators.

In this article, following Gorgi and Yazdanpanah, we define two new concepts of the ideal amenability for a Banach algebra A. We compare these notions with J-weak amenability and ideal amenability, where J is a closed two-sided ideal in A. We also study the hereditary properties of quotient ideal amenability for Banach algebras. Some examples show that the concepts of A/J-weak amenability and of J-weak amenability do not coincide for Banach algebras in general.

The objective of this article is to discuss the existence and uniqueness of mild solutions for a class of non-autonomous semilinear differential equations with nonlocal condition via monotone iterative method with upper and lower solutions in an ordered complete norm space X, using evolution system and measure of noncompactness.

The solvability of the problem with local nonhomogeneous two-point in time conditions for a homogeneous PDE of the second order in time and infinite order in spatial variable in the case when the set of zeroes of the characteristic determinant is not empty and does not coincide with C is investigated. The existence of a solution of the problem in which the right-hand sides of the two-point conditions are quasi-polynomials is proved. We propose the differential-symbol method of constructing the solutions of the problem.

Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some ℓp−type fractional difference sets via the gamma function. Although, we characterize compactness conditions on those sets using the main key of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is ℓ∞. However, we use some recent results to exactly characterize the classes of compact matrix operators when the final space is the set of bounded sequences.

In this paper we will present the pointwise and normwise estimations of the deviations considered by W. Łenski, B. Szal, [Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24] and S. Saini, U. Singh, [Boll. Unione Mat. Ital., 2016, 9, 495-504] under general assumptions on the class considered sequences defining the method of the summability. We show that the obtained estimations are the best possible for some subclasses of L p by constructing the suitable type of functions.

In 2011, Shabir and Naz [1] employed the notion of soft sets to introduce the concept of soft topologies; and in 2014, El-Sheikh and Abd El-Latif [2] relaxed the conditions of soft topologies to construct a wider and more general class, namely supra soft topologies. In this disquisition, we continue studying supra soft topologies by presenting two kinds of supra soft separation axioms, namely supra soft T i -spaces and supra p-soft T i -spaces for i = 0, 1, 2, 3, 4. These two types are formulated with respect to the ordinary points; and the difference between them is attributed to the applied non belong relations in their definitions.We investigate the relationships between them and their parametric supra topologies; and we provide many examples to separately elucidate the relationships among spaces of each type. Then we elaborate that supra p-soft T i -spaces are finer than supra soft T i -spaces in the case of i = 0, 1, 4; and we demonstrate that supra soft T 3 -spaces are finer than supra p-soft T 3 -spaces.We point out that supra p-soft T i -axioms imply supra p-soft T i −1 , however, this characterization does not hold for supra soft T i -axioms (see, Remark (3.30)). Also, we give a complete description of the concepts of supra p-soft T i -spaces ( i = 1, 2) and supra p-soft regular spaces. Moreover,we define the finite product of supra soft spaces and manifest that the finite product of supra soft T i (supra p-soft T i ) is supra soft T i (supra p-soft T i ) for i = 0, 1, 2, 3. After investigating some properties of these axioms in relation with some of the supra soft topological notions such as supra soft subspaces and enriched supra soft topologies, we explore the images of these axioms under soft S * -continuous mappings. Ultimately, we provide an illustrative diagram to show the interrelations between the initiated supra soft spaces.

In this paper, we provide some generalizations of the Darbo’s fixed point theorem associated with the measure of noncompactness and present some results on the existence of the coupled fixed point theorems for a special class of operators in a Banach space. To acquire this result, we define α - ψ and β - ψ condensing operators and using them we propose new fixed point results. Our results generalize and extend some comparable results from the literature. Additionally, as an application, we apply the obtained fixed point theorems to study the nonlinear functional integral equations.

We consider a new subgradient extragradient iterative algorithm with inertial extrapolation for approximating a common solution of variational inequality problems and fixed point problems of a multivalued demicontractive mapping in a real Hilbert space. We established a strong convergence theorem for our proposed algorithm under some suitable conditions and without prior knowledge of the Lipschitz constant of the underlying operator. We present numerical examples to show that our proposed algorithm performs better than some recent existing algorithms in the literature.

In this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; vt+vvx-vxx=0,   x>0,   t>0,v(x,0)=u+,   x>0,v(0,t)=ub,  t>0,$\matrix{ {{v_t} + v{v_x} - {v_{xx}} = 0,\,\,\,x > 0,\,\,\,t > 0,} \cr {v\left( {x,0} \right) = {u_ + },\,\,\,x > 0,} \cr {v\left( {0,t} \right) = {u_b},\,\,t > 0,} \cr }$ where x and t represent distance and time, respectively, and u + is an initial condition, ub is a boundary condition which are constants ( u + ≠ ub ). Analytic solution of above problem is solved depending on parameters ( u + and ub ) then compared with numerical solutions to show there is a good agreement with each solutions.

The neutrix composition F ( f ( x )) of a distribution F ( x ) and a locally summable function f ( x ) is said to exist and be equal to the distribution h ( x ) if the neutrix limit of the sequence { F n ( f ( x ))} is equal to h ( x ), where F n ( x ) = F ( x ) * δ n ( x ) and {δ n ( x )} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function ( x ). The function cosh+-1(x+1)$\cosh _ + ^{ - 1}\left( {x + 1} \right)$ is defined by cosh+-1(x+1)=H(x)cosh-1(|x|+1),$$\cosh _ + ^{ - 1}\left( {x + 1} \right) = H\left( x \right){\cosh ^{ - 1}}\left( {\left| x \right| + 1} \right),$$ where H ( x ) denotes Heaviside’s function. It is then proved that the neutrix composition δ(s)[cosh+-1(x1/r+1)]${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right]$] exists and δ(s)[cosh+-1(x1/r+1)]=∑k=0s-1∑j=0kr+r-1∑i=0j(-1)kr+r+s-j-1r2j+2(kr+r-1j)(ji)[(j-2i+1)s-(i-2i-1)s]δ(k)(x),$${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right] = \sum\limits_{k = 0}^{s - 1} {\sum\limits_{j = 0}^{kr + r - 1} {\sum\limits_{i = 0}^j {{{{{( - 1)}^{kr + r + s - j - 1}}r} \over {{2^{j + 2}}}}\left( {\matrix{{kr + r - 1} \cr j \cr } } \right)} } } \left( {\matrix{j \cr i \cr } } \right)\left[ {{{\left( {j - 2i + 1} \right)}^s} - {{\left( {i - 2i - 1} \right)}^s}} \right]{\delta ^{(k)}}(x),$$ for r , s = 1, 2, . . . . Further results are also proved. Our results improve, extend and generalize the main theorem of [Fisher B., Al-Sirehy F., Some results on the neutrix composition of distributions involving the delta function and the function cosh −1 + ( x + 1), Appl. Math. Sci. (Ruse), 2014, 8(153), 7629–7640].

We consider the following second order evolution equation modelling a nonlinear oscillator with damping ü(t)+γu˙(t)+Au(t)=f(t),             (SEE)$$\ddot{u} (t) + \gamma \dot u(t) + Au\left( t \right) = f\left( t \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\rm{SEE}}} \right)$$ where A is a maximal monotone and α -inverse strongly monotone operator in a real Hilbert space H . With suitable assumptions on γ and f ( t ) we show that A −1 (0) ≠ ∅, if and only if (SEE) has a bounded solution and in this case we provide approximation results for elements of A −1 (0) by proving weak and strong convergence theorems for solutions to (SEE) showing that the limit belongs to A −1 (0). As a discrete version of (SEE), we consider the following second order difference equation un+1-un-αn(un-un-1)+λnAun+1∋f(t),$${u_{n + 1}} - {u_n} - {\alpha _n}\left( {{u_n} - {u_{n - 1}}} \right) + {\lambda _n}A{u_{n + 1}\ni} f\left( t \right),$$ where A is assumed to be only maximal monotone (possibly multivalued). By using the results in [Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417], we prove ergodic, weak and strong convergence theorems for the sequence u n , and show that the limit is the asymptotic center of u n and belongs to A −1 (0). This again shows that A −1 (0) ≠ ∅ if and only if u n is bounded. Also these results solve an open problem raised in [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11], namely the study of the convergence results for the inexact inertial proximal algorithm. Our paper is motivated by the previous results in [Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl., 1990, 147, 465–476; Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226–235; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math., 2010, 40, 1289–1311; Djafari Rouhani B., Khatibzadeh H., A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation, J. Math. Anal. Appl., 2010, 363, 648–654; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations, Nonlinear Anal., 2009, 70, 4369–4376; Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417] and significantly improves upon the results of [Attouch H., Maingé P. E., Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects, ESAIM Control Optim. Calc. Var., 2011, 17(3), 836–857], and [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11].

A numerical technique for one-dimensional Bratu’s problem is displayed in this work. The technique depends on Bernstein polynomial approximation. Numerical examples are exhibited to verify the efficiency and accuracy of the proposed technique. In this sequel, the obtained error was shown between the proposed technique, Chebyshev wavelets, and Legendre wavelets. The results display that this technique is accurate.

We prove strong convergence of the sequence generated by implicit viscosity approximation method involving a multivalued nonexpansive mapping in framework of CAT(0) space. Under certain appropriate conditions on parameters, we show that such a sequence converges strongly to a fixed point of the mapping which solves a variational inequality. We also present some convergence results for the implicit viscosity approximation method in complete ℝ-trees without the endpoint condition.

Here, we estimate the degree of approximation of a conjugate function g˜{\tilde g} and a derived conjugate function g˜′{\tilde g'} , of a 2 π -periodic function g∈Zrλg \in Z_r^\lambda , r ≥ 1, using Hausdorff means of CFS (conjugate Fourier series) and CDFS (conjugate derived Fourier series) respectively. Our main theorems generalize four previously known results. Some important corollaries are also deduced from our main theorems. We also partially review the earlier work of the authors in respect of order of the Euler-Hausdorff product method.

In this paper, we prove strong convergence and Δ−convergence of Fibonacci-Mann iteration for a monotone non-Lipschitzian mapping (i.e. nearly asymptotically nonexpansive mapping) in partially ordered hyperbolic metric space. Moreover, we prove stability of Fibonacci-Mann iteration. Further, we construct a numerical example to illustrate results. Our results simultaneously generalize the results of Alfuraidan and Khamsi [Bull. Aust. Math. Soc., 2017, 96, 307–316] and Schu [J. Math. Anal. Appl., 1991, 58, 407–413].

Let 𝒜 be a unital C * -algebra with a faithful state ϕ . We study the geometry of the unit sphere 𝕊 ϕ = { x ∈ 𝒜 : ϕ ( x * x ) = 1} and the projective space ℙ ϕ = 𝕊 ϕ /𝕋. These spaces are shown to be smooth manifolds and homogeneous spaces of the group 𝒰 ϕ (𝒜) of isomorphisms acting in 𝒜 which preserve the inner product induced by ϕ , which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in ℙ ϕ , and prove the existence of minimal geodesics, both with given initial data, and given endpoints.

We introduce the ( q , h )-blossom of bivariate polynomials, and we define the bivariate ( q , h )-Bernstein polynomials and ( q , h )-Bézier surfaces on rectangular domains using the tensor product. Using the ( q , h )-blossom, we construct recursive evaluation algorithms for ( q , h )-Bézier surfaces and we derive a dual functional property, a Marsden identity, and a number of other properties for bivariate ( q , h )-Bernstein polynomials and ( q , h )-Bézier surfaces. We develop a subdivision algorithm for ( q , h )-Bézier surfaces with a geometric rate of convergence. Recursive evaluation algorithms for quantum ( q , h )-partial derivatives of bivariate polynomials are also derived.

In the paper we investigate slice holomorphic functions F : ℂ n → ℂ having bounded L -index in a direction, i.e. these functions are entire on every slice {z 0 + t b : t ∈ ℂ } for an arbitrary z 0 ∈ ℂ n and for the fixed direction b ∈ℂ n \ {0}, and ( ∃m 0 ∈ ℤ + ) ( ∀m ∈ ℤ + ) ( ∀z ∈ ℂ n ) the following inequality holds |∂bmF(z)|m!Lm(z)≤max0≤k≤m0|∂bkF(z)|k!Lk(z),{{\left| {\partial _{\bf{b}}^mF(z)} \right|} \over {m!{L^m}(z)}} \le \mathop {\max }\limits_{0 \le k \le {m_0}} {{\left| {\partial _{\bf{b}}^kF(z)} \right|} \over {k!{L^k}(z)}}, where L : ℂ n → ℝ + is a positive continuous function, ∂bF(z)=ddtF(z+tb)|t=0,∂bpF=∂b(∂bp-1F){\partial _{\bf{b}}}F(z) = {d \over {dt}}F\left( {z + t{\bf{b}}} \right){|_{t = 0}},\partial _{\bf{b}}^pF = {\partial _{\bf{b}}}\left( {\partial _{\bf{b}}^{p - 1}F} \right)for p ≥ 2. Also, we consider index boundedness in the direction of slice holomorphic solutions of some partial differential equations with partial derivatives in the same direction. There are established sufficient conditions providing the boundedness of L -index in the same direction for every slie holomorphic solutions of these equations.

In this paper we prove an Ozguç, Yurdakadim and Taş version of the Korovkin-type approximation by operators in the sense of the power series method. That is, we try to extend the Korovkin approximation theorems, obtained by Ozguç and Taş in 2016, and Taş and Yurdakadim in 2017, for concrete classes of Banach spaces to the class of Riesz spaces. Some applications are presented.

This paper introduces the more general result on existence, uniqueness and boundedness for solutions of nonstandard Volterra type integral equation on an arbitrary time scales. We use Lipschitz type function and the Banach’s fixed point theorem at functional space endowed with a suitable Bielecki type norm. Furthermore, it allows to get new sufficient conditions for boundedness and continuous dependence of solutions.

This paper aims to show new algebraic properties from the q -generalized Bernoulli polynomials Bn[m-1](x;q)B_n^{[m - 1]}(x;q) of level m , as well as some others identities which connect this polynomial class with the q -generalized Bernoulli polynomials of level m , as well as the q -gamma function, and the q -Stirling numbers of the second kind and the q -Bernstein polynomials.