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In this paper, we introduce an interesting kind of convergence for a double sequence called the uniform convergence at a point. We give an example and demonstrate a Korovkin-type approximation theorem for a double sequence of functions using the uniform convergence at a point. Then we show that our result is stronger than the Korovkin theorem given by Volkov and present several graphs. Finally, in the last section, we compute the rate of convergence.

In this paper, we study the Hyers-Ulam stability of a nonautonomous semilinear reaction-diffusion equation. More precisely, we consider a nonautonomous parabolic equation with a diffusion given by the fractional Laplacian. We see that such a stability is a consequence of a Gronwall-type inequality.

The quantum groups nowadays attract a considerable interest of mathematicians and physicists. The theory of q-special functions has received a group-theoretic interpretation using the techniques of quantum groups and quantum algebras. This paper focuses on introducing the q-Tricomi functions and 2D q-Tricomi functions through the generating function and series expansion and for the first time establishing a connecting relation between the q-Tricomi and q-Bessel functions. The behavior of these functions is described through shapes, and the contrast between them is observed using mathematical software. Further, the problem of framing the q-Tricomi and 2D q-Tricomi functions in the context of the irreducible representation ( ω ) {(\omega)} of the two-dimensional quantum algebra ℰ q ⁢ ( 2 ) {\mathcal{E}_{q}(2)} is addressed, and certain relations involving these functions are obtained. 2-Variable 1-parameter q-Tricomi functions and their relationship with the 2-variable 1-parameter q-Bessel functions are also explored.

The combined systems of integral equations have become of great importance in various fields of sciences such as electromagnetic and nuclear physics. New classes of the merged type of Urysohn Volterra-Chandrasekhar quadratic integral equations are proposed in this paper. This proposed system involves fractional Urysohn Volterra kernels and also Chandrasekhar kernels. The solvability of a coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type is studied. To realize the existence of a solution of those mixed systems, we use the Perov’s fixed point combined with the Leray-Schauder fixed point approach in generalized Banach algebra spaces.

We consider iterated integrals of $\log\zeta(s)$ on certain vertical and horizontal lines. Here, the function $\zeta(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of $\int_{0}^{t} \log \zeta(1/2 + it')dt'$ under the Riemann Hypothesis. Moreover, we show that, for any $m\geq 2$, the denseness of the values of an $m$-times iterated integral on the critical line is equivalent to the Riemann Hypothesis., Comment: 15 pages

In this paper, we introduce the generalized Tetranacci hybrid numbers and, as special cases, Tetranacci and Tetranacci-Lucas hybrid numbers. Moreover, we present Binet’s formulas, generating functions, and the summation formulas for those hybrid numbers.

In this paper we describe families of commuting invertible formal power series in one indeterminate over ℂ, using the method of formal functional equations. We give a characterization of such families where the set of multipliers (first coefficients) σ of its members F (x) = σx + . . . is infinite, in particular of such families which are maximal with respect to inclusion, so called families of type I. The description of these families is based on Aczél–Jabotinsky differential equations, iteration groups, and on some results on normal forms of invertible series with respect to conjugation.

In this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-valued functions analytic in the unit ball B 2 = { z ∈ C 2 : | z | = | z 1 | 2 + | z 2 | 2 < 1 } , $\begin{array}{} \mathbb{B}^2\! = \!\{z\!\in\!\mathbb{C}^2: |z|\! = \!\small\sqrt{|z_1|^2+|z_2|^2}\! \lt \! 1\}, \end{array} $ where L = (l 1, l 2): 𝔹2 → R + 2 $\begin{array}{} \mathbb{R}^2_+ \end{array} $ is a positive continuous vector-valued function. Particularly, we deduce analog of Hayman’s theorem for this class of functions. The theorem shows that in the definition of boundedness of L-index in joint variables for vector-valued functions we can replace estimate of norms of all partial derivatives by the estimate of norm of (p + 1)-th order partial derivative. This form of criteria could be convenient to investigate analytic vector-valued solutions of system of partial differential equations because it allow to estimate higher-order partial derivatives by partial derivatives of lesser order. Also, we obtain sufficient conditions for index boundedness in terms of estimate of modulus of logarithmic derivative in each variable for every component of vector-valued function outside some exceptional set by the vector-valued function L(z).

In [Rahimi, M.: Entropy as an integral operator, Math. Slovaca 69(1) (2019), 139–146], we assigned an integral operator on a Hilbert space to any topological dynamical system of finite entropy and stated the entropy of the system in terms of the spectrum of the defined operator. Unfortunately, there is a mistake in the proof of the main theorem of the paper which makes the result incorrect. So, we can not extract the entropy of a topological dynamical system in terms of the spectrum of the introduced operator. In this note, we modify the main theorem of [11] by giving a modification to the proof of the theorem. Then, replacing the integral operator introduced in [11] by another linear operator, we will state the entropy of the system in terms of the spectrum of the new operator.

In this paper, we mainly consider the existence and Hyers-Ulam stability of solutions for a class of fractional differential equations involving time-varying delays and non-instantaneous impulses. By the Krasnoselskii’s fixed point theorem, we present the new constructive existence results for the addressed equation. In addition, we deduce that the equations have Hyers-Ulam stable solutions by utilizing generalized Grönwall’s inequality. Some results in this literature are new and improve some early conclusions.

In this paper, sufficient conditions for oscillation of unbounded solutions of a class of fourth order neutral delay differential equations of the form ( r ( t ) ( y ( t ) + p ( t ) y ( t − τ ) ) ″ ) ″ + q ( t ) G ( y ( t − α ) ) − h ( t ) H ( y ( t − σ ) ) = 0 $$\begin{array}{} \displaystyle (r(t)(y(t)+p(t)y(t-\tau))'')''+q(t)G(y(t-\alpha))-h(t)H(y(t-\sigma))=0 \end{array}$$ are discussed under the assumption ∫ 0 ∞ t r ( t ) d t = ∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}\text{d}~~ t=\infty \end{array}$$

In this paper we investigate some fundamental properties of unital topology on a lattice ordered group with order unit. We show that some essential properties of order unit norm on a vector lattice with order unit, are valid for unital l-groups. For instance we show that for an Archimedean Riesz space G with order unit u, the unital topology and the strong link topology are the same.

In this paper, we find all the solutions (X, Y, Z) = (FI , FJ , FK ), where FI , FJ , and FK represent nonzero Fibonacci numbers, satisfying a generalization of Markoff equation called the Jin-Schmidt equation: AX 2 + BY 2 + CZ 2 = DXYZ + 1.

The aim of this paper is to generalize several differential subordination results, involving arithmetic, geometric and harmonic means of the expressions p(z) and p ( z ) + z p ′ ( z ) p ( z ) . $\begin{array}{} p(z)+\frac{zp'(z)}{p(z)}. \end{array} $ Are also given certain applications of the main results.

In this paper, we mainly give some quadratic refinements of Young type inequalities. Namely: ( v a + ( 1 − v ) b ) 2 − v ∑ j = 1 N 2 j ( b − a b 2 j − 1 2 j ) 2 ≤ ( a v b 1 − v ) 2 + v 2 ( a − b ) 2 $$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-v{{\sum\limits_{j=1}^N}}2^{j}\Big(b- \sqrt[2^{j}]{ab^{2^{j}-1} }\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+v^{2}(a-b)^{2} \end{array}$$ for v ∉ [0, 1 2 N + 1 $\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$ ], N ∈ ℕ, a, b > 0; and ( v a + ( 1 − v ) b ) 2 − ( 1 − v ) ∑ j = 1 N 2 j ( a − a 2 j − 1 b 2 j ) 2 ≤ ( a v b 1 − v ) 2 + ( 1 − v ) 2 ( a − b ) 2 $$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-(1-v){{\sum\limits_{j=1}^N}}2^{j}\Big(a- \sqrt[2^{j}]{a^{2^{j}-1}b}\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+(1-v)^{2}(a-b)^{2} \end{array}$$ for v ∉ [1 − 1 2 N + 1 $\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$ , 1], N ∈ ℕ, a, b > 0. As an application of these scalars results, we obtain some matrix inequalities for operators and Hilbert-Schmidt norms.

This paper is devoted to the study of the class of decomposable double MS-algebras. Necessary and sufficient conditions for a decomposable MS-algebra to be a decomposable double MS-algebra are deduced. We construct decomposable double MS-algebras by means of decomposable MS-quadruples and we prove that there exists a one-to-one correspondence between decomposable double MS-algebras and decomposable MS-quadruples. Moreover, a construction of decomposable K 2-algebras (Stone algebras) by means of K 2-quadruples (Stone quadruples) is given. We conclude by introducing and characterizing isomorphisms of decomposable double MS-algebras in terms of decomposable MS-quadruples.

In this paper we define the monadic pseudo BE-algebras and investigate their properties. We prove that the existential and universal quantifiers of a monadic pseudo BE-algebra form a residuated pair. Special properties are studied for the particular case of monadic bounded commutative pseudo BE-algebras. Monadic classes of pseudo BE-algebras are investigated and it is proved that the quantifiers on bounded commutative pseudo BE-algebras are also quantifiers on the corresponding pseudo MV-algebras. The monadic deductive systems and monadic congruences of monadic pseudo BE-algebras are defined and their properties are studied. It is proved that, in the case of a monadic distributive commutative pseudo BE-algebra there is a one-to-one correspondence between monadic congruences and monadic deductive systems, and the monadic quotient pseudo BE-algebra algebra is also defined.

Recently, the problem of bounding the sup norms of L 2 {L^{2}} -normalized cuspidal automorphic newforms ϕ on GL 2 {\mathrm{GL}_{2}} in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character χ of ϕ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general χ. If the level N is a square, our result reduces to ∥ ϕ ∥ ∞ ≪ N 1 4 + ϵ , \|\phi\|_{\infty}\ll N^{\frac{1}{4}+\epsilon}, at least under the Ramanujan Conjecture. In particular, when χ has conductor N, this improves upon the previous best known bound ∥ ϕ ∥ ∞ ≪ N 1 2 + ϵ {\|\phi\|_{\infty}\ll N^{\frac{1}{2}+\epsilon}} in this setup (due to [A. Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 2017, 1009–1045]) and matches a lower bound due to [N. Templier, Large values of modular forms, Camb. J. Math. 2 2014, 1, 91–116], thus our result is essentially optimal in this case.

This paper concerns the study of the Schwartz differential equation { h , τ } = s ⁢ E 4 ⁡ ( τ ) {\{h,\tau\}=s\operatorname{E}_{4}(\tau)} , where E 4 {\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of SL 2 ⁡ ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of SL 2 ⁡ ( ℤ ) {\operatorname{SL}_{2}({\mathbb{Z}})} . This also leads to the solutions to the Fuchsian differential equation y ′′ + s ⁢ E 4 ⁡ y = 0 {y^{\prime\prime}+s\operatorname{E}_{4}y=0} .

The Cauchy problem of the 2D Zakharov–Kuznetsov equation ∂ t ⁡ u + ∂ x ⁡ ( ∂ x ⁢ x + ∂ y ⁢ y ) ⁡ u + u ⁢ u x = 0 {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space H - 1 / 4 {H^{-1/4}} , and it is globally well-posed in H - 1 / 4 {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.

In this paper, we study the Cauchy problem of the parabolic Monge–Ampère equation - u t ⁢ det ⁡ D 2 ⁢ u = f ⁢ ( x , t ) -u_{t}\det D^{2}u=f(x,t) and obtain the existence and uniqueness of viscosity solutions with asymptotic behavior by using the Perron method.

In 1977, Colville, Davis, and Keimel [Positive derivations on f-rings, J. Aust. Math. Soc. Ser. A 23 1977, 3, 371–375] proved that a positive derivation on an Archimedean f-algebra A has its range in the set of nilpotent elements of A. The main objective of this paper is to obtain a generalization of the above Colville, Davis and Keimel result to general derivations. Moreover, we give a new version of the Singer–Wermer conjecture for the class of second-order derivations acting on uniformly complete almost f-algebras.

This paper proposes a new extended Lindley distribution, which has a more flexible density and hazard rate shapes than the Lindley and Power Lindley distributions, based on the mixture distribution structure in order to model with new distribution characteristics real data phenomena. Its some distributional properties such as the shapes, moments, quantile function, Bonferonni and Lorenz curves, mean deviations and order statistics have been obtained. Characterizations based on two truncated moments, conditional expectation as well as in terms of the hazard function are presented. Different estimation procedures have been employed to estimate the unknown parameters and their performances are compared via Monte Carlo simulations. The flexibility and importance of the proposed model are illustrated by two real data sets.

In this paper, we are mainly interested to study the generalization of typically real functions in the unit disk. We study some coefficient inequalities concerning this class of functions. In particular, we find the Zalcman conjecture for generalized typically real functions.

In this paper, we develop a new method based on Newton polygon and graded polynomials, similar to the known one based on Newton polygon and residual polynomials. This new method allows us the factorization of any monic polynomial in any henselian valued field. As applications, we give a new proof of Hensel’s lemma and a theorem on prime ideal factorization.

In this paper, we prove the boundedness of the Bn maximal operator and Bn singular integral operators associated with the Laplace-Bessel differential operator Δ Bn on variable exponent Lebesgue spaces.

In the paper, we give new improvements of the reverse Hölder and Minkowski integral inequalities. These new results in special case yield the Pólya-Szegö’s inequality and reverse Minkowski’s inequality, respectively.

Dynamics of composition of entire functions is well related to it's factors, as it is known that for entire functions f and g, fog has wandering domain if and only if gof has wandering domain. However the Fatou components may have different structures and properties. In this paper we have shown the existence of domains with all possibilities of wandering and periodic in given angular region θ.

In this paper, we prove some conditional results about the order of zero at central point s = 1/2 of the Rankin-Selberg L-function L(s, πf × π͠′ f ). Then, we give an upper bound for the height of the first zero with positive imaginary part of L(s, πf × π͠′ f ). We apply our results to automorphic L-functions.

In this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H 3(1) and H 2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: | a 3 2 − a 5 | $\begin{array}{} |a_3^2-a_5| \end{array}$ for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

The paper deals with the exploration of those subclasses of the variable exponent Lebesgue space L p ⁢ ( ⋅ ) {L^{p(\,\cdot\,)}} with min ⁡ p ⁢ ( ⋅ ) = 1 {\min p(\,\cdot\,)=1} , which are invariant with respect to Cauchy singular integral operators.

Let ℛ {\mathscr{R}} be a prime ring, 𝒬 r {\mathscr{Q}_{r}} the right Martindale quotient ring of ℛ {\mathscr{R}} and 𝒞 {\mathscr{C}} the extended centroid of ℛ {\mathscr{R}} . In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e., ( [ δ ⁢ ( f ⁢ ( x 1 , … , x n ) ) , f ⁢ ( x 1 , … , x n ) ] ) m = [ δ ⁢ ( f ⁢ ( x 1 , … , x n ) ) , f ⁢ ( x 1 , … , x n ) ] , \big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})], where 1 < m ∈ ℤ + {1 , f ⁢ ( x 1 , x 2 , … , x n ) {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over 𝒞 {\mathscr{C}} and δ is a skew derivation of ℛ {\mathscr{R}} .

In this paper, we study the L q {L^{q}} - L r {L^{r}} boundedness of bi-parameter Fourier integral operators defined by general rough Hörmander class amplitudes.

In this paper, we establish an improved variable coefficient version of the square function inequality, by which the local smoothing estimate L α p → L p {L^{p}_{\alpha}\to L^{p}} for the Fourier integral operators satisfying cinematic curvature condition is further improved. In particular, we establish almost sharp results for 2 < p ⩽ 3 {2 and push forward the estimate for the critical point p = 4 {p=4} . As a consequence, the local smoothing estimate for the wave equation on the manifold is refined. We generalize the results in [S. Lee and A. Vargas, On the cone multiplier in ℝ 3 \mathbb{R}^{3} , J. Funct. Anal. 263 2012, 4, 925–940; J. Lee, A trilinear approach to square function and local smoothing estimates for the wave operator, preprint 2018, https://arxiv.org/abs/1607.08426v5] to its variable coefficient counterpart. The main ingredients in the argument includes multilinear oscillatory integral estimate [J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 2006, 2, 261–302] and decoupling inequality [D. Beltran, J. Hickman and C. D. Sogge, Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds, Anal. PDE 13 2020, 2, 403–433].

This paper studies the two-weight estimates of variation and oscillation operators for commutators of singular integrals with weighted BMO {\mathrm{BMO}} functions. A new characterization of weighted BMO {\mathrm{BMO}} spaces via the boundedness of variation and oscillation operators for the iterated commutators of Calderón–Zygmund singular integrals in the two-weight setting is given.

In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is convex, inspired by Colding-Naber's original idea of parabolic smoothing of the distance functions., Comment: revised version, to appear in J. Reine Angew. Math

In this paper we establish some trapezoid type inequalities for the Riemann-Liouville fractional integrals of functions of bounded variation and of Hölder continuous functions. Applications for the g-mean of two numbers are provided as well. Some particular cases for Hadamard fractional integrals are also provided.

The purpose of this paper is to introduce a new type of φ -implicit relation in S - metric spaces and to prove a general fixed point for a pair of weakly compatible mappings, which generalize Theorems 1, 2, 4 [23], Theorems 1-7 [13], Corollary 2.19 [13], Theorems 2.2, 2.4 [19], Theorems 3.2, 3.3, 3.4 [20] and other known results.

In the present paper we establish necessary and sufficient conditions under which two functions can be separated by a delta-convex function. This separation will be understood as a separation with respect to the partial order generated by the Lorentz cone. An application to a stability problem for delta-convexity is also given.

The main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential equations of the type dX t = S ( X t ) dt + ε dB t H , K , X 0 = x 0 , 0 ≤ t ≤ T , {\rm{d}}{{\rm{X}}_{\rm{t}}} = {\rm{S}}\left( {{{\rm{X}}_{\rm{t}}}} \right){\rm{dt + }}\varepsilon {\rm{dB}}_{\rm{t}}^{{\rm{H,K}}},\,{{\rm{X}}_{\rm{0}}} = {{\rm{x}}_{\rm{0}}},\,0 \le {\rm{t}} \le {\rm{T,}} where { B t H , K , t ≥ 0 {\rm{B}}_{\rm{t}}^{{\rm{H,K}}},{\rm{t}} \ge {\rm{0}} } is a bifractional Brownian motion with known parameters H ∈ (0, 1), K ∈ (0, 1] and HK ∈ (1/2, 1). We estimate the unknown function S(xt) by a kernel estimator ̂St and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.

In this paper a new class of sets termed as ω ∗ μ-open sets has been introduced and studied. Using these concept, a unified theory for decomposition of (μ, λ)-continuity has been given.

Let ℜ {\mathfrak{R}} be a ring with center Z ⁢ ( ℜ ) {Z(\mathfrak{R})} . In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation 𝔉 {\mathfrak{F}} such that ( [ 𝔉 ⁢ ( s m ) ⁢ s n + s n ⁢ 𝔉 ⁢ ( s m ) , s r ] k ) l ∈ Z ⁢ ( ℜ ) {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every s ∈ ℜ {s\in\mathfrak{R}} , then either 𝔉 ⁢ ( s ) = p ⁢ s {\mathfrak{F}(s)=ps} for every s ∈ ℜ {s\in\mathfrak{R}} or ℜ {\mathfrak{R}} satisfies s 4 {s_{4}} and 𝔉 ⁢ ( s ) = s ⁢ p {\mathfrak{F}(s)=sp} for every s ∈ ℜ {s\in\mathfrak{R}} and p ∈ 𝔘 {p\in\mathfrak{U}} , the Utumi quotient ring of ℜ {\mathfrak{R}} . As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.

In this paper, we consider a natural extension in the context of d-orthogonality for asymptotic analysis of orthogonal polynomials. We introduce, for several d-orthogonal polynomials, asymptotic expansions in terms of d-Hermite ones. From these expansions, several limits between d-orthogonal polynomials are obtained.

In this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth: { ( - Δ p ) s ⁢ u = Q u ⁢ ( u , v ) + H u ⁢ ( u , v ) in ⁢ Ω , ( - Δ p ) s ⁢ v = Q v ⁢ ( u , v ) + H v ⁢ ( u , v ) in ⁢ Ω , u = v = 0 in ⁢ ℝ N ∖ Ω , u , v ≥ 0 , u , v ≠ 0 in ⁢ Ω , \left\{\begin{aligned} \displaystyle{}(-\Delta_{p})^{s}u&\displaystyle=Q_{u}(u% ,v)+H_{u}(u,v)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta_{p})^{s}v&\displaystyle=Q_{v}(u,v)+H_{v}(u,v)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{N}\setminus\Omega,\\ \displaystyle u,v&\displaystyle\geq 0,\quad u,v\neq 0&&\displaystyle\phantom{}% \text{in }\Omega,\end{aligned}\right. where ( - Δ p ) s {(-\Delta_{p})^{s}} denotes the fractional p-Laplacian operator, p > 1 {p>1} , s ∈ ( 0 , 1 ) {s\in(0,1)} , p ⁢ s < N {ps , p s * = N ⁢ p N - p ⁢ s {p_{s}^{*}=\frac{Np}{N-ps}} is the critical Sobolev exponent, Ω is a bounded domain in ℝ N {\mathbb{R}^{N}} with Lipschitz boundary, and Q and H are homogeneous functions of degrees p and q with p < q ≤ p s ∗ {p and Q u {Q_{u}} and Q v {Q_{v}} are the partial derivatives with respect to u and v, respectively. To establish our existence result, we need to prove a concentration-compactness principle associated with the fractional p-Laplacian system for the fractional order Sobolev spaces in bounded domains which is significantly more difficult to prove than in the case of single fractional p-Laplacian equation and is of its independent interest (see Lemma 5.1). Our existence results can be regarded as an extension and improvement of those corresponding ones both for the nonlinear system of classical p-Laplacian operators (i.e., s = 1 {s=1} ) and for the single fractional p-Laplacian operator in the literature. Even a special case of our main results on systems of fractional Laplacian ( - Δ ) s {(-\Delta)^{s}} (i.e., p = 2 {p=2} and 0 < s < 1 {0 ) has not been studied in the literature before.

In this paper we discuss coefficient problems for functions in the class 𝒞 0 ⁢ ( k ) {{\mathcal{C}}_{0}(k)} . This family is a subset of 𝒞 {{\mathcal{C}}} , the class of close-to-convex functions, consisting of functions which are convex in the positive direction of the real axis. Our main aim is to find some bounds of the difference of successive coefficients depending on the fixed second coefficient. Under this assumption we also estimate | a n + 1 | - | a n | {|a_{n+1}|-|a_{n}|} and | a n | {|a_{n}|} . Moreover, it is proved that Re ⁡ { a n } ≥ 0 {\operatorname{Re}\{a_{n}\}\geq 0} for all f ∈ 𝒞 0 ⁢ ( k ) {f\in{\mathcal{C}}_{0}(k)} .

In this paper, we introduce a new contraction, namely, almost {\mathcal{Z}} contraction with respect to \zeta \in {\mathcal{Z}} , in the setting of complete metric spaces. We proved that such contraction possesses a fixed point and the given theorem covers several existing results in the literature. We consider an example to illustrate our result.

In this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly exploit sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity theory, as well as fixed point arguments. A uniqueness result is also presented.

In 2018, Das et al. [Characterization of rough weighted statistical statistical limit set, Math. Slovaca68(4) (2018), 881–896] (or, Ghosal et al. [Effects on rough𝓘-lacunary statistical convergence to induce the weighted sequence, Filomat32(10) (2018), 3557–3568]) established the result: The diameter of rough weighted statistical limit set (or, rough weighted 𝓘-lacunary limit set) of a sequencex= {xn}n∈ℕis2rlim infn∈Atn$\begin{array}{} \frac{2r}{{\liminf\limits_{n\in A}} t_n} \end{array}$if the weighted sequence {tn}n∈ℕis statistically bounded (or, self weighted 𝓘-lacunary statistically bounded), whereA= {k∈ ℕ :tk

In this paper, I will construct three families of Sidon sequences of certain subsets of ℝ, in particular I will study Farey sequences, square roots, and reciprocals. It will be shown that Sidon sequences over them have cardinality of between c 1 N 3 / 4 log N $\begin{array}{} \displaystyle c_1\frac{N^{3/4}} {\log{N}} \end{array}$ and c 2 N 3/4, c 3 N, and c 4 N log log N log N . $\begin{array}{} \displaystyle c_4 \frac{N\log{\log{N}}}{\log{N}}. \end{array}$

Given a finite graph H, the n th member Gn of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in G n−1 by adding edges or identifying vertices, always in the same way. The genus polynomial Γ G (z) of a graph G is the generating function enumerating all orientable embeddings of G by genus. Over the past 30 years, most calculations of genus polynomials Γ Gn (z) for the graphs in a linear family have been obtained by partitioning the embeddings of Gn into types 1, 2, …, k with polynomials Γ G n j $\begin{array}{} \Gamma_{G_n}^j \end{array}$ (z), for j = 1, 2, …, k; from these polynomials, we form a column vector V n ( z ) = [ Γ G n 1 ( z ) , Γ G n 2 ( z ) , … , Γ G n k ( z ) ] t $\begin{array}{} V_n(z) = [\Gamma_{G_n}^1(z), \Gamma_{G_n}^2(z), \ldots, \Gamma_{G_n}^k(z)]^t \end{array}$ that satisfies a recursion Vn (z) = M(z)V n−1(z), where M(z) is a k × k matrix of polynomials in z. In this paper, the Cayley-Hamilton theorem is used to derive a k th degree linear recursion for Γ n (z), allowing us to avoid the partitioning, thereby yielding a reduction from k 2 multiplications of polynomials to k such multiplications. Moreover, that linear recursion can facilitate proofs of real-rootedness and log-concavity of the polynomials. We illustrate with examples.