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The main result of this paper states that if $${\mathcal {F}}$$ is a subgroup-closed saturated formation of full characteristic, then the $${\mathcal {F}}$$ -residual of a K- $${\mathcal {F}}$$ -subnormal subgroup S of a finite group G is a large subgroup of G provided that the $${\mathcal {F}}$$ -hypercentre of every subgroup X of G containing S is contained in the $${\mathcal {F}}$$ -residual of X. This extends a recent result of Aivazidis, Safonova and Skiba.

Let X and Y be two Banach spaces. A bounded operator $$T: X \longrightarrow Y$$ is said to be a BS-compact operator whenever T sends Banach-Saks subsets of X onto norm compact sets of Y. In this paper, our central focus is upon introducing the class of almost BS-compact operators. The paper rests essentially on two parts. The first is devoted to the connection of this new class of operators with classical notions of operators, such as BS-compact operators, AM-compact operators, and Dunfort-Pettis operators. The second part is dedicated to the domination problem within the framework of (almost) BS-compact operators.

This paper deals with a fourth-order nonlinear parabolic equation with logarithmic nonlinearity coming from the modeling of epitaxial growth. First, by establishing a new infinite time blow-up condition which is independent of the mountain-pass level, we show the solution can be extended over time (the whole half line) and then blows up at $$\infty $$ ; second, we prove that the solution can blow up at $$\infty $$ with arbitrary initial energy using this new infinite time blow-up condition; thirdly, some numerical simulations are presented to verify and illustrate the theoretical results. The results of this paper complete and extend the previous studies on this type of model.

Mathematics Subject Classification. 42C05, 33C50., The authors are grateful to the referee for his/her valuable comments and careful reading, which allowed us to improve this paper. The work of the first author (MEM) has been supported by Ministerio de Ciencia, Innovaci´on y Universidades (MICINN) grant PGC2018-096504-B-C33. Second and third authors (TEP and MAP) thank FEDER/Ministerio de Ciencia, Innovaci´on y Universidades—Agencia Estatal de Investigaci´on/PGC2018-094932-B-I00 and Research Group FQM-384 by Junta de Andaluc´ıa. This work is supported in part by the IMAG-Mar´ıa de Maeztu grant CEX2020-001105-M/AEI/10. 13039/501100011033., The so-called Koornwinder bivariate orthogonal polynomials are generated by means of a non-trivial procedure involving two families of univariate orthogonal polynomials and a function ρ(t) such that ρ(t)2 is a polynomial of degree less than or equal to 2. In this paper, we extend the Koornwinder method to the case when one of the univariate families is orthogonal with respect to a Sobolev inner product. Therefore, we study the new Sobolev bivariate families obtaining relations between the classical original Koornwinder polynomials and the Sobolev one, deducing recursive methods in order to compute the coefficients. The case when one of the univariate families is classical is analysed. Finally, some useful examples are given., Ministerio de Ciencia, Innovación y Universidades (MICINN) grant PGC2018-096504-B-C33, EDER/Ministerio de Ciencia, Innovación y Universidades—Agencia Estatal de Investigación/PGC2018-094932-B-I00, Research Group FQM-384 by Junta de Andalucía, IMAG-María de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033, Universidad de Granada/CBUA

In this paper, first we prove the Crandall–Liggett exponential theorem in nonlinear semigroup theory on Hadamard manifolds. This theorem states that a semigroup of contractions can be constructed by the resolvent of a monotone vector field on Hadamard manifolds. Then, we show that the generated semigroup satisfies the evolution equation governed by the monotone vector field. The results of this paper are extensions of the classical results of Crandall and Liggett (Am J Math 93:265–298, 1971) and Brezis and Pazy (Israel J Math 8:367–383, 1970) to Hadamard manifolds. Some examples are also presented in the last part of the paper.

In this paper, we present some fixed point theorems of Krasnosel’skii’s type in Banach spaces. The involved operators need not to be compact nor weakly continuous. The results are obtained and formulated with the use of the measures of weak noncompactness and a large classes of contractions (strict contractions, nonlinear contractions, as well as nonexpansive or pseudocontractive mappings). Throughout the paper, we use the hypothesis $$\mathsf {(H1)}$$ and $$\mathsf {(H2)}$$ , which are one of the main ingredients of the proofs. Finally, with the obtained fixed point results, we discuss the existence of solutions to a stationary transport equation with delayed neutrons.

In the present paper, we investigate the existence and regularity of weak solutions for $$\psi $$ -Hilfer fractional boundary value problem in $$\mathbb {C}^{\alpha ,\beta ;\psi }_{2}$$ and $$\mathcal {H}$$ (Hilbert space) spaces, using extension of the Lax–Milgram theorem. In this sense, to finalize the paper, we discuss the integration by parts for $$\psi $$ -Riemann–Liouville fractional integral and $$\psi $$ -Hilfer fractional derivative.

The aim of this paper is to investigate the stability problem for a nonlinear dispersive equation with memory. More precisely, the equation under consideration combines the well-known Korteweg-de Vries and Kuramoto–Sivashinsky equations, subject to the presence of a boundary memory term. The problem is shown to be well-posed for small initial data and provided that reasonable conditions hold for the parameters of the system and the memory kernel. In addition, we prove that the trivial solution is exponentially stable in spite of the memory effect. It is noteworthy that these outcomes are obtained under numerous instances of the physical parameters of the system.

Funding for open access charge: Universidad de Granada / CBUA., The authors are grateful to Margarita Arias, Jos´e Antonio G´alvez and Francisco Martín for helpful comments during the preparation of this manuscript., In this paper, we study phi-minimal surfaces in R-3 when the function phi is invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in R-2. We describe a full classification of complete flat-embedded phi-minimal surfaces if phi is strictly monotone and characterize rotational phi-minimal surfaces by its behavior at infinity when phi has a quadratic growth., Universidad de Granada / CBUA

A canonical factorization is given for a quadratic pencil of accretive operators in a Hilbert space. Also, we establish some relationships between an m-accretive operator and its Moore-Penorse inverse. As an application, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation in the non-homogeneous case. The paper is concluded with some questions left open from the preceding discussions.

In this paper, we study the atomic decomposition for variable exponent Bergman spaces. We also give some characterizations for the boundedness and compactness of composition operators on these spaces, which corrects a result by Morovatpoor and Abkar in (Mediterr. J. Math. 17:9, 2020). Furthermore, we investigate the difference of two composition operators on variable exponent Bergman spaces.

Let N be a normal subgroup of a finite group G, and x an element of N. Objective that $$|x^G|=|G:C_G(x)|$$ , so $$|x^G|$$ is called “minimal” when $$C_G(x)$$ is a maximal subgroup of G. In this paper, we characterize the structure of N when $$|x^G|$$ is minimal for every non-G-central element x of N.

In this paper, we deal with the quasilinear Schrodinger equation $$\begin{aligned} -\epsilon ^{2}\Delta u+V(x)u-\epsilon ^2u\Delta (u^2)=h(u)+ u^{22^*-1},\ u>0,\ x\in \mathbb {R}^{N}, \end{aligned}$$ where $$\epsilon >0$$ is a small parameter, $$N\ge 3$$ , V is continuous and h is of subcritical growth. When V satisfies a local condition and h is merely continuous, we obtain the multiplicity and concentration of solutions using the method of Nehari manifold, penalization techniques and Ljusternik–Schnirelmann category theory.

Let h be a non-vanishing convex univalent function and p be an analytic function in $$\mathbb {D}$$ . We consider the differential subordination $$\psi _i(p(z), z p'(z)) \prec h(z)$$ with the admissible functions $$ \psi _1:=(\beta p(z)+\gamma )^{-\alpha }\left( \tfrac{(\beta p(z)+\gamma )}{\beta (1-\alpha )}+ z p'(z)\right) $$ and $$\psi _2:=\tfrac{1}{\sqrt{\gamma \beta }}\arctan \left( \sqrt{\tfrac{\beta }{\gamma }}p^{1-\alpha }(z)\right) +\left( \tfrac{1-\alpha }{\beta p^{2 (1-\alpha )}(z)+\gamma }\right) \tfrac{z p'(z)}{p^{\alpha }(z)}$$ . The objective of this paper is to find the dominants, preferably the best dominant (say q) of the solution of the above differential subordination satisfying $$\psi _i(q(z), n zq'(z))= h(z)$$ . Furthermore, we show that $$\psi _i(q(z),zq'(z))= h(z)$$ is an exact differential equation and q is a convex univalent function in $$\mathbb {D}$$ . In addition, we estimate the sharp lower bound of $${{\,\mathrm{Re}\,}}p$$ for different choices of h and derive a univalence criterion for functions in $$\mathcal {H}$$ (class of analytic normalized functions) as an application to our results.

In this paper, we consider the nonlocal double phase problems in unbounded domains. At first, we establish a maximum principle for these problems. Then we prove the monotonicity and uniqueness of positive solutions by developing a sliding method for nonlocal double phase equations.

This paper analyses a class of parabolic linear cooperative systems in a cylindrical domain with degenerate spatio-temporal potentials. In other words, potentials vanish in some non-empty connected subdomains which are disjoint and increase in size temporally. Then, the vanishing subdomains for the potentials are not cylindrical. Following a similar idea to the semiclassical analysis behaviour, but done here for parabolic problems, under these geometrical assumptions, the asymptotic behaviour of the system is ascertained when a parameter, in front of these potentials, goes to infinity. In particular, the strong convergence of the solutions of the system is obtained using energy methods and the theory associated with the $$\Gamma $$ Γ -convergence. Also, the exponential decay of the solutions to zero in the exterior of the subdomains where the potentials vanish is achieved.

In this paper, we present a parameterized inexact Uzawa type method for solving a class of large sparse generalized saddle point problems, and analyze its convergence using a different approach from those utilized in the existing literature. Some numerical results are given to illustrate the effectiveness of the proposed method.

Extending derivability of a mapping from one point of a $$\mathrm C^*$$ -algebra to the entire space is one of the interesting problems in derivation theory. In this paper, by considering a $$\mathrm C^*$$ -algebra A as a Jordan triple with triple product $$\{a,b,c\}=(ab^*c+cb^*a)/2$$ , and its dual space as a ternary A-module, we prove that a continuous conjugate linear mapping T from A into its dual space is a ternary derivation whenever it is ternary derivable at zero and the element T(1) is skew-symmetric.

The purpose of this paper is to present a new version of the Bihari inequality with singular kernel and give a simple proof of the fractional Gronwall lemma. Our new ideas rest on the use of Young’s and Holder’s inequalities to simplify the complex inequalities. Based on this new type of Bihari inequality we can relax many results of fractional differential equations and inclusions and stochastic differential equations. Also, the obtained inequalities can be used to analyze a specific class of fractional differential equations, both linear and nonlinear. Using the Caputo fractional derivative, the study of an initial valued problem for a fractional differential equation provides some topological proprieties for the solution set, and shows it is the intersection of a decreasing sequence of compact nonempty contractible spaces. We extend the classical Kneser’s theorem on the solution structure of the ordinary differential equation and relax some results about the fractional differential equation. Also, we establish existence results for Caputo fractional stochastic differential equations. Finally, we study the existence of solution for fractional differential inclusion in Banach lattice.

In this paper, the authors investigate the oscillatory behavior of quasilinear second order delay difference equations of the form $$\begin{aligned} \Delta (b(n)(\Delta u(n))^{\alpha })+p(n)u^{\beta }(n-\sigma )=0. \end{aligned}$$ By obtaining new monotonic properties of the nonoscillatory solutions and using them to linearize the equation leads to new oscillation criteria. The criteria obtained improve existing ones in the literature. Two examples are included to show the importance of the main results.

The interest of this paper is to deal with long-time behavior of the solutions to the following Von Karman equation involving variable sources and clamped boundary conditions: $$\begin{aligned} \quad u_{tt}+\Delta ^{2} u+a|u_t|^{m(x)-2}u_{t}=[u,F(u)]+b|u|^{p(x)-2}u,\quad \Delta ^{2}F(u)=-[u,u]. \end{aligned}$$ First of all, the authors construct a new control function and apply the Sobolev embedding inequality to establish some qualitative relationships among initial energy value, the term $$\int _{\Omega }\frac{1}{p(x)}|u|^{p(x)}\mathrm{d}x$$ and the Airy stress functions, which ensure that the energy functional are nonnegative with respect to time variable. And then, some energy estimates and Komornik inequality is used to prove a uniform estimate of decay rates of the solution which provides an estimation of long-time behavior of solutions. As we know, such results are seldom seen for the variable exponent case.

In this paper, we are concerned by the study of a discontinuous elliptic problem involving a fractional p-Laplacian arising in differents context. Under suitable conditions, we provide the existence and multiplicity result via the nonsmooth critical point theory.

In this paper, we study the nonexistence of nontrivial global solutions for a nonlinear fractional differential. The equation involves two Hadamard fractional derivatives of different orders and a nonlinear source term. Our results are obtained using the test-function method, several properties of fractional derivatives and some integral inequalities. An example is provided to illustrate our findings.

The aim of this paper is to study the following non-homogeneous Neumann-type problem 0.1 $$\begin{aligned} \left\{ \begin{array} {ll} - \mathrm{{div}}(\alpha (\vert \nabla u\vert )\nabla u) + u = u \vert u\vert ^{p - 2}, &{} \quad \mathrm{in } \, B_{1}, \\ \dfrac{\partial u}{\partial \nu } = 0, &{} \quad \mathrm{on } \, \partial B_{1} , \end{array}\right. \end{aligned}$$ where $$B_{1}$$ is the unit ball in $$\mathbb {R}^{n}$$ and $$p > 2$$ . We establish the existence of a non-constant, positive, radially non-decreasing weak solution for (0.1), under certain assumptions on $$\alpha $$ . Our approach relies on the theory of Orlicz spaces combined with a new variational method that allows one to deal with problems beyond the usual locally compactness structure and a variant of Mountain Pass Theorem.

In the last decade, many authors essentially contributed to the attractive theory of multiple zeta values. Nevertheless, since their introduction in 1992, there are still many hypotheses and open problems waiting to be solved. The aim of this paper is to develop a method for transforming the multiple zeta-star values $$\zeta ^\star (\{2\}_K,c)$$ leading to a new sum formula for alternating multiple zeta-star values. Its most simple case has the intelligible form $$\begin{aligned} \sum _{t=0}^{c-2}(-2)^{t+1} \sum _{\begin{array}{c} i\ge 2,\,\varvec{s}\in \mathbb {N}^t\\ i+|\varvec{s}\!|=c \end{array}} \zeta ^\star ({\overline{i}},\varvec{s}) =(-1)^c\cdot \zeta (c). \end{aligned}$$ As a by-product, we also establish a closed form for a new harmonic-like finite summation containing binomial coefficients.

We study the product weighted Hardy–Littlewood average $$\mathcal {H}_{\varphi }$$ in this paper. More precisely, we first give the sufficient and necessary condition for the boundedness of $$\mathcal {H}_{\varphi }$$ on the product central Morrey space $$\vec {\dot{B}}^{p,\lambda }(\mathbb {R}^n\times \mathbb {R}^m)$$ , and obtain the sharp constant at the same time. Then we obtain a characterization of the boundedness for the commutator formed by $$\mathcal {H}_{\varphi }$$ and a product central bounded mean oscillation function b. As a consequence, we give a complete answer to a question posed by Fu et al. (Forum Math 27(5):2825–2851, 2015).

In this paper, the existence, uniqueness and $$L^{p}$$ -regularity properties of solutions of initial value problem for improved abstract Boussinesq equation is obtained. The equation includes a linear operator A in a Banach space E. We can obtain the existence, uniqueness and qualitative properties a different classes improved Boussinesq equations by choosing the space E and linear operator A, which occur in a wide variety of physical systems. By applying this result, initial value problem for nonlocal Boussinesq equations and mixed problem for degenerate Boussinesq equations are studied.

Given a graph G and a subset of vertices $$D\subseteq V(G)$$ D ⊆ V ( G ) , the external neighbourhood of D is defined as $$N_e(D)=\{u\in V(G){\setminus } D:\, N(u)\cap D\ne \varnothing \}$$ N e ( D ) = { u ∈ V ( G ) \ D : N ( u ) ∩ D ≠ ∅ } , where N(u) denotes the open neighbourhood of u. Now, given a subset $$D\subseteq V(G)$$ D ⊆ V ( G ) and a vertex $$v\in D$$ v ∈ D , the external private neighbourhood of v with respect to D is defined to be $$\mathrm{epn}(v,D)=\{u\in V(G){\setminus } D: \, N(u)\cap D=\{v\}\}.$$ epn ( v , D ) = { u ∈ V ( G ) \ D : N ( u ) ∩ D = { v } } . The strong differential of a set $$D\subseteq V(G)$$ D ⊆ V ( G ) is defined as $$\partial _s(D)=|N_e(D)|-|D_w|,$$ ∂ s ( D ) = | N e ( D ) | - | D w | , where $$D_w=\{v\in D:\, \mathrm{epn}(v,D)\ne \varnothing \}$$ D w = { v ∈ D : epn ( v , D ) ≠ ∅ } . In this paper, we focus on the study of the strong differential of a graph, which is defined as $$\begin{aligned} \partial _s(G)=\max \{\partial _s(D):\, D\subseteq V(G)\}. \end{aligned}$$ ∂ s ( G ) = max { ∂ s ( D ) : D ⊆ V ( G ) } . Among other results, we obtain general bounds on $$\partial _s(G)$$ ∂ s ( G ) and we prove a Gallai-type theorem, which states that $$\partial _s(G)+\gamma _{_I}(G)=\mathrm{n}(G)$$ ∂ s ( G ) + γ I ( G ) = n ( G ) , where $$\gamma _{_I}G)$$ γ I G ) denotes the Italian domination number of G. Therefore, we can see the theory of strong differential in graphs as a new approach to the theory of Italian domination. One of the advantages of this approach is that it allows us to study the Italian domination number without the use of functions. As we can expect, we derive new results on the Italian domination number of a graph.

In this paper, we study the constrained minimization problem (1.1) of the Kirchhoff–Schrodinger energy functional under an $$L^{2}$$ -subcritical perturbation. The existence and nonexistence of constraint minimizers are completely classified in terms of the $$L^{2}$$ -subcritical exponent q. Especially for $$q\in (\frac{4}{3},\frac{8}{3})$$ , we prove that there exists a critical value $$\beta ^{*}$$ such that (1.1) has no minimizer if the coefficient $$\beta $$ of $$L^{2}$$ -critical term satisfies $$\beta =\beta ^{*}$$ . For $$q\in (\frac{4}{3},\frac{8}{3})$$ , the blow-up behavior of minimizers as $$\beta \nearrow \beta ^{*}$$ are also analyzed rigorously if the coefficient $$\lambda $$ of $$L^{2}$$ -subcritical term satisfies $$\lambda >\lambda _{0}$$ , where $$\lambda _{0}$$ is a positive constant.

In this paper, we propose a new three term conjugate gradient method, in which the search direction is close to the direction in the memoryless BFGS method. The global convergence of the method is established under a modified Wolfe line search. Results of numerical experiments presented confirm that the three term method is effective and superior to some other conjugate gradient methods if the standard Wolfe line search strategy is used. Furthermore, the method produces a set of even better results when implemented under a modified Wolfe line search.

In this paper, (dual) purely Rickart objects are introduced as generalizations of (dual) Rickart objects in Grothendieck categories. Examples showing the relations between (dual) relative Rickart objects and (dual) relative purely Rickart objects are given. It is shown that in a spectral category, (dual) relative purely Rickart objects coincide with (dual) relative Rickart objects. (Co)products of (dual) relative purely Rickart objects are studied. Classes all of whose objects are (dual) relative purely Rickart are identified. It is shown how this theory may be employed to study (dual) relative purely Baer objects in Grothendieck categories. Also applications to module and comodule categories are given.

In this paper, we exhibit that non-trivial concircular vector fields play an important role in characterizing spheres, as well as Euclidean spaces. Given a non-trivial concircular vector field $$\xi $$ on a connected Riemannian manifold (M, g), two smooth functions $$\sigma $$ and $$\rho $$ called potential function and connecting function are naturally associated to $$\xi $$ . We use non-trivial concircular vector fields on n-dimensional compact Riemannian manifolds to find four different characterizations of spheres $$ {\mathbf {S}}^{n}(c)$$ . In particular, we prove an interesting result namely an n-dimensional compact Riemannian manifold (M, g) that admits a non-trivial concircular vector field $$\xi $$ such that the Ricci operator is invariant under the flow of $$\xi $$ , if and only if, (M, g) is isometric to a sphere $$ {\mathbf {S}}^{n}(c)$$ . Similarly, we find two characterizations of Euclidean spaces $${\mathbf {E}}^{n}$$ . In particular, we show that an n-dimensional complete and connected Riemannian manifold (M, g) admits a non-trivial concircular vector field $$\xi $$ that annihilates the Ricci operator, if and only if, (M, g) is isometric to the Euclidean space $${\mathbf {E}}^{n}$$ .

In this paper, the subadditivity and superadditivity of convex functions and concave functions on positive operators are proved by applying the function orders preserving or reversing operator inequalities. Moreover, a version of Jensen’s operator inequality without operator convexity is applied to obtain some results on subadditivity or superadditivity of a class of non-negative functions which are not necessarily convex or concave.

Functional encryption (FE) is an exciting new public key paradigm that provides solutions to most of the security challenges of cloud computing in a non-interactive manner. In the context of FE, inner product functional encryption (IPFE) is a widely useful cryptographic primitive. It enables a user with secret key $$usk_\mathbf {y}$$ associated to a vector $$\mathbf {y}$$ to retrieve only $$\langle \mathbf {x},\mathbf {y}\rangle $$ from a ciphertext encrypting a vector $$\mathbf {x}$$ , not beyond that. In the last few decades, several constructions of IPFE have been designed based on traditional classical cryptosystems, which are vulnerable to large enough quantum computers. However, there are few quantum computer resistants i.e., post-quantum IPFE. Multivariate cryptography is one of the promising candidates of post-quantum cryptography. In this paper, we propose for the first-time multivariate cryptography-based IPFE. Our work achieves non-adaptive simulation-based security under the hardness of the MQ problem.

In this paper, we obtain some basic inequalities for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians.

In this paper, we define a dual $$\phi $$ -combination $${\widetilde{Q}}_{\phi , \xi }$$ . Using the log-convexity of strictly decreasing convex function $$\phi ^{-1}$$ , we give the dual $$\phi $$ -Brunn–Minkowski inequality. Moreover, the equivalence between the dual $$\phi $$ -Brunn–Minkowski inequality and the dual $$\phi $$ -Minkowski mixed volume inequality is demonstrated.

Let $$(M^n, g(t))$$ be a compact Riemannian manifold. In this paper, we derive the evolution formula for the geometric constant $$\lambda _{a}^{b} (g)$$ as an infimum of a certain energy function when the following partial differential equation: $$\begin{aligned} -\Delta _{\phi } u + a u \log u + b S u = \lambda _{a}^{b}(g) u \end{aligned}$$ with $$\int _M u^2 d\mu = 1$$ , has positive solutions, where a and b are real constants along the extended Ricci flow and the normalized extended Ricci flow. In addition, we derive some monotonicity formulas by imposing some conditions along both the extended Ricci flow and the normalized extended Ricci flow.

In the present article, we make use of the set partitions and the generating functions to give new combinatorial relations for the generalized central factorial numbers. In the second part of the paper, we present a relationship between the Bernoulli polynomials and the Stirling numbers with higher level.

In this paper, we obtain several versions of the real Paley-Wiener theorems for a generalized Fourier transform associated to a Dunkl-type differential-difference operator on the real line, which is related to Lions’ second-order singular differential operator, and includes Dunkl operator associated with the reflection group $${\mathbb {Z}}_2$$ on $${\mathbb {R}}$$ as a special case.

The aim of this paper is twofold. First, we adapt the Burgeth’s spherical cap method [Manuscripta Math. 77:283–291, 1992 by Burgeth and Proceedings of the NATO Advanced Research Workshop on Classical and Modem Potential Theory and Applications, pp 133–147, 1994 by Burgeth] to the planar case to establish some Schwarz type lemmas for mappings admitting general Poisson type representations on the unit disk. Second, we prove a Landau type theorem for $$T_\alpha $$ -harmonic functions introduced by Olofsson (J Anal Math 123:227–249, 2014).

We study the existence and multiplicity of weak solutions for a Kirchhoff–Schrodinger type problem in $$\mathbb R^4$$ involving a critical nonlinearity and a suitable small perturbation. When $$N=4$$ , the Sobolev exponent is $$2^*=4$$ and, as a consequence, there is a tie between the growth for the nonlocal term and critical nonlinearity. Such behaviour causes new difficulties to treat our study from an exclusively variational point of view, besides those already known for the local operators. Some tools we used in this paper are the mountain-pass and Ekeland’s Theorems and the Lions’ Concentration Compactness Principle.

In this paper, using the Weierstrass–Enneper formula and the hodographic coordinate system, we find the relationships between the Ramanujan identity and a generalized class of minimal translation surfaces, known as affine minimal translation surfaces. We find the Dirichlet series expansion of the affine Scherk surface. We also obtain some of the probability measures of affine Scherk surface with respect to its logarithmic distribution. Next, we classify the affine minimal translation surfaces in $${\mathbb {L}}^3$$ and remark the analogous forms in $${\mathbb {L}}^3.$$

The goal of the present paper is to study the viscoelastic wave equation with variable exponents $$\begin{aligned} u_{tt}-\Delta _{p(x)}u-\Delta u+\int _0^tg(t-s)\Delta u(s)\mathrm{{d}}s-\Delta u_t=|u|^{q(x)-2}u \end{aligned}$$ under initial-boundary value conditions, where the exponents of nonlinearity p(x) and q(x) are given functions. To be more precise, blow-up in finite time is proved, upper and lower bounds of the blow-up time are obtained as well. The global existence of weak solutions is presented, moreover, a general stability of solutions is obtained. This work generalizes and improves earlier results in the literature.

Let $$(F_n)_{n\ge 0}$$ be the Fibonacci sequence given by $$F_{n+2}=F_{n+1}+F_n$$ for $$n\ge 0$$ , where $$F_0=0$$ and $$F_1=1$$ . In this paper, we explicitly find all solutions of the title Diophantine equation using lower bounds for linear forms in logarithms and properties of continued fractions. Further, we use a version of the Baker–Davenport reduction method in Diophantine approximation due to Dujella and Peth $$\ddot{\text {o}}$$ .

Renormalization Group (RG) method has been recognized as an unified and effective approach for singularly perturbed problems in many fields, such as oscillation and boundary layer problems in ordinary differential equations (ODEs), center manifolds in dynamical systems, and long-time asymptotic behavior in partial differential equations (PDEs), etc. In this paper, we are going to investigate the application of RG method to the asymptotic behavior of a class of stochastic differential equation with multiplicative noise. We will first show how to formulate the RG equation and the approximate solution. Then, the rigorous estimation of the error between the approximate solution and the real one will be presented.

In this paper we prove $$L^p$$ estimates for Stein’s square functions associated with Fourier–Bessel expansions. Furthermore, we prove transference results for square functions from Fourier–Bessel series to Hankel transforms. Actually, these are transference results for vector-valued multipliers from discrete to continuous in the Bessel setting. As a consequence, we deduce the sharpness of the range of p for the $$L^p$$ -boundedness of Fourier–Bessel Stein’s square functions from the corresponding property for Hankel–Stein square functions. Finally, we deduce $$L^p$$ estimates for Fourier–Bessel multipliers from that ones we have got for our Stein square functions.

In this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.

In this paper, we study some existence and uniqueness results for systems of differential equations in which each of equations of the system involves a different Stieltjes derivative. Specifically, we show that this problems can only have one solution under the Osgood condition, or even, the Montel–Tonelli condition. We also explore some results guaranteeing the existence of solution under these conditions. Along the way, we obtain some interesting properties for the Lebesgue– Stieltjes integral associated to a finite sum of nondecreasing and left–continuous maps, as well as a characterization of the pseudometric topologies defined by this type of maps Ignacio Márquez Albés was partially supported by Xunta de Galicia under grant ED481A-2017/095 and project ED431C 2019/02. F. Adrián F. Tojo was partially supported by Xunta de Galicia, project ED431C 2019/02, and by the Agencia Estatal de Investigación (AEI) of Spain under Grant MTM2016-75140-P, co-financed by the European Community fund FEDER SI

Let $${\mathcal {H}}$$ be a separable Hilbert space. It is known that the finite sum of Bessel sequences in $${\mathcal {H}}$$ is still a Bessel sequence. But the finite sum of generalized notions of frames does not necessarily remain stable in its initial form. In this paper, for a prescribed Bessel sequence $$F=\{f_n\}_{n=1}^\infty $$ , we introduce and study $${\mathcal {KF}}$$ , the set consisting of all operators $$K\in {\mathcal {B}}({\mathcal {H}})$$ , such that $$\{f_n\}_{n=1}^\infty $$ is a K-frame. We show that $${\mathcal {KF}}$$ is a right ideal of $${\mathcal {B}}({\mathcal {H}})$$ . We indicate by an example that $${\mathcal {KF}}$$ is not necessarily a left ideal. Moreover, we provide some sufficient conditions for the finite sum of K-frames to be a K-frame. We also use some examples to compare our results with existing ones. These examples demonstrate that our achievements do not depend on the available results. Furthermore, we study the same subject for K-g-frames and controlled frames and get some similar significant results.

In this paper, we study the stationary incompressible power law fluid flow in a thin porous medium. The media under consideration is a bounded perforated 3D domain confined between two parallel plates, where the distance between the plates is very small. The perforation consists in an array solid cylinders, which connect the plates in perpendicular direction, distributed periodically with diameters of small size compared to the period. For a specific choice of the thickness of the domain, we found that the homogenization of the power law Stokes system results a lower-dimensional nonlinear Brinkman type law.