In a bounded domain $\mathrm{\Omega }\subset {ℝ}^N$, $N\ge 2$, we study the solvability of the boundary value problem

$$-{\mathrm{\Delta }}_{p}u=H\left(x,u,\nabla u\right)\text{ in}\mathrm{\Omega },u=0\text{ on}\partial \mathrm{\Omega },$$

where $-{\mathrm{\Delta }}_{p}u=-div\left(|\nabla u{|}^{p-2}\nabla u\right)$ is the $p$-Laplace operator with $1<p<N$ and the nonlinearity $H$ satisfies $\left|H\left(x,u,\nabla u\right)\right|\le a|\nabla u{|}^q+b|u{|}^r+f\left(x\right)$ with $a,b\ge 0$ and $q,r>0$. Assuming $f\in L^m\left(\mathrm{\Omega }\right)$ for a suitable exponent $m>1$, we prove the existence of solutions for different sets of values $\left(q,r\right)$. To this end, we apply the classical Schauder fixed point theorem, relying on some well-known uniqueness, regularity estimates and stability results for renormalized and weak solutions. In the last section we make some examples to illustrate the applicability of our results to a large class of equations.

In this paper, we study the existence (uniqueness) of solutions for nonlinear fractional differential equations with different orders of Hadamard fractional derivatives involving associated with nonlocal boundary conditions. Several fixed point theorems are used for sufficient conditions of the existence (uniqueness) of solutions to nonlinear differential equations such as Banach’s contraction principle, the Leray–Schauder nonlinear alternative, and Krasnoselskii’s fixed point theorem. Applications of the main results are also presented.

We obtain generalized inequalities for wavelet frames with composite dilations by virtue of Fourier transform.

A characterization is obtained for the Fourier transform of functions belonging to ${\mathcal{ℒ}}^2\left(ℝ,H\right)$, where $H$ denotes a Hilbert $C^{\ast }$-module. But in the case of functions belonging to $L^1\left(ℝ,H\right)$ a similar result is proved when $H$ is a separable Hilbert space.

Weaving frame theory was recently introduced to deal with some problems in signal processing and wireless sensor networks. Also, fusion frames are an important extension of frames, applied in many areas especially in wireless sensor networks. In this paper, we survey the notion of weaving Hilbert space fusion frames. This concept has potential applications in wireless sensor networks which require distributed processing using different fusion frames. Indeed, we present several approaches for identifying and constructing weaving fusion frames in terms of local frames, bounded operators in Hilbert spaces and also dual fusion frames. To this end, we present some conditions under which a fusion frame with its duals constitute pairs of woven fusion frames. As a result, we show that fusion Riesz bases are woven with all of their duals. Finally, we obtain some new results on fusion frames and weaving fusion frames under operator perturbations.

We study a nonlinear Steklov boundary-value problem involving the weighted $p\left(\cdot \right)$-Laplacian. Using the Ricceri’s variational principle, we obtain the existence of at least three weak solutions in double weighted variable exponent Sobolev space.

The aim of this work is to study oscillatory behavior of solutions for a fourth-order neutral nonlinear differential equation ${\left(b\left(x\right){\left(w^{m-1}\left(x\right)\right)}^{\gamma }\right)}^{\prime }+{\sum }_{i=1}^j{q}_i\left(x\right)f\left(w\left(g_i\left(x\right)\right)\right)=0$, $x\ge x_0$. The results obtained are based on the Riccati transformation, integral averaging technique and the theory of comparison with second-order delay equations. The obtained results complements and generalize the earlier ones. Some examples are illustrated to show the applicability of the obtained results.

We focus on the Bohr radius problem for the derivatives of analytic functions, along with a technique of establishing Bohr inequalities in classical and generalized settings.

In [4], class number 2 is characterized among algebraic number rings using basic factorization tools. In this note, we extend these characterizations using finer factorization invariants.

The Köthe–Bochner spaces $L_{\rho }\left(X\right)$ are the vector valued version of the scalar Köthe spaces $L_{\rho }$, which generalize the Lebesgue spaces $L^p$, the Orlicz spaces and many other functional spaces. We study the possible identification between the dual ${\left(L_{\rho }\left(X\right)\right)}^{\prime }$ and the associated space $L_{{\rho }^{\prime }}\left(X^{\prime }\right)$ in connection with the Radon–Nikodým property of $X^{\prime }$.

We introduce the concept of a $k$-almost Yamabe soliton which extends naturally from Yamabe solitons. Our aim is to study the $k$-almost Yamabe soliton $\left(g,V,k,\lambda \right)$ on a contact metric manifold $M^{2n+1}$. Firstly, for a general contact metric manifold, it is proved that $V$ is Killing if the potential vector field $V$ is a contact vector field and that $M$ is $K$-contact if $V$ is collinear with Reeb vector field. Secondly, we prove that a compact $K$-contact manifold, admitting a $k$-almost Yamabe gradient soliton, is isometric to a standard unit sphere. Moreover, for a complete Sasakian manifold admitting a $k$-almost Yamabe soliton, we show that it is isometric to a standard unit sphere ${\mathbb{𝕊}}^{2n+1}\left(1\right)$ when $n>1$ and for $n=1$, $M$ is also isometric to a standard unit sphere if it admits a closed $k$-almost Yamabe soliton. Finally, we consider a contact metric $\left(\kappa ,\mu \right)$-manifold with a nontrivial $k$-almost Yamabe gradient soliton and show that it is flat in dimension $3$ and in higher dimension $M$ is locally isometric to $E^{n+1}\times {\mathbb{𝕊}}^n\left(4\right)$. In the end, we construct two examples of contact metric manifolds with a $k$-almost Yamabe soliton.

The authors consider the higher order complementary Lidstone boundary value problem. Some upper and lower bounds for positive solutions of the problem are obtained. Sufficient conditions for the existence and nonexistence of positive solutions are established.

Although the Bruhat order on a Weyl group is closely related to the singularities of the Schubert varieties for the corresponding Kac–Moody group, it can be difficult to use this information to prove general theorems. This paper uses the action of the affine Weyl group of type ${Ã}_2$ on a Euclidean space $V\cong {ℝ}^2$ to study the Bruhat order on $W$. We believe that these methods can be used to study the Bruhat order on arbitrary affine Weyl groups. Our motivation for this study was to extend the lookup conjecture of Boe and Graham (which is a conjectural simplification of the Carrell–Peterson criterion for rational smoothness) to type ${Ã}_2$. Computational evidence suggests that the only Schubert varieties in type ${Ã}_2$ where the “nontrivial” case of the lookup conjecture occurs are the spiral Schubert varieties, and as a step towards the lookup conjecture, we prove it for a spiral Schubert variety $X\left(w\right)$ of type ${Ã}_2$. The proof uses descriptions we obtain of the elements $x\le w$ and of the rationally smooth locus of $X\left(w\right)$ in terms of the $W$-action on $V$. As a consequence we describe the maximal nonrationally smooth points of $X\left(w\right)$. The results of this paper are used in a sequel to describe the smooth locus of $X\left(w\right)$, which is different from the rationally smooth locus.

In this survey article, we describe imploded cross-sections, which were developed in order to solve the problem that the cross-section of a Hamiltonian $K$-space is usually not symplectic. In some specific examples we contrast the intersection homology of some imploded cross-sections with their homology intersection spaces. Moreover, we compute the homology of intersection spaces associated to the open cone of a simply connected, smooth, oriented manifold and the suspension of such a manifold.

Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_1^k+p_2^2+p_3^2$, where $p_1$, $p_2$, $p_3$ are prime numbers, holds in intervals shorter than the ones previously known.

We define and study left and right generalized Drazin inverses for a closed densely defined linear operator in a Hilbert space. These operators are characterized by means of the generalized Kato decomposition and the single-valued extension property. We prove that they are invariant under commuting quasinilpotent perturbations. Finally, we give an application to solve singular linear equations.

In this paper we prove the existence of at least one nontrivial ground state solution for fourth-order elliptic equations of the form

$${\mathrm{\Delta }}^{2}u-\mathrm{\Delta }u+u=K\left(x\right)\left[f\left(u\right)+g\left(u\right)\right],x\in {ℝ}^4,u\in H^2\left({ℝ}^4\right),$$($P$)

where ${\mathrm{\Delta }}^2:=\mathrm{\Delta }\left(\mathrm{\Delta }\right)$ is the biharmonic operator, $f$ is a continuous nonnegative function with polynomial growth at infinity, $g$ is a continuous nonnegative function with exponential growth and $K$ is a positive bounded continuous function that can vanish at infinity. Our results complete the analysis made in F. Sani (Comm. Pure Appl. Anal. 12 (2013), 405–428), where the author studied Schrödinger equations involving the biharmonic operator with coercive potentials. Our approach is based on various techniques such as the mountain-pass theorem, Trundinger–Moser type inequalities and compactness results.

In the signal-processing literature, a frame is a mechanism for performing analysis and reconstruction in a Hilbert space. By contrast, in quantum theory, a positive operator-valued measure (POVM) decomposes a Hilbert-space vector for the purpose of computing measurement probabilities. Frames and their most common generalizations can be seen to give rise to POVMs, but does every reasonable POVM arise from a type of frame? We answer this question using a Radon–Nikodym-type result.

We study the Schur convexity of functions which are obtained from the differences of a refinement of Hermite–Hadamard inequality.

A formula for regularized sums of eigenvalues of a Sturm–Liouville problem with delayed argument at the point of discontinuity is obtained. Moreover, nodal points of the related problem are investigated.

We investigate the representations of integrals involving the product of the Legendre-chi function and the ${tanh}^{-1}x$ or $arctanx$ functions. The investigation will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.

Let $\mathcal{𝒜}$ be an arbitrary $C^{\ast }$-algebra which contains a nontrivial projection $P_1$ and let $\phi :\mathcal{𝒜}\to \mathcal{𝒜}$ be a surjective map which satisfies

$$\phi \left(A_1\right){\bullet }_{\eta }\phi \left(A_2\right){\bullet }_{\eta }\cdots {\bullet }_{\eta }\phi \left(A_{n-1}\right){\bullet }_{\eta }\phi \left(P\right)=A_1{\bullet }_{\eta }{A}_2{\bullet }_{\eta }\cdots {\bullet }_{\eta }{A}_{n-1}{\bullet }_{\eta }P$$

for every $A_1,A_2,\dots ,A_{n-1}\in \mathcal{𝒜}$ $\left(n\ge 3\right)$, $P\in \left\{P_1,I-P_1\right\}$, and some $\eta \in \mathbf{C},$ such that $\left|\eta \right|=1$, $\eta \ne \pm 1$, and $A_1{\bullet }_{\eta }{A}_2{\bullet }_{\eta }\cdots {\bullet }_{\eta }{A}_{n-1}{\bullet }_{\eta }P$ is the Jordan multiple $\eta$-$\ast$-product where $A_1{\bullet }_{\eta }{A}_2=A_1{A}_2+\eta A_2{A}_1^{\ast }.$ We determine the concrete form of map $\phi$ on any arbitrary $C^{\ast }$-algebra. Also, if $n\ge 3$, $\eta =-1$ and $\mathcal{𝒜}$ is an arbitrary $\ast$-algebra (with unit $I$), over the complex field $\mathbf{C}$ which contains a nontrivial projection $P_1$, then $\phi$ is of the form $\phi \left(T\right)=T\phi \left(I\right)$ for all $T\in \mathcal{𝒜}$.

We consider numerical methods for solving the modified Cahn–Hilliard equation involving strong nonlinearities. A fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme is proposed to overcome the time-consuming computation of the nonlinear terms. The TT-M FE algorithm includes three main steps: Firstly, a nonlinear FE scheme is solved on a coarse time-mesh ${\tau }_c$. Here, the FE method is used for spatial discretization and the implicit second-order $𝜃$ scheme (containing both implicit Crank–Nicolson scheme and second-order backward difference method) is used for temporal discretization. Secondly, the Lagrange’s interpolation is used to obtain the interpolation result on the fine time-mesh. Finally, a linearized FE system is solved on a fine time-mesh $\tau$ $\left(\tau <{\tau }_c\right)$. The stability analysis and priori error estimates are provided in detail. Numerical examples are given to demonstrate the validity of the proposed scheme. The TT-M FE method is compared with the traditional Galerkin FE method and it is evident that the TT-M FE method can save the calculation time.

This paper investigates the optimal recovery of Sobolev spaces $W_{\infty }^r\left[-1,1\right],r\in \mathbb{N}$ in space $L_{\infty }\left[-1,1\right]$ and weighted spaces $L_{p,\omega }\left[-1,1\right],1\le p<\infty$ with $\omega$ a continuous integrable weight function in $\left(-1,1\right)$. We obtain the values of the sampling numbers of $W_{\infty }^r\left[-1,1\right]$ in $L_{\infty }\left[-1,1\right]$ and $L_{p,\omega }\left[-1,1\right],1\le p<\infty$. We prove that the Lagrange interpolation algorithms based on the Chebyshev nodes of the first kind are optimal for $p=\infty$. Meanwhile, we prove that the Lagrange interpolation algorithms based on the zeros of polynomial of degree $r$ with the leading coefficient $1$ of the least deviation from zero in $L_{p,\omega }\left[-1,1\right]$ are optimal for $1\le p<\infty$. We also give the optimal Lagrange interpolation algorithms when we ask the endpoints to be included in the nodes.

We study the fractional $p$-Laplacian problem with Hardy–Sobolev exponents. We prove: there is a ${\lambda }_0>0$ such that for any $\lambda \in \left(0,{\lambda }_0\right)$, the above problem possesses infinitely many solutions. We achieve our goal by making use of variational methods, more specifically, the Nehari manifold and Lusternik–Schnirelmann theory.

This paper studies the blow-up criteria of strong solutions to the 3-dimensional incompressible micropolar fluid equations with partial viscosities. Two blow-up criteria are given to the local-in-time strong solution. The proof is based on energy estimates and harmonic analysis methods.