Volume 21, Issue 2

Abstract. In this paper, we extend the coupled coincidence point theorems for a mixed g -monotone operator F:X×X→X$F:X\times X\rightarrow X$ obtained by Alotaibi and Alsulami [Fixed Point Theory Appl. (2011), article ID 44], by weakening the involved contractive condition. Two examples are given to illustrate the effectiveness of our generalizations. Our result also generalizes some recent results announced in the literature. Moreover, some applications to integral equations are presented.

Abstract. In this paper, we introduce two symmetric q -Dunkl-classical orthogonal polynomial sets generalizing the Hermite polynomials as range of discrete q -Hermite polynomials I and II by a suitable orthogonality preserving operator. Then, we state some of their properties: explicit expressions, recurrence relations, generating functions, inversion formulas, linear functionals and components. We conclude by locating these polynomials among orthogonal polynomials generalizing Hermite polynomials and classified according to lowering operators.

Abstract. Let LE2$L^2_E$ be a Banach space of all ℙ-square integrable random variables defined on some probability space (Ω,𝒜,ℙ)$(\Omega ,\mathcal {A},\mathbb {P})$ with values in the complex separable Banach space E . For an LE2$L_{E}^{2}$-valued stationary continuous random function 𝔛=(Xg)g∈G$\mathfrak {X}=(X_{g})_{g\in G}$ on the locally compact abelian group G , we provide a condition of existence of a random measure Z taking values in LE2$L^{2}_{E}$ such that (Xg)g∈G$(X_{g})_{g\in G}$ has a spectral representation as a Fourier transform of Z .

Abstract. Suppose that S is a sublinear operator such as the Littlewood–Paley g function, the Lusin area integral or the intrinsic square function. We prove several sharp weighted inequalities for the m -th order commutator of the operator S with b∈ BMO $b\in \mathrm {BMO}$. Finally, we construct several examples showing that our bounds are sharp.

Abstract. It is shown that every infinite-dimensional real Banach space X contains a sequence (xn)n∈ℕ$(x_n)_{n\in \mathbb {N}}$ with the following properties: (a) Some subsequence of (∑k=1nxk)n∈ℕ$(\sum _{k=1}^nx_k)_{n\in \mathbb {N}}$ converges in X and supn∈ℕ∥∑k=1nxk∥≤1$\sup _{n\in \mathbb {N}}\Vert \sum _{k=1}^nx_k\Vert \le 1$; (b) ∑k=1∞∥xk∥p<∞$\sum _{k=1}^{\infty }\Vert x_k\Vert ^p&lt;\infty $ for every p∈]2,+∞[$p\in {]2,+\infty [}$; (c) for any permutation π:ℕ→ℕ$\pi :\mathbb {N}\rightarrow \mathbb {N}$ and any sequence (θn)n∈ℕ$(\theta _n)_{n\in \mathbb {N}}$ with θn∈{-1,1}$\theta _n\in \lbrace -1,1\rbrace $, n=1,2,...$n=1,2,\ldots $, the series ∑k=1∞θkxπ(k)$\sum _{k=1}^{\infty }\theta _k x_{\pi (k)}$ diverges in X . This result implies, in particular, that the rearrangement theorem and the Dvoretzky–Hanani theorem fail drastically for infinite-dimensional Banach spaces.

Abstract. In the present paper we investigate a three-dimensional boundary-contact problem of dynamics for a homogeneous hemitropic elastic medium with regard to friction. We prove the uniqueness theorem using the corresponding Green formulas and positive definiteness of the potential energy. To analyze the existence of solutions we reduce equivalently the problem under consideration to a spatial variational inequality. We consider a special parameter-dependent regularization of this variational inequality which is equivalent to the relevant regularized variational equation depending on a real parameter and study its solvability by the Faedo–Galerkin method. Some a priori estimates for solutions of the regularized variational equation are established and with the help of an appropriate limiting procedure the existence theorem for the original contact problem with friction is proved.

Abstract. In this paper, we apply two theorems from triple critical points theory to establish the existence of at least three solutions for the quasilinear second order differential equations on a compact interval [a,b]⊂ℝ$[a,b]\subset \mathbb {R}$, -(pi-1)|ui'(x)|pi-2ui''(x)=λFui(x,u1,⋯,un)hi(x,ui'),x∈(a,b),ui(a)=ui(b)=0,$ {\left\lbrace \begin{array}{ll} -(p_{i}-1)|u_{i}^{\prime }(x)|^{p_{i}-2}u_{i}^{\prime \prime }(x)=\lambda F_{u_{i}}(x,u_{1},\dots ,u_{n})h_{i}(x,u_{i}^{\prime }), &amp; x\in (a,b), u_{i}(a)=u_{i}(b)=0, \end{array}\right.} $ for 1≤i≤n$1\le i\le n$, under appropriate hypotheses. The obtained results extend some existing results on the subject.

Abstract. New results on the existence, uniqueness and maximal regularity of a solution are given for a parabolic equation set in a non-regular domain Q=(t,x1)∈ℝ2:0<t<T;ϕ1(t)<x1<ϕ2(t)×∏i=1N-1]0,bi[$ Q=\bigl \lbrace (t,x_1) \in \mathbb {R}^{2}:0&lt;t&lt;T;\; \varphi _1(t) &lt;x_1&lt;\varphi _{2}(t) \bigr \rbrace \times \prod _{i=1}^{N-1}{]0,b_i[} $ of ℝ N+1 , with Cauchy–Dirichlet boundary conditions, under some assumptions on the functions ϕ1$\varphi _1$ and ϕ2$\varphi _2$. The right-hand side term of the equation is taken in L2(Q)$L^{2}(Q)$. The method used is based on the approximation of the domain Q by a sequence of sub-domains (Qn)$(Q_n)$ which can be transformed into regular domains. This work is an extension of the two space variables case studied in [Differ. Equ. Appl. 2 (2010), no. 2, 251–263].

Abstract. For singular in a phase variable second order differential inequalities, a priori estimates of positive solutions, satisfying nonlinear nonlocal boundary conditions, are established.

Abstract. The behavior of the quasiconformal reflection coefficient and of the Fredholm eigenvalue of a plane analytic Jordan curve is studied under the square root transformation.

Abstract. In this paper, the expansion of a basic confluent hypergeometric function in terms of an incomplete q -gamma function is investigated and shown to be an asymptotic expansion for a basic confluent hypergeometric function and a q -analogue of the Hadamard expansion. The general expansion associated with a basic hypergeometric function is introduced. In special cases, this provides alternative derivations of the q -Hadamard expansion. Expansions and asymptotics for modified q -Bessel functions are derived as special cases from the general expansion.