Band 23, Heft 3

Existence results of positive solutions for a nonlocal elliptic problem of Kirchhoff-type are established. The approach is based on variational methods.

We study the existence result of a renormalized solution for a class of nonlinear parabolic equations of the form ∂⁡b⁢(x,u)∂⁡t-div⁡(a⁢(x,t,u,∇⁡u))+g⁢(x,t,u,∇⁡u)+H⁢(x,t,∇⁡u)=μ in ⁢Ω×(0,T),${\partial b(x,u)\over\partial t}-\operatorname{div}(a(x,t,u,\nabla u))+g(x,t,u% ,\nabla u)+H(x,t,\nabla u)=\mu\quad\text{in }\Omega\times(0,T),$ where the right-hand side belongs to L1⁢(QT)+Lp′⁢(0,T;W-1,p′⁢(Ω))${L^{1}(Q_{T})+L^{p^{\prime}}(0,T;W^{-1,p^{\prime}}(\Omega))}$ and b⁢(x,u)${b(x,u)}$ is unbounded function of u , -div⁡(a⁢(x,t,u,∇⁡u))${{-}\operatorname{div}(a(x,t,u,\nabla u))}$ is a Leray–Lions type operator with growth |∇⁡u|p-1${|\nabla u|^{p-1}}$ in ∇⁡u${\nabla u}$. The critical growth condition on g is with respect to ∇⁡u${\nabla u}$ and there is no growth condition with respect to u , while the function H⁢(x,t,∇⁡u)${H(x,t,\nabla u)}$ grows as |∇⁡u|p-1${|\nabla u|^{p-1}}$.

In this paper we consider the modification of Szász–Durrmeyer operators based on the Jain basis function. Voronovskaya-type estimates of point-wise convergence along with its quantitative version based on the weighted modulus of smoothness are given. Moreover, a direct approximation theorem for the operators is proved.

The transformation of the original characteristic equation of the multivelocity linear transport theory was carried out by expanding the scattering function for the problem to be solved as a spectral integral over a complete set of eigenfunctions for the previously solved transport problem. The obtained equation represents a singular integral equation containing a spectral integral over the spectrum of the solved problem, whose kernel depends on the difference between the scattering of the problem to be solved and that of the already solved problem. We consider also the examples illustrating the validity of such a transformation. M. Kanal and J. Davies made a similar transformation of the characteristic equation of the one-velocity transport theory.

In this paper, we prove a Tauberian theorem for the product of the Abel method and the Cesàro method of order α, which improves some classical Tauberian theorems for the Abel and Cesàro summability methods.

Fix a nonzero window g∈𝒮⁢(ℝn)${g\in\mathcal{S}(\mathbb{R}^{n})}$, a weight function w on ℝ2⁢n${\mathbb{R}^{2n}}$ and 1≤p,q≤∞${1\leq p,q\leq\infty}$. The weighted Lorentz type modulation space M⁢(p,q,w)⁢(ℝn)${M(p,q,w)(\mathbb{R}^{n})}$ consists of all tempered distributions f∈𝒮′⁢(ℝn)${f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})}$ such that the short time Fourier transform Vg⁢f${V_{g}f}$ is in the weighted Lorentz space L⁢(p,q,w⁢d⁢μ)⁢(ℝ2⁢n)${L(p,q,w\,d\mu)(\mathbb{R}^{2n})}$. The norm on M⁢(p,q,w)⁢(ℝn)${M(p,q,w)(\mathbb{R}^{n})}$ is ∥f∥M⁢(p,q,w)=∥Vg⁢f∥p⁢q,w${\|f\/\|_{M(p,q,w)}=\|V_{g}f\/\|_{pq,w}}$. This space was firstly defined and some of its properties were investigated for the unweighted case by Gürkanlı in [9] and generalized to the weighted case by Sandıkçı and Gürkanlı in [16]. Let 1<p1,p2<∞${1<p_{1},p_{2}<\infty}$, 1≤q1,q2<∞${1\leq q_{1},q_{2}<\infty}$, 1≤p3,q3≤∞${1\leq p_{3},q_{3}\leq\infty}$, ω1,ω2${\omega_{1},\omega_{2}}$ be polynomial weights and ω3${\omega_{3}}$ be a weight function on ℝ2⁢n${\mathbb{R}^{2n}}$. In the present paper, we define the bilinear multiplier operator from M⁢(p1,q1,ω1)⁢(ℝn)×M⁢(p2,q2,ω2)⁢(ℝn)${M(p_{1},q_{1},\omega_{1})(\mathbb{R}^{n})\times M(p_{2},q_{2},\omega_{2})(% \mathbb{R}^{n})}$ to M⁢(p3,q3,ω3)⁢(ℝn)${M(p_{3},q_{3},\omega_{3})(\mathbb{R}^{n})}$ in the following way. Assume that m⁢(ξ,η)${m(\xi,\eta)}$ is a bounded function on ℝ2⁢n${\mathbb{R}^{2n}}$, and define Bm⁢(f,g)⁢(x)=∫ℝn∫ℝnf^⁢(ξ)⁢g^⁢(η)⁢m⁢(ξ,η)⁢e2⁢π⁢i⁢〈ξ+η,x〉⁢𝑑ξ⁢𝑑η for all f,g∈𝒮⁢(ℝn). $B_{m}(f,g)(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\hat{f}(\xi)\hat{g}(% \eta)m(\xi,\eta)e^{2\pi i\langle\xi+\eta,x\rangle}\,d\xi\,d\eta\quad\text{for % all ${f,g\in\mathcal{S}(\mathbb{R}^{n})}$. }$ The function m is said to be a bilinear multiplier on ℝn${\mathbb{R}^{n}}$ of type (p1,q1,ω1;p2,q2,ω2;p3,q3,ω3)${(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})}$ if Bm${B_{m}}$ is the bounded bilinear operator from M⁢(p1,q1,ω1)⁢(ℝn)×M⁢(p2,q2,ω2)⁢(ℝn)${M(p_{1},q_{1},\omega_{1})(\mathbb{R}^{n})\times M(p_{2},q_{2},\omega_{2})(% \mathbb{R}^{n})}$ to M⁢(p3,q3,ω3)⁢(ℝn)${M(p_{3},q_{3},\omega_{3})(\mathbb{R}^{n})}$. We denote by BM⁢(p1,q1,ω1;p2,q2,ω2)⁢(ℝn)${\mathrm{BM}(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2})(\mathbb{R}^{n})}$ the space of all bilinear multipliers of type (p1,q1,ω1;p2,q2,ω2;p3,q3,ω3)${(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})}$, and define ∥m∥(p1,q1,ω1;p2,q2,ω2;p3,q3,ω3)=∥Bm∥${\|m\|_{(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})% }=\|B_{m}\|}$. We discuss the necessary and sufficient conditions for Bm${B_{m}}$ to be bounded. We investigate the properties of this space and we give some examples.

In this note we present a new proof of certain closed expressions derived by Lorch and Szego [5] for the Schlömilch series containing Struve functions 𝐇ν${\mathbf{H}_{\nu}}$ and show their validity for an essentially wider range of variables when ν>-12${\nu>-\frac{1}{2}}$.

The purpose of this paper is to study a nondifferentiable minimax semi-infinite programming problem in a complex space. For such a semi-infinite programming problem, necessary and sufficient optimality conditions are established by utilizing the invexity assumptions. Subsequently, these optimality conditions are utilized as a basis for formulating dual problems. In order to relate the primal and dual problems, we have also derived appropriate duality theorems.

In this paper we combine the Lie algebraic methods and the monomiality principle techniques to obtain new results concerning generalized Laguerre polynomials. Also, we derive generating relations involving modified relativistic Laguerre polynomials into the context of the representation D⁢(u,m0)${D(u,m_{0})}$ of a three-dimensional Lie algebra sl⁢(2)${\mathrm{sl}(2)}$.

It is shown that the difference between a Vitali–Bernstein selector and a partial Vitali–Bernstein selector can be of Lebesgue measure zero and of first Baire category. This result gives an answer to a question posed by G. Lazou.

It is well known that if the Hardy–Littlewood maximal operator is bounded in the variable exponent Lebesgue space Lp⁢(⋅)⁢[0;1]${L^{p(\,\cdot\,)}[0;1]}$, then p⁢(⋅)∈BMO1/log${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$. On the other hand, there exists an exponent p⁢(⋅)∈BMO1/log${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$, 1<p-≤p+<∞${1<p_{-}\leq p_{+}<\infty}$, such that the Hardy–Littlewood maximal operator is not bounded in Lp⁢(⋅)⁢[0;1]${L^{p(\,\cdot\,)}[0;1]}$. But for any exponent p⁢(⋅)∈BMO1/log${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$, 1<p-≤p+<∞${1<p_{-}\leq p_{+}<\infty}$, there exists a constant c>0${c>0}$ such that the Hardy–Littlewood maximal operator is bounded in Lp⁢(⋅)+c⁢[0;1]${L^{p(\,\cdot\,)+c}[0;1]}$. In this paper, we construct an exponent p⁢(⋅)${p(\,\cdot\,)}$, 1<p-≤p+<∞${1<p_{-}\leq p_{+}<\infty}$, 1/p⁢(⋅)∈BLO1/log${1/p(\,\cdot\,)\in\mathrm{BLO}^{1/\log}}$ such that the Hardy–Littlewood maximal operator is not bounded in Lp⁢(⋅)⁢[0;1]${L^{p(\,\cdot\,)}[0;1]}$.

A graph is bicritical if the removal of any pair of vertices decreases the domination number. We study the properties of bicritical graphs and their relation with critical graphs, and we obtain results for bicritical graphs with edge connectivity two or three. We also generalize the notion of the coalescence of two graphs and investigate the bicriticality of such graphs.

An exact asymptotic value of the logarithm for a counting process in the max-scheme is obtained.

The idea of functions of bounded differential variation was introduced by Bhatt, Dabhi and Kachhia in [2]. In the present paper, we introduce functions of bounded fractional differential variation using the Caputo-type fractional derivative instead of the commonly used first-order derivative. Various properties and relation with some known results of classical analysis are also studied. We prove that the space BFDV⁢[a,b]${\mathrm{BFDV}[a,b]}$ of all functions of bounded fractional differential variation on [a,b]${[a,b]}$ is a normed algebra under certain type of norms.

We consider the American put problem in a general one-dimensional diffusion model. The risk-free interest rate is constant, and volatility is assumed to be a function of time and stock price. We use the well-known parabolic obstacle problem and establish the continuity estimate of the optional exercise boundaries of the American put option with respect to the local volatilities, which may be considered as a generalization of the Achdou results [1].

In this paper, we are interested in the study of the asymptotic analysis of a dynamical problem in elasticity with nonlinear friction of Tresca type. The Lamé coefficients of a thin layer are assumed to vary with respect to the thin layer parameter ε and to depend on the temperature. We prove the existence and uniqueness of a weak solution for the limit problem. The proof is carried out by the use of the asymptotic behavior when the dimension of the domain tends to zero.

In this paper, we study a coupled system of nonlinear fractional differential equations involving m nonlinear terms, m∈ℕ*${m\in\mathbb{N^{*}}}$. We begin by introducing a new Banach space. Then, we establish new existence and uniqueness results using the Banach contraction principle. We also prove an existence result using the Schaefer fixed point theorem. Finally, we give some illustrative examples.