Volume 25, Issue 1

In this paper, we investigate a Meijer–Laplace transform enfolding Fox’s H -functions on a class of Boehmians. The extended Meijer–Laplace transform of a Boehmian is determined and executed to preserve certain properties of the classical transform. The inverse problem and related theorems are also discussed in some details.

In the present paper, we introduce the notion of generalized (σ,τ){(\sigma,\tau)}- n -derivations in a near-ring N and investigate a property involving generalized (σ,τ){(\sigma,\tau)}- n -derivations of a prime near-ring N , which makes N a commutative ring. Additive commutativity of a prime near-ring N satisfying certain identities involving generalized (σ,τ){(\sigma,\tau)}- n -derivations is also obtained.

In this paper, we consider an eigenvalue problem for the biharmonic operator that describes the transverse vibrations of the plate. Under the imposed boundary conditions, the eigenvalues of this operator are indeed eigenfrequencies of the clamped plate. The domain of the plate is taken variable and the domain functional, involving an eigenfrequency, is studied. A new formula for an eigenfrequency is proved, the first variation of the functional with respect to the domain is calculated, and the necessary condition for an optimal shape is derived. New explicit formulas are obtained for the eigenfrequency in the optimal domain in some particular cases.

In the present paper, we propose a sequence of generalized genuine Baskakov–Durrmeyer-type link operators. In terms of ordinary approximation, we estimate local and global direct results and also study the weighted approximation result. In terms of simultaneous approximation, we establish an asymptotic formula of Voronovskaja kind. In the last section, we prove convergence in Lp{L_{p}}-norm.

For certain groups of isometric transformations of the Euclidean plane ℝ2{\mathbb{R}^{2}}, negligible and absolutely negligible subsets of this plane are considered and compared with each other.

In this paper, we first establish some refinements of the weighted generalized trapezoid inequality for functions of bounded variation in terms of a cumulative variation function, and then obtain some perturbed versions of the weighted generalized trapezoid inequality. Several applications for selfadjoint operators on complex Hilbert spaces are also given.

In this paper, we present various forms of the inverse of a special Schur complement C⁢D-1⁢B{CD^{-1}B} in the case where B , C , D are operators on Hilbert spaces. We show that when C⁢D-1⁢B{CD^{-1}B} is invertible, there always exist operators X and Y which belong to some special classes of generalized inverses of B and C , for which (C⁢D-1⁢B)-1=X⁢D⁢Y{(CD^{-1}B)^{-1}=XDY}. Some explicit expressions for such X and Y are given.

In this note we show that if f∈Hp⁢(ℝn)∩Ls⁢(ℝn){f\in H^{p}(\mathbb{R}^{n})\cap L^{s}(\mathbb{R}^{n})}, where 0<p≤1<s<∞{0<p\leq 1<s<\infty}, then there exists a (p,∞){(p,\infty)}-atomic decomposition which converges to f in Ls⁢(ℝn){L^{s}(\mathbb{R}^{n})}. From this result, we obtain that a bounded linear operator T on Ls⁢(ℝn){L^{s}(\mathbb{R}^{n})} can be extended to a bounded operator from Hp⁢(ℝn){H^{p}(\mathbb{R}^{n})} into Lp⁢(ℝn){L^{p}(\mathbb{R}^{n})} if and only if T is bounded uniformly in Lp{L^{p}} norm on all (p,∞){(p,\infty)}-atoms. A similar result is also obtained from Hp⁢(ℝn){H^{p}(\mathbb{R}^{n})} into Hp⁢(ℝn){H^{p}(\mathbb{R}^{n})}.

In the present paper, we use the perturbation algorithm to reduce a purely implicit four-layer semi-discrete scheme for an abstract evolutionary equation to two-layer schemes. An approximate solution of the original problem is constructed using the solutions of these schemes. Estimates of the approximate solution error are proved in a Hilbert space.

The aim of this paper is to present a comprehensive study of operator m -convex functions. Let m∈[0,1]{m\in[0,1]}, and J=[0,b]{J=[0,b]} for some b∈ℝ{b\in\mathbb{R}} or J=[0,∞){J=[0,\infty)}. A continuous function φ:J→ℝ{\varphi\colon J\to\mathbb{R}} is called operator m -convex if for any t∈[0,1]{t\in[0,1]} and any self-adjoint operators A,B∈𝔹⁢(ℋ){A,B\in\mathbb{B}({\mathscr{H}})}, whose spectra are contained in J , we have φ⁢(t⁢A+m⁢(1-t)⁢B)≤t⁢φ⁢(A)+m⁢(1-t)⁢φ⁢(B){\varphi(tA+m(1-t)B)\leq t\varphi(A)+m(1-t)\varphi(B)}. We first generalize the celebrated Jensen inequality for continuous m -convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements. Introducing the notion of operator m -convexity, we extend the Choi–Davis–Jensen inequality for operator m -convex functions. We also present an operator version of the Jensen–Mercer inequality for m -convex functions and generalize this inequality for operator m -convex functions involving continuous fields of operators and unital fields of positive linear mappings. Employing the Jensen–Mercer operator inequality for operator m -convex functions, we construct the m -Jensen operator functional and obtain an upper bound for it.

In this paper, we establish Ostrowski-type inequalities for strongly convex functions, by using some classical inequalities and elementary analysis. We also give some results for the product of two strongly convex functions.

We show, under an iterative condition which is similar to but stronger than that of Ambrosetti and Rabinowitz and by using a variational method, the existence of a T -periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential z¨+V′⁢(z)=0,z∈ℝ,\ddot{z}+V^{\prime}(z)=0,\quad z\in\mathbb{R}, for any T>0{T>0}. Moreover, such a solution has T/k{T/k} as a minimal period for some integer 1≤k≤3{1\leq k\leq 3}.

The computational complexity of an algorithm is traditionally measured for the worst and the average case. The worst-case estimation guarantees a certain worst-case behavior of a given algorithm, although it might be rough, since in “most instances” the algorithm may have a significantly better performance. The probabilistic average-case analysis claims to derive an average performance of an algorithm, say, for an “average instance” of the problem in question. That instance may be far away from the average of the problem instances arising in a given real-life application, and so the average case analysis would also provide a non-realistic estimation. We suggest that, in general, a wider use of probabilistic models for a more accurate estimation of the algorithm efficiency could be possible. For instance, the quality of the solutions delivered by an approximation algorithm may also be estimated in the “average” probabilistic case. Such an approach would deal with the estimation of the quality of the solutions delivered by the algorithm for the most common (for a given application) problem instances. As we illustrate, the probabilistic modeling can also be used to derive an accurate time complexity performance measure, distinct from the traditional probabilistic average-case time complexity measure. Such an approach could, in particular, be useful when the traditional average-case estimation is still rough or is not possible at all.

We establish the boundedness of the vector-valued Hardy–Littlewood maximal operator in variable exponent Herz spaces, which were introduced by Samko in [33]. We also introduce variable exponent Herz type Besov and Triebel–Lizorkin spaces and give characterizations of these new spaces by maximal functions.

In this work, we define a quaternionic osculating curves and give the conditions for curves to be osculating with the help of curvature functions.