scholarly journals NECESSARY AND SUFFICIENT CONDITIONS AND OPTIMAL CONSTANT FACTORS FOR THE VALIDITY OF MULTIPLE INTEGRAL HALF-DISCRETE HILBERT TYPE INEQUALITIES WITH A CLASS OF QUASI-HOMOGENEOUS KERNELS

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Bing He ◽  
◽  
Yong Hong ◽  
Zhen Li ◽  
Bicheng Yang ◽  
...  
Keyword(s):  
Sufficient Conditions ◽  
Multiple Integral ◽  
Necessary And Sufficient Conditions ◽  
Optimal Constant ◽  
Necessary And Sufficient ◽  
Hilbert Type

scholarly journals Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with nonhomogeneous kernels

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bing He ◽  
Yong Hong ◽  
Zhen Li
Keyword(s):  
Integral Inequality ◽  
Function Method ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Multiple Integral ◽  
Constant Factor ◽  
Best Constant ◽  
Necessary And Sufficient ◽  
Best Constant Factor ◽  
Hilbert Type

AbstractFor the Hilbert type multiple integral inequality $$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m,\rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } $$ ∫ R + n ∫ R + m K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) f ( x ) g ( y ) d x d y ≤ M ∥ f ∥ p , α ∥ g ∥ q , β with a nonhomogeneous kernel $K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/ \|y\|^{\lambda _{2}}_{n, \rho })$ K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) = G ( ∥ x ∥ m , ρ λ 1 / ∥ y ∥ n , ρ λ 2 ) ($\lambda _{1}\lambda _{2}> 0$ λ 1 λ 2 > 0 ), in this paper, by using the weight function method, necessary and sufficient conditions that parameters p, q, $\lambda _{1}$ λ 1 , $\lambda _{2}$ λ 2 , α, β, m, and n should satisfy to make the inequality hold for some constant M are established, and the expression formula of the best constant factor is also obtained. Finally, their applications in operator boundedness and operator norm are also considered, and the norms of several integral operators are discussed.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

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