The aim of this study is to investigate the dynamic response of axially moving two-layer laminated plates on the Winkler and Pasternak foundations. The upper and lower layers are formed from a bidirectional functional...The aim of this study is to investigate the dynamic response of axially moving two-layer laminated plates on the Winkler and Pasternak foundations. The upper and lower layers are formed from a bidirectional functionally graded(FG) layer and a graphene platelet(GPL) reinforced porous layer, respectively. Henceforth, the combined layers will be referred to as a two-dimensional(2D) FG/GPL plate. Two types of porosity and three graphene dispersion patterns, each of which is distributed through the plate thickness,are investigated. The mechanical properties of the closed-cell layers are used to define the variation of Poisson’s ratio and the relationship between the porosity coefficients and the mass density. For the GPL reinforced layer, the effective Young’s modulus is derived with the Halpin-Tsai micro-system model, and the rule of mixtures is used to calculate the effective mass density and Poisson’s ratio. The material of the upper 2D-FG layer is graded in two directions, and its effective mechanical properties are also derived with the rule of mixtures. The dynamic governing equations are derived with a first-order shear deformation theory(FSDT) and the von Kármán nonlinear theory. A combination of the dynamic relaxation(DR) and Newmark’s direct integration methods is used to solve the governing equations in both time and space. A parametric study is carried out to explore the effects of the porosity coefficients, porosity and GPL distributions, material gradients, damping ratios, boundary conditions, and elastic foundation stiffnesses on the plate response. It is shown that both the distributions of the porosity and graphene nanofillers significantly affect the dynamic behaviors of the plates. It is also shown that the reduction in the dynamic deflection of the bilayer composite plates is maximized when the porosity and GPL distributions are symmetric.展开更多
In this article, we extend some estimates of the right-hand side of the Hermite-Hadamard type inequality for preinvex functions with fractional integral.The notion of logarithmically s-Godunova-Levin-preinvex function...In this article, we extend some estimates of the right-hand side of the Hermite-Hadamard type inequality for preinvex functions with fractional integral.The notion of logarithmically s-Godunova-Levin-preinvex function in second sense is introduced and then a new Herrnite-Hadarnard inequality is derived for the class of logarithmically s-Godunova-Levin-preinvex function.展开更多
The second-order temporal interference of two independent single-mode continuous-wave lasers is discussed by em- ploying two-photon interference in Feynman's path integral theory. It is concluded that whether the sec...The second-order temporal interference of two independent single-mode continuous-wave lasers is discussed by em- ploying two-photon interference in Feynman's path integral theory. It is concluded that whether the second-order temporal interference pattern can or cannot be retrieved via two-photon coincidence counting rate is dependent on the resolution time of the detection system and the frequency difference between these two lasers. Two identical and tunable single-mode continuous-wave diode lasers are employed to verify the predictions. These studies are helpful to understand the physics of two-photon interference with photons of different spectra.展开更多
This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡ ■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z =...This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡ ■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture.展开更多
When dynamic force is applied to a saturated porous soil, drainage is common. In this paper, the saturated porous soil with a two-phase saturated medium is simulated, and Lamb's integral formulas with drainage and st...When dynamic force is applied to a saturated porous soil, drainage is common. In this paper, the saturated porous soil with a two-phase saturated medium is simulated, and Lamb's integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot's equation and Betti's theorem (the reciprocal theorem). According to the basic solution to Biot's equation, Green's function Gij and three terms of Green's function G4i, Gi4, and G44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented. The displacement field with drainage, the magnitude of drainage, and the pore pressure of the center explosion source are obtained in computation. The results of the classical Sharpe's solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared. Good agreement is observed.展开更多
The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of...The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of a hypersingular integral equation by a suitable application of Green's integral theorem in terms of difference of potential functicns across the barrier. This integral equation is solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients are obtained numerically and presented graphically for various values of the wave number and ice-cover parameter.展开更多
In this paper, by introducing the norm ||x|| (x ∈ Rn), a multiple Hardy- Hilbert's integral inequality with the best constant factor and it's equivalent form are given.
Some new inequalities of Hermite-Hadamard's integration are established. As for as inequalities about the righthand side of the classical Hermite-Hadamard's integral inequality refined by S Qaisar in [3], a ne...Some new inequalities of Hermite-Hadamard's integration are established. As for as inequalities about the righthand side of the classical Hermite-Hadamard's integral inequality refined by S Qaisar in [3], a new upper bound is given. Under special conditions,the bound is smaller than that in [3].展开更多
The second-order temporal interference of classical and nonclassical light at an asymmetrical beam splitter is discussed based on two-photon interference in Feynman's path integral theory. The visibility of the secon...The second-order temporal interference of classical and nonclassical light at an asymmetrical beam splitter is discussed based on two-photon interference in Feynman's path integral theory. The visibility of the second-order interference pattern is determined by the properties of the superposed light beams, the ratio between the intensities of these two light beams, and the reflectivity of the asymmetrical beam splitter. Some requirements about the asymmetrical beam splitter have to be satisfied in order to ensure that the visibility of the second-order interference pattern of nonclassical light beams exceeds the classical limit. The visibility of the second-order interference pattern of photons emitted by two independent single-photon sources is independent of the ratio between the intensities. These conclusions are important for the researches and applications in quantum optics and quantum information when an asymmetrical beam splitter is employed.展开更多
ZTE's integrated solutions for CDMA network coverage and value- added services are presented.Different schemes for specific appli- cation requirements are described in detail.The integrated service platforms conce...ZTE's integrated solutions for CDMA network coverage and value- added services are presented.Different schemes for specific appli- cation requirements are described in detail.The integrated service platforms concerning positioning service,multimedia Email,mobile office,etc.are overviewed.展开更多
The establishment and rapid rise of the European Union (EU) have increasingly made Europe an independent force. Objectively, the enlargement of both the EU and NATO has speeded up the process of European integration, ...The establishment and rapid rise of the European Union (EU) have increasingly made Europe an independent force. Objectively, the enlargement of both the EU and NATO has speeded up the process of European integration, and "One important link of Russia’s diplomacy展开更多
The integrated traditional Chinese and western medicine (ICWM) of obstetrics and gynecology (OBS/GYN) emerged eventually with more than 40 years’ hard struggle, and one of the most promising articles is the integrati...The integrated traditional Chinese and western medicine (ICWM) of obstetrics and gynecology (OBS/GYN) emerged eventually with more than 40 years’ hard struggle, and one of the most promising articles is the integration of the masterpieces of menstrual disorder and infertility in TCM and the virtual explosion of new knowledge and methods in展开更多
The causes and the problems of integration of scientific & technical innovation (ISTI) and inevitability are proposed. The status of domestic scientific & technical innovation is analysed. A series of emergent man...The causes and the problems of integration of scientific & technical innovation (ISTI) and inevitability are proposed. The status of domestic scientific & technical innovation is analysed. A series of emergent managerial problems caused by the system ISTI (SISTI) are pointed out. Because of Hall three dimensions structure's enlightenment, a three dimensional logical net diagram in which the system integration of innovation based on problem in real world but not on disciplinary logic is drawn. The intelligent group' s need span is greater under the conditions of relatively low material level than common groups who have lower degree of education. The SISTI is composed by multi-rule and intelligent multi-agent behavior. It is concluded that the logical relationships of integrated technology which based on the author's experience and observation must be considerded for a valid management of SISTI.展开更多
Nowadays fuzzy concepts are frequently used as statistical parameters, while the traditional normal distribution can only accept determinate variable. In order to design a practical model for fuzzy statistic events, t...Nowadays fuzzy concepts are frequently used as statistical parameters, while the traditional normal distribution can only accept determinate variable. In order to design a practical model for fuzzy statistic events, this paper combines the fuzzy number, like “may-occur”, “very-likely-occur”, “rarely-occur”, to optimize the normal distribution probability density function, to provide a significant method in statistics.展开更多
Since China began implementing reform and opening-up policies, along with a fast development of the national economy and great changes of social life, an increasing number of peasan's have kept flowing into cities an...Since China began implementing reform and opening-up policies, along with a fast development of the national economy and great changes of social life, an increasing number of peasan's have kept flowing into cities and economically more developed regions. Aspiring to improve their life and yearning for a better urban life, they defy the restraint of the household registry system and, in the process of moving from one place to another, ignore economic and psychological costs. As early as in year 2000 China's migrant population reached 140 million, more than 10% of the total population. Since 2000 the migrant poptilation in Zhejiang Province, east China, has kept growing by 20% annually, the second fastest in the country for seven consecutive years.展开更多
In this paper, we apply Hirota's discretisation to a three-dimensional integrable Lotka- Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is vol...In this paper, we apply Hirota's discretisation to a three-dimensional integrable Lotka- Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is volume-preserving, and has two independent first integrals. Moreover, it can be formally reduced to a system in one dimension via a volume- preserving transformation. If the given initial value is located in the positive octant, we prove that the numerical solution is confined to a one-dimensional connected and compact space which is diffeomorphic to a circle.展开更多
When concentrated forces are applied at any points of the outer region of an ellipse in an infinite plate, the complex potentials are determined using the conformal mapping method and Cauchy's integral formula. And t...When concentrated forces are applied at any points of the outer region of an ellipse in an infinite plate, the complex potentials are determined using the conformal mapping method and Cauchy's integral formula. And then, based on the superposition principle, the analyt- ical solutions for stress around an elliptical hole in an infinite plate subjected to a uniform far-field stress and concentrated forces, are obtained. Tangential stress concentration will occur on the hole boundary when only far-field uniform loads are applied. When concen- trated forces are applied in the reversed directions of the uniform loads, tangential stress concentration on the hole boundary can be released significantly. In order to minimize the tangential stress concentration, we need to determine the optimum positions and values of the concentrated forces. Three different optimization methods are applied to achieve this aim. The results show that the tangential stress can be released significantly when the op- timized concentrated forces are applied.展开更多
Nowadays the self-healing techniques in ATM networks are highly considered. However, in mostof networks, the normal control and sell-healing control are separately considered. In this paper, intenttedATM VP-based cont...Nowadays the self-healing techniques in ATM networks are highly considered. However, in mostof networks, the normal control and sell-healing control are separately considered. In this paper, intenttedATM VP-based control strategies for ATM survivable networks are prOPosed, which combine the normalcontrol and self-healing control in a unified wad, being able to operate under both normal conditions andcatastrophic failure situations. These strategies allow a cost-effective network plan and design.展开更多
In this paper, some new generalizations of inverse type Hilbert-Pachpatte integral inequalities are proved. The results of this paper reduce to those of Pachpatte (1998, J. Math. Anal. Appl. 226, 166–179) and Zhao an...In this paper, some new generalizations of inverse type Hilbert-Pachpatte integral inequalities are proved. The results of this paper reduce to those of Pachpatte (1998, J. Math. Anal. Appl. 226, 166–179) and Zhao and Debnath (2001, J. Math. Anal. Appl. 262, 411–418).展开更多
文摘The aim of this study is to investigate the dynamic response of axially moving two-layer laminated plates on the Winkler and Pasternak foundations. The upper and lower layers are formed from a bidirectional functionally graded(FG) layer and a graphene platelet(GPL) reinforced porous layer, respectively. Henceforth, the combined layers will be referred to as a two-dimensional(2D) FG/GPL plate. Two types of porosity and three graphene dispersion patterns, each of which is distributed through the plate thickness,are investigated. The mechanical properties of the closed-cell layers are used to define the variation of Poisson’s ratio and the relationship between the porosity coefficients and the mass density. For the GPL reinforced layer, the effective Young’s modulus is derived with the Halpin-Tsai micro-system model, and the rule of mixtures is used to calculate the effective mass density and Poisson’s ratio. The material of the upper 2D-FG layer is graded in two directions, and its effective mechanical properties are also derived with the rule of mixtures. The dynamic governing equations are derived with a first-order shear deformation theory(FSDT) and the von Kármán nonlinear theory. A combination of the dynamic relaxation(DR) and Newmark’s direct integration methods is used to solve the governing equations in both time and space. A parametric study is carried out to explore the effects of the porosity coefficients, porosity and GPL distributions, material gradients, damping ratios, boundary conditions, and elastic foundation stiffnesses on the plate response. It is shown that both the distributions of the porosity and graphene nanofillers significantly affect the dynamic behaviors of the plates. It is also shown that the reduction in the dynamic deflection of the bilayer composite plates is maximized when the porosity and GPL distributions are symmetric.
基金The Key Scientific and Technological Innovation Team Project(2014KCT-15)in Shaanxi Province
文摘In this article, we extend some estimates of the right-hand side of the Hermite-Hadamard type inequality for preinvex functions with fractional integral.The notion of logarithmically s-Godunova-Levin-preinvex function in second sense is introduced and then a new Herrnite-Hadarnard inequality is derived for the class of logarithmically s-Godunova-Levin-preinvex function.
基金Project supported by the National Natural Science Foundation of China(Grant No.11404255)the Doctor Foundation of Education Ministry of China(Grant No.20130201120013)
文摘The second-order temporal interference of two independent single-mode continuous-wave lasers is discussed by em- ploying two-photon interference in Feynman's path integral theory. It is concluded that whether the second-order temporal interference pattern can or cannot be retrieved via two-photon coincidence counting rate is dependent on the resolution time of the detection system and the frequency difference between these two lasers. Two identical and tunable single-mode continuous-wave diode lasers are employed to verify the predictions. These studies are helpful to understand the physics of two-photon interference with photons of different spectra.
文摘This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡ ■C f (t)dt/(t z) taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x + iy plane consisted of the simply connected open domain D + bounded by C and the open domain D outside C. (1) With f (z) assumed to be C n (n ∞-times continuously differentiable) z ∈ D + and in a neighborhood of C, f (z) and its derivatives f (n) (z) are proved uniformly continuous in the closed domain D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) The same claims (as for f (z) and J[ f (z)]) are shown extended to hold for the complement function F(z), defined to be C n z ∈ D and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four general- ized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f (z) in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f (z) in domain D , based on the continuous numerical value of f (z) z ∈ D + = [D + + C], is presented for resolution as a conjecture.
基金Project supported by the National Natural Science Foundation of China(No.10572129)
文摘When dynamic force is applied to a saturated porous soil, drainage is common. In this paper, the saturated porous soil with a two-phase saturated medium is simulated, and Lamb's integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot's equation and Betti's theorem (the reciprocal theorem). According to the basic solution to Biot's equation, Green's function Gij and three terms of Green's function G4i, Gi4, and G44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented. The displacement field with drainage, the magnitude of drainage, and the pore pressure of the center explosion source are obtained in computation. The results of the classical Sharpe's solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared. Good agreement is observed.
基金supported by the Department of Science and Technology of New Delhi (No.SR/SY/MS:521/08)
文摘The present paper is concerned with scattering of water waves from a vertical plate, modeled as an elastic plate, submerged in deep water covered with a thin uniform sheet of ice. The problem is formulated in terms of a hypersingular integral equation by a suitable application of Green's integral theorem in terms of difference of potential functicns across the barrier. This integral equation is solved by a collocation method using a finite series involving Chebyshev polynomials. Reflection and transmission coefficients are obtained numerically and presented graphically for various values of the wave number and ice-cover parameter.
文摘In this paper, by introducing the norm ||x|| (x ∈ Rn), a multiple Hardy- Hilbert's integral inequality with the best constant factor and it's equivalent form are given.
基金Supported by the Key Scientific and Technological Innovation Team Project in Shaanxi Province(2014KCT-15)
文摘Some new inequalities of Hermite-Hadamard's integration are established. As for as inequalities about the righthand side of the classical Hermite-Hadamard's integral inequality refined by S Qaisar in [3], a new upper bound is given. Under special conditions,the bound is smaller than that in [3].
基金supported by the National Natural Science Foundation of China(Grant No.11404255)the Doctor Foundation of Education Ministry of China(Grant No.20130201120013)+1 种基金the Programme of Introducing Talents of Discipline to Universities,China(Grant No.B14040)the Fundamental Research Funds for the Central Universities,China
文摘The second-order temporal interference of classical and nonclassical light at an asymmetrical beam splitter is discussed based on two-photon interference in Feynman's path integral theory. The visibility of the second-order interference pattern is determined by the properties of the superposed light beams, the ratio between the intensities of these two light beams, and the reflectivity of the asymmetrical beam splitter. Some requirements about the asymmetrical beam splitter have to be satisfied in order to ensure that the visibility of the second-order interference pattern of nonclassical light beams exceeds the classical limit. The visibility of the second-order interference pattern of photons emitted by two independent single-photon sources is independent of the ratio between the intensities. These conclusions are important for the researches and applications in quantum optics and quantum information when an asymmetrical beam splitter is employed.
文摘ZTE's integrated solutions for CDMA network coverage and value- added services are presented.Different schemes for specific appli- cation requirements are described in detail.The integrated service platforms concerning positioning service,multimedia Email,mobile office,etc.are overviewed.
文摘The establishment and rapid rise of the European Union (EU) have increasingly made Europe an independent force. Objectively, the enlargement of both the EU and NATO has speeded up the process of European integration, and "One important link of Russia’s diplomacy
文摘The integrated traditional Chinese and western medicine (ICWM) of obstetrics and gynecology (OBS/GYN) emerged eventually with more than 40 years’ hard struggle, and one of the most promising articles is the integration of the masterpieces of menstrual disorder and infertility in TCM and the virtual explosion of new knowledge and methods in
基金Sponsored bythe National Innovation Basein Management on National Defense Science Technology National Economy Mobilization of the SecondPhase of"985"Project
文摘The causes and the problems of integration of scientific & technical innovation (ISTI) and inevitability are proposed. The status of domestic scientific & technical innovation is analysed. A series of emergent managerial problems caused by the system ISTI (SISTI) are pointed out. Because of Hall three dimensions structure's enlightenment, a three dimensional logical net diagram in which the system integration of innovation based on problem in real world but not on disciplinary logic is drawn. The intelligent group' s need span is greater under the conditions of relatively low material level than common groups who have lower degree of education. The SISTI is composed by multi-rule and intelligent multi-agent behavior. It is concluded that the logical relationships of integrated technology which based on the author's experience and observation must be considerded for a valid management of SISTI.
文摘Nowadays fuzzy concepts are frequently used as statistical parameters, while the traditional normal distribution can only accept determinate variable. In order to design a practical model for fuzzy statistic events, this paper combines the fuzzy number, like “may-occur”, “very-likely-occur”, “rarely-occur”, to optimize the normal distribution probability density function, to provide a significant method in statistics.
文摘Since China began implementing reform and opening-up policies, along with a fast development of the national economy and great changes of social life, an increasing number of peasan's have kept flowing into cities and economically more developed regions. Aspiring to improve their life and yearning for a better urban life, they defy the restraint of the household registry system and, in the process of moving from one place to another, ignore economic and psychological costs. As early as in year 2000 China's migrant population reached 140 million, more than 10% of the total population. Since 2000 the migrant poptilation in Zhejiang Province, east China, has kept growing by 20% annually, the second fastest in the country for seven consecutive years.
基金The authors would like to thank J. Niesen for his helpful suggestions in improving the presentation of the paper. The first author was supported by the Fundamental Research Funds for the Central Universities (WK2030040057). The second author was sup- ported by the National Natural Science Foundation of China (11271357), the Foundation for Innovative Research Groups of the NNSFC (11321061) and the ITER-China Program (2014G- B124005). The third author was supported by the Foundation of the NNSFC (10990012) and the Marine Public Welfare Project of China (201105032).
文摘In this paper, we apply Hirota's discretisation to a three-dimensional integrable Lotka- Volterra system. By analyzing the three-dimensional modified equation of the resulting numerical method, we show that it is volume-preserving, and has two independent first integrals. Moreover, it can be formally reduced to a system in one dimension via a volume- preserving transformation. If the given initial value is located in the positive octant, we prove that the numerical solution is confined to a one-dimensional connected and compact space which is diffeomorphic to a circle.
基金supported by the National Natural Science Foundation of China [grant numbers 11172101, 11572126]
文摘When concentrated forces are applied at any points of the outer region of an ellipse in an infinite plate, the complex potentials are determined using the conformal mapping method and Cauchy's integral formula. And then, based on the superposition principle, the analyt- ical solutions for stress around an elliptical hole in an infinite plate subjected to a uniform far-field stress and concentrated forces, are obtained. Tangential stress concentration will occur on the hole boundary when only far-field uniform loads are applied. When concen- trated forces are applied in the reversed directions of the uniform loads, tangential stress concentration on the hole boundary can be released significantly. In order to minimize the tangential stress concentration, we need to determine the optimum positions and values of the concentrated forces. Three different optimization methods are applied to achieve this aim. The results show that the tangential stress can be released significantly when the op- timized concentrated forces are applied.
文摘Nowadays the self-healing techniques in ATM networks are highly considered. However, in mostof networks, the normal control and sell-healing control are separately considered. In this paper, intenttedATM VP-based control strategies for ATM survivable networks are prOPosed, which combine the normalcontrol and self-healing control in a unified wad, being able to operate under both normal conditions andcatastrophic failure situations. These strategies allow a cost-effective network plan and design.
文摘In this paper, some new generalizations of inverse type Hilbert-Pachpatte integral inequalities are proved. The results of this paper reduce to those of Pachpatte (1998, J. Math. Anal. Appl. 226, 166–179) and Zhao and Debnath (2001, J. Math. Anal. Appl. 262, 411–418).