In this paper, by using the Nevanlinna Theory on angular domain, we establish a theorem which concerns the growth of entire function and his zero. As an application, we survey the location of zero of higher order diff...In this paper, by using the Nevanlinna Theory on angular domain, we establish a theorem which concerns the growth of entire function and his zero. As an application, we survey the location of zero of higher order differential equation, which can be regarded as an alternating but precise version of Wu and Yi.展开更多
Wavelength-dependent mathematical modelling of the differential energy change of a photon has been performed inside a proposed hypothetical optical medium.The existence of this medium demands certain mathematical cons...Wavelength-dependent mathematical modelling of the differential energy change of a photon has been performed inside a proposed hypothetical optical medium.The existence of this medium demands certain mathematical constraints,which have been derived in detail.Using reverse modelling,a medium satisfying the derived conditions is proven to store energy as the photon propagates from the entry to exit point.A single photon with a given intensity is considered in the analysis and hypothesized to possess a definite non-zero probability of maintaining its energy and velocity functions analytic inside the proposed optical medium,despite scattering,absorption,fluorescence,heat generation,and other nonlinear mechanisms.The energy and velocity functions are thus singly and doubly differentiable with respect to wavelength.The solution of the resulting second-order differential equation in two variables proves that energy storage or energy flotation occurs inside a medium with a refractive index satisfying the described mathematical constraints.The minimum-value-normalized refractive index profiles of the modelled optical medium for transformed wavelengths both inside the medium and for vacuum have been derived.Mathematical proofs,design equations,and detailed numerical analyses are presented in the paper.展开更多
Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable...Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Probenius method of power series.展开更多
In this paper we establish Levin type comparison theorems for certain second order differential equations. The results obtained here generalize and extend some of the earlier ones related to the Levin's comparison...In this paper we establish Levin type comparison theorems for certain second order differential equations. The results obtained here generalize and extend some of the earlier ones related to the Levin's comparison theorems.展开更多
Some new oscillation criteria are given for forced second order differential equations with mixed nonlinearities by using the generalized variational principle and Riccati technique. Our results generalize and extend ...Some new oscillation criteria are given for forced second order differential equations with mixed nonlinearities by using the generalized variational principle and Riccati technique. Our results generalize and extend some known oscillation results in the literature.展开更多
This paper investigates equation (1) in twocases:(i)P=0, (ii)P satisfies|P (i,x,y,z,w)<(A+|y|+|z|+|w|)q(t).whereq(t) is a nonnegative function of t. For case (i) the asymptotic stability in the large of the trivial...This paper investigates equation (1) in twocases:(i)P=0, (ii)P satisfies|P (i,x,y,z,w)<(A+|y|+|z|+|w|)q(t).whereq(t) is a nonnegative function of t. For case (i) the asymptotic stability in the large of the trivial solution x=0 is investigated and for case (ii) the boundedness result is obtained for solutions of equation (1). These results improve and include several well-known results.展开更多
A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous f...A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous functions on[t0,∞). This criterion extends and unifies some of the results obtained in [1]- [5].展开更多
By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation wit...By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation with deviating arguments.展开更多
A thorough investigation of the systemd^2y(x):dx^2+p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤x<x_0(2π>0> -η, x_0≤x<2π(η>p(x)=p(x+2π),-∞<x<∞is given, and the method can be appl...A thorough investigation of the systemd^2y(x):dx^2+p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤x<x_0(2π>0> -η, x_0≤x<2π(η>p(x)=p(x+2π),-∞<x<∞is given, and the method can be applied to one with other periodic impulse coefficients.展开更多
In this paper we apply fractional calculus to solve the 3rd order ordinary differential equation of the following form: (z-a)(z-b)(z-c)φ 3+(βz 2+γz+D)φ 2+(α(2β-3α-3)z+αγ+α(α+1)(a+b+c))φ 1+α(α-...In this paper we apply fractional calculus to solve the 3rd order ordinary differential equation of the following form: (z-a)(z-b)(z-c)φ 3+(βz 2+γz+D)φ 2+(α(2β-3α-3)z+αγ+α(α+1)(a+b+c))φ 1+α(α-1)(β-2α-2)φ=f.展开更多
The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, wher...The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.展开更多
This paper deals with even order nonlinear differential equations. Applying the Schauder fixed point theorem and the Schwartz inequality technique, we give a criterion to ensure the existence and uniqueness of periodi...This paper deals with even order nonlinear differential equations. Applying the Schauder fixed point theorem and the Schwartz inequality technique, we give a criterion to ensure the existence and uniqueness of periodic solutions. This criterion is a complement of the well known Lazer type results.展开更多
A class of nonlinear fractional order differential equations with delay is investigated in this paper. Using Leray-Schauder fixed point theorem and the contraction mapping theorem, we obtain some sufficient conditions...A class of nonlinear fractional order differential equations with delay is investigated in this paper. Using Leray-Schauder fixed point theorem and the contraction mapping theorem, we obtain some sufficient conditions for the existence and uniqueness of solutions to the fractional order differential equations.展开更多
Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 ...Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.展开更多
In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermit...In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.展开更多
By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established re...By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.展开更多
Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized higher order algebraic differential equations.
By the generalized Riccati transformation and the integral averaging technique, some sufficient conditions of oscillation of the solutions for second order nonlinear differential equations with damping were discussed....By the generalized Riccati transformation and the integral averaging technique, some sufficient conditions of oscillation of the solutions for second order nonlinear differential equations with damping were discussed.Some sufficient oscillation criteria for previous equations were built up.Some oscillation criteria have been expanded and strengthened in some other known results.展开更多
文摘In this paper, by using the Nevanlinna Theory on angular domain, we establish a theorem which concerns the growth of entire function and his zero. As an application, we survey the location of zero of higher order differential equation, which can be regarded as an alternating but precise version of Wu and Yi.
文摘Wavelength-dependent mathematical modelling of the differential energy change of a photon has been performed inside a proposed hypothetical optical medium.The existence of this medium demands certain mathematical constraints,which have been derived in detail.Using reverse modelling,a medium satisfying the derived conditions is proven to store energy as the photon propagates from the entry to exit point.A single photon with a given intensity is considered in the analysis and hypothesized to possess a definite non-zero probability of maintaining its energy and velocity functions analytic inside the proposed optical medium,despite scattering,absorption,fluorescence,heat generation,and other nonlinear mechanisms.The energy and velocity functions are thus singly and doubly differentiable with respect to wavelength.The solution of the resulting second-order differential equation in two variables proves that energy storage or energy flotation occurs inside a medium with a refractive index satisfying the described mathematical constraints.The minimum-value-normalized refractive index profiles of the modelled optical medium for transformed wavelengths both inside the medium and for vacuum have been derived.Mathematical proofs,design equations,and detailed numerical analyses are presented in the paper.
文摘Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Probenius method of power series.
文摘In this paper we establish Levin type comparison theorems for certain second order differential equations. The results obtained here generalize and extend some of the earlier ones related to the Levin's comparison theorems.
文摘Some new oscillation criteria are given for forced second order differential equations with mixed nonlinearities by using the generalized variational principle and Riccati technique. Our results generalize and extend some known oscillation results in the literature.
文摘This paper investigates equation (1) in twocases:(i)P=0, (ii)P satisfies|P (i,x,y,z,w)<(A+|y|+|z|+|w|)q(t).whereq(t) is a nonnegative function of t. For case (i) the asymptotic stability in the large of the trivial solution x=0 is investigated and for case (ii) the boundedness result is obtained for solutions of equation (1). These results improve and include several well-known results.
文摘A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous functions on[t0,∞). This criterion extends and unifies some of the results obtained in [1]- [5].
文摘By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation with deviating arguments.
基金This work is supported by the National Science Fund of Peop1e's Republic of China
文摘A thorough investigation of the systemd^2y(x):dx^2+p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤x<x_0(2π>0> -η, x_0≤x<2π(η>p(x)=p(x+2π),-∞<x<∞is given, and the method can be applied to one with other periodic impulse coefficients.
文摘In this paper we apply fractional calculus to solve the 3rd order ordinary differential equation of the following form: (z-a)(z-b)(z-c)φ 3+(βz 2+γz+D)φ 2+(α(2β-3α-3)z+αγ+α(α+1)(a+b+c))φ 1+α(α-1)(β-2α-2)φ=f.
基金The Natural Science Foundation of Department ofEducation of Jiangsu Province (No06KJD110087)
文摘The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.
文摘This paper deals with even order nonlinear differential equations. Applying the Schauder fixed point theorem and the Schwartz inequality technique, we give a criterion to ensure the existence and uniqueness of periodic solutions. This criterion is a complement of the well known Lazer type results.
文摘A class of nonlinear fractional order differential equations with delay is investigated in this paper. Using Leray-Schauder fixed point theorem and the contraction mapping theorem, we obtain some sufficient conditions for the existence and uniqueness of solutions to the fractional order differential equations.
基金Supported by the Natural Science Foundation of Hunan Province（06JJ50008） Supported by the Natural Science Foundation of Guangdong Province（7004569）
文摘Suffcient conditions for the existence of at least one solution of two-point boundary value problems for second order nonlinear differential equations [φ(x(t))] + kx(t) + g(t,x(t)) = p(t),t ∈(0,π) x(0) = x(π) = 0 are established,where [φ(x)] =(|x |p-2x) with p > 1.Our result is new even when [φ(x)] = x in above problem,i.e.p = 2.Examples are presented to illustrate the effciency of the theorem in this paper.
文摘In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.
文摘By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.
文摘Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized higher order algebraic differential equations.
文摘By the generalized Riccati transformation and the integral averaging technique, some sufficient conditions of oscillation of the solutions for second order nonlinear differential equations with damping were discussed.Some sufficient oscillation criteria for previous equations were built up.Some oscillation criteria have been expanded and strengthened in some other known results.
