For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential ...For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential equations of the second-order. So it is important to develop Asgeirsson mean value theorem. The mean value of solution for the higher order equation hay been discussed primarily and has no exact result at present. The mean value theorem for the higher order equation can be deduced and satisfied generalized biaxial symmetry potential equation by using the result of Asgeirsson mean value theorem and the properties of derivation and integration. Moreover, the mean value formula can be obtained by using the regular solutions of potential equation and the special properties of Jacobi polynomials. Its converse theorem is also proved. The obtained results make it possible to discuss on continuation of the solutions and well posed problem.展开更多
The mean value theorem for derivatives says that for a given function over a closed and bounded interval, there is a point <em>P</em> on the graph such that the tangent at <em>P</em> is paralle...The mean value theorem for derivatives says that for a given function over a closed and bounded interval, there is a point <em>P</em> on the graph such that the tangent at <em>P</em> is parallel to the secant through the two endpoints. The mean value theorem for definite integrals says that the area under the function is equal to the area of a rectangle whose base is the length of the interval and height of some point <em>Q</em> on the graph. These two theorems have been studied and utilized extensively and they form the backbone of many important theorems in different branches of mathematics. In this note, we pose the question: for what functions do the two points <em>P </em>and <em>Q</em> always coincide? We find that the only analytic functions satisfying this condition are linear or exponential functions.展开更多
Let (Ω, F, P) be a probability space and L0(F,R) the algebra of equivalence classes of real- valued random variables on (Ω, F, P). When L0(F,R) is endowed with the topology of convergence in probability, we ...Let (Ω, F, P) be a probability space and L0(F,R) the algebra of equivalence classes of real- valued random variables on (Ω, F, P). When L0(F,R) is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0(F, R) to L0(F,R). As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module (S, ||·||) is random uniformly convex iff LP(S) is uniformly convex for each fixed positive number p such that 1 〈 p 〈 +∞.展开更多
In this paper we present a mean value theorem derived from Flett's mean value theorem. It turns out that cubic polynomials have the midpoint of the interval as their mean value point.To answer what class of functi...In this paper we present a mean value theorem derived from Flett's mean value theorem. It turns out that cubic polynomials have the midpoint of the interval as their mean value point.To answer what class of functions have this property,we consider a functional equation associated with this mean value theorem.This equation is then solved in a general setting on abelian groups.展开更多
In this note,we consider a new unknown input observer design for nonlinear systems.The main idea consists in determining the estimation error and mean value theorem parameters(β)to introduce them into proposed observ...In this note,we consider a new unknown input observer design for nonlinear systems.The main idea consists in determining the estimation error and mean value theorem parameters(β)to introduce them into proposed observer structure.This process is designed on the basis of mean value theorem and genetic algorithm.The stability study relies on the use of a classical quadratic Lyapunov function.The observer’s gains are determined systematically.For the validation of theoretical development proposed in this paper,we consider two practical realizations that deals with the secure communication problem.展开更多
The problem of state feedback controllers for a class of Takagi-Sugeno (T-S) Lipschitz nonlinear systems is investigated. A simple systematic and useful synthesis method is proposed based on the use of the different...The problem of state feedback controllers for a class of Takagi-Sugeno (T-S) Lipschitz nonlinear systems is investigated. A simple systematic and useful synthesis method is proposed based on the use of the differential mean value theorem (DMVT) and convex theory. The proposed design approach is based on the mean value theorem (MVT) to express the nonlinear error dynamics as a convex combination of known matrices with time varying coefficients as linear parameter varying (LPV) systems. Using the Lyapunov theory, stability conditions are obtained and expressed in terms of linear matrix inequalities (LMIs). The controller gains are then obtained by solving linear matrix inequalities. The effectiveness of the proposed approach for closed loop-field oriented control (CL-FOC) of permanent magnet synchronous machine (PMSM) drives is demonstrated through an illustrative simulation for the proof of these approaches. Furthermore, an extension for controller design with parameter uncertainties and perturbation performance is discussed.展开更多
We mainly study the existence of positive solutions for the following third order singular multi-point boundary value problem{x^(3)(t) + f(t, x(t), x′(t)) = 0, 0 〈 t 〈 1,x(0)-∑i=1^m1 αi x(ξi) = 0...We mainly study the existence of positive solutions for the following third order singular multi-point boundary value problem{x^(3)(t) + f(t, x(t), x′(t)) = 0, 0 〈 t 〈 1,x(0)-∑i=1^m1 αi x(ξi) = 0, x′(0)-∑i=1^m2 βi x′(ηi) = 0, x′(1)=0,where 0 ≤ ai≤∑i=1^m1 αi 〈 1, i = 1, 2, ···, m1, 0 〈 ξ1〈 ξ2〈 ··· 〈 ξm1〈 1, 0 ≤βj≤∑i^m2=1βi〈1,J=1,2, ···, m2, 0 〈 η1〈 η2〈 ··· 〈 ηm2〈 1. And we obtain some necessa βi 〈=11, j = 1,ry and sufficient conditions for the existence of C^1[0, 1] and C^2[0, 1] positive solutions by constructing lower and upper solutions and by using the comparison theorem. Our nonlinearity f(t, x, y)may be singular at x, y, t = 0 and/or t = 1.展开更多
This paper deals with the existence of triple positive solutions for the 1-dimensional equation of Laplace-type (φ(x′(t)))′+q(t)f(t,x(t),x′(t))=0,t∈(0,1),subject to the following boundary condit...This paper deals with the existence of triple positive solutions for the 1-dimensional equation of Laplace-type (φ(x′(t)))′+q(t)f(t,x(t),x′(t))=0,t∈(0,1),subject to the following boundary condition:a1φ(x(0))-a2φ(x'(0))=0,a3φ(x(1))+a4φ(x'(1))=0,where φ is an odd increasing homogeneous homeomorphism. By using a new fixed point theorem, sufficient conditions are obtained that guarantee the existence of at least three positive solu- tions. The emphasis here is that the nonlinear term f is involved with the first order derivative explicitly.展开更多
The existence of positive solutions for second order m-point boundary value problemx″-q(t)f(x,x′)x′=0, x(0)= m i=2 b ix(ξ i),x′(1)=αx′(0)are investigated,where ξ i,b i and α are constants satisfying...The existence of positive solutions for second order m-point boundary value problemx″-q(t)f(x,x′)x′=0, x(0)= m i=2 b ix(ξ i),x′(1)=αx′(0)are investigated,where ξ i,b i and α are constants satisfying 0=ξ 1<ξ 2<...<ξ m-1 <ξ m=1,b i≥0 for i=2,...,m with β∶= m i=2 b i∈[0,1), and α>1. Our approach is based on the fixed point theorem in cones.展开更多
The paper deals with the existence of three- solutions for the second- order differential equations with nonlinear boundary value conditions x″=f(t,x,x′) , t∈ [a,b], g1(x(a) ,x′(a) ) =0 , g2 (x(b) ,x′(...The paper deals with the existence of three- solutions for the second- order differential equations with nonlinear boundary value conditions x″=f(t,x,x′) , t∈ [a,b], g1(x(a) ,x′(a) ) =0 , g2 (x(b) ,x′(b) ) =0 , where f :[a,b]× R1× R1→ R1,gi:R1× R1→ R1(i=1 ,2 ) are continuous functions.The methods employed are the coincidence degree theory.As an application,the sufficient conditions under which there are arbitrary odd solutions for the BVP are obtained展开更多
In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdif...In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdifferentials.展开更多
Fixed point theory is one of the most important subjects in the setting of metric spaces since fixed point theorems can be used to determine the existence and the uniqueness of solutions of such mathematical problems....Fixed point theory is one of the most important subjects in the setting of metric spaces since fixed point theorems can be used to determine the existence and the uniqueness of solutions of such mathematical problems.It is known that many problems in applied sciences and engineering can be formulated as functional equations.Such equations can be transferred to fixed point theorems in an easy manner.Moreover,we use the fixed point theory to prove the existence and uniqueness of solutions of such integral and differential equations.Let X be a non-empty set.A fixed point for a self-mapping T on X is a point𝑒𝑒∈𝑋𝑋that satisfying T e=e.One of the most challenging problems in mathematics is to construct some iterations to faster the calculation or approximation of the fixed point of such problems.Some mathematicians constructed and generated some new iteration schemes to calculate or approximate the fixed point of such problems such as Mann et al.[Mann(1953);Ishikawa(1974);Sintunavarat and Pitea(2016);Berinde(2004b);Agarwal,O’Regan and Sahu(2007)].The main purpose of the present paper is to introduce and construct a new iteration scheme to calculate or approximate the fixed point within a fewer number of steps as much as we can.We prove that our iteration scheme is faster than the iteration schemes given by Sintunavarat et al.[Sintunavarat and Pitea(2016);Agarwal,O’Regan and Sahu(2007);Mann(1953);Ishikawa(1974)].We give some numerical examples by using MATLAB to compare the efficiency and effectiveness of our iterations scheme with the efficiency of Mann et al.[Mann(1953);Ishikawa(1974);Sintunavarat and Pitea(2016);Abbas and Nazir(2014);Agarwal,O’Regan and Sahu(2007)]schemes.Moreover,we introduce a problem raised from Newton’s law of cooling as an application of our new iteration scheme.Also,we support our application with a numerical example and figures to illustrate the validity of our iterative scheme.展开更多
In this paper, the solution, more general than [1], of a weak singular integral equation integral(0)(pi)integral(-infinity)(infinity) p(s,psi)d sk(psi)d psi=F(r,theta), (r,theta)epsilon (Q) over bar=Q+partial derivati...In this paper, the solution, more general than [1], of a weak singular integral equation integral(0)(pi)integral(-infinity)(infinity) p(s,psi)d sk(psi)d psi=F(r,theta), (r,theta)epsilon (Q) over bar=Q+partial derivative Q subject to constraint p(s,psi)=0, for (s,psi)=(r,theta)is not an element of Q={r,theta)/F(r,theta)>c*} is found p=2/pi[root w g'(0)+integral(0)(w) root w-u g '(u)du] where k and F are given continuous functions; (s,psi) is a local polar coordinating with origin at M(r,theta); (r,theta) is the global polar coordinating with origin at O(0,0) F(r,theta)=c* (const.) is the boundary contour partial derivative Q of the considered range Q; g(w)=F(r,theta)/[pi k(psi(0))]; g'=dg/dw; w=N-r(2)sin(2)(theta+psi(0)); psi(0) and N are mean values. The solution shown in type (2.19) of [1] is a special case of the above solution and only suits F(r,theta)=w. The solution of a rigid cone contact with elastic half space, more simple and clear than Love's (1939), is given as an example of application.展开更多
Let N be a sufficiently large even integer. In this paper it is proved that the equation N=p+P_2, p≤N^(0.95), is solvable, where p denotes a prime and P_2 denotes an almost prime with at most two prime factors.
This paper deals with the problem of delay size stability analysis of single input-delayed linear and nonlinear systems. Conventional reduction, reduction linked by sliding mode, and linear memoryless control approach...This paper deals with the problem of delay size stability analysis of single input-delayed linear and nonlinear systems. Conventional reduction, reduction linked by sliding mode, and linear memoryless control approaches are used for simple input-delayed systems to obtain the stability conditions. Several first order examples are investigated systematically to demonstrate the capabilities and limitations of the advanced stability analysis techniques including Lyapunov-Krasovskii functionals, Newton-Leibniz formula, and a newly addressed Lagrange mean value theorem. Numerical comparative results show the usefulness and effectiveness of the advanced delay size analysis techniques proposed in this paper.展开更多
An adaptation of the active disturbance rejection control(AD RC)approach is applied to a fractional-order system with a flat output.Albeit the rather scarce information about the system,the conditions to establish an ...An adaptation of the active disturbance rejection control(AD RC)approach is applied to a fractional-order system with a flat output.Albeit the rather scarce information about the system,the conditions to establish an ultimate bound for a specific configuration of the system are found and compiled in a guideline for the tuning of the observer implemented in the ADRC.展开更多
文摘For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential equations of the second-order. So it is important to develop Asgeirsson mean value theorem. The mean value of solution for the higher order equation hay been discussed primarily and has no exact result at present. The mean value theorem for the higher order equation can be deduced and satisfied generalized biaxial symmetry potential equation by using the result of Asgeirsson mean value theorem and the properties of derivation and integration. Moreover, the mean value formula can be obtained by using the regular solutions of potential equation and the special properties of Jacobi polynomials. Its converse theorem is also proved. The obtained results make it possible to discuss on continuation of the solutions and well posed problem.
文摘The mean value theorem for derivatives says that for a given function over a closed and bounded interval, there is a point <em>P</em> on the graph such that the tangent at <em>P</em> is parallel to the secant through the two endpoints. The mean value theorem for definite integrals says that the area under the function is equal to the area of a rectangle whose base is the length of the interval and height of some point <em>Q</em> on the graph. These two theorems have been studied and utilized extensively and they form the backbone of many important theorems in different branches of mathematics. In this note, we pose the question: for what functions do the two points <em>P </em>and <em>Q</em> always coincide? We find that the only analytic functions satisfying this condition are linear or exponential functions.
基金Supported by National Natural Science Foundation of China (Grant No. 10871016)
文摘Let (Ω, F, P) be a probability space and L0(F,R) the algebra of equivalence classes of real- valued random variables on (Ω, F, P). When L0(F,R) is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0(F, R) to L0(F,R). As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module (S, ||·||) is random uniformly convex iff LP(S) is uniformly convex for each fixed positive number p such that 1 〈 p 〈 +∞.
文摘In this paper we present a mean value theorem derived from Flett's mean value theorem. It turns out that cubic polynomials have the midpoint of the interval as their mean value point.To answer what class of functions have this property,we consider a functional equation associated with this mean value theorem.This equation is then solved in a general setting on abelian groups.
文摘In this note,we consider a new unknown input observer design for nonlinear systems.The main idea consists in determining the estimation error and mean value theorem parameters(β)to introduce them into proposed observer structure.This process is designed on the basis of mean value theorem and genetic algorithm.The stability study relies on the use of a classical quadratic Lyapunov function.The observer’s gains are determined systematically.For the validation of theoretical development proposed in this paper,we consider two practical realizations that deals with the secure communication problem.
文摘The problem of state feedback controllers for a class of Takagi-Sugeno (T-S) Lipschitz nonlinear systems is investigated. A simple systematic and useful synthesis method is proposed based on the use of the differential mean value theorem (DMVT) and convex theory. The proposed design approach is based on the mean value theorem (MVT) to express the nonlinear error dynamics as a convex combination of known matrices with time varying coefficients as linear parameter varying (LPV) systems. Using the Lyapunov theory, stability conditions are obtained and expressed in terms of linear matrix inequalities (LMIs). The controller gains are then obtained by solving linear matrix inequalities. The effectiveness of the proposed approach for closed loop-field oriented control (CL-FOC) of permanent magnet synchronous machine (PMSM) drives is demonstrated through an illustrative simulation for the proof of these approaches. Furthermore, an extension for controller design with parameter uncertainties and perturbation performance is discussed.
基金supported by the National Science Foundation of Shandong Province(ZR2009AM004)
文摘We mainly study the existence of positive solutions for the following third order singular multi-point boundary value problem{x^(3)(t) + f(t, x(t), x′(t)) = 0, 0 〈 t 〈 1,x(0)-∑i=1^m1 αi x(ξi) = 0, x′(0)-∑i=1^m2 βi x′(ηi) = 0, x′(1)=0,where 0 ≤ ai≤∑i=1^m1 αi 〈 1, i = 1, 2, ···, m1, 0 〈 ξ1〈 ξ2〈 ··· 〈 ξm1〈 1, 0 ≤βj≤∑i^m2=1βi〈1,J=1,2, ···, m2, 0 〈 η1〈 η2〈 ··· 〈 ηm2〈 1. And we obtain some necessa βi 〈=11, j = 1,ry and sufficient conditions for the existence of C^1[0, 1] and C^2[0, 1] positive solutions by constructing lower and upper solutions and by using the comparison theorem. Our nonlinearity f(t, x, y)may be singular at x, y, t = 0 and/or t = 1.
基金Supported by the NNSF of China(10371006) Tianyuan Youth Grant of China(10626033).
文摘This paper deals with the existence of triple positive solutions for the 1-dimensional equation of Laplace-type (φ(x′(t)))′+q(t)f(t,x(t),x′(t))=0,t∈(0,1),subject to the following boundary condition:a1φ(x(0))-a2φ(x'(0))=0,a3φ(x(1))+a4φ(x'(1))=0,where φ is an odd increasing homogeneous homeomorphism. By using a new fixed point theorem, sufficient conditions are obtained that guarantee the existence of at least three positive solu- tions. The emphasis here is that the nonlinear term f is involved with the first order derivative explicitly.
基金Natural Scince Foundation of China and Foundation for University Key Teacher by the Ministry of Education
文摘The existence of positive solutions for second order m-point boundary value problemx″-q(t)f(x,x′)x′=0, x(0)= m i=2 b ix(ξ i),x′(1)=αx′(0)are investigated,where ξ i,b i and α are constants satisfying 0=ξ 1<ξ 2<...<ξ m-1 <ξ m=1,b i≥0 for i=2,...,m with β∶= m i=2 b i∈[0,1), and α>1. Our approach is based on the fixed point theorem in cones.
文摘The paper deals with the existence of three- solutions for the second- order differential equations with nonlinear boundary value conditions x″=f(t,x,x′) , t∈ [a,b], g1(x(a) ,x′(a) ) =0 , g2 (x(b) ,x′(b) ) =0 , where f :[a,b]× R1× R1→ R1,gi:R1× R1→ R1(i=1 ,2 ) are continuous functions.The methods employed are the coincidence degree theory.As an application,the sufficient conditions under which there are arbitrary odd solutions for the BVP are obtained
文摘In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdifferentials.
文摘Fixed point theory is one of the most important subjects in the setting of metric spaces since fixed point theorems can be used to determine the existence and the uniqueness of solutions of such mathematical problems.It is known that many problems in applied sciences and engineering can be formulated as functional equations.Such equations can be transferred to fixed point theorems in an easy manner.Moreover,we use the fixed point theory to prove the existence and uniqueness of solutions of such integral and differential equations.Let X be a non-empty set.A fixed point for a self-mapping T on X is a point𝑒𝑒∈𝑋𝑋that satisfying T e=e.One of the most challenging problems in mathematics is to construct some iterations to faster the calculation or approximation of the fixed point of such problems.Some mathematicians constructed and generated some new iteration schemes to calculate or approximate the fixed point of such problems such as Mann et al.[Mann(1953);Ishikawa(1974);Sintunavarat and Pitea(2016);Berinde(2004b);Agarwal,O’Regan and Sahu(2007)].The main purpose of the present paper is to introduce and construct a new iteration scheme to calculate or approximate the fixed point within a fewer number of steps as much as we can.We prove that our iteration scheme is faster than the iteration schemes given by Sintunavarat et al.[Sintunavarat and Pitea(2016);Agarwal,O’Regan and Sahu(2007);Mann(1953);Ishikawa(1974)].We give some numerical examples by using MATLAB to compare the efficiency and effectiveness of our iterations scheme with the efficiency of Mann et al.[Mann(1953);Ishikawa(1974);Sintunavarat and Pitea(2016);Abbas and Nazir(2014);Agarwal,O’Regan and Sahu(2007)]schemes.Moreover,we introduce a problem raised from Newton’s law of cooling as an application of our new iteration scheme.Also,we support our application with a numerical example and figures to illustrate the validity of our iterative scheme.
文摘In this paper, the solution, more general than [1], of a weak singular integral equation integral(0)(pi)integral(-infinity)(infinity) p(s,psi)d sk(psi)d psi=F(r,theta), (r,theta)epsilon (Q) over bar=Q+partial derivative Q subject to constraint p(s,psi)=0, for (s,psi)=(r,theta)is not an element of Q={r,theta)/F(r,theta)>c*} is found p=2/pi[root w g'(0)+integral(0)(w) root w-u g '(u)du] where k and F are given continuous functions; (s,psi) is a local polar coordinating with origin at M(r,theta); (r,theta) is the global polar coordinating with origin at O(0,0) F(r,theta)=c* (const.) is the boundary contour partial derivative Q of the considered range Q; g(w)=F(r,theta)/[pi k(psi(0))]; g'=dg/dw; w=N-r(2)sin(2)(theta+psi(0)); psi(0) and N are mean values. The solution shown in type (2.19) of [1] is a special case of the above solution and only suits F(r,theta)=w. The solution of a rigid cone contact with elastic half space, more simple and clear than Love's (1939), is given as an example of application.
基金Project supported by The National Natural Science Foundation of China (Grant no. 10171060. 19801021)by MCSEC
文摘Let N be a sufficiently large even integer. In this paper it is proved that the equation N=p+P_2, p≤N^(0.95), is solvable, where p denotes a prime and P_2 denotes an almost prime with at most two prime factors.
文摘This paper deals with the problem of delay size stability analysis of single input-delayed linear and nonlinear systems. Conventional reduction, reduction linked by sliding mode, and linear memoryless control approaches are used for simple input-delayed systems to obtain the stability conditions. Several first order examples are investigated systematically to demonstrate the capabilities and limitations of the advanced stability analysis techniques including Lyapunov-Krasovskii functionals, Newton-Leibniz formula, and a newly addressed Lagrange mean value theorem. Numerical comparative results show the usefulness and effectiveness of the advanced delay size analysis techniques proposed in this paper.
文摘An adaptation of the active disturbance rejection control(AD RC)approach is applied to a fractional-order system with a flat output.Albeit the rather scarce information about the system,the conditions to establish an ultimate bound for a specific configuration of the system are found and compiled in a guideline for the tuning of the observer implemented in the ADRC.