一类Laplace双曲型方程广义Cauchy问题的反问题

本文讨论了确定Ｌａｐｌｉｔｃｅ双曲型方程u_xy(x,y)+a(x,y)u_x(x,y)+b(x,y)+u_y(x,y)+q(x)u(x,y)=f(x,y)的广义Ｃａｕｃｈｙ问题中系数ｑ（ｘ）的反问题。文中利用特征法线及不动点理论，导出了与反问题等价的非线性积分方程组，证明了反问题局部解的存在唯一性，最后给出了反问题整体用的唯一性定理。     

Numerical Solutions to the Darboux Problem with Functional Dependence

The paper deals with the Darboux problem for the equation D _{xy z} (x,y) = f(x,y,z( _x,y ) where z( _x,y ) is a function defined by . We construct a general class of difference methods for this problem. We prove the existence and uniqueness of solutions to implicit functional difference equations by means of a comparison method; moreover we give an error estimate. The convergence of explicit difference schemes is proved under a general assumption that given functions satisfy nonlinear estimates of the Perron type. Our results are illustrated by a numerical example.     

Backward stochastic differential equations and applications to optimal control

We study the existence and uniqueness of the following kind of backward stochastic differential equation, $$x(t) + \int_t^T {f(x(s),y(s),s)ds + \int_t^T {y(s)dW(s) = X,} }$$ under local Lipschitz condition, where (Ω, ?,P, W(·), ?_t) is a standard Wiener process, for any given (x, y),f(x, y, ·) is an ?_t-adapted process, andX is ?_t-measurable. The problem is to look for an adapted pair (x(·),y(·)) that solves the above equation. A generalized matrix Riccati equation of that type is also investigated. A new form of stochastic maximum principle is obtained.     

Uniqueness implies existence for (n,p) boundary value problems

The uniqueness of solutions of certain boundary value problems implies their existence for the nth order nonlinear ordinary differential equation y⁽ⁿ⁾=f(x,y,y^',y^'',....,y^(n-1)) This existence is established by using shooting methods and yields results for (n,p) boundary value problems.     

A periodic problem for the inhomogeneous equation of string oscillations

We study a periodic problem for the equation u _tt−u_xx=g(x, t), u(x, t+T)=u(x, t), u(x+ω, t)= =u(x, t), ℝ² and establish conditions of the existence and uniqueness of the classical solution. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 558–565, April, 1997.     

Almost Automorphic Functions in Frechet Spaces and Applications to Differential Equations

In this paper we first develop a theory of almost automorphic functions with values in Frechet spaces. Then, we consider the semilinear differential equation x'(t) = A x(t) + f(t, x(t)), t ∈ ℝ in a Frechet space X, where A is the infinitesimal generator of a C₀-semigroup satisfying some conditions of exponential stability. Under suitable conditions on f, we prove the existence and uniqueness of an almost automorphic mild solution to the equation.     

The Connection between the Characteristic Roots and the Corresponding Solutions of a Single Linear Differential Equation with Comparable Coefficients

Given a linear differential equation of the form x (ⁿ) + a₁ (t) x (n-1) + …+ a_n (t) x = 0 with variable coefficients defined on the positive semi -axis for t ? 1. We denote its fundamental set of solutions (FSS) by {exp [∫ ri (t) dt] } (i = 1, 2,…,n). In this paper we look for the asymptotic connection (as t → ∞) between the logarithmic derivatives ri (t) of an FSS and of the roots of the characteristic equation yⁿ + a₁ (t) y^n-1 +… + a_n (t) = 0. We mainly consider the case when the coefficients of the equation and the characteristic roots are comparable and have the power order of growth for t → ∞. We discuss the conditions when the functions λ_ii(t) are equivalent to the corresponding roots λ_ii(t) of the characteristic equation as t → ∞.     

Large solutions to an anisotropic quasilinear elliptic problem

In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ${\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)$ in a bounded domain ${\Omega \subset \mathbb{R}^N \times \mathbb{R}^M}In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem

divx (|?x u|p-2?xu)(x,y) + divy (|?y u|q-2?y u) (x, y) = ur(x, y){\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)      9. The Second Boundary-Value Problem for Pseudoparabolic Equations in Noncylindrical Domains      Ivanova  M. V.  Ushakov  V. I. 《Mathematical Notes》2002,72(1-2):43-47 This paper is devoted to the study of the solvability of the second mixed problem in a noncylindrical domain for the nonstationary equation $${\text{div}}(k(x){\text{ grad }}u_t ) - c(x)u_t - b(x)u(x,t) = f(x,t)$$ called the pseudoparabolic equation. We prove existence and uniqueness theorems for the solution in the case of contracting (as time t increases) domains.      10. A coupled non-linear hyperbolic-sobolev system   总被引：1，自引：0，他引：1   Richard E. Ewing 《Annali di Matematica Pura ed Applicata》1977,114(1):331-349 Summary A boundary-initial value problem for a quasilinear hyperbolic system in one space variable is coupled to a boundary-initial value problem for a quasilinear equation of Sobolev type in two space variables of the form Mut(x, t)+L(t) u (x, t)=f(x, t, u(x, t)) where M and L(t) are second order elliptic spacial operators. The coupling occurs through one of the boundary conditions for the hyperbolic system and the source term in the equation of Sobolev type. Such a coupling can arise in the consideration of oil flowing in a fissured medium and out of that medium via a pipe. Barenblatt, Zheltov, and Kochina[2] have modeled flow in a fissured medium via a special case of the above equation. A local existence and uniqueness theorem is demonstrated. The proof involves the method of characteristics, some applications of results of R. Showalter and the contraction mapping theorem. Entrata in Redazione il 28 luglio 1976.      11. Approximation Methods for Solving the Cauchy Problem      Cristinel Mortici 《Czechoslovak Mathematical Journal》2005,55(3):709-718 In this paper we give some new results concerning solvability of the 1-dimensional differential equation y′ = f(x, y) with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if f is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.      12. Multidimensional BSDEs with weak monotonicity and general growth generators      Sheng Jun Fan  Long Jiang 《数学学报(英文版)》2013,29(10):1885-1906 This paper aims at solving a multidimensional backward stochastic differential equation （BSDE） whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonieity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonieity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results.      13. Weak solutionsu(x,t) to parabolic partial differential equations with coefficients that depend uponu(yl, σ, (t,u(x,t))), 1 = 1, … k      John R. Cannon  Paul C. DuChateau 《Journal of Differential Equations》1981,42(3):438-446 The existence of weak solutionsu(x, t) to parabolic partial differential equations with coefficients that depend onu(yl, σl(t, u(x, t))), l = 1,… k, is demonstrated using a retardation of the time arguments in the coefficients along with regularity and compactness results for solutions of linear parabolic partial differential equations.      14. 一类广义单种群Logistic模型正周期解的存在性和唯一性   总被引：1，自引：0，他引：1       下载免费PDF全文 杨鑫松  芮伟国 《数学物理学报(A辑)》2009,29(5):1442-1452 该文利用Krasnoselskii不动点定理和Schwarz不等式, 获得了关于非自治的广义单种群Logistic模型 x=x(t){a(t)-b(t)x(t)-∑ni=1ci (t)x(t-τi(t))-∫0-∞k(t, s)x(t+s)ds} 的正周期解的存在性和唯一性的一些新的结果.      15. Existence and uniqueness result for multidimensional BSDEs with generators of Osgood type      Shengjun FAN  Long JIANG  Matt DAVISON 《Frontiers of Mathematics in China》2013,8(4):811-824 This paper is interested in solving a multidimensional backward stochastic differential equation (BSDE) whose generator satisfies the Osgood condition in y and the Lipschitz condition in z. We establish an existence and uniqueness result of solutions for this kind of BSDEs, which generalizes some known results.      16. Smooth solution of one boundary-value problem      N. H. Khoma  P. V. Tsynalko 《Ukrainian Mathematical Journal》1997,49(12):1932-1937 We study the boundary value problem for the quasilinear equation u u ? uxx=F[u, ut], u(x, 0)= u(x, π)=0, u(x+w, t)=u(x, t), x ε ®, t ε [0, π], and establish conditions under which a theorem on the uniqueness of a smooth solution is true.      17. Constructive existence and uniqueness for some nonlinear two-point boundary value problems      James A. Pennline 《Journal of Mathematical Analysis and Applications》1983,96(2):584-598 Constructive existence and uniqueness theorems are presented for the problem $\text{y″ = ?(x, y), y(0) = y}{}_0\text{, y(1) = y}{}_1$. Applications to several problems are also given including one in which the boundary values are y′(0) = y′0, y(1) = y1.      18. On the global existence and uniqueness of solutions to Prandtl’s system      Xin Ying Xu  Jun Ning Zhao 《数学学报(英文版)》2009,25(1):109-132 In this paper, we consider the Prandtl system for the non-stationary boundary layer in the vicinity of a point where the outer flow has zero velocity. It is assumed that U（t, x, y） = x^mU1（t, x）, where 0 〈 x 〈 L and m 〉 1. We establish the global existence of the weak solution to this problem. Moreover the uniqueness of the weak solution is proved.      19. Holomorphic Solutions of a Functional Equation Related to Nonlinear Difference Systems      Mami Suzuki 《Results in Mathematics》2010,58(1-2):17-35 Here we consider the following functional equation, $$\Psi(X(x,\Psi(x)))=Y(x, \Psi(x)),$$ where X(x, y) and Y(x, y) are holomorphic functions in |x| < δ 1, |y| < δ 1. When we consider a nonlinear simultaneous system of two variables difference equations, we can reduce it to a single difference equation of first order by a solution Ψ of the above functional equation. We obtain a matrix by the linear terms of functions X and Y. When the all eigenvalues of the matrix are equal to 1, it is difficult to have a solution of the above functional equation. In the present paper, we derive a formal solution of the above functional equation under the condition. Further we prove the existence of a solution which is holomorphic and have an asymptotically expansion of the formal solution. Moreover, we will show an example of nonlinear difference system such that our results are applicable.      20. A generalized jacobi theta function      S. H. Lehnigk  G. F. Roach 《Mathematical Methods in the Applied Sciences》1984,6(1):327-344 The delta function initial condition solution v*(x,t;y) at x = y ≥ 0 of the generalized Feller equation is used to define a generalized Jacobi Theta function \documentclass{article}\pagestyle{empty}\begin{document}$ \Theta (x,t) = \upsilon *(x,t;0) + 2\sum\limits_{n = 1}^\infty {v*(x,t;y_n)} $\end{document} for a sufficiently rapidly increasing and unbounded positive sequence {yy}. It is shown that Θ(x,t) is analytic in each variable in certain regions of the complex x and t planes and that it is a solution of the generalized Feller equation. For those parameters for which this equation reduces to the heat equation, Θ(x,t) reduces to the third Jacobi Theta function.

The Second Boundary-Value Problem for Pseudoparabolic Equations in Noncylindrical Domains

This paper is devoted to the study of the solvability of the second mixed problem in a noncylindrical domain for the nonstationary equation $${\text{div}}(k(x){\text{ grad }}u_t ) - c(x)u_t - b(x)u(x,t) = f(x,t)$$ called the pseudoparabolic equation. We prove existence and uniqueness theorems for the solution in the case of contracting (as time t increases) domains.     

A coupled non-linear hyperbolic-sobolev system

Summary A boundary-initial value problem for a quasilinear hyperbolic system in one space variable is coupled to a boundary-initial value problem for a quasilinear equation of Sobolev type in two space variables of the form Mu_t(x, t)+L(t) u (x, t)=f(x, t, u(x, t)) where M and L(t) are second order elliptic spacial operators. The coupling occurs through one of the boundary conditions for the hyperbolic system and the source term in the equation of Sobolev type. Such a coupling can arise in the consideration of oil flowing in a fissured medium and out of that medium via a pipe. Barenblatt, Zheltov, and Kochina[2] have modeled flow in a fissured medium via a special case of the above equation. A local existence and uniqueness theorem is demonstrated. The proof involves the method of characteristics, some applications of results of R. Showalter and the contraction mapping theorem. Entrata in Redazione il 28 luglio 1976.     

Approximation Methods for Solving the Cauchy Problem

In this paper we give some new results concerning solvability of the 1-dimensional differential equation y′ = f(x, y) with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if f is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.     

Multidimensional BSDEs with weak monotonicity and general growth generators

This paper aims at solving a multidimensional backward stochastic differential equation （BSDE） whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonieity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonieity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results.     

Weak solutionsu(x,t) to parabolic partial differential equations with coefficients that depend uponu(yl, σ, (t,u(x,t))), 1 = 1, … k

The existence of weak solutionsu(x, t) to parabolic partial differential equations with coefficients that depend onu(y_l, σ_l(t, u(x, t))), l = 1,… k, is demonstrated using a retardation of the time arguments in the coefficients along with regularity and compactness results for solutions of linear parabolic partial differential equations.     

一类广义单种群Logistic模型正周期解的存在性和唯一性

该文利用Krasnoselskii不动点定理和Schwarz不等式, 获得了关于非自治的广义单种群Logistic模型 x=x(t){a(t)-b(t)x(t)-∑ⁿ_i=1c_i (t)x(t-τ_i(t))-∫⁰_-∞k(t, s)x(t+s)ds} 的正周期解的存在性和唯一性的一些新的结果.     

Existence and uniqueness result for multidimensional BSDEs with generators of Osgood type

This paper is interested in solving a multidimensional backward stochastic differential equation (BSDE) whose generator satisfies the Osgood condition in y and the Lipschitz condition in z. We establish an existence and uniqueness result of solutions for this kind of BSDEs, which generalizes some known results.     

Smooth solution of one boundary-value problem

We study the boundary value problem for the quasilinear equation u _u ? u_xx=F[u, u_t], u(x, 0)= u(x, π)=0, u(x+w, t)=u(x, t), x ε ®, t ε [0, π], and establish conditions under which a theorem on the uniqueness of a smooth solution is true.     

Constructive existence and uniqueness for some nonlinear two-point boundary value problems

Constructive existence and uniqueness theorems are presented for the problem $\text{y″ = ?(x, y), y(0) = y}{}_0\text{, y(1) = y}{}_1$. Applications to several problems are also given including one in which the boundary values are y′(0) = y′₀, y(1) = y₁.     

On the global existence and uniqueness of solutions to Prandtl’s system

In this paper, we consider the Prandtl system for the non-stationary boundary layer in the vicinity of a point where the outer flow has zero velocity. It is assumed that U（t, x, y） = x^mU1（t, x）, where 0 〈 x 〈 L and m 〉 1. We establish the global existence of the weak solution to this problem. Moreover the uniqueness of the weak solution is proved.     

Holomorphic Solutions of a Functional Equation Related to Nonlinear Difference Systems

Here we consider the following functional equation, $$\Psi(X(x,\Psi(x)))=Y(x, \Psi(x)),$$ where X(x, y) and Y(x, y) are holomorphic functions in |x| < δ ₁, |y| < δ ₁. When we consider a nonlinear simultaneous system of two variables difference equations, we can reduce it to a single difference equation of first order by a solution Ψ of the above functional equation. We obtain a matrix by the linear terms of functions X and Y. When the all eigenvalues of the matrix are equal to 1, it is difficult to have a solution of the above functional equation. In the present paper, we derive a formal solution of the above functional equation under the condition. Further we prove the existence of a solution which is holomorphic and have an asymptotically expansion of the formal solution. Moreover, we will show an example of nonlinear difference system such that our results are applicable.     

A generalized jacobi theta function

The delta function initial condition solution v^*(x,t;y) at x = y ≥ 0 of the generalized Feller equation is used to define a generalized Jacobi Theta function \documentclass{article}\pagestyle{empty}\begin{document}$ \Theta (x,t) = \upsilon *(x,t;0) + 2\sum\limits_{n = 1}^\infty {v*(x,t;y_n)} $\end{document} for a sufficiently rapidly increasing and unbounded positive sequence {y_y}. It is shown that Θ(x,t) is analytic in each variable in certain regions of the complex x and t planes and that it is a solution of the generalized Feller equation. For those parameters for which this equation reduces to the heat equation, Θ(x,t) reduces to the third Jacobi Theta function.