Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper:

Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

We consider the Hill operator

Multiplicative type functional equations arising from characterization problems

We give the general and the so-called density function solutions of equation

Nonoscillation theorems for certain second order nonlinear differential equations

Some necessary and sufficient conditions for nonoscillation are established for the second order nonlinear differential equation (r(t)y(x(t))|x^￠(t)|^p-1x^￠(t))^￠+c(t)f(x(t))=0,    t 3 t₀,(r(t)\psi(x(t))\vert x^{\prime}(t)\vert^{p-1}x^{\prime}(t))^{\prime}+c(t)f(x(t))=0,\quad t\ge t_0,     

Semiconcavity estimates for viscous Hamilton–Jacobi equations

We present sharp Hessian estimates of the form D² S^e(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation

Numerical verification of condition for approximately midconvex functions

Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR₊ ? \mathbbR₊{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if

An inverse spectral theorem for Kre?n strings with a negative eigenvalue

A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L²(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D_\mathfrakmD_x{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as

On Lipschitz selections of affine-set valued mappings

We prove a Helly-type theorem for the family of all k-dimensional affine subsets of a Hilbert space H. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M,r) ({\cal M},\rho) into this family.¶Let F be such a mapping satisfying the following condition: for every subset M￠ ì M {\cal M'} \subset {\cal M} consisting of at most 2^k+1 points, the restriction F|_M￠ F|_{\cal M'} of F to M￠ {\cal M'} has a selection f_M￠ (i.e. f_M￠(x) ? F(x) for all x  ? M￠) f_{\cal M'}\,({\rm i.e.}\,f_{\cal M'}(x) \in F(x)\,{\rm for\,all}\,x\,\in {\cal M'}) satisfying the Lipschitz condition ||f_M￠(x) - f_M￠(y)||  ￡ r(x,y ), x,y ? M￠ \parallel f_{\cal M'}(x) - f_{\cal M'}(y)\parallel\,\le \rho(x,y ),\,x,y \in {\cal M'} . Then F has a Lipschitz selection f : M ? H f : {\cal M} \to H such that ||f(x) - f(y) ||  ￡ gr(x,y ), x,y ? M \parallel f(x) - f(y) \parallel\,\le \gamma \rho (x,y ),\,x,y \in {\cal M} where g = g(k) \gamma = \gamma(k) is a constant depending only on k. (The upper bound of the number of points in M￠ {\cal M'} , 2^k+1, is sharp.)¶The proof is based on a geometrical construction which allows us to reduce the problem to an extension property of Lipschitz mappings defined on subsets of metric trees.     

Differential Invariants of Second Order ODEs,I

This paper is devoted to differential invariants of equations

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      11. Functional equations on semigroups      P. Sinopoulos 《Aequationes Mathematicae》2000,59(3):255-261 Summary. We determine the general solution g:S? F g:S\to F of the d'Alembert equation¶¶g(x+y)+g(x+sy)=2g(x)g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)g(y)\qquad (x,y\in S) ,¶the general solution g:S? G g:S\to G of the Jensen equation¶¶g(x+y)+g(x+sy)=2g(x)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)\qquad (x,y\in S) ,¶and the general solution g:S? H g:S\to H of the quadratic equation¶¶g(x+y)+g(x+sy)=2g(x)+2g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)+2g(y)\qquad (x,y\in S) ,¶ where S is a commutative semigroup, F is a quadratically closed commutative field of characteristic different from 2, G is a 2-cancellative abelian group, H is an abelian group uniquely divisible by 2, and s \sigma is an endomorphism of S with s(s(x)) = x \sigma(\sigma(x)) = x .      12. On quasimonotone increasing homeomorphisms      Gerd Herzog 《Archiv der Mathematik》2001,77(3):215-221 Let f ? C(\Bbb Rn,\Bbb Rn) f\in C(\Bbb R^n,\Bbb R^n) be quasimonotone increasing such that Y(f(y)-f(x)) ￡ -c Y(y-x) (x << y) \Psi (f(y)-f(x)) \!\le -c \Psi (y-x) (x\ll y) for a linear and strictly positive functional Y \Psi and c > 0. We prove that f is a homeomorphism with decreasing and Lipschitz continuous inverse and we prove the global asymptotic stability of the equilibrium solution of x￠=f(x) x'=f(x) .      13. On some composite functional inequalities      Włodzimierz Fechner 《Aequationes Mathematicae》2010,79(3):307-314 The aim of the paper is to deal with the following composite functional inequalities f(f(x)-f(y)) ￡ f(x+y) + f(f(x-y)) -f(x) - f(y), f(f(x)-f(y)) ￡ f(f(x+y)) + f(x-y) -f(x) - f(y), f(f(x)-f(y)) ￡ f(f(x+y)) + f(f(x-y)) -f(f(x)) - f(y),\begin{gathered}f(f(x)-f(y)) \leq f(x+y) + f(f(x-y)) -f(x) - f(y), \hfill \\ f(f(x)-f(y)) \leq f(f(x+y)) + f(x-y) -f(x) - f(y), \hfill \\ f(f(x)-f(y)) \leq f(f(x+y)) + f(f(x-y)) -f(f(x)) - f(y),\end{gathered}      14. Criteria of exponential decay for eigenfunctions of some classes of integral operators      V. M. Kaplitsky 《Journal of Mathematical Sciences》2010,165(4):455-462 We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded domains in ℝ n . We consider integral operators K whose kernels have the form k( x,y ) = c( x,y )\frace - a| x - y || x - y |b , ( x,y ) ? W×W, k\left( {x,y} \right) = c\left( {x,y} \right)\frac{{{e^{ - \alpha \left| {x - y} \right|}}}}{{{{\left| {x - y} \right|}^\beta }}},\,\left( {x,y} \right) \in \Omega \times \Omega,      15. Large solutions to an anisotropic quasilinear elliptic problem      Jorge García-Melián  Julio D. Rossi  José C. Sabina de Lis 《Annali di Matematica Pura ed Applicata》2010,189(4):689-712 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)      16. A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}      Bart De Bruyn 《Annals of Combinatorics》2010,14(3):307-318 Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2n - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2n - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.      17. On the problem of determining the parameter of a parabolic equation      A. Ashyralyev 《Ukrainian Mathematical Journal》2011,62(9):1397-1408 We study the boundary-value problem of determining the parameter p of a parabolic equation v￠(t) + Av(t) = f(t) + p,    0 \leqslant t \leqslant 1,    v(0) = j,     v(1) = y, v^{\prime}(t) + Av(t) = f(t) + p,\quad 0 \leqslant t \leqslant 1,\quad v(0) = \varphi, \quad v(1) = \psi,      18. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    19. Lagrangian-Penalization Algorithm for Constrained Optimization and Variational Inequalities      Pierre Frankel  Juan Peypouquet 《Set-Valued and Variational Analysis》2012,20(2):169-185 Let X, Y be real Hilbert spaces. Consider a bounded linear operator A : X → Y and a nonempty closed convex set C ì Y\mathcal{C}\subset Y. In this paper we propose an inexact proximal-type algorithm to solve constrained optimization problems (P)                   inf{f(x) : Ax ? C},(\mathcal{P})\qquad\qquad\qquad \inf \{f(x)\ :\ Ax\in \mathcal{C}\},      20. Existence results for second order boundary value problems using the barrier strip technique      P. S. Kelevedjiev  D. O’Regan  N. Popivanov  R. P. Agarwal 《Aequationes Mathematicae》2010,79(3):281-291 Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem ${x''=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,}Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem x"=f(t,x),  t ? (0,1),  x(0)=A,  x￠(1)=0,{x'=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,} where the scalar function f(t, x) may be singular at x = A.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Functional equations on semigroups

Summary. We determine the general solution g:S? F g:S\to F of the d'Alembert equation¶¶g(x+y)+g(x+sy)=2g(x)g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)g(y)\qquad (x,y\in S) ,¶the general solution g:S? G g:S\to G of the Jensen equation¶¶g(x+y)+g(x+sy)=2g(x)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)\qquad (x,y\in S) ,¶and the general solution g:S? H g:S\to H of the quadratic equation¶¶g(x+y)+g(x+sy)=2g(x)+2g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)+2g(y)\qquad (x,y\in S) ,¶ where S is a commutative semigroup, F is a quadratically closed commutative field of characteristic different from 2, G is a 2-cancellative abelian group, H is an abelian group uniquely divisible by 2, and s \sigma is an endomorphism of S with s(s(x)) = x \sigma(\sigma(x)) = x .     

On quasimonotone increasing homeomorphisms

Let f ? C(\Bbb Rⁿ,\Bbb Rⁿ) f\in C(\Bbb R^n,\Bbb R^n) be quasimonotone increasing such that Y(f(y)-f(x)) ￡ -c Y(y-x) (x << y) \Psi (f(y)-f(x)) \!\le -c \Psi (y-x) (x\ll y) for a linear and strictly positive functional Y \Psi and c > 0. We prove that f is a homeomorphism with decreasing and Lipschitz continuous inverse and we prove the global asymptotic stability of the equilibrium solution of x￠=f(x) x'=f(x) .     

On some composite functional inequalities

The aim of the paper is to deal with the following composite functional inequalities

Criteria of exponential decay for eigenfunctions of some classes of integral operators

We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded domains in ℝ ⁿ . We consider integral operators K whose kernels have the form

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)      16. A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}      Bart De Bruyn 《Annals of Combinatorics》2010,14(3):307-318 Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2n - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2n - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.      17. On the problem of determining the parameter of a parabolic equation      A. Ashyralyev 《Ukrainian Mathematical Journal》2011,62(9):1397-1408 We study the boundary-value problem of determining the parameter p of a parabolic equation v￠(t) + Av(t) = f(t) + p,    0 \leqslant t \leqslant 1,    v(0) = j,     v(1) = y, v^{\prime}(t) + Av(t) = f(t) + p,\quad 0 \leqslant t \leqslant 1,\quad v(0) = \varphi, \quad v(1) = \psi,      18. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    19. Lagrangian-Penalization Algorithm for Constrained Optimization and Variational Inequalities      Pierre Frankel  Juan Peypouquet 《Set-Valued and Variational Analysis》2012,20(2):169-185 Let X, Y be real Hilbert spaces. Consider a bounded linear operator A : X → Y and a nonempty closed convex set C ì Y\mathcal{C}\subset Y. In this paper we propose an inexact proximal-type algorithm to solve constrained optimization problems (P)                   inf{f(x) : Ax ? C},(\mathcal{P})\qquad\qquad\qquad \inf \{f(x)\ :\ Ax\in \mathcal{C}\},      20. Existence results for second order boundary value problems using the barrier strip technique      P. S. Kelevedjiev  D. O’Regan  N. Popivanov  R. P. Agarwal 《Aequationes Mathematicae》2010,79(3):281-291 Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem ${x''=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,}Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem x"=f(t,x),  t ? (0,1),  x(0)=A,  x￠(1)=0,{x'=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,} where the scalar function f(t, x) may be singular at x = A.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}

Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2ⁿ - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2ⁿ - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.     

On the problem of determining the parameter of a parabolic equation

We study the boundary-value problem of determining the parameter p of a parabolic equation

Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations

This paper studies the existence of a positive solution to the second-order periodic boundary value problem

Lagrangian-Penalization Algorithm for Constrained Optimization and Variational Inequalities

Let X, Y be real Hilbert spaces. Consider a bounded linear operator A : X → Y and a nonempty closed convex set C ì Y\mathcal{C}\subset Y. In this paper we propose an inexact proximal-type algorithm to solve constrained optimization problems

Existence results for second order boundary value problems using the barrier strip technique

Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem ${x''=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,}Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem x"=f(t,x),  t ? (0,1),  x(0)=A,  x￠(1)=0,{x'=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,} where the scalar function f(t, x) may be singular at x = A.