Solvability of boundary-value problems for nonlinear fractional differential equations

We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations

Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      3. Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function      V. A. Kofanov 《Ukrainian Mathematical Journal》2011,63(7):1118-1135 For arbitrary [α, β] ⊂ R and p > 0, we solve the extremal problem òab | x(k)(t) |qdt ? sup,     q 3 p,    k = 0    \textor    q 3 1,    k 3 1, \int\limits_\alpha^\beta {{{\left| {{x^{(k)}}(t)} \right|}^q}dt \to \sup, \quad q \geq p,\quad k = 0\quad {\text{or}}\quad q \geq 1,\quad k \geq 1},      4. Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion      M. N. Mishra  B. L. S. Prakasa Rao 《Statistical Inference for Stochastic Processes》2011,14(2):101-109 Consider a stochastic process {X t , 0 ≤ t ≤ T} governed by a stochastic differential equation given by dXt = S(Xt)   dt + e  dWtH,    X0=x0,    0 ￡ t ￡ T dX_t= S(X_t) \;dt + \epsilon \; dW_t^H,\quad X_0=x_0,\quad 0 \leq t \leq T      5. Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials      Plamen Djakov  Boris Mityagin 《Mathematische Annalen》2011,351(3):509-540 We consider the Hill operator Ly = - y￠￠ + v(x)y,    0 ￡ x ￡ p,Ly = - y^{\prime \prime} + v(x)y, \quad0 \leq x \leq \pi,      6. Multiple solutions for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}      Fukun Zhao  Leiga Zhao  Yanheng Ding 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,15(6):495-511 This paper is concerned with the following periodic Hamiltonian elliptic system {l-Du+V(x)u=g(x,v) in  \mathbbRN,-Dv+V(x)v=f(x,u) in  \mathbbRN,u(x)? 0 and v(x)?0 as  |x|?￥,\left \{\begin{array}{l}-\Delta u+V(x)u=g(x,v)\, {\rm in }\,\mathbb{R}^N,\\-\Delta v+V(x)v=f(x,u)\, {\rm in }\, \mathbb{R}^N,\\ u(x)\to 0\, {\rm and}\,v(x)\to0\, {\rm as }\,|x|\to\infty,\end{array}\right.      7. On the Euler–Lagrange equation for a variational problem: the general case II      Stefano Bianchini  Matteo Gloyer 《Mathematische Zeitschrift》2010,265(4):889-923 In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L￥loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      8. 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    9. Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations      Mark C. Veraar 《Journal of Evolution Equations》2010,10(1):85-127 In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: (SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.      10. Degenerate differential operators in weighted Hölder spaces      V. P. Orlov 《Mathematical Notes》1977,21(6):428-433 A differential operator ?, arising from the differential expression $$lv(t) \equiv ( - 1)^r v^{[n]} (t) + \sum\nolimits_{k = 0}^{n - 1} {p_k } (t)v^{[k]} (t) + Av(t),0 \leqslant t \leqslant 1,$$ , and system of boundary value conditions $$P_v [v] = \sum\nolimits_{k = 0}^{n_v } {\alpha _{vk} } r^{[k]} (1) = 0.v - 1, \ldots ,\mu ,0 \leqslant \mu< n$$ is considered in a Banach space E. Herev [k](t)=(a(t) d/dt) k v(t)a(t) being continuous fort?0, α(t) >0 for t > 0 and \(\int_0^1 {\frac{{dz}}{{a(z)}} = + \infty ;}\) the operator A is strongly positive in E. The estimates , are obtained for ?: n even, λ varying over a half plane.      11. Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth      Belgacem Rebiai  Saïd Benachour 《Journal of Evolution Equations》2010,10(3):511-527 The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C 1 in \mathbbRn{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)eahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and limh? +￥hb-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.      12. On the Asymptotic Behavior of Solutions of Differential Systems      Tuan Vu  Van Pham 《Ukrainian Mathematical Journal》2005,57(1):165-172 There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let ϕ (t) be a scalar continuous monotonically increasing positive function tending to ∞ as t → ∞. It is established that if all solutions of a differential system satisfy the inequality then the solution x(t; t 0, x 0) of this differential system tends to 0 faster than .__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 137–142, January, 2005.      13. On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type      Zhiting Xu 《Monatshefte für Mathematik》2007,57(5):157-171 Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q1(t)| y(t-s1)|a sgn y(t-s1) +q2(t)| y(t-s2)|b sgn y(t-s2)=0,    t 3 t0,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q 1, q2 ? C([t0, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.      14. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      15. Drift estimation for a periodic mean reversion process      Herold Dehling  Brice Franke  Thomas Kott 《Statistical Inference for Stochastic Processes》2010,13(3):175-192 In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form dXt=(L(t)-a Xt) dt + s dBt,     t 3 0, dX_t=(L(t)-\alpha\, X_t)\, dt + \sigma\, dB_t, \quad t\geq 0,      16. On the existence and uniqueness of minimizers for a class of integral functionals      Graziano Crasta  Annalisa Malusa 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(2):129-150 We study the solvability of the minimization problem minh ? Ka ò0T a(t)[ f( |h￠(t)| ) + g( h(t) ) ]  dt,\mathop {\min }\limits_{\eta \in \mathcal{K}_\alpha } \int_0^T {\alpha (t)\left[ {f\left( {|\eta '(t)|} \right) + g\left( {\eta (t)} \right)} \right]} \,dt,      17. Best constant in a three-parameter Poincaré inequality      I. V. Gerasimov  A. I. Nazarov 《Journal of Mathematical Sciences》2011,179(1):80-99 We study the problem of finding the best constant in the generalized Poincaré inequality lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1],        ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0,      18. Differential Operators Associated to the Cauchy-Fueter Operator in Quaternion Algebra      Thanh Van Nguyen 《Advances in Applied Clifford Algebras》2011,21(3):591-605 This paper deals with the initial value problem of the type ${\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)$ $w(0,x)=\varphi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$ where t is the time, L is a linear first order operator (matrix-type) in Quaternionic Analysis and ${\varphi}This paper deals with the initial value problem of the type \frac?w?t=L(t,x,w,\frac?w?xi)                                 (1){\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)      19. Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>      M. Sh. Shabozov  G. A. Yusupov 《Siberian Mathematical Journal》2011,52(6):1124-1136 We find the exact values of the n-widths for the classes of periodic differentiable functions in L 2[0, 2π] satisfying the constraint ò0h t[(W)\tilde] m1/m (f(r) ;t)dt \leqslant F(h) ,\int\limits_0^h {t\tilde \Omega _m^{1/m} (f^{(r)} ;t)dt \leqslant \Phi (h)} ,      20. Rate of Convergence of Positive Series      O. B. Skaskiv 《Ukrainian Mathematical Journal》2004,56(12):1975-1988 We investigate the rate of convergence of series of the form where λ = (λn), 0 = λ0 < λn ↑ + ∞, n → + ∞, β = {βn: n ≥ 0} ⊂ ℝ+, and τ(x) is a nonnegative function nondecreasing on [0; +∞), and where the sequence λ = (λn) is the same as above and f (x) is a function decreasing on [0; +∞) and such that f (0) = 1 and the function ln f(x) is convex on [0; +∞).__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1665 – 1674, December, 2004.      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

Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

For arbitrary [α, β] ⊂ R and p > 0, we solve the extremal problem

Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion

Consider a stochastic process {X _t , 0 ≤ t ≤ T} governed by a stochastic differential equation given by

Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

We consider the Hill operator

Multiple solutions for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}

This paper is concerned with the following periodic Hamiltonian elliptic system

On the Euler–Lagrange equation for a variational problem: the general case II

In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L^￥_loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem

inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      8. 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    9. Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations      Mark C. Veraar 《Journal of Evolution Equations》2010,10(1):85-127 In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: (SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.      10. Degenerate differential operators in weighted Hölder spaces      V. P. Orlov 《Mathematical Notes》1977,21(6):428-433 A differential operator ?, arising from the differential expression $$lv(t) \equiv ( - 1)^r v^{[n]} (t) + \sum\nolimits_{k = 0}^{n - 1} {p_k } (t)v^{[k]} (t) + Av(t),0 \leqslant t \leqslant 1,$$ , and system of boundary value conditions $$P_v [v] = \sum\nolimits_{k = 0}^{n_v } {\alpha _{vk} } r^{[k]} (1) = 0.v - 1, \ldots ,\mu ,0 \leqslant \mu< n$$ is considered in a Banach space E. Herev [k](t)=(a(t) d/dt) k v(t)a(t) being continuous fort?0, α(t) >0 for t > 0 and \(\int_0^1 {\frac{{dz}}{{a(z)}} = + \infty ;}\) the operator A is strongly positive in E. The estimates , are obtained for ?: n even, λ varying over a half plane.      11. Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth      Belgacem Rebiai  Saïd Benachour 《Journal of Evolution Equations》2010,10(3):511-527 The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C 1 in \mathbbRn{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)eahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and limh? +￥hb-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.      12. On the Asymptotic Behavior of Solutions of Differential Systems      Tuan Vu  Van Pham 《Ukrainian Mathematical Journal》2005,57(1):165-172 There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let ϕ (t) be a scalar continuous monotonically increasing positive function tending to ∞ as t → ∞. It is established that if all solutions of a differential system satisfy the inequality then the solution x(t; t 0, x 0) of this differential system tends to 0 faster than .__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 137–142, January, 2005.      13. On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type      Zhiting Xu 《Monatshefte für Mathematik》2007,57(5):157-171 Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q1(t)| y(t-s1)|a sgn y(t-s1) +q2(t)| y(t-s2)|b sgn y(t-s2)=0,    t 3 t0,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q 1, q2 ? C([t0, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.      14. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      15. Drift estimation for a periodic mean reversion process      Herold Dehling  Brice Franke  Thomas Kott 《Statistical Inference for Stochastic Processes》2010,13(3):175-192 In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form dXt=(L(t)-a Xt) dt + s dBt,     t 3 0, dX_t=(L(t)-\alpha\, X_t)\, dt + \sigma\, dB_t, \quad t\geq 0,      16. On the existence and uniqueness of minimizers for a class of integral functionals      Graziano Crasta  Annalisa Malusa 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(2):129-150 We study the solvability of the minimization problem minh ? Ka ò0T a(t)[ f( |h￠(t)| ) + g( h(t) ) ]  dt,\mathop {\min }\limits_{\eta \in \mathcal{K}_\alpha } \int_0^T {\alpha (t)\left[ {f\left( {|\eta '(t)|} \right) + g\left( {\eta (t)} \right)} \right]} \,dt,      17. Best constant in a three-parameter Poincaré inequality      I. V. Gerasimov  A. I. Nazarov 《Journal of Mathematical Sciences》2011,179(1):80-99 We study the problem of finding the best constant in the generalized Poincaré inequality lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1],        ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0,      18. Differential Operators Associated to the Cauchy-Fueter Operator in Quaternion Algebra      Thanh Van Nguyen 《Advances in Applied Clifford Algebras》2011,21(3):591-605 This paper deals with the initial value problem of the type ${\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)$ $w(0,x)=\varphi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$ where t is the time, L is a linear first order operator (matrix-type) in Quaternionic Analysis and ${\varphi}This paper deals with the initial value problem of the type \frac?w?t=L(t,x,w,\frac?w?xi)                                 (1){\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)      19. Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>      M. Sh. Shabozov  G. A. Yusupov 《Siberian Mathematical Journal》2011,52(6):1124-1136 We find the exact values of the n-widths for the classes of periodic differentiable functions in L 2[0, 2π] satisfying the constraint ò0h t[(W)\tilde] m1/m (f(r) ;t)dt \leqslant F(h) ,\int\limits_0^h {t\tilde \Omega _m^{1/m} (f^{(r)} ;t)dt \leqslant \Phi (h)} ,      20. Rate of Convergence of Positive Series      O. B. Skaskiv 《Ukrainian Mathematical Journal》2004,56(12):1975-1988 We investigate the rate of convergence of series of the form where λ = (λn), 0 = λ0 < λn ↑ + ∞, n → + ∞, β = {βn: n ≥ 0} ⊂ ℝ+, and τ(x) is a nonnegative function nondecreasing on [0; +∞), and where the sequence λ = (λn) is the same as above and f (x) is a function decreasing on [0; +∞) and such that f (0) = 1 and the function ln f(x) is convex on [0; +∞).__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1665 – 1674, December, 2004.      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

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

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E:

(SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.      10. Degenerate differential operators in weighted Hölder spaces      V. P. Orlov 《Mathematical Notes》1977,21(6):428-433 A differential operator ?, arising from the differential expression $$lv(t) \equiv ( - 1)^r v^{[n]} (t) + \sum\nolimits_{k = 0}^{n - 1} {p_k } (t)v^{[k]} (t) + Av(t),0 \leqslant t \leqslant 1,$$ , and system of boundary value conditions $$P_v [v] = \sum\nolimits_{k = 0}^{n_v } {\alpha _{vk} } r^{[k]} (1) = 0.v - 1, \ldots ,\mu ,0 \leqslant \mu< n$$ is considered in a Banach space E. Herev [k](t)=(a(t) d/dt) k v(t)a(t) being continuous fort?0, α(t) >0 for t > 0 and \(\int_0^1 {\frac{{dz}}{{a(z)}} = + \infty ;}\) the operator A is strongly positive in E. The estimates , are obtained for ?: n even, λ varying over a half plane.      11. Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth      Belgacem Rebiai  Saïd Benachour 《Journal of Evolution Equations》2010,10(3):511-527 The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C 1 in \mathbbRn{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)eahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and limh? +￥hb-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.      12. On the Asymptotic Behavior of Solutions of Differential Systems      Tuan Vu  Van Pham 《Ukrainian Mathematical Journal》2005,57(1):165-172 There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let ϕ (t) be a scalar continuous monotonically increasing positive function tending to ∞ as t → ∞. It is established that if all solutions of a differential system satisfy the inequality then the solution x(t; t 0, x 0) of this differential system tends to 0 faster than .__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 137–142, January, 2005.      13. On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type      Zhiting Xu 《Monatshefte für Mathematik》2007,57(5):157-171 Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q1(t)| y(t-s1)|a sgn y(t-s1) +q2(t)| y(t-s2)|b sgn y(t-s2)=0,    t 3 t0,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q 1, q2 ? C([t0, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.      14. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      15. Drift estimation for a periodic mean reversion process      Herold Dehling  Brice Franke  Thomas Kott 《Statistical Inference for Stochastic Processes》2010,13(3):175-192 In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form dXt=(L(t)-a Xt) dt + s dBt,     t 3 0, dX_t=(L(t)-\alpha\, X_t)\, dt + \sigma\, dB_t, \quad t\geq 0,      16. On the existence and uniqueness of minimizers for a class of integral functionals      Graziano Crasta  Annalisa Malusa 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(2):129-150 We study the solvability of the minimization problem minh ? Ka ò0T a(t)[ f( |h￠(t)| ) + g( h(t) ) ]  dt,\mathop {\min }\limits_{\eta \in \mathcal{K}_\alpha } \int_0^T {\alpha (t)\left[ {f\left( {|\eta '(t)|} \right) + g\left( {\eta (t)} \right)} \right]} \,dt,      17. Best constant in a three-parameter Poincaré inequality      I. V. Gerasimov  A. I. Nazarov 《Journal of Mathematical Sciences》2011,179(1):80-99 We study the problem of finding the best constant in the generalized Poincaré inequality lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1],        ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0,      18. Differential Operators Associated to the Cauchy-Fueter Operator in Quaternion Algebra      Thanh Van Nguyen 《Advances in Applied Clifford Algebras》2011,21(3):591-605 This paper deals with the initial value problem of the type ${\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)$ $w(0,x)=\varphi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$ where t is the time, L is a linear first order operator (matrix-type) in Quaternionic Analysis and ${\varphi}This paper deals with the initial value problem of the type \frac?w?t=L(t,x,w,\frac?w?xi)                                 (1){\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)      19. Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>      M. Sh. Shabozov  G. A. Yusupov 《Siberian Mathematical Journal》2011,52(6):1124-1136 We find the exact values of the n-widths for the classes of periodic differentiable functions in L 2[0, 2π] satisfying the constraint ò0h t[(W)\tilde] m1/m (f(r) ;t)dt \leqslant F(h) ,\int\limits_0^h {t\tilde \Omega _m^{1/m} (f^{(r)} ;t)dt \leqslant \Phi (h)} ,      20. Rate of Convergence of Positive Series      O. B. Skaskiv 《Ukrainian Mathematical Journal》2004,56(12):1975-1988 We investigate the rate of convergence of series of the form where λ = (λn), 0 = λ0 < λn ↑ + ∞, n → + ∞, β = {βn: n ≥ 0} ⊂ ℝ+, and τ(x) is a nonnegative function nondecreasing on [0; +∞), and where the sequence λ = (λn) is the same as above and f (x) is a function decreasing on [0; +∞) and such that f (0) = 1 and the function ln f(x) is convex on [0; +∞).__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1665 – 1674, December, 2004.      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

Degenerate differential operators in weighted Hölder spaces

A differential operator ?, arising from the differential expression $$lv(t) \equiv ( - 1)^r v^{[n]} (t) + \sum\nolimits_{k = 0}^{n - 1} {p_k } (t)v^{[k]} (t) + Av(t),0 \leqslant t \leqslant 1,$$ , and system of boundary value conditions $$P_v [v] = \sum\nolimits_{k = 0}^{n_v } {\alpha _{vk} } r^{[k]} (1) = 0.v - 1, \ldots ,\mu ,0 \leqslant \mu< n$$ is considered in a Banach space E. Herev ^[k](t)=(a(t) d/dt) ^k v(t)a(t) being continuous fort?0, α(t) >0 for t > 0 and \(\int_0^1 {\frac{{dz}}{{a(z)}} = + \infty ;}\) the operator A is strongly positive in E. The estimates , are obtained for ?: n even, λ varying over a half plane.     

Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth

The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C ¹ in \mathbbRⁿ{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)e^ahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and lim_{h? +￥}h^b-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.     

On the Asymptotic Behavior of Solutions of Differential Systems

There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let ϕ (t) be a scalar continuous monotonically increasing positive function tending to ∞ as t → ∞. It is established that if all solutions of a differential system satisfy the inequality

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q₁(t)| y(t-s₁)|^a sgn y(t-s₁) +q₂(t)| y(t-s₂)|^b sgn y(t-s₂)=0,    t 3 t₀,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, q₂ ? C([t₀, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.     

Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions

In this paper, we consider the following nonlinear fractional three-point boundary-value problem:

Drift estimation for a periodic mean reversion process

In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form

On the existence and uniqueness of minimizers for a class of integral functionals

We study the solvability of the minimization problem

Best constant in a three-parameter Poincaré inequality

We study the problem of finding the best constant in the generalized Poincaré inequality

Differential Operators Associated to the Cauchy-Fueter Operator in Quaternion Algebra

This paper deals with the initial value problem of the type ${\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)$ $w(0,x)=\varphi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$ where t is the time, L is a linear first order operator (matrix-type) in Quaternionic Analysis and ${\varphi}This paper deals with the initial value problem of the type

\frac?w?t=L(t,x,w,\frac?w?xi)                                 (1){\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)      19. Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>      M. Sh. Shabozov  G. A. Yusupov 《Siberian Mathematical Journal》2011,52(6):1124-1136 We find the exact values of the n-widths for the classes of periodic differentiable functions in L 2[0, 2π] satisfying the constraint ò0h t[(W)\tilde] m1/m (f(r) ;t)dt \leqslant F(h) ,\int\limits_0^h {t\tilde \Omega _m^{1/m} (f^{(r)} ;t)dt \leqslant \Phi (h)} ,      20. Rate of Convergence of Positive Series      O. B. Skaskiv 《Ukrainian Mathematical Journal》2004,56(12):1975-1988 We investigate the rate of convergence of series of the form where λ = (λn), 0 = λ0 < λn ↑ + ∞, n → + ∞, β = {βn: n ≥ 0} ⊂ ℝ+, and τ(x) is a nonnegative function nondecreasing on [0; +∞), and where the sequence λ = (λn) is the same as above and f (x) is a function decreasing on [0; +∞) and such that f (0) = 1 and the function ln f(x) is convex on [0; +∞).__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1665 – 1674, December, 2004.

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Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We find the exact values of the n-widths for the classes of periodic differentiable functions in L ₂[0, 2π] satisfying the constraint

Rate of Convergence of Positive Series

We investigate the rate of convergence of series of the form