Abdellatif Ghendir Aoun

In this paper, we study a fractional differential equation $$^{c}D^{\alpha}_{0^{+}}u(t)+f(t,u(t))=0,\quad t\in(0, +\infty)$$ satisfying the boundary conditions: $$u^{\prime}(0)=0,\quad \lim_{t\rightarrow +\infty}\,^{c}D^{\alpha-1}_{0^{+}}u(t)=g(u),$$ where $1<\alpha\leqslant2$, $^{c}D^{\alpha}_{0^{+}}$ is the standard Caputo fractional derivative of order $\alpha$. The main tools used in the paper is contraction principle in the Banach space and the fixed point theorem due to D. O''Regan. Some the compactness criterion and existence of solutions are established.     

多维非退化扩散过程的象集与图集的一致Packing维数

设B(t)=(B(t))=(B1(t),B2(t),…,BN(t))为N维Brown运动,设α(x)=(αij(x),1(≤)I(≤)d,1(≤)j(≤)N),β(x)=(βi(x),1(≤)I(≤)d),x∈Rd,1(≤)d(≤)N,α(x)和β(x)有界连续和满足Lipchitz条件,且存在常数c0＞0,使得对每个x∈Rd,a(x)=α(x)α(x)*的每个特征根都不小于c0.设dX(t)=α(X(t))dB(t) β(X(t))dt,设d(≥)3.可以证明P(ωDimX(E,ω)=DimGRX(E,ω)=2DimE,(A)E∈B[0,∞))=1.这里X(E,ω)={X(t,ω)t∈E},GRX(E,ω)={(t,X(t,ω))t∈E},DimF表示F的Packing维数.     

Uniqueness and existence of solutions for a singular system with nonlocal operator via perturbation method

In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows: $$ \left\{\begin{array}{ll} (-\Delta_p)^su = a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\ (-\Delta_p)^s v= b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega,\ u=v = 0 ,\;\;\mbox{ in }\,\mathbb{R}^N\setminus\Omega, \end{array} \right. $$ where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary, $0<\alpha <1,$ $0<\beta <1,$ $2-\alpha -\beta 相似文献   

Existence Theory for an Arbitrary Order Fractional Differential Equation with Deviating Argument

In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument

Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }We study the global in time existence of small classical solutions to the nonlinear Schr?dinger equation with quadratic interactions of derivative type in two space dimensions

$\left\{{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \right.\quad\quad\quad\quad\quad\quad (0.1)$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)      6. Oscillation Theory of $h$-fractional Difference Equations          下载免费PDF全文 Fanfan Li  Zhenlai Han 《Journal of Nonlinear Modeling and Analysis》2021,3(1):105-113 In this paper, we initiate the oscillation theory for $h$-fractional difference equations of the form \begin{equation*} \begin{cases} _{a}\Delta^{\alpha}_{h}x(t)+r(t)x(t)=e(t)+f(t,x(t)),\ \ \ t\in\mathbb{T}_{h}^{a},\ \ 1<\alpha<2,\x(a)=c_{0},\ \ \Delta_{h}x(a)=c_{1},\ \ \ c_{0}, c_{1}\in\mathbb{R}, \end{cases} \end{equation*} where $_{a}\Delta^{\alpha}_{h}$ is the Riemann-Liouville $h$-fractional difference of order $\alpha,$ $\mathbb{T}_{h}^{a}:=\{a+kh, k\in\mathbb{Z^{+}}\cup\{0\}\},$ and $a\geqslant0,$ $h>0.$ We study the oscillation of $h$-fractional difference equations with Riemann-Liouville derivative, and obtain some sufficient conditions for oscillation of every solution. Finally, we give an example to illustrate our main results.      7. An Existence Result for a Class of Chemically Reacting Systems with Sign-Changing Weights          下载免费PDF全文 S. H. Rasouli & H. Norouzi 《偏微分方程(英文版)》2015,28(1):1-8 We prove the existence of positive solutions for the system$$\begin{align*}\begin{cases}-\Delta_{p} u =\lambda a(x){f(v)}{u^{-\alpha}},\qquad x\in \Omega,\\-\Delta_{q} v = \lambda b(x){g(u)}{v^{-\beta}},\qquad x\in \Omega,\\u = v =0, \qquad x\in\partial \Omega,\end{cases}\end{align*}$$where $\Delta_{r}z={\rm div}(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator and $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\alpha, \beta \in (0,1).$ Here $ a(x)$ and $ b(x)$ are $C^{1}$ sign-changingfunctions that maybe negative near the boundary and $f,g $ are $C^{1}$ nondecreasing functions, such that $f, g :\ [0,\infty)\to [0,\infty);$ $f(s)>0,$ $g(s)>0$ for $s> 0$, $\lim_{s\to\infty}g(s)=\infty$ and$$\lim_{s\to\infty}\frac{f(Mg(s)^{\frac{1}{q-1}})}{s^{p-1+\alpha}}=0,\qquad \forall M>0.$$We discuss the existence of positive weak solutions when $f$, $g$, $a(x)$ and $b(x)$ satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.      8. 一类分数阶微分方程多点边值问题解的存在性      郝晓红  周宗福 《数学研究及应用》2013,33(2):175-188 本文讨论下面一类分数阶微分方程多点边值问题 $$\align &D^{\alpha}_{0+}u(t) = f(t, u(t),~D^{\alpha-1}_{0+}u(t), D^{\alpha-2}_{0+}u(t), D^{\alpha-3}_{0+}u(t)),~~t\in(0,1), \\&I^{4-\alpha}_{0+}u(0) = 0, ~D^{\alpha-1}_{0+}u(0)=\displaystyle{\sum_{i=1}^{m}}\alpha_{i}D^{\alpha-1}_{0+}u(\xi_{i}),\\&D^{\alpha-2}_{0+}u(1)=\sum\limits_ {j=1}^{n}\beta_{j} D^{\alpha-2}_{0+}u(\eta_{j}),~D^{\alpha-3}_{0+}u(1)-D^{\alpha-3}_{0+}u(0)=D^{\alpha-2}_{0+}u(\frac{1}{2}),\endalign$$其中$3<\alpha \leq 4$是一个实数.通过应用Mawhin重合度理论和构建适当的算子,得到了该边值问题解的存在性结果.      9. On the asymptotic behavior of solutions of certain integro-differential equations          下载免费PDF全文 Said R. Grace  John R. Graef  Ercan Tunc 《Journal of Applied Analysis & Computation》2019,9(4):1305-1318 The authors present conditions under which every positive solution $x(t)$ of the integro--differential equation $x^{\prime \prime }(t)=a(t)+\int_{c}^{t}(t-s)^{\alpha-1}[e(s)+k(t,s)f(s,x(s))]ds, \quad c>1, \ \alpha >0,$ satisfies $x(t)=O(tA(t))\textrm{ as }t\rightarrow \infty,$ i.e, $\limsup_{t\rightarrow \infty }\frac{x(t)}{tA(t)}<\infty, \textrm{where} \ A(t)=\int_{c}^{t}a(s)ds.$ From the results obtained, they derive a technique that can be applied to some related integro--differential equations that are equivalent to certain fractional differential equations of Caputo type of any order.      10. Remarks on the regularity criteria of the solutions of the 3D micropolar fluid equations          下载免费PDF全文 Liu Qiao 《Journal of Applied Analysis & Computation》2014,4(4):355-365 We provide two regularity criteria for the weak solutions of the 3D micropolar fluid equations, the first one in terms of one directional derivative of the velocity, i.e., $\partial_{3}u$, while the second one is is in terms of the behavior of the direction of the velocity $\frac{u}{|u|}$. More precisely, we prove that if \begin{equation*} \partial_{3}u \in L^{\beta}(0,T;L^{\alpha}(\mathbb{R}^{3}))\quad\text{ with }\frac{2}{\beta}+\frac{3}{\alpha}\leq 1+\frac{1}{\alpha}, 2&lt; \alpha \leq\infty, 2\leq\beta&lt; \infty; \end{equation*} or \begin{equation*} \operatorname{div}\left(\frac{u}{|u|}\right)\in L^{\frac{4}{1-2r}}(0,T;\dot{X}_{r}(\mathbb{R}^{3}))\quad \text{ with } 0\leq r&lt; \frac{1}{2}, \end{equation*} then the weak solution $(u(x,t),\omega(x,t))$ is regular on $\mathbb{R}^{3}\times [0,T]$. Here $\dot{X}_{r}(\mathbb{R}^{3})$ is the multiplier space.      11. Parabolic power concavity and parabolic boundary value problems      Kazuhiro Ishige  Paolo Salani 《Mathematische Annalen》2014,358(3-4):1091-1117 This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem $$\begin{aligned} (P)\quad \left\{ \begin{array}{l@{\quad }l} \partial _t u=\varDelta u +f(x,t,u,\nabla u) &{} \text{ in }\quad \varOmega \times (0,\infty ),\\ u(x,t)=0 &{} \text{ on }\quad \partial \varOmega \times (0,\infty ),\\ u(x,0)=0 &{} \text{ in }\quad \varOmega , \end{array} \right. \end{aligned}$$ where $\varOmega $ is a bounded convex domain in $\mathbf{R}^n$ and $f$ is a nonnegative continuous function in $\varOmega \times (0,\infty )\times \mathbf{R}\times \mathbf{R}^n$ . We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\overline{\varOmega }\times [0,\infty )$ .      12. Convergence of Solutions of General Dispersive Equations Along Curve      Yong DING  Yaoming NIU 《数学年刊B辑(英文版)》2019,40(3):363-388 In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.      13. Uniqueness of solutions for an integral boundary value problem with fractional $q$-differences          下载免费PDF全文 Yaqiong Cui  Shugui Kang  Huiqin Chen 《Journal of Applied Analysis & Computation》2018,8(2):524-531 This paper deals with uniqueness of solutions for integral boundary value problem$\left\{\begin{array}{l}(D_q^{\alpha}u)(t)+f(t, u(t))=0,\ \ \ t\in(0,1),\ u(0)=D_qu(0)=0,\ \ u(1)=\lambda\int_0^1u(s){\mbox d}_qs, \end{array}\right.$ where $\alpha\in(2,3]$, $\lambda\in (0,[\alpha]_q)$, $D_q^{\alpha}$ denotes the $q$-fractional differential operator of order $\alpha$. By using the iterative method and one new fixed point theorem, we obtain that there exist a unique nontrivial solution and a unique positive solution.      14. 半拟线性p-Laplacian问题结点解的存在性      沈文国 《数学研究及应用》2017,37(2):242-252 In this paper, we study the existence of nodal solutions for the following problem:-(φ_p(x′))′= α(t)φ_p(x~+) + β(t)φ_p(x~-) + ra(t)f(x), 0 t 1,x(0) = x(1) = 0,where φ_p(s) = |s|~(p-2)s, a ∈ C([0, 1],(0, ∞)), x~+= max{x, 0}, x~-=- min{x, 0}, α(t), β(t) ∈C[0, 1]; f ∈ C(R, R), sf(s) 0 for s ≠ 0, and f_0, f_∞∈(0, ∞), where f_0 = lim_|s|→0f(s)/φ_p(s), f_∞ = lim|s|→+∞f(s)/φ_p(s).We use bifurcation techniques and the approximation of connected components to prove our main results.      15. Existence Result for a Superlinear Fractional Navier Boundary Value Problems      Habib?Maagli  Abdelwaheb?DhifliEmail author  Abdulah?Khamis?Alzahrani 《Mediterranean Journal of Mathematics》2018,15(2):68 In this paper, we study the following fractional Navier boundary value problem $$\begin{aligned} \left\{ \begin{array}{lllc} D^{\beta }(D^{\alpha }u)(x)=u(x)g(u(x)),\quad x\in (0,1), \\ \displaystyle \lim _{x\longrightarrow 0}x^{1-\beta }D^{\alpha }u(x)=-a,\quad \,\,u(1)=b, \end{array} \right. \end{aligned}$$ where \(\alpha ,\beta \in (0,1]\) such that \(\alpha +\beta >1\), \(D^{\beta }\) and \(D^{\alpha }\) stand for the standard Riemann–Liouville fractional derivatives and a, b are nonnegative constants such that \(a+b>0\). The function g is a nonnegative continuous function in \([0,\infty )\) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.       16. On Fatou Type Convergence of Convolution Type Double Singular Integral Operators      H. Karsli 《分析论及其应用》2015,31(3):307-320 In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].      17. Multiplicity of Weak Solutions for a $(p(x), q(x))$-Kirchhoff Equation with Neumann Boundary Conditions          下载免费PDF全文 Ahmed Ahme  Mohamed Saad Bouh ELEMINE VALL 《Journal of Applied Analysis & Computation》2024,14(4) The aim of this study is to investigate the existence of infinitely many weak solutions for the $(p(x), q(x))$-Kirchhoff Neumann problem described by the following equation : \begin{equation*} \left\{\begin{array}{ll} -\left(a_{1}+a_{2}\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(\cdot)}u-\left(b_{1}+b_{2}\int_{\Omega}\frac{1}{q(x)}|\nabla u|^{q(x)}dx\right)\Delta_{q(\cdot)}u\+\lambda(x)\Big(|u|^{p(x)-2} u+|u|^{q(x)-2} u\Big)= f_1(x,u)+f_2(x,u) &\mbox{ in } \Omega, \\frac{\partial u}{\partial \nu} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right. \end{equation*} By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.      18. 弱半正三阶三点边值问题的正解      姚庆六 《数学研究及应用》2010,30(1):173-180 The positive solutions are studied for the nonlinear third-order three-point boundary value problem u′″（t）=f（t,u（t））,a.e,t∈[0,1],u（0）=u′（η）=u″（1）=0, where the nonlinear term f（t, u） is a Caratheodory function and there exists a nonnegative function h ∈ L^1[0, 1] such that f（t, u） 〉 ≥-h（t）. The existence of n positive solutions is proved by considering the integrations of ＂height functions＂ and applying the Krasnosel＇skii fixed point theorem on cone.      19. Eigenvalues of Fourth-order Singular Sturm-Liouville Boundary Value Problems          下载免费PDF全文 Lina Zhou  Weihua Jiang  Qiaoluan Li 《Journal of Nonlinear Modeling and Analysis》2020,2(4):485-493 In this paper, by using Krasnoselskii''s fixed-point theorem, some sufficient conditions of existence of positive solutions for the following fourth-order nonlinear Sturm-Liouville eigenvalue problem:\begin{equation*}\left\{\begin{array}{lll} \frac{1}{p(t)}(p(t)u'')''(t)+ \lambda f(t,u)=0, t\in(0,1), \\ u(0)=u(1)=0, \\ \alpha u''(0)- \beta \lim_{t \rightarrow 0^{+}} p(t)u''(t)=0, \\ \gamma u''(1)+\delta\lim_{t \rightarrow 1^{-}} p(t)u''(t)=0, \end{array}\right.\end{equation*} are established, where $\alpha,\beta,\gamma,\delta \geq 0,$ and $~\beta\gamma+\alpha\gamma+\alpha\delta >0$. The function $p$ may be singular at $t=0$ or $1$, and $f$ satisfies Carath\''{e}odory condition.      20. Fujita exponents for evolution problems with nonlocal diffusion   总被引：1，自引：0，他引：1   Jorge García-Melián  Fernando Quirós 《Journal of Evolution Equations》2010,10(1):147-161 We prove the existence of a critical exponent of Fujita type for the nonlocal diffusion problem $\left\{{l@{\quad}l}u_t(x, t) = J*u(x, t)-u(x, t) + u^p(x, t), & \qquad x \in \mathbb{R}^N,\; t > 0,\\ u(x, 0) = u_0(x), & \qquad x \in\mathbb{R}^N,\right.$\left\{\begin{array}{l@{\quad}l}u_t(x, t) = J*u(x, t)-u(x, t) + u^p(x, t), & \qquad x \in \mathbb{R}^N,\; t > 0,\\ u(x, 0) = u_0(x), & \qquad x \in\mathbb{R}^N,\end{array}\right.

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Oscillation Theory of $h$-fractional Difference Equations

In this paper, we initiate the oscillation theory for $h$-fractional difference equations of the form \begin{equation*} \begin{cases} _{a}\Delta^{\alpha}_{h}x(t)+r(t)x(t)=e(t)+f(t,x(t)),\ \ \ t\in\mathbb{T}_{h}^{a},\ \ 1<\alpha<2,\x(a)=c_{0},\ \ \Delta_{h}x(a)=c_{1},\ \ \ c_{0}, c_{1}\in\mathbb{R}, \end{cases} \end{equation*} where $_{a}\Delta^{\alpha}_{h}$ is the Riemann-Liouville $h$-fractional difference of order $\alpha,$ $\mathbb{T}_{h}^{a}:=\{a+kh, k\in\mathbb{Z^{+}}\cup\{0\}\},$ and $a\geqslant0,$ $h>0.$ We study the oscillation of $h$-fractional difference equations with Riemann-Liouville derivative, and obtain some sufficient conditions for oscillation of every solution. Finally, we give an example to illustrate our main results.     

An Existence Result for a Class of Chemically Reacting Systems with Sign-Changing Weights

We prove the existence of positive solutions for the system$$\begin{align*}\begin{cases}-\Delta_{p} u =\lambda a(x){f(v)}{u^{-\alpha}},\qquad x\in \Omega,\\-\Delta_{q} v = \lambda b(x){g(u)}{v^{-\beta}},\qquad x\in \Omega,\\u = v =0, \qquad x\in\partial \Omega,\end{cases}\end{align*}$$where $\Delta_{r}z={\rm div}(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator and $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\alpha, \beta \in (0,1).$ Here $ a(x)$ and $ b(x)$ are $C^{1}$ sign-changingfunctions that maybe negative near the boundary and $f,g $ are $C^{1}$ nondecreasing functions, such that $f, g :\ [0,\infty)\to [0,\infty);$ $f(s)>0,$ $g(s)>0$ for $s> 0$, $\lim_{s\to\infty}g(s)=\infty$ and$$\lim_{s\to\infty}\frac{f(Mg(s)^{\frac{1}{q-1}})}{s^{p-1+\alpha}}=0,\qquad \forall M>0.$$We discuss the existence of positive weak solutions when $f$, $g$, $a(x)$ and $b(x)$ satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.     

一类分数阶微分方程多点边值问题解的存在性

本文讨论下面一类分数阶微分方程多点边值问题 $$\align &D^{\alpha}_{0+}u(t) = f(t, u(t),~D^{\alpha-1}_{0+}u(t), D^{\alpha-2}_{0+}u(t), D^{\alpha-3}_{0+}u(t)),~~t\in(0,1), \\&I^{4-\alpha}_{0+}u(0) = 0, ~D^{\alpha-1}_{0+}u(0)=\displaystyle{\sum_{i=1}^{m}}\alpha_{i}D^{\alpha-1}_{0+}u(\xi_{i}),\\&D^{\alpha-2}_{0+}u(1)=\sum\limits_ {j=1}^{n}\beta_{j} D^{\alpha-2}_{0+}u(\eta_{j}),~D^{\alpha-3}_{0+}u(1)-D^{\alpha-3}_{0+}u(0)=D^{\alpha-2}_{0+}u(\frac{1}{2}),\endalign$$其中$3<\alpha \leq 4$是一个实数.通过应用Mawhin重合度理论和构建适当的算子,得到了该边值问题解的存在性结果.     

On the asymptotic behavior of solutions of certain integro-differential equations

The authors present conditions under which every positive solution $x(t)$ of the integro--differential equation $x^{\prime \prime }(t)=a(t)+\int_{c}^{t}(t-s)^{\alpha-1}[e(s)+k(t,s)f(s,x(s))]ds, \quad c>1, \ \alpha >0,$ satisfies $x(t)=O(tA(t))\textrm{ as }t\rightarrow \infty,$ i.e, $\limsup_{t\rightarrow \infty }\frac{x(t)}{tA(t)}<\infty, \textrm{where} \ A(t)=\int_{c}^{t}a(s)ds.$ From the results obtained, they derive a technique that can be applied to some related integro--differential equations that are equivalent to certain fractional differential equations of Caputo type of any order.     

Remarks on the regularity criteria of the solutions of the 3D micropolar fluid equations

We provide two regularity criteria for the weak solutions of the 3D micropolar fluid equations, the first one in terms of one directional derivative of the velocity, i.e., $\partial_{3}u$, while the second one is is in terms of the behavior of the direction of the velocity $\frac{u}{|u|}$. More precisely, we prove that if \begin{equation*} \partial_{3}u \in L^{\beta}(0,T;L^{\alpha}(\mathbb{R}^{3}))\quad\text{ with }\frac{2}{\beta}+\frac{3}{\alpha}\leq 1+\frac{1}{\alpha}, 2&lt; \alpha \leq\infty, 2\leq\beta&lt; \infty; \end{equation*} or \begin{equation*} \operatorname{div}\left(\frac{u}{|u|}\right)\in L^{\frac{4}{1-2r}}(0,T;\dot{X}_{r}(\mathbb{R}^{3}))\quad \text{ with } 0\leq r&lt; \frac{1}{2}, \end{equation*} then the weak solution $(u(x,t),\omega(x,t))$ is regular on $\mathbb{R}^{3}\times [0,T]$. Here $\dot{X}_{r}(\mathbb{R}^{3})$ is the multiplier space.     

Parabolic power concavity and parabolic boundary value problems

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem $$\begin{aligned} (P)\quad \left\{ \begin{array}{l@{\quad }l} \partial _t u=\varDelta u +f(x,t,u,\nabla u) &{} \text{ in }\quad \varOmega \times (0,\infty ),\\ u(x,t)=0 &{} \text{ on }\quad \partial \varOmega \times (0,\infty ),\\ u(x,0)=0 &{} \text{ in }\quad \varOmega , \end{array} \right. \end{aligned}$$ where $\varOmega $ is a bounded convex domain in $\mathbf{R}^n$ and $f$ is a nonnegative continuous function in $\varOmega \times (0,\infty )\times \mathbf{R}\times \mathbf{R}^n$ . We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\overline{\varOmega }\times [0,\infty )$ .     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

Uniqueness of solutions for an integral boundary value problem with fractional $q$-differences

This paper deals with uniqueness of solutions for integral boundary value problem$\left\{\begin{array}{l}(D_q^{\alpha}u)(t)+f(t, u(t))=0,\ \ \ t\in(0,1),\ u(0)=D_qu(0)=0,\ \ u(1)=\lambda\int_0^1u(s){\mbox d}_qs, \end{array}\right.$ where $\alpha\in(2,3]$, $\lambda\in (0,[\alpha]_q)$, $D_q^{\alpha}$ denotes the $q$-fractional differential operator of order $\alpha$. By using the iterative method and one new fixed point theorem, we obtain that there exist a unique nontrivial solution and a unique positive solution.     

半拟线性p-Laplacian问题结点解的存在性

In this paper, we study the existence of nodal solutions for the following problem:-(φ_p(x′))′= α(t)φ_p(x~+) + β(t)φ_p(x~-) + ra(t)f(x), 0 t 1,x(0) = x(1) = 0,where φ_p(s) = |s|~(p-2)s, a ∈ C([0, 1],(0, ∞)), x~+= max{x, 0}, x~-=- min{x, 0}, α(t), β(t) ∈C[0, 1]; f ∈ C(R, R), sf(s) 0 for s ≠ 0, and f_0, f_∞∈(0, ∞), where f_0 = lim_|s|→0f(s)/φ_p(s), f_∞ = lim|s|→+∞f(s)/φ_p(s).We use bifurcation techniques and the approximation of connected components to prove our main results.     

Existence Result for a Superlinear Fractional Navier Boundary Value Problems

In this paper, we study the following fractional Navier boundary value problem

$$\begin{aligned} \left\{ \begin{array}{lllc} D^{\beta }(D^{\alpha }u)(x)=u(x)g(u(x)),\quad x\in (0,1), \\ \displaystyle \lim _{x\longrightarrow 0}x^{1-\beta }D^{\alpha }u(x)=-a,\quad \,\,u(1)=b, \end{array} \right. \end{aligned}$$

where \(\alpha ,\beta \in (0,1]\) such that \(\alpha +\beta >1\), \(D^{\beta }\) and \(D^{\alpha }\) stand for the standard Riemann–Liouville fractional derivatives and a, b are nonnegative constants such that \(a+b>0\). The function g is a nonnegative continuous function in \([0,\infty )\) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.

     

On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].     

Multiplicity of Weak Solutions for a $(p(x), q(x))$-Kirchhoff Equation with Neumann Boundary Conditions

The aim of this study is to investigate the existence of infinitely many weak solutions for the $(p(x), q(x))$-Kirchhoff Neumann problem described by the following equation : \begin{equation*} \left\{\begin{array}{ll} -\left(a_{1}+a_{2}\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(\cdot)}u-\left(b_{1}+b_{2}\int_{\Omega}\frac{1}{q(x)}|\nabla u|^{q(x)}dx\right)\Delta_{q(\cdot)}u\+\lambda(x)\Big(|u|^{p(x)-2} u+|u|^{q(x)-2} u\Big)= f_1(x,u)+f_2(x,u) &\mbox{ in } \Omega, \\frac{\partial u}{\partial \nu} =0 \quad &\mbox{on} \quad \partial\Omega.\end{array}\right. \end{equation*} By employing a critical point theorem proposed by B. Ricceri, which stems from a more comprehensive variational principle, we have successfully established the existence of infinitely many weak solutions for the aforementioned problem.     

弱半正三阶三点边值问题的正解

The positive solutions are studied for the nonlinear third-order three-point boundary value problem u′″（t）=f（t,u（t））,a.e,t∈[0,1],u（0）=u′（η）=u″（1）=0, where the nonlinear term f（t, u） is a Caratheodory function and there exists a nonnegative function h ∈ L^1[0, 1] such that f（t, u） 〉 ≥-h（t）. The existence of n positive solutions is proved by considering the integrations of ＂height functions＂ and applying the Krasnosel＇skii fixed point theorem on cone.     

Eigenvalues of Fourth-order Singular Sturm-Liouville Boundary Value Problems

In this paper, by using Krasnoselskii''s fixed-point theorem, some sufficient conditions of existence of positive solutions for the following fourth-order nonlinear Sturm-Liouville eigenvalue problem:\begin{equation*}\left\{\begin{array}{lll} \frac{1}{p(t)}(p(t)u'')''(t)+ \lambda f(t,u)=0, t\in(0,1), \\ u(0)=u(1)=0, \\ \alpha u''(0)- \beta \lim_{t \rightarrow 0^{+}} p(t)u''(t)=0, \\ \gamma u''(1)+\delta\lim_{t \rightarrow 1^{-}} p(t)u''(t)=0, \end{array}\right.\end{equation*} are established, where $\alpha,\beta,\gamma,\delta \geq 0,$ and $~\beta\gamma+\alpha\gamma+\alpha\delta >0$. The function $p$ may be singular at $t=0$ or $1$, and $f$ satisfies Carath\''{e}odory condition.     

Fujita exponents for evolution problems with nonlocal diffusion

We prove the existence of a critical exponent of Fujita type for the nonlocal diffusion problem