Positive solutions for a boundary-value problem with Riemann–Liouville fractional derivative*

In this work, we are mainly concerned with the existence of positive solutions for the fractional boundary-value problem $$ \left\{ {\begin{array}{*{20}{c}} {D_{0+}^{\alpha }D_{0+}^{\alpha }u=f\left( {t,u,{u}^{\prime},-D_{0+}^{\alpha }u} \right),\quad t\in \left[ {0,1} \right],} \hfill \\ {u(0)={u}^{\prime}(0)={u}^{\prime}(1)=D_{0+}^{\alpha }u(0)=D_{0+}^{{\alpha +1}}u(0)=D_{0+}^{{\alpha +1}}u(1)=0.} \hfill \\ \end{array}} \right. $$ Here ?? ?? (2, 3] is a real number, $ D_{0+}^{\alpha } $ is the standard Riemann?CLiouville fractional derivative of order ??. By virtue of some inequalities associated with the fractional Green function for the above problem, without the assumption of the nonnegativity of f, we utilize the Krasnoselskii?CZabreiko fixed-point theorem to establish our main results. The interesting point lies in the fact that the nonlinear term is allowed to depend on u, u??, and $ -D_{0+}^{\alpha } $ .     

A singular boundary value problem for nonlinear differential equations of fractional order

We are concerned with the nonlinear differential equation of fractional order $$\mathcal{D}^{\alpha}_{0+}u(t)=f(t,u(t),u'(t)),\quad \mbox{a.\,e.}\ t\in (0,1),$$ where $\mathcal{D}^{\alpha}_{0+}$ is the Riemann-Liouville fractional order derivative, subject to the boundary conditions $$u(0)=u(1)=0.$$ We obtain the existence of at least one solution using the Leray-Schauder Continuation Principle.     

Existence results of fractional differential equations with irregular boundary conditions and p-Laplacian operator

In this paper, we discuss the existence of solutions for irregular boundary value problems of nonlinear fractional differential equations with p-Laplacian operator $$\left \{ \begin{array}{l} {\phi}_p(^cD_{0+}^{\alpha}u(t))=f(t,u(t),u'(t)), \quad 0< t<1, \ 1< \alpha \leq2, \\ u(0)+(-1)^{\theta}u'(0)+bu(1)=\lambda, \qquad u(1)+(-1)^{\theta}u'(1)=\int_0^1g(s,u(s))ds,\\ \quad \theta=0,1, \ b \neq \pm1, \end{array} \right . $$ where \(^{c}D_{0+}^{\alpha}\) is the Caputo fractional derivative, ? _p (s)=|s| ^p?2 s, p>1, \({\phi}_{p}^{-1}={\phi}_{q}\) , \(\frac {1}{p}+\frac{1}{q}=1\) and \(f: [0,1] \times\mathbb{R} \times\mathbb {R} \longrightarrow\mathbb{R}\) . Our results are based on the Schauder and Banach fixed point theorems. Furthermore, two examples are also given to illustrate the results.     

Multiple positive solutions for fractional differential systems

In this paper, we study the existence of positive solution to boundary value problem for fractional differential system $$\left\{\begin{array}{ll}D_{0^+}^\alpha u (t) + a_1 (t) f_1 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1),\\D_{0^+}^\alpha v (t) + a_2 (t) f_2 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1), \;\; 2 < \alpha < 3,\\u (0)= u' (0) = 0, \;\;\;\; u' (1) - \mu_1 u' (\eta_1) = 0,\\v (0)= v' (0) = 0, \;\;\;\; v' (1) - \mu_2 v' (\eta_2) = 0,\end{array}\right.$$ where ${D_{0^+}^\alpha}$ is the Riemann-Liouville fractional derivative of order ??. By using the Leggett-Williams fixed point theorem in a cone, the existence of three positive solutions for nonlinear singular boundary value problems is obtained.     

Existence of positive solutions for fractional differential systems with multi point boundary conditions

In this paper, we study the existence of positive solutions to the boundary value problem for the fractional differential system $$\left\{\begin{array}{lll} D_{0^+}^\beta \phi_p(D_{0^+}^\alpha u) (t) = f_1 (t, u (t), v (t)),\quad t \in (0, 1),\\ D_{0^+}^\beta \phi_p(D_{0^+}^\alpha v) (t) = f_2 (t, u (t), v(t)), \quad t \in (0, 1),\\ D_{0^+}^\alpha u(0)= D_{0^+}^\alpha u(1)=0,\; u (0) = 0, \quad u (1)-\Sigma_{i=1}^{m-2} a_{1i}\;u(\xi_{1i})=\lambda_1,\\ D_{0^+}^\alpha v(0)= D_{0^+}^\alpha v(1)=0,\; v (0) = 0, \quad v (1)-\Sigma_{i=1}^{m-2} a_{2i}\; v(\xi_{2i})=\lambda_2, \end{array}\right. $$ where ${1<\alpha,\beta\leq 2, 2 <\alpha + \beta\leq 4, D_{0^+}^\alpha}$ is the Riemann–Liouville fractional derivative of order α. By using the Leggett–Williams fixed point theorem in a cone, the existence of three positive solutions for nonlinear singular boundary value problems is obtained.     

On the Generalized Feynman–Kac Transformation for?Nearly Symmetric Markov Processes

Suppose that X is a right process which is associated with a non-symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on L ²(E;m). For $u\in D(\mathcal{E})$ , we have Fukushima??s decomposition: $\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$ . In this paper, we investigate the strong continuity of the generalized Feynman?CKac semigroup defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$ . Let $Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E})_{b}$ . Denote by J ₁ the dissymmetric part of the jumping measure J of $(\mathcal{E},D(\mathcal{E}))$ . Under the assumption that J ₁ is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if and only if there exists a constant ?? ₀??0 such that $\|P^{u}_{t}\|_{2}\leq e^{\alpha_{0}t}$ for every t>0. If one of these conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on L ²(E;m). If X is equipped with a differential structure, then this result also holds without assuming that J ₁ is finite.     

Existence and uniqueness of nontrivial solutions for eigenvalue boundary value problem of nonlinear fractional differential equation

In this paper, we study the existence and uniqueness of a nontrivial solution to eigenvalue problems for the following nonlinear fractional differential equation of the form $$\begin{aligned} \left\{ \begin{array}{l} -D^{\alpha }_{0^{+}}u(t)=\lambda [f(t, u(t), D^{\beta }_{0^{+}}u(t))+g(t)],~~ 0 where \(\lambda \) is a parameter, \(D^{\alpha }_{0^{+}},D^{\beta }_{0^{+}}\) are two standard Riemann–Liouville fractional derivatives, \(0<\beta <1<\alpha \le 2,\alpha -\beta >1,f: [0,1]\times {\mathbb{R }}\times {\mathbb{R }}\rightarrow {\mathbb{R }}\) is continuous, and \(g(t): (0, 1)\rightarrow [0, +\infty )\) is Lebesgue integrable. We obtain several sufficient conditions of the existence and uniqueness of nontrivial solution of the above eigenvalue problems when \(\lambda \) is in some interval. Our approach is based on the Leray–Schauder nonlinear alternative. In addition, some examples are included to demonstrate the main result.     

Convergence of bounded solutions of nonlinear parabolic problems on a bounded interval: the singular case

In this article we prove that for every ${1 < p \le 2}$ and for every continuous function ${f: [0,1]\times\mathbb{R}\to\mathbb{R}}$ , which is Lipschitz continuous in the second variable, uniformly with respect to the first one, each bounded solution of the one-dimensional heat equation $$\begin{array}{ll}u_{t}-\{|u_{x}|^{p-2}u_{x} \}_{x}+f(x,u)=0\qquad{\rm in} \quad (0,1)\times (0,+\infty) \end{array}$$ with homogeneous Dirichlet boundary conditions converges as ${t\to+\infty}$ to a stationary solution. The proof follows an idea of Matano which is based on a comparison principle. Thus, a key step is to prove a comparison principle on non-cylindrical open sets.     

The reverse H?lder inequality for power means

For a function ?? non-negative on the interval [0, 1], the power mean of order ??????0 is defined by the equality $ \mathcal{M}_{\alpha \varphi} (t) = {\left( {\frac{1}{t}\int_0^t {{\varphi^\alpha }(u)du} } \right)^{1/\alpha }},\,0 < t \leqslant 1 $ . We consider the class $ {\widetilde{{RH}}^{\alpha, \beta }}(B) $ of functions ?? satisfying the reverse H?lder inequality $$ {\mathcal{M}_\beta }_\varphi \leqslant B \cdot {\mathcal{M}_\alpha }_\varphi $$ at some ???<???,??·??????0,???>?1. The sharp estimates for the summability exponents of the compositions of power means are established. As a result, we determine the properties of self-improvement of the summability exponents of functions from $ {\widetilde{{RH}}^{\alpha, \beta }}(B) $ .     

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 $\overline{\varOmega }\times [0,\infty )$ .     

Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities

Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z ^d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = e^{γ t} , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = e^{γ t} , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,\eta(1))$      

Convergence to steady states of solutions to nonlinear integral evolution equations

We prove that any global bounded solution of the nonlinear evolutionary integral equation $$\dot{u}(t) + \int\limits_0^t a(t-s)\mathcal{E}'(u(s))ds =f(t), \quad t >0 $$ tends to a single equilibrium state for long time (i.e., ${\mathcal{E}'(\vartheta)=0}$ where ${\vartheta= \lim_{t \rightarrow \infty} u(t)}$ on a real Hilbert space), where ${\mathcal{E}'}$ is the Fréchet derivative of a functional ${\mathcal{E}}$ , which satisfies the ?ojasiewicz?CSimon inequality near ${\vartheta}$ . The vector-valued function f and the scalar kernel a satisfy suitable conditions.     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

On the asymptotic behavior of solutions of Emden–Fowler equations on time scales

Consider the Emden-Fowler dynamic equation $$ x^{\Delta\Delta}(t)+p(t)x^\alpha(t)=0,\:\:\alpha >0 , \qquad \qquad \qquad \qquad (0.1) $$ where ${p\in C_{rd}([t_0,\infty)_{\mathbb{T}},\mathbb{R}), \alpha}$ is the quotient of odd positive integers, and ${\mathbb{T}}$ denotes a time scale which is unbounded above and satisfies an additional condition (C) given below. We prove that if ${\int^\infty_{t_0}t^\alpha |p(t)|\Delta t<\infty}$ (and when ???=?1 we also assume lim _t???? tp(t)??(t)?=?0), then (0.1) has a solution x(t) with the property that $$ \lim_{t\rightarrow\infty} \frac{x(t)}{t}=A\neq 0.$$      

Existence and Multiplicity of Positive Solutions to a Quasilinear Elliptic Equation with Strong Allee Effect Growth Rate

In this paper we consider a p-Laplacian equation with strong Allee effect growth rate and Dirichlet boundary condition $$\left\{\begin{array}{ll} {\rm div} (|\nabla u|^{p-2} \nabla u) + \lambda f(x,u)=0, &\quad x \in \Omega, \\ u=0, &\quad x \in \partial \Omega, \qquad \qquad ^ {(P_\lambda)} \end{array}\right.$$ where Ω is a bounded smooth domain in ${\mathbb{R}^N}$ for ${N \ge 1, p > 1}$ , and λ is a positive parameter. By using variational methods and a suitable truncation technique, we prove that problem (P _λ) has at least two positive solutions for large parameter and it has no positive solutions for small parameter. In addition, a nonexistence result is investigated.     

Global solutions in higher dimensions to a fourth-order parabolic equation modeling epitaxial thin-film growth

The initial-value problem for $$u_t=-\Delta^2 u - \mu\Delta u - \lambda \Delta |\nabla u|^2 + f(x)\qquad \qquad (\star)$$ is studied under the conditions ${{\frac{\partial}{\partial\nu}} u={\frac{\partial}{\partial\nu}} \Delta u=0}$ on the boundary of a bounded convex domain ${\Omega \subset {\mathbb{R}}^n}$ with smooth boundary. This problem arises in the modeling of the evolution of a thin surface when exposed to molecular beam epitaxy. Correspondingly the physically most relevant spatial setting is obtained when n?=?2, but previous mathematical results appear to concentrate on the case n?=?1. In this work, it is proved that when n??? 3,??? ?? 0, ???>?0 and ${f \in L^\infty(\Omega)}$ satisfies ${{\int_\Omega} f \ge 0}$ , for each prescribed initial distribution ${u_0 \in L^\infty(\Omega)}$ fulfilling ${{\int_\Omega} u_0 \ge 0}$ , there exists at least one global weak solution ${u \in L^2_{loc}([0,\infty); W^{1,2}(\Omega))}$ satisfying ${{\int_\Omega} u(\cdot,t) \ge 0}$ for a.e. t?>?0, and moreover, it is shown that this solution can be obtained through a Rothe-type approximation scheme. Furthermore, under an additional smallness condition on??? and ${\|f\|_{L^\infty(\Omega)}}$ , it is shown that there exists a bounded set ${S\subset L^1(\Omega)}$ which is absorbing for ${(\star)}$ in the sense that for any such solution, we can pick T?>?0 such that ${e^{2\lambda u(\cdot,t)}\in S}$ for all t?>?T, provided that ?? is a ball and u ₀ and f are radially symmetric with respect to x?=?0. This partially extends similar absorption results known in the spatially one-dimensional case. The techniques applied to derive appropriate compactness properties via a priori estimates include straightforward testing procedures which lead to integral inequalities involving, for instance, the functional ${{\int_\Omega} e^{2\lambda u}dx}$ , but also the use of a maximum principle for second-order elliptic equations.     

Positive solutions to m-point boundary value problem of fractional differential equation

In this paper, we study the multiplicity of positive solutions to the following m-point boundary value problem of nonlinear fractional differential equations: Dqu(t) + f(t, u(t)) = 0, 0 < t < 1, u(0) = 0, u(1) =sum (μiDpu(t)|t = ξi ) from i =1 to ∞ m-2, where q ∈R , 1 相似文献