Existence and iteration of positive solutions for a three-point boundary value problem of second order integro-differential equation with p-Laplace operator

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for the following three-point boundary value problem $$\left\{\begin{array}{l}(\phi_p(u'))'(t)+q(t)f\left(u(t),u'(t),Tu(t),Su(t)\right)=0,\quad0 < t< 1,\\u'(0)=\alpha u'(\eta),\quad u(1)=g(u'(1)),\end{array}\right.$$ where ? _p (s)=|s| ^p?2 s,p>1,α∈[0,1),η∈(0,1), T and S are all linear operators, g(t) is continuous and nonincreasing on (?∞,0]. The main tools are monotone iterative technique and numerical simulation. We illustrate our results by one example, and give its numerical results by iterative scheme.     

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.     

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.     

Existence of triple symmetric positive solutions for four-point boundary-value problem with one-dimensional p-Laplacian

In this paper, we consider the four-point boundary value problem for one-dimensional p-Laplacian $$\bigl(\phi_{p}(u'(t))\bigr)'+q(t)f\bigl(t,u(t),u'(t)\bigr)=0,\quad t\in(0,1),$$ subject to the boundary conditions $$u(0)-\beta u'(\xi)=0,\qquad u(1)+\beta u'(\eta)=0,$$ where φ _p (s)=|s| ^p?2 s. Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three symmetric positive solutions to the above boundary value problem. The interesting point is the nonlinear term f is involved with the first-order derivative explicitly.     

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.     

Solvability for Some Higher Order Multi-Point Boundary Value Problems at Resonance

In this paper, by using the Mawhin’s continuation theorem, we obtain an existence theorem for some higher order multi-point boundary value problems at resonance in the following form: $$\begin{array}{lll}x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))+e(t),\ t\in(0,1),\\x^{(i)}(0) = 0, i=0,1,\ldots,n-1,\ i\neq p, \\x^{(k)}(1) = \sum\limits_{j=1}^{m-2}{\beta_j}x^{(k)}(\eta_j),\end{array}$$ where ${f:[0,1]\times \mathbb{R}^n \to \mathbb{R}=(-\infty,+\infty)}$ is a continuous function, ${e(t)\in L^1[0,1], p, k\in\{0,1,\ldots,n-1\}}$ are fixed, m ≥ 3 for p ≤ k (m ≥ 4 for p > k), ${\beta_j \in \mathbb{R}, j=1,2,\ldots,m-2, 0 < \eta_1 < \eta_2 < \cdots < \eta_{m-2} <1 }$ . We give an example to demonstrate our results.     

Positive solutions of a Lidstone boundary value problem with variable coefficient function

We establish the existence of positive solutions of the Lidstone boundary value problem $$\begin{array}{rcl}(-1)^{n}u^{(2n)}&=&\lambda a(t)f(u),\quad 0<t<1,\\[3pt]u^{(2i)}(0)&=&u^{(2i)}(1)=0,\quad 0\leq i\leq n-1\end{array}$$ for all sufficiently small positive real λ, where the function a may change sign in [0,1] and the function f:[0,∞)→R satisfies f(0)>0. We also show that our assumption is not vacuous.     

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.     

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 $(0,\eta(1))$      

Numerical extremal solutions for a mixed problem with singular {\phi} -Laplacian

We are concerned with extremal solutions for the mixed boundary value problem $$-\left(r^{N-1}\phi(u')\right)' = r^{N-1} g(r, u), \quad u'(0) = 0 = u(R),$$ where ${g : [0, R] \times \mathbb{R} \to \mathbb{R}}$ is a continuous function and ${\phi : (-\eta, \eta) \to \mathbb{R}}$ is an increasing homeomorphism with ${\phi(0) = 0.}$ We prove the existence of minimal and maximal solutions in presence of well-ordered lower and upper solutions and develop a numerical algorithm for theirs approximation. Also, we provide numerical experiments confirming the theoretical aspects.     

Constant-Sign Lp Solutions for a System of Integral Equations

We consider the following system of integral equations $$u_i(t)=\int^1_0g_i(t,s)f(s,u_1(s),u_2(s),\cdots,u_n(s))ds,\quad t\in \lbrack 0,1\rbrack,1\leq i\leq n.$$ Our aim is to establish criteria such that the above system has a constant-sign solution (u₁, u₂, …, u _n) ∈ (L^p[0, 1])ⁿ, where the integer 1 ≤ p < ∞ is fixed. We shall tackle the case when f is ‘nonnegative’ as well as the case when f is ‘semipositone’. The above problem is also extended to that on the half-line [0, ∞) $$u_i(t)=\int^1_0g_i(t,s)f(s,u_1(s),u_2(s),\cdots,u_n(s))ds,\quad t\in \lbrack 0,\infty ),1\leq i\leq n.$$      

Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.

     

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation \(T\) of a probability space \((X,\mathcal{F },\mu )\) and some \(d \ge 1\) we investigate almost sure and distributional convergence of random variables of the form $$\begin{aligned} x \rightarrow \frac{1}{C_n} \sum _{0\le i_1,\ldots ,\,i_d where \(C_1, C_2,\ldots \) are normalizing constants and the kernel \(f\) belongs to an appropriate subspace in some \(L_p(X^d\!,\, \mathcal{F }^{\otimes d}\!,\,\mu ^d)\) . We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with \(T\) and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and \(d=2\) , we also show that the convergence holds in distribution towards a quadratic form \(\sum _{m=1}^{\infty } \lambda _m\eta ^2_m\) in independent standard Gaussian variables \(\eta _1, \eta _2, \ldots \) .     

Existence and multiplicity of periodic solutions for the ordinary p-Laplacian systems

Some existence and multiplicity results are obtained for periodic solutions of the ordinary p-Laplacian systems: $$\left\{\begin{array}{@{}l@{\quad{}}l}(|u'(t)|^{p-2}u'(t))'=\nabla F(t,u(t)),&\mbox{a.e. }t\in[0,T],\\[4pt]u(0)-u(T)=u'(0)-u'(T)=0\end{array}\right.$$ by using the Saddle Point Theorem, the least action principle and the Three-critical-point Theorem.     

The general iterative methods for nonexpansive semigroups in Banach spaces

Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let ${\mathcal{S} = \{T(s): 0 \leq s < \infty\}}$ be a nonexpansive semigroup on E such that ${Fix(\mathcal{S}) := \cap_{t\geq 0}Fix( T(t) ) \not= \emptyset}$ , and f is a contraction on E with coefficient 0 <  α <  1. Let F be δ-strongly accretive and λ-strictly pseudo-contractive with δ + λ >  1 and ${0 < \gamma < \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }$ . When the sequences of real numbers {α _n } and {t _n } satisfy some appropriate conditions, the three iterative processes given as follows : $${\left.\begin{array}{ll}{x_{n+1} = \alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1} = \alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$ and $$ z_{n+1} = T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$ converge strongly to ${\tilde{x}}$ , where ${\tilde{x}}$ is the unique solution in ${Fix(\mathcal{S})}$ of the variational inequality $${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$ Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065–3071, 2009) and Chen and He (Appl Math Lett 20:751–757, 2007) and many others.     

A property of an autonomous minimax optimal control problem

A control system \(\dot x = f\left( {x,u} \right)\) ,u) with cost functional $$\mathop {ess \sup }\limits_{T0 \leqslant t \leqslant T1} G\left( {x\left( t \right),u\left( t \right)} \right)$$ is considered. For an optimal pair \(\left( {\bar x\left( \cdot \right),\bar u\left( \cdot \right)} \right)\) ,ū(·)), there is a maximum principle of the form $$\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right) = \mathop {\max }\limits_{u \in \Omega \left( t \right)} \eta \left( t \right)f\left( {\bar x\left( t \right),u} \right).$$ By means of this fact, it is shown that \(\eta \left( t \right)f\left( {\bar x\left( t \right),\bar u\left( t \right)} \right)\) is equal to a constant almost everywhere.     

Existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equations

In this paper, we consider the existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equation boundary-value problem $$\left\{ \begin{array}{@{}l}-D^{\alpha}_{0+}u(t)=f(t,u(t)), \quad t\in[0,1]\\[3pt]u(0)=u(1)=u''(0)=0\end{array} \right.$$ where 2<????3 is a real number, and $D^{\alpha}_{0+}$ is the Caputo??s fractional derivative, and f:[0,1]×[0,+??)??[0,+??) is continuous. By means of a fixed-point theorem on cones, some existence, nonexistence and multiplicity of positive solutions are obtained.     

On Nonlocal Boundary Value Problems for Nonlinear Integro-differential Equations of Arbitrary Fractional Order

In this paper, we prove the existence of solutions of a nonlocal boundary value problem for nonlinear integro-differential equations of fractional order given by $$ \begin{array}{ll} ^cD^qx(t) = f(t,x(t),(\phi x)(t),(\psi x)(t)), \quad 0 < t < 1,\\x(0) = \beta x(\eta), x'(0) =0, x''(0) =0, \ldots, x^{(m-2)}(0) =0, x(1)= \alpha x(\eta), \end{array}$$ where $${q \in (m-1, m], m \in \mathbb{N}, m \ge 2}$, $0< \eta <1$$ , and ${\phi x}$ and ${\psi x}$ are integral operators. The existence results are established by means of the contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative example is also presented.     

Nonscattering solutions and blowup at infinity for the critical wave equation

We consider the critical focusing wave equation $(-\partial _t^2+\Delta )u+u^5=0$ in ${\mathbb{R }}^{1+3}$ and prove the existence of energy class solutions which are of the form $$\begin{aligned} u(t,x)=t^\frac{\mu }{2}W(t^\mu x)+\eta (t,x) \end{aligned}$$ in the forward lightcone $\{(t,x)\in {\mathbb{R }}\times {\mathbb{R }}^3: |x|\le t, t\gg 1\}$ where $W(x)=(1+\frac{1}{3} |x|^2)^{-\frac{1}{2}}$ is the ground state soliton, $\mu $ is an arbitrary prescribed real number (positive or negative) with $|\mu |\ll 1$ , and the error $\eta $ satisfies $$\begin{aligned} \Vert \partial _t \eta (t,\cdot )\Vert _{L^2(B_t)} +\Vert \nabla \eta (t,\cdot )\Vert _{L^2(B_t)}\ll 1,\quad B_t:=\{x\in {\mathbb{R }}^3: |x|<t\} \end{aligned}$$ for all $t\gg 1$ . Furthermore, the kinetic energy of $u$ outside the cone is small. Consequently, depending on the sign of $\mu $ , we obtain two new types of solutions which either concentrate as $t\rightarrow \infty $ (with a continuum of rates) or stay bounded but do not scatter. In particular, these solutions contradict a strong version of the soliton resolution conjecture.