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.     

Solvability of a Kind of Sturm–Liouville Boundary Value Problems with Impulses via Variational Methods

The purpose of this paper is to use a three critical point theorem due to Ricceri to obtain the existence of at least three solutions for the following Sturm–Liouville boundary value problem with impulses

$\begin{cases}(\phi_{p}(x'(t)))'=(a(t)\phi_{p}(x)+\lambda f(t,x)+\mu h(x))g(x'(t)),\quad \mbox{a.e. }t\in[0,1],\\\Delta G(x'(t_{i}))=I_{i}(x(t_{i})),\quad i=1,2,\ldots,k,\\\alpha_{1}x(0)-\alpha_{2}x'(0)=0,\\\beta_{1}x(1)+\beta_{2}x'(1)=0,\end{cases}$

where p>1, φ _p (x)=|x|^p?2 x, λ, μ are positive parameters, \(G(x)=\int_{0}^{x}\frac{(p-1)|s|^{p-2}}{g(s)}\,ds\). The interest is that the nonlinear term includes x′. We exhibit the existence of at least three solutions and h(x) can be an arbitrary C ¹ functional with compact derivative. As an application, an example is given to illustrate the results.

     

Positive Solutions of a Third-Order Boundary Value Problem with Full Nonlinearity

This paper is concerned with the existence of positive solutions of the third-order boundary value problem with full nonlinearity

$$\begin{aligned} \left\{ \begin{array}{lll} u'''(t)&{}=f(t,u(t),u'(t),u''(t)),\quad t\in [0,1],\\ u(0)&{}=u'(1)=u''(1)=0, \end{array}\right. \end{aligned}$$

where \(f:[0,1]\times \mathbb {R}^+\times \mathbb {R}^+\times \mathbb {R}^-\rightarrow \mathbb {R}^+\) is continuous. Under some inequality conditions on f as |(x, y, z)| small or large enough, the existence results of positive solution are obtained. These inequality conditions allow that f(t, x, y, z) may be superlinear, sublinear or asymptotically linear on x, y and z as \(|(x,y,z)|\rightarrow 0\) and \(|(x,y,z)|\rightarrow \infty \). For the superlinear case as \(|(x,y,z)|\rightarrow \infty \), a Nagumo-type growth condition is presented to restrict the growth of f on y and z. Our discussion is based on the fixed point index theory in cones.

     

Positive solutions for mixed problems of singular fractional differential equations

We investigate the existence of positive solutions to the singular fractional boundary value problem: $^c\hspace{-1.0pt}D^{\alpha }u +f(t,u,u^{\prime },^c\hspace{-2.0pt}D^{\mu }u)=0$, u′(0) = 0, u(1) = 0, where 1 < α < 2, 0 < μ < 1, f is a L^q‐Carathéodory function, $q > \frac{1}{\alpha -1}$, and f(t, x, y, z) may be singular at the value 0 of its space variables x, y, z. Here $^c \hspace{-1.0pt}D$ stands for the Caputo fractional derivative. The results are based on combining regularization and sequential techniques with a fixed point theorem on cones.     

A Class of Nonlocal Impulsive Problems for Integrodifferential Equations in Banach Spaces

In this paper, we study the existence and uniqueness of the PC-mild solution for a class of nonlinear integrodifferential impulsive differential equations with nonlocal conditions $$\left\{\begin{array}{l} x'(t)=Ax(t)+f\left(t,x(t), \int_{0}^{t}k(t,s,x(s))ds\right), \quad t\in J=[0,b], \,\, t\neq t_{i},\\ x(0)=g(x)+x_{0},\\ \Delta x(t_{i})=I_{i}(x(t_{i})), \quad i=1,2,\ldots,p, \,\, 0=t_{0} < t_{1} < \cdots < t_{p} < t_{p+1}=b.\end{array} \right.$$ Using the generalized Ascoli-Arzela theorem given by us, some fixed point technique including Schaefer fixed point theorem and Krasnoselskii fixed point theorem, and theory of operators semigroup, some new results are obtained. At last, some examples are given to illustrate the theory.     

Positive solutions of higher-order singular boundary value problems

Some results of existence of positive solutions for singular boundary value problem $$\left\{\begin{array}{l}\displaystyle (-1)^{m}u^{(2m)}(t)=p(t)f(u(t)),\quad t\in(0,1),\\[2mm]\displaystyle u^{(i)}(0)=u^{(i)}(1)=0,\quad i=0,\ldots,m-1,\end{array}\right.$$ are given, where the function p(t) may be singular at t=0,1. Our analysis relies on the variational method.     

Existence of solutions for a fully nonlinear fourth-order two-point boundary value problem

In this paper, we investigate the existence of solutions of a fully nonlinear fourth-order differential equation $$x^{(4)}=f(t,x,x',x'',x'''),\quad t\in [0,1]$$ with one of the following sets of boundary value conditions; $$x'(0)=x(1)=a_{0}x''(0)-b_{0}x'''(0)=a_{1}x''(1)+b_{1}x'''(1)=0,$$ $$x(0)=x'(1)=a_{0}x''(0)-b_{0}x'''(0)=a_{1}x''(1)+b_{1}x'''(1)=0.$$ By using the Leray-Schauder degree theory, the existence of solutions for the above boundary value problems are obtained. Meanwhile, as an application of our results, an example is given.     

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

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

Oscillation of second order quasilinear neutral delay differential equations

In this paper, by employing Riccati transformation technique, some new sufficient conditions for the oscillation criteria are given for the second order quasilinear neutral delay differential equations with delayed argument in the form $$\bigl(r(t)\bigl|z'(t)\bigr|^{\alpha-1}z'(t)\bigr)'+q(t)f\bigl(x\bigl(\sigma(t)\bigr)\bigr)=0,\quad t\geq t_0,$$ where z(t)=x(t)?p(t)x(??(t)), 0??p(t)??p<1, lim _t???? p(t)=p ₁<1, q(t)>0, ??>0. Two examples are considered to illustrate the main results.     

On solvability of derivative dependent doubly singular boundary value problems

In this paper we establish existence of solutions of singular boundary value problem ?(p(x)y ^′(x))^′=q(x)f(x,y,py′) for 0<x≤b and $\lim_{x\rightarrow0^{+}}p(x)y^{\prime}(x)=0$ , α ₁ y(b)+β ₁ p(b)y ^′(b)=γ ₁ with p(0)=0 and q(x) is allowed to have integrable discontinuity at x=0. So the problem may be doubly singular. Here we consider $\lim_{x\rightarrow0^{+}}\frac{q(x)}{p'(x)}\neq0$ therefore $\lim_{x\rightarrow0^{+}}p(x)y'(x)=0$ does not imply y′(0)=0 unless $\lim_{x\rightarrow0^{+}}f(x,y(x),p(x)y'(x))=0$ .     

Carleman estimates for one-dimensional degenerate heat equations

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=x^α (1 − x)^β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=x^α with α ∈[0, 2). Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Multi-point boundary value problems for one-dimensionalp-Laplacian at resonance

In this paper, we consider the multi-point boundary value problems for one-dimensional p-Laplacian at resonance: $(\phi _p (x'(t)))' = f(t,x(t),x'(t))$ subject to the boundary value conditions: $(\phi _p (x'(0)) = \sum\limits_{i = 1}^{n - 2} {\alpha _i \phi _p (x'(\xi _i ))} $ , $(\phi _p (x'(1)) = \sum\limits_{j = 1}^{m - 2} {\beta _j \phi _p (x'(\eta _i ))} $ where ? _p (s)=|s|^p-2 s, p>1,α_i(1≤i≤n-2)∈R,β_{jit}(1≤j≤m-2)∈R, 0<ξ₁<ξ₂<...<ξ_n-2<1, 0<η₁<η₂<...<η_m-2<1, By applying the extension of Mawhin’s continuation theorem, we prove the existence of at least one solution. Our result is new.     

On global transformations of ordinary differential equations of the second order

The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form

A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics

We consider the ordinary differential equation $$x^2 u''=axu'+bu-c \bigl(u'-1\bigr)^2, \quad x\in(0,x_0), $$ with $a\in\mathbb{R}, b\in\mathbb{R}$ , c>0 and the singular initial condition u(0)=0, which in financial economics describes optimal disposal of an asset in a market with liquidity effects. It is shown in the paper that if a+b<0 then no continuous solutions exist, whereas if a+b>0 then there are infinitely many continuous solutions with indistinguishable asymptotics near 0. Moreover, it is proved that in the latter case there is precisely one solution u corresponding to the choice x ₀=∞ which is such that 0≤u(x)≤x for all x>0, and that this solution is strictly increasing and concave.     

Regularity lifting of weak solutions for nonlinear sub-Laplace equations on homogeneous groups

Let G be a homogeneous group, and let X ₁, X ₂, … , X _m be left invariant real vector fields being homogeneous of degree one on G. We consider the following Dirichlet boundary value problem of the sub-Laplace equation involving the critical exponent and singular term: $$\left\{\begin{array}{ll}-\sum_{j=1}^{m}X_j^2u(x)-\frac{a}{\|x\|^\nu}u(x)=u^{\frac{Q+2}{Q-2}}(x), x\in\Omega,\\ u(x)=0, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\, x\in \partial\Omega,\end{array}\right.$$ where ${\Omega\subset G}$ is a bounded domain with smooth boundary and ${\mathbf{0}\in\Omega}$ , Q is the homogeneous dimension of G, ${a\in \mathbb{R},\ \nu <2 }$ . We boost u to ${L^p(\Omega)}$ for any ${1\leq p < \infty}$ if ${u\in S^{1,2}_0(\Omega)}$ is a weak solution of the problem above.     

On The Generalized Cocycle Equation of Ebanks and Ng

I show that in order to solve the functional equation $$F_{1}(x+y,z)+F_{2}(y+z,x)F_{3}(z+x,\ y)+F_{4}(x,y)+F_{5}(y,z)+F_{6}(z,x)=0$$ for six unknown functions (x,y,z are elements of an abelian monoid, and the codomain of each F _j is the same divisible abelian group) it is necessary and sufficient to solve each of the following equations in a single unknown function $$\matrix{\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad G(x+y,\ z)- G(x,z)- G(y,z)=G(y+z,x)- G(y,x)- G(z,x)\cr \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad H(x+y,\ z)- H(x,z)- H(y,x)+H(y+z,\ x)- H(y,x)- H(z,x)\cr +H(z+x,\ y)- H(z,y)- H(x,y)=0.}$$      

Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in {\mathbb {R}^{3}}

We consider a class of semilinear elliptic equations of the form $$ \label{eq:abs}-\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\mathbb {R}^{3},$$ where ${a:\mathbb {R} \to \mathbb {R}}$ is a periodic, positive, even function and, in the simplest case, ${W : \mathbb {R} \to \mathbb {R}}$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W^{\prime}(q(x))=0,\ x\in\mathbb {R},\quad q(x)\to\pm1\,{\rm as}\, x\to \pm\infty,$$ we show, via variational methods the existence of infinitely many geometrically distinct solutions u of (0.1) verifying u(x, y, z) → ± 1 as x → ± ∞ uniformly with respect to ${(y, z) \in \mathbb {R}^{2}}$ and such that ${\partial_{y}u \not \equiv0, \partial_{z}u \not\equiv 0}$ in ${\mathbb {R}^{3}}$ .     

Positive Solutions for Second Order Singular Boundary Value Problems with Derivative Dependence on Infinite Intervals

The existence of at least one positive solution and the existence of multiple positive solutions are established for the singular second-order boundary value problem

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.     

Triple positive solutions for a class of third-order p-Laplacian singular boundary value problems

In this work, we study the existence of triple positive solutions for one-dimensional p-Laplacian singular boundary value problems $$\begin{array}{l}(\phi_p(y''(t)))'+f(t)g(t,\,y(t),\,y'(t),\,y''(t))=0,\quad 0<t<1,\\[3pt]ay(0)-by'(0)=0,\qquad cy(1)+dy'(1)=0,\qquad y''(0)=0,\end{array}$$ where φ _p (s)=|s| ^p?2 s,?p>1, g:[0,?1]×[0,?+∞)×R ²?[0,?+∞) and f:(0,?1)?[0,?+∞) are continuous. The nonlinear term f may be singular at t=0 and/or t=1. Firstly, Green’s function for the associated linear boundary value problem is constructed. Then, by making use of a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of triple positive solutions to the above boundary value problem. The interesting point is that the nonlinear term g involved with the first-order and second-order derivatives explicitly.