Abstract:
We study the fourth order difference equation $$u(m+4) = f(m, u(m), u(m+1),u(m+2), u(m+3)),,$$ where $f: mathbb {Z} imes {mathbb R} ^4 o {mathbb R}$ is continuous and the equation $u_5 = f(m, u_1, u_2, u_3,$ $ u_4)$ can be solved for $u_1$ as a continuous function of $u_2, u_3, u_4, u_5$ for each $m in {mathbb Z}$. It is shown that the uniqueness of solutions implies the existence of solutions for Lidstone boundary-value problems on ${mathbb Z}$. To this end we use shooting and topological methods.

Abstract:
We investigate the following fourth-order four-point nonhomogeneous Sturm-Liouville boundary value problem: , , , where and are nonnegative parameters. Some sufficient conditions are given for the existence and uniqueness of a positive solution. The dependence of the solution on the parameters is also studied.

Abstract:
We investigate the following fourth-order four-point nonhomogeneous Sturm-Liouville boundary value problem: u(4)=f(t,u),t∈[0,1], αu(0) βu′(0)=λ1,γu(1)+δu′(1)=λ2, au′′(ξ1) bu′′′(ξ1)= λ3,cu′′(ξ2)+du′′′(ξ2)= λ4, where 0≤ξ1<ξ2≤1 and λi(i=1,2,3,4) are nonnegative parameters. Some sufficient conditions are given for the existence and uniqueness of a positive solution. The dependence of the solution on the parameters λi(i=1,2,3,4) is also studied.

Abstract:
Let G and f:[0,1]×ℝ4→ℝ be two functions satisfying Caratheodory conditions. This paper is concerned with the problems of existence and uniqueness of solutions for the nonlinear fourth-order ordinary differential equation y′′′′

Abstract:
This article concerns the fourth-order boundary-value problem $$displaylines{ x^{(4)}(t)=f(t,x(t),x''(t)),quad tin (0,1),cr x(0)=x(1)=x''(0)=x''(1)= heta, }$$ in a Banach space. We present some spectral conditions, on the nonlinearity $f(t,u,v)$, to guarantee the existence and uniqueness of solutions. Our method is different from the one used in the references, even the above problem in a scalar space.

By mixed monotone method, we establish the existence and uniqueness of
positive solutions for fourth-order nonlinear singular Sturm-Liouville
problems. The theorems obtained are very general and complement previously
known results.

Abstract:
It is well-known that the first and second Painlev\'e equations admit solutions characterised by divergent asymptotic expansions near infinity in specified sectors of the complex plane. Such solutions are pole-free in these sectors and called tronqu\'ee solutions by Boutroux. In this paper, we show that similar solutions exist for the third and fourth Painlev\'e equations as well.

Abstract:
We examine the equation \[\Delta^2 u = \lambda f(u) \qquad \Omega, \] with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $ \lambda$. We obtain similar results for the sytem {equation*} \{{array}{rrl} -\Delta u &=& \lambda f(v) \qquad \Omega, -\Delta v &=& \gamma g(u) \qquad \Omega, u&=& v = 0 \qquad \partial Omega. {array}. {equation*}

Abstract:
We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial of degree , which involves interpolating data at the odd-order derivatives. For we will provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound. Then, these results will be used to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problems which consist of a th order differential equation and the complementary Lidstone boundary conditions.