The boundary layer equations for the non-Newtonian power law fluid are examined under the classical conditions of uniform flow past a semi infinite flat plate. We investigate the behavior of the similarity solution and employing the Crocco-like transformation we establish the power series representation of the solution near the plate.

We consider a Lidstone boundary value problem in ${ℝ}^k$ at resonance. We prove the existence of a solution under the assumption that the nonlinear part is a Carathéodory map and conditions similar to those of Landesman-Lazer are satisfied.

We consider boundary value problems for second order differential equations of the form ${(x^{\text{'}}+g(t,x,x^{\text{'}}))}^{\text{'}}=f(t,x,x^{\text{'}})$ with the boundary conditions $r(x\left(0\right),x^{\text{'}}\left(0\right),x\left(T\right))+\varphi \left(x\right)=0$, $w(x\left(0\right),x\left(T\right),x^{\text{'}}\left(T\right))+\psi \left(x\right)=0$, where $g,r,w$ are continuous functions, $f$ satisfies the local Carathéodory conditions and $\varphi ,\psi$ are continuous and nondecreasing functionals. Existence results are proved by the method of lower and upper functions and applying the degree theory for $\alpha$-condensing operators.

We investigate two boundary value problems for the second order differential equation with $p$-Laplacian $${\left(a\left(t\right){\Phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}=b\left(t\right)F\left(x\right),t\in I=[0,\infty ),$$ where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: $$\mathrm{i})x\left(0\right)=c>0,\underset{t\to \infty }{lim}x\left(t\right)=0;\mathrm{ii})x^{\text{'}}\left(0\right)=d<0,\underset{t\to \infty }{lim}x\left(t\right)=0.$$

This paper is concerned with existence and uniqueness of solutions of the three-point problem $u^{\text{'}\text{'}\text{'}}=f(t,u,u^{\text{'}},u^{\text{'}\text{'}}),u\left(c\right)=0,u^{\text{'}}\left(a\right)=u^{\text{'}}\left(b\right).u^{\text{'}\text{'}}\left(a\right)=u^{\text{'}\text{'}}\left(b\right),a\le c\le b$. The problem is at resonance, in the sense that the associated linear problem has non-trivial solutions. We use the method of lower and upper solutions.