If $Y$ is a subset of the space ${ℝ}^n\times {ℝ}^n$, we call a pair of continuous functions $U$, $V$ $Y$-compatible, if they map the space ${ℝ}^n$ into itself and satisfy $Ux·Vy\ge 0$, for all $(x,y)\in Y$ with $x·y\ge 0$. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential $n$-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its...

By using the well-known Leggett–Williams multiple fixed point theorem for cones, some new criteria are established for the existence of three positive periodic solutions for a class of n-dimensional functional differential equations with impulses of the form ⎧y’(t) = A(t)y(t) + g(t,yt), $t\ne t_j$, j ∈ ℤ, ⎨ ⎩$y\left(t{⁺}_j\right)=y\left(t{¯}_j\right)+I_j\left(y\left(t_j\right)\right)$, where $A\left(t\right)={\left(a_{ij}\left(t\right)\right)}_{n\times n}$ is a nonsingular matrix with continuous real-valued entries.

The fourth order periodic boundary value problem $u^{\left(4\right)}-m⁴u+F(t,u)=0$, 0 < t < 2π, with $u^{\left(i\right)}\left(0\right)=u^{\left(i\right)}\left(2\pi \right)$, i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of $\pm {10}^{-7}$.

We study the Dirichlet boundary value problem for the $p$-Laplacian of the form $$-{\Delta }_{p}u-{\lambda }_1{\left|u\right|}^{p-2}u=f\text{in}\Omega ,u=0\text{on}\partial \Omega ,$$ where $\Omega \subset {ℝ}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $N\ge 1$, $p>1$, $f\in C\left(\overline{\Omega }\right)$ and ${\lambda }_1>0$ is the first eigenvalue of ${\Delta }_p$. We study the geometry of the energy functional $$E_p\left(u\right)=\frac{1}{p}{\int }_{\Omega }{\left|\nabla u\right|}^p-\frac{{\lambda }_1}{p}{\int }_{\Omega }{\left|u\right|}^p-{\int }_{\Omega }fu$$ and show the difference between the case $1<p<2$ and the case $p>2$. We also give the characterization of the right hand sides $f$ for which the above Dirichlet problem is solvable and has multiple solutions.

The paper deals with the periodic boundary value problem (1) $L_{4}x\left(t\right)+a\left(t\right)x\left(t\right)\in F(t,x\left(t\right))$, $t\in J=[a,b]$, (2) $L_{i}x\left(a\right)=L_{i}x\left(b\right)$, $i=0,1,2,3$, where $L_{0}x\left(t\right)=a_{0}x\left(t\right)$, $L_{i}x\left(t\right)=a_i\left(t\right)L_{i-1}x\left(t\right)$, $i=1,2,3,4$, $a_0\left(t\right)=a_4\left(t\right)=1$, $a_i\left(t\right)$, $i=1,2,3$ and $a\left(t\right)$ are continuous on $J$, $a\left(t\right)\ge 0$, $a_i\left(t\right)>0$, $i=1,2$, $a_1\left(t\right)=a_3\left(t\right)·F(t,x):J\times R\to${nonempty convex compact subsets of $R$}, $R=(-\infty ,\infty )$. The existence of such periodic solution is proven via Ky Fan’s fixed point theorem.

We study in the space of continuous functions defined on [0,T] with values in a real Banach space E the periodic boundary value problem for abstract inclusions of the form ⎧ $x\in S(x\left(0\right),se{l}_F\left(x\right))$ ⎨ ⎩ x (T) = x(0), where, $F:[0,T]\times \to 2^E$ is a multivalued map with convex compact values, ⊂ E, $se{l}_F$ is the superposition operator generated by F, and S: × L¹([0,T];E) → C([0,T]; ) an abstract operator. As an application, some results are given to the periodic boundary value problem for nonlinear differential inclusions governed...