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.

In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that...

We shall consider periodic problems for ordinary differential equations of the form $$\left\{\begin{array}{c}{x}^{\text{'}}\left(t\right)=f(t,x\left(t\right)),\hfill \\ x\left(0\right)=x\left(a\right),\hfill \end{array}\right.$$ where $f:[0,a]\times R^n\to R^n$ satisfies suitable assumptions. To study the above problem we shall follow an approach based on the topological degree theory. Roughly speaking, if on some ball of $R^n$, the topological degree of, associated to (), multivalued Poincaré operator $P$ turns out to be different from zero, then problem () has solutions. Next by using the multivalued version of the classical Liapunov-Krasnoselskǐ guiding potential...

In this paper, we study the existence of periodic solutions to a class of functional differential system. By using Schauder's fixed point theorem, we show that the system has aperiodic solution under given conditions. Finally, four examples are given to demonstrate the validity of our main results.

By means of the Krasnoselskii fixed piont theorem, periodic solutions are found for a neutral type delay differential system of the form $$x^{\text{'}}\left(t\right)+c{x}^{\text{'}}\left(t-\tau \right)=A\left(t,x\left(t\right)\right)x\left(t\right)+f\left(t,x\left(t-r_1\left(t\right)\right),\cdots ,x\left(t-r_k\left(t\right)\right)\right).$$

We consider first order neutral functional differential equations with multiple deviating arguments of the form ${(x\left(t\right)+Bx(t-\delta ))}^{\text{'}}=g₀(t,x\left(t\right))+{\sum }_{k=1}^n{g}_k(t,x(t-{\tau }_k\left(t\right)))+p\left(t\right)$. By using coincidence degree theory, we establish some sufficient conditions on the existence and uniqueness of periodic solutions for the above equation. Moreover, two examples are given to illustrate the effectiveness of our results.

By using the coincidence degree theory of Mawhin, we study the existence of periodic solutions for $n$ th order delay differential equations with damping terms $x^{\left(n\right)}\left(t\right)=\sum _{i=1}^s{b}_i{\left[x^{\left(i\right)}\left(t\right)\right]}^{2k-1}+f\left(x(t-\tau \left(t\right))\right)+p\left(t\right)$. Some new results on the existence of periodic solutions of the investigated equation are obtained.

We study the existence of spatial periodic solutions for nonlinear elliptic equations $-\Delta u+g(x,u\left(x\right))=0,x\in {ℝ}^N$ where $g$ is a continuous function, nondecreasing w.r.t. $u$. We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions $g$ are investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical simulations....

We study the existence of spatial periodic solutions for nonlinear elliptic equations $-\Delta u+g(x,u\left(x\right))=0,x\in {ℝ}^N$ where g is a continuous function, nondecreasing w.r.t. u. We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions g are investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical simulations. ...

In this paper, we consider periodic solutions for a class of nonlinear evolution equations with non-instantaneous impulses on Banach spaces. By constructing a Poincaré operator, which is a composition of the maps and using the techniques of a priori estimate, we avoid assuming that periodic solution is bounded like in [1-4] and try to present new sufficient conditions on the existence of periodic mild solutions for such problems by utilizing semigroup theory and Leray-Schauder's fixed point theorem....