In this paper, we study differential equations arising from the generating functions of the generalized Bell polynomials.We give explicit identities for the generalized Bell polynomials. Finally, we investigate the zeros of the generalized Bell polynomials by using numerical simulations.

New existence results are presented for the two point singular “resonant” boundary value problem $\frac{1}{p}{\left(p{y}^{\text{'}}\right)}^{\text{'}}+ry+{\lambda }_{m}qy=f(t,y,p{y}^{\text{'}})$ a.eȯn $[0,1]$ with $y$ satisfying Sturm Liouville or Periodic boundary conditions. Here ${\lambda }_m$ is the ${(m+1)}^{st}$ eigenvalue of $\frac{1}{pq}[{\left(p{u}^{\text{'}}\right)}^{\text{'}}+rpu]+\lambda u=0$ a.eȯn $[0,1]$ with $u$ satisfying Sturm Liouville or Periodic boundary data.

This paper aims to provide a systematic approach to the treatment of differential equations of the typedyt = Σi fi(yt) dxti where the driving signal xt is a rough path. Such equations are very common and occur particularly frequently in probability where the driving signal might be a vector valued Brownian motion, semi-martingale or similar process.However, our approach is deterministic, is totally independent of probability and permits much rougher paths than the Brownian paths usually discussed....

We extend the method of quasilinearization to differential equations in abstract normal cones. Under some assumptions, corresponding monotone iterations converge to the unique solution of our problem and this convergence is superlinear or semi–superlinear

In this paper, using the properties of the Henstock-Kurzweil integral and corresponding theorems, we prove the existence theorem for the equation x' = f(t,x) and inclusion x' ∈ F(t,x) in a Banach space, where f is Henstock-Kurzweil integrable and satisfies some conditions.

We give a meaning to derivative of a function $uℝ\to X$, where $X$ is a complete metric space. This enables us to investigate differential equations in a metric space. One can prove in particular Gronwall’s Lemma, Peano and Picard Existence Theorems, Lyapunov Theorem or Nagumo Theorem in metric spaces. The main idea is to define the tangent space ${𝒯}_{x}X$ of $x\in X$. Let $u,v[0,1)\to X$, $u\left(0\right)=v\left(0\right)$ be continuous at zero. Then by the definition $u$ and $v$ are in the same equivalence class if they are tangent at zero, that is if $$\underset{h\to 0^{+}}{lim}\frac{d\left(u\right(h),v(h\left)\right)}{h}=0.$$ By ${𝒯}_{x}X$ we denote...

In this article, we describe the differential equations on functions from R into real Banach space. The descriptions are based on the article [20]. As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article [21] and the article [32]. And applying the theorems of Riemann integral introduced in the article [22], we proved the ordinary differential equations on real...

The aim of this paper is to construct the analytic vector fields with given as trajectories or solutions. In particular we construct the polynomial vector field from given conics (ellipses, hyperbola, parabola, straight lines) and determine the differential equations from a finite number of solutions.

We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra which suits well to the fact that, in accordance with Willems' standpoint, flatness and defect are best defined without...

Two models of microbial growth are derived as a resuslt of a discussion of the models of Monod and Hinshelwood types. The approach takes account of the lyse of dead cells in inhibitory products as well as in those which stimulate the growth. The asymptotic behaviour of the models is proved and the models applied to a chemostat.

In this paper we consider the nonlocal (nonstandard) Cauchy problem for differential inclusions in Banach spaces x'(t) ∈ F(t,x(t)), x(0)=g(x), t ∈ [0,T] = I. Investigation over some multivalued integrals allow us to prove the existence of solutions for considered problem. We concentrate on the problems for which the assumptions are expressed in terms of the weak topology in a Banach space. We recall and improve earlier papers of this type. The paper is complemented...

We prove the existence of solutions of differential inclusions on a half-line. Our results are based on an approximation method combined with a diagonalization method.