We deal with the problems of four boundary points conditions for both differential inclusions and differential equations with and without moving constraints. Using a very recent result we prove existence of generalized solutions for some differential inclusions and some differential equations with moving constraints. The results obtained improve the recent results obtained by Papageorgiou and Ibrahim-Gomaa. Also by means of a rather different approach based on an existence theorem due to O. N. Ricceri...

We show the existence of solutions to a boundary-value problem for fourth-order differential inclusions in a Banach space, under Lipschitz’s contractive conditions, Carathéodory conditions and lower semicontinuity conditions.

We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations - sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order...

The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $$f(t,vy,wy+uvz)=f(x,y,z)u^{2}v+g(t,x,u,v,w)vz+h(t,x,u,v,w)y+2uwz$$ is solved on $ℝ$ for $y\ne 0$, $v\ne 0.$

Perceptions about function changes are represented by rules like “If X is SMALL then Y is QUICKLY INCREASING.” The consequent part of a rule describes a granule of directions of the function change when X is increasing on the fuzzy interval given in the antecedent part of the rule. Each rule defines a granular differential and a rule base defines a granular derivative. A reconstruction of a fuzzy function given by the granular derivative and the initial value given by the rule is similar to Euler’s...

Let (P,ω) be a symplectic manifold. We find an integrability condition for an implicit differential system D' which is formed by a Lagrangian submanifold in the canonical symplectic tangent bundle (TP,ὡ).

We introduce and study the sheaf of Deligne to describe singular points of a linear differential operator $D$ and we develop a technique based on homological algebra to prove index theorems for $D$.As particular cases, we obtain index theorems for $D$ acting in spaces of multisummable series and a new proof of the index theorem of Malgrange in the space of convergent power series and of the index theorems of Ramis in the spaces of Gevrey series.We compute the values of these indices in terms of the formal...

We investigate the existence of solutions to first order initial value problems for differential inclusions subject to impulsive effects. We shall rely on a fixed point theorem for condensing maps to prove our results.

The paper deals with the existence of a Kneser solution of the $n$-th order nonlinear differential inclusion $$\begin{array}{cc}\hfill x^{\left(n\right)}\left(t\right)\in -A_1(t,x\left(t\right),...,x^{(n-1)}\left(t\right))x^{(n-1)}\left(t\right)-...-A_n(t,x\left(t\right),...,& x^{(n-1)}\left(t\right))x\left(t\right)\\ & \text{for}\text{a.a.}t\in [a,\infty ),\end{array}$$ where $a\in (0,\infty )$, and $A_i:[a,\infty )\times {ℝ}^n\to ℝ$, $i=1,...,n,$ are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

This paper is concerned with square integrable quasi-derivatives for any solution of a general quasi-differential equation of $n$th order with complex coefficients $M\left[y\right]-\lambda wy=wf(t,y^{\left[0\right]},...,y^{[n-1]})$, $t\in [a,b)$ provided that all $r$th quasi-derivatives of solutions of $M\left[y\right]-\lambda wy=0$ and all solutions of its normal adjoint $M^{+}\left[z\right]-\overline{\lambda }wz=0$ are in $L_w^2(a,b)$ and under suitable conditions on the function $f$.