In this paper, the existence and the localization result will be proven for vector Dirichlet problem with an upper-Carathéodory right-hand side. The result will be obtained by combining the continuation principle with bound sets technique.

The paper deals with the existence of a Kneser solution of the $n$-th order nonlinear differential inclusion $$\begin{array}{cc}\hfill {x}^{\left(n\right)}\left(t\right)\in -{A}_{1}(t,x\left(t\right),...,{x}^{(n-1)}\left(t\right)){x}^{(n-1)}\left(t\right)-...-{A}_{n}(t,x\left(t\right),...,& {x}^{(n-1)}\left(t\right))x\left(t\right)\\ & \text{for}\phantom{\rule{4.0pt}{0ex}}\text{a.a.}\phantom{\rule{4pt}{0ex}}t\in [a,\infty ),\end{array}$$
where $a\in (0,\infty )$, and ${A}_{i}:[a,\infty )\times {\mathbb{R}}^{n}\to \mathbb{R}$, $i=1,...,n,$ are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

Sufficient conditions are given for the solvability of an impulsive Dirichlet boundary value problem to forced nonlinear differential equations involving the combination of viscous and dry frictions. Apart from the solvability, also the explicit estimates of solutions and their derivatives are obtained. As an application, an illustrative example is given, and the corresponding numerical solution is obtained by applying Matlab software.

We study the existence of a mild solution to the nonlocal initial value problem for semilinear second-order differential inclusions in abstract spaces. The result is obtained by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables getting the result without any requirements for compactness of the right-hand side or of the cosine family generated by the linear operator.

The weighted average is a well-known aggregation operator that is widely applied in various mathematical models. It possesses some important properties defined for aggregation operators, like monotonicity, continuity, idempotency, etc., that play an important role in practical applications. In the paper, we reveal whether and in which way such properties can be observed also for the fuzzy weighted average operator where the weights as well as the weighted values are expressed by noninteractive fuzzy...

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