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In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions y = y”” - (A² + B²)y” + A²B²y ∈ F(t,y), (*) with the initial conditions y(0) = y’(0) = y”(0) = y”’(0) = 0, (**) where the matrices $A,B\in {ℝ}^{d\times d}$ are commutative and the multifunction $F:[0,1]\times {ℝ}^d⇝cl\left({ℝ}^d\right)$ is Lipschitz continuous in y with a t-independent constant l < ||A||²||B||². Main theorem. Assume that $F:[0,1]\times {ℝ}^d⇝cl\left({ℝ}^d\right)ismeasurableintandintegrablybounded.Let$y₀ ∈ W4,1$beanarbitraryfunctionsatisfying(**)andsuchthat$ $d_H(y₀\left(t\right),F(t,y₀\left(t\right)))\le p₀\left(t\right)$ a.e. in [0,1], where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W4,1 of (*) with (**) such...

In the paper we give an analogue of the Filippov Lemma for the second order differential inclusions with the initial conditions y(0) = 0, y′(0) = 0, where the matrix A ∈ ℝd×d and multifunction is Lipschitz continuous in y with a t-independent constant l. The main result is the following: Assume that F is measurable in t and integrably bounded. Let y 0 ∈ W 2,1 be an arbitrary function fulfilling the above initial conditions and such that where p 0 ∈ L 1[0, 1]. Then there exists a solution y ∈ W 2,1...

A new approach to differentiation on a time scale is presented. We give a suitable generalization of the Vitali Lemma and apply it to prove that every increasing function f: → ℝ has a right derivative f₊’(x) for ${\mu }_{\Delta }$-almost all x ∈ . Moreover, ${\int }_{[a,b)}f{₊}^{\text{'}}\left(x\right)d{\mu }_{\Delta }\le f\left(b\right)-f\left(a\right)$.