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1
Content available remote

Extremal selections of multifunctions generating a continuous flow

100%
Bressan A., Crasta G.
Annales Polonici Mathematici
|
1994-1995
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tom 60
|
nr 2
101-117
EN
Let $F:[0,T] × ℝ^n → 2^{ℝ^n}$ be a continuous multifunction with compact, not necessarily convex values. In this paper, we prove that, if F satisfies the following Lipschitz Selection Property: (LSP) For every t,x, every y ∈ c̅o̅F(t,x) and ε > 0, there exists a Lipschitz selection ϕ of c̅o̅F, defined on a neighborhood of (t,x), with |ϕ(t,x)-y| < ε, then there exists a measurable selection f of ext F such that, for every x₀, the Cauchy problem ẋ(t) = f(t,x(t)), x(0) = x₀, has a unique Carathéodory solution, depending continuously on x₀. We remark that every Lipschitz multifunction with compact values satisfies (LSP). Another interesting class for which (LSP) holds consists of those continuous multifunctions F whose values are compact and have convex closure with nonempty interior.
2
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Nonlinear boundary value problems for differential inclusions y'' ∈ F(t,y,y')

80%
Erbe L., Krawcewicz W.
Annales Polonici Mathematici
|
1991
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tom 54
|
nr 3
195-226
EN
Applying the topological transversality method of Granas and the a priori bounds technique we prove some existence results for systems of differential inclusions of the form y'' ∈ F(t,y,y'), where F is a Carathéodory multifunction and y satisfies some nonlinear boundary conditions.
3

Optimal feedback control of Ginzburg-Landau equation for superconductivity via differential inclusion

80%
You Y.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1996
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tom 16
|
nr 1
5-41
EN
Slightly below the transition temperatures, the behavior of superconducting materials is governed by the Ginzburg-Landau (GL) equation which characterizes the dynamical interaction of the density of superconducting electron pairs and the exited electromagnetic potential. In this paper, an optimal control problem of the strength of external magnetic field for one-dimensional thin film superconductors with respect to a convex criterion functional is considered. It is formulated as a nonlinear coefficient control problem in Hilbert spaces. The global existence and regularity of solutions of the state equation, the existence of optimal control, and the maximum principle as a necessary condition satisfied by optimal control are proved. By proving the local Lipschitz continuity of the value functions and by using lower Dini derivatives, an optimal synthesis (i.e. optimal feedback control) is obtained via solving a differential inclusion.
4
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The method of upper and lower solutions for perturbed nth order differential inclusions

80%
Dhage B., Petruşel A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2006
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tom 26
|
nr 1
57-76
EN
In this paper, an existence theorem for nth order perturbed differential inclusion is proved under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions is also obtained under certain monotonicity conditions on the multi-functions involved in the inclusion. Our results extend the existence results of Dhage et al. [7,8] and Agarwal et al. [1].
5
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On boundary value problems of second order differential inclusions

80%
Dhage B.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2004
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tom 24
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nr 1
73-96
EN
This paper presents sufficient conditions for the existence of solutions to boundary-value problems of second order multi-valued differential inclusions. The existence of extremal solutions is also obtained under certain monotonicity conditions.
6
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Existence of three anti-periodic solutions for second-order impulsive differential inclusions with two parameters

80%
Heidarkhani S., Afrouzi G., Hadjian A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2013
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tom 33
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nr 2
115-133
EN
Applying two three critical points theorems, we prove the existence of at least three anti-periodic solutions for a second-order impulsive differential inclusion with a perturbed nonlinearity and two parameters.
7
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The solution set of a differential inclusionon a closed set of a Banach space

80%
Wen S.
Applicationes Mathematicae
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1995-1996
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tom 23
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nr 1
13-23
EN
We consider differential inclusions with state constraints in a Banach space and study the properties of their solution sets. We prove a relaxation theorem and we apply it to prove the well-posedness of an optimal control problem.
8
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Representation of the set of mild solutions to the relaxed semilinear differential inclusion

70%
Benedetti I., Panasenko E.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2006
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tom 26
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nr 1
143-158
EN
We study the relation between the solutions set to a perturbed semilinear differential inclusion with nonconvex and non-Lipschitz right-hand side in a Banach space and the solutions set to the relaxed problem corresponding to the original one. We find the conditions under which the set of solutions for the relaxed problem coincides with the intersection of closures (in the space of continuous functions) of sets of δ-solutions to the original problem.
9
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Boundary value problems for differential inclusions with fractional order

70%
Benchohra M., Hamani S.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2008
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tom 28
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nr 1
147-164
EN
In this paper, we shall establish sufficient conditions for the existence of solutions for a boundary value problem for fractional differential inclusions. Both cases of convex valued and nonconvex valued right hand sides are considered.
10

Existence and relaxation results for nonlinear second order evolution inclusions

70%
Migórski S.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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1995
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tom 15
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nr 2
129-148
EN
In this paper we study nonlinear evolution inclusions associated with second order equations defined on an evolution triple. We prove two existence theorems for the cases where the orientor field is convex valued and nonconvex valued, respectively. We show that when the orientor field is Lipschitzean, then the set of solutions of the nonconvex problem is dense in the set of solutions of the convexified problem.
11
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Positive integrable solutions for nonlinear integral and differential inclusions of fractional-orders

70%
Al-Issa S., El-Sayed A. M. A.
Commentationes Mathematicae
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2009
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tom 49
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nr 2
EN
In this paper we study the global existence of positive integrable solution for the nonlinear integral inclusion of fractional order \[ x(t) \in p(t) + I^\alpha F_1 (t, I^\beta f_2 (t, x(\varphi(t)))),\quad t \in (0, 1). \] As an application the global existence of the solution for the initial-value problem of the arbitrary (fractional) orders differential inclusion \[ \frac{dx(t)}{dt}\in p(t)+ I^\alphaF_1(t,D^\gammax(t))),\quad \text{a.e.}\ t gt 0 \] will be studied.
12
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On the topological dimension of the solutions sets for some classes of operator and differential inclusions

51%
Bader R., Gel'man B., Kamenskii M., Obukhovskii V.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2002
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tom 22
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nr 1
17-32
EN
In the present paper, we give the lower estimation for the topological dimension of the fixed points set of a condensing continuous multimap in a Banach space. The abstract result is applied to the fixed point set of the multioperator of the form $𝓕 = S 𝓟_F$ where $𝓟_F$ is the superposition multioperator generated by the Carathéodory type multifunction F and S is the shift of a linear injective operator. We present sufficient conditions under which this set has the infinite topological dimension. In the last section of the paper, we consider the applications of the solutions sets for Cauchy and periodic problems for semilinear differential inclusions in a Banach space.
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