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Near viability for fully nonlinear differential inclusions

100%
Căpraru I., Lazu A.
Open Mathematics
|
2014
|
tom 12
|
nr 10
1447-1459
EN
We consider the nonlinear differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A is an m-dissipative operator on a separable Banach space X and F is a multi-function. We establish a viability result under Lipschitz hypothesis on F, that consists in proving the existence of solutions of the differential inclusion above, starting from a given set, which remain arbitrarily close to that set, if a tangency condition holds. To this end, we establish a kind of set-valued Gronwall’s lemma and a compactness theorem, which are extensions to the nonlinear case of similar results for semilinear differential inclusions. As an application, we give an approximate null controllability result.
2
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Global Existence and Stability for Neutral Functional Evolution Equations with State-Dependent Delay

76%
Baliki A., Benchohra M.
Nonautonomous Dynamical Systems
|
2014
|
tom 1
|
nr 1
EN
In this paper we prove the global existence and attractivity of mild solutions for neutral semilinear evolution equations with state-dependent delay in a Banach space.
3
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Existence of solutions for delay evolution equations with nonlocal conditions

76%
Zhang X., Li Y.
Open Mathematics
|
2017
|
tom 15
|
nr 1
616-627
EN
In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.
4
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On the nonlocal Cauchy problem for semilinear fractional order evolution equations

64%
Wang J., Zhou Y., Fečkan M.
Open Mathematics
|
2014
|
tom 12
|
nr 6
911-922
EN
In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.
5
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Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

46%
Migórski S.
Open Mathematics
|
2012
|
tom 10
|
nr 6
1953-1968
EN
We survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdifferential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas leading to inequality problems with multivalued and nonmonotone boundary conditions encountered in mechanics.
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