A Scorza-Dragoni approach to Dirichlet problem with an upper-Carathéodory right-hand side

Pavlačková, Martina

doi:10.12775/tmna.2014.045

Cited by 9 publications

(10 citation statements)

References 6 publications

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“…For the basic theory on impulsive differential equations and inclusions in finitedimensional spaces, the reader is referred to the literature [1][2][3]. The recent results about impulsive differential equations and inclusions can be found, e.g., in [4][5][6][7] or [8] and see also the references therein.…”

Section: Introductionmentioning

confidence: 99%

Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

Pavlačková

Taddei

2022

Mathematics

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In this paper, the existence of a mild solution to the Cauchy problem for impulsive semilinear second-order differential inclusion in a Banach space is investigated in the case when the nonlinear term also depends on the first derivative. This purpose is achieved by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables obtaining the result under easily verifiable and not restrictive conditions on the impulsive terms, the cosine family generated by the linear operator and the right-hand side while avoiding any requirement for compactness. Firstly, the problems without impulses are investigated, and then their solutions are glued together to construct the solution to the impulsive problem step by step. The paper concludes with an application of the obtained results to the generalized telegraph equation with a Balakrishnan–Taylor-type damping term.

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Section: Introductionmentioning

confidence: 99%

Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

Pavlačková

Taddei

2022

Mathematics

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“…Bound sets approach which is used in Sections 3 and 4 of our paper was initiated by Gaines and Mawhin in [17] for proving the existence of periodic solutions of first-order as well as second-order systems of differential equations (see also the references therein). Bound sets theory for multivalued Dirichlet or Floquet problems without impulses was developed in [2]- [7], [24], [28]. Recently, bound sets approach has been applied also for multivalued impulsive Dirichlet problem in [25] and [26].…”

Section: Corresponding Authormentioning

confidence: 99%

“…Bound sets theory for multivalued Dirichlet or Floquet problems without impulses was developed in [2]- [7], [24], [28]. Recently, bound sets approach has been applied also for multivalued impulsive Dirichlet problem in [25] and [26]. In this paper, it will be shown how changing the boundary conditions from Dirichlet to Floquet will affect the assumptions guaranteeing the existence of a bound set for the considered impulsive boundary value problem.…”

Section: Corresponding Authormentioning

confidence: 99%

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Bound Sets Approach to Impulsive Floquet Problems for Vector Second-Order Differential Inclusions

Pavlačková

Taddei

2022

Differ Equ Dyn Syst

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In this paper, the existence and the localization of a solution of an impulsive vector multivalued second-order Floquet boundary value problem are investigated. The method used in the paper is based on the combination of a fixed point index technique with bound sets approach. At first, problems with upper-Carathéodory right-hand sides are investigated and it is shown afterwards how can the conditions be simplified in more regular case of upper semi-continuous right hand side. In this more regular case, the conditions ensuring the existence and the localization of a solution are put directly on the boundary of the considered bound set. This strict localization of the sufficient conditions is very significant since it allows some solutions to escape from the set of candidate solutions. In both cases, the C 1 -bounding functions with locally Lipschitzian gradients are considered at first and it is shown afterwards how the conditions change in case of C 2 -bounding functions. The paper concludes with an application of obtained results to Liénard-type equations and inclusions and the comparisons of our conclusions with the few results related to impulsive periodic and antiperiodic Liénard equations are obtained.

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“…Extensions of these methods have been obtained by several authors in various directions such as Floquet BVPs, Carathéodory conditions, non-smooth bounding functions and non-smooth boundaries, differential inclusions, extensions to Banach spaces, PDEs, integro-differential equations, differential equations with impulses, nonlocal BVPs. A constantly growing literature in this area shows the persistent influence of the bound sets and bounding functions approach; see [2,3,6,7,37,38,39,44] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Bound sets for a class of $ϕ$-Laplacian operators

Feltrin,

Zanolin

2020

Preprint

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We provide an extension of the Hartman-Knobloch theorem for periodic solutions of vector differential systems to a general class of φ-Laplacian differential operators. Our main tool is a variant of the Manásevich-Mawhin continuation theorem developed for this class of operator equations, together with the theory of bound sets. Our results concern the case of convex bound sets for which we show some new connections using a characterisation of sublevel sets due to Krantz and Parks. We also extend to the φ-Laplacian vector case a classical theorem of Reissig for scalar periodically perturbed Liénard equations.

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A Scorza-Dragoni approach to Dirichlet problem with an upper-Carathéodory right-hand side

Cited by 9 publications

References 6 publications

Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

Bound Sets Approach to Impulsive Floquet Problems for Vector Second-Order Differential Inclusions

Bound sets for a class of $ϕ$-Laplacian operators

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