Existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces with anti-periodic boundary conditions

Sahbi Boussandel

Displaying similar documents to “Existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces with anti-periodic boundary conditions”

Nonlinear multivalued boundary value problems

Ralf Bader, Nikolaos S. Papageorgiou (2001)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we study nonlinear second order differential inclusions with a multivalued maximal monotone term and nonlinear boundary conditions. We prove existence theorems for both the convex and nonconvex problems, when $d o m A \neq ℝ^{N}$ and $d o m A = ℝ^{N}$ , with A being the maximal monotone term. Our formulation incorporates as special cases the Dirichlet, Neumann and periodic problems. Our tools come from multivalued analysis and the theory of nonlinear monotone operators.

Existence and uniqueness of positive periodic solutions for a class of integral equations with parameters

Shu-Gui Kang, Bao Shi, Sui Sun Cheng (2009)

Annales Polonici Mathematici

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Existence of periodic solutions of functional differential equations with parameters such as Nicholson’s blowflies model call for the investigation of integral equations with parameters defined over spaces with periodic structures. In this paper, we study one such equation $ϕ (x) = λ \int_{[x, x + ω] \cap Ω} K (x, y) h (y) f (y, ϕ (y - τ (y))) d y$ , x ∈ Ω, by means of the proper value theory of operators in Banach spaces with cones. Existence, uniqueness and continuous dependence of proper solutions are established.

Group actions on monotone skew-product semiflows with applications

Feng Cao, Mats Gyllenberg, Yi Wang (2016)

Journal of the European Mathematical Society

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We discuss a general framework of monotone skew-product semiflows under a connected group action. In a prior work, a compact connected group $G$ -action has been considered on a strongly monotone skew-product semiflow. Here we relax the strong monotonicity and compactness requirements, and establish a theory concerning symmetry or monotonicity properties of uniformly stable 1-cover minimal sets. We then apply this theory to show rotational symmetry of certain stable entire solutions for...

Monotone extenders for bounded c-valued functions

Kaori Yamazaki (2010)

Studia Mathematica

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Let c be the Banach space consisting of all convergent sequences of reals with the sup-norm, $C_{\infty} (A, c)$ the set of all bounded continuous functions f: A → c, and $C_{A} (X, c)$ the set of all functions f: X → c which are continuous at each point of A ⊂ X. We show that a Tikhonov subspace A of a topological space X is strong Choquet in X if there exists a monotone extender $u : C_{\infty} (A, c) \to C_{A} (X, c)$ . This shows that the monotone extension property for bounded c-valued functions can fail in GO-spaces, which provides a negative answer...

On the existence of viable solutions for a class of second order differential inclusions

Aurelian Cernea (2002)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We prove the existence of viable solutions to the Cauchy problem x” ∈ F(x,x’), x(0) = x₀, x’(0) = y₀, where F is a set-valued map defined on a locally compact set $M \subset R^{2 n}$ , contained in the Fréchet subdifferential of a ϕ-convex function of order two.

Abstract inclusions in Banach spaces with boundary conditions of periodic type

Lahcene Guedda, Ahmed Hallouz (2014)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We study in the space of continuous functions defined on [0,T] with values in a real Banach space E the periodic boundary value problem for abstract inclusions of the form ⎧ $x \in S (x (0), s e l_{F} (x))$ ⎨ ⎩ x (T) = x(0), where, $F : [0, T] \times \to 2^{E}$ is a multivalued map with convex compact values, ⊂ E, $s e l_{F}$ is the superposition operator generated by F, and S: × L¹([0,T];E) → C([0,T]; ) an abstract operator. As an application, some results are given to the periodic boundary value problem for nonlinear differential inclusions governed...

Periodic solutions for quasilinear vector differential equations with maximal monotone terms

Nikolaos C. Kourogenis, Nikolaos S. Papageorgiou (1997)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We consider a quasilinear vector differential equation with maximal monotone term and periodic boundary conditions. Approximating the maximal monotone operator with its Yosida approximation, we introduce an auxiliary problem which we solve using techniques from the theory of nonlinear monotone operators and the Leray-Schauder principle. To obtain a solution of the original problem we pass to the limit as the parameter λ > 0 of the Yosida approximation tends to zero.

Existence of two solutions for quasilinear periodic differential equations with discontinuities

Nikolaos S. Papageorgiou, Francesca Papalini (2002)

Archivum Mathematicum

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In this paper we examine a quasilinear periodic problem driven by the one- dimensional $p$ -Laplacian and with discontinuous forcing term $f$ . By filling in the gaps at the discontinuity points of $f$ we pass to a multivalued periodic problem. For this second order nonlinear periodic differential inclusion, using variational arguments, techniques from the theory of nonlinear operators of monotone type and the method of upper and lower solutions, we prove the existence of at least two non trivial...

Periodic solutions to a non-linear differential equation of the order $2n+1$

Monika Kubicova (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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A criterion for the existance of periodic solutions of an ordinary differential equation of order k proved by J. Andres and J. Vorâcek for k = 3 is extended to an arbitrary odd k.

Can we assign the Borel hulls in a monotone way?

Márton Elekes, András Máthé (2009)

Fundamenta Mathematicae

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A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/ $G_{δ}$ hull to every negligible/measurable subset of [0,1]? Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone $G_{δ}$ hull operation for all measurable sets is consistent. It remains open whether existence here is also...

Periodic solutions to a non-linear differential equation of the order $2n+1$

Monika Kubicova (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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A criterion for the existance of periodic solutions of an ordinary differential equation of order k proved by J. Andres and J. Vorâcek for k = 3 is extended to an arbitrary odd k.

What is a monotone Lagrangian cobordism?

François Charette (2012-2014)

Séminaire de théorie spectrale et géométrie

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We explain the notion of Lagrangian cobordism. A flexibility/rigidity dichotomy is illustrated by considering Lagrangian tori in $ℂ^{2}$ . Towards the end, we present a recent construction by Cornea and the author [8], of monotone cobordisms that are not trivial in a suitable sense.

Multiple positive solutions of a nonlinear fourth order periodic boundary value problem

Lingbin Kong, Daqing Jiang (1998)

Annales Polonici Mathematici

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The fourth order periodic boundary value problem $u^{(4)} - m ⁴ u + F (t, u) = 0$ , 0 < t < 2π, with $u^{(i)} (0) = u^{(i)} (2 π)$ , i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of $\pm 10^{- 7}$ .

Monotone operators in divergence form with $x$ -dependent multivalued graphs

Gilles Francfort, François Murat, Luc Tartar (2004)

Bollettino dell'Unione Matematica Italiana

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We prove the existence of solutions to $- div a (x, grad u) = f$ , together with appropriate boundary conditions, whenever $a (x, e)$ is a maximal monotone graph in $e$ , for every fixed $x$ . We propose an adequate setting for this problem, in particular as far as measurability is concerned. It consists in looking at the graph after a $45^{\circ}$ rotation, for every fixed $x$ ; in other words, the graph $d \in a (x, e)$ is defined through $d - e = φ (x, d + e)$ , where $φ$ is a Carathéodory contraction in $R^{N}$ . This definition is shown to be equivalent to the fact that $a (x, \cdot)$ is pointwise...