Nonlinear multivalued boundary value problems

Ralf Bader; Nikolaos S. Papageorgiou

Displaying similar documents to “Nonlinear multivalued boundary value problems”

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...

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...

Existence of positive solutions for a nonlinear fourth order boundary value problem

Ruyun Ma (2003)

Annales Polonici Mathematici

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We study the existence of positive solutions of the nonlinear fourth order problem $u^{(4)} (x) = λ a (x) f (u (x))$ , u(0) = u’(0) = u”(1) = u”’(1) = 0, where a: [0,1] → ℝ may change sign, f(0) < 0, and λ < 0 is sufficiently small. Our approach is based on the Leray-Schauder fixed point theorem.

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...

Multiplicity of positive solutions for a nonlinear fourth order equation

D. R. Dunninger (2001)

Annales Polonici Mathematici

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We study the existence and multiplicity of positive solutions of the nonlinear fourth order problem ⎧ $u^{(4)} = λ f (u)$ in (0,1), ⎨ ⎩u(0) = a ≥ 0, u’(0) = a’ ≥ 0, u(1) = b ≥ 0, u(1) = -b’ ≤ 0 The methods employed are upper and lower solutions and degree theory arguments.

Three solutions for a nonlinear Neumann boundary value problem

Najib Tsouli, Omar Chakrone, Omar Darhouche, Mostafa Rahmani (2014)

Applicationes Mathematicae

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The aim of this paper is to establish the existence of at least three solutions for the nonlinear Neumann boundary-value problem involving the p(x)-Laplacian of the form $- Δ_{p (x)} u + a (x) | u^{| p (x) - 2} u = μ g (x, u)$ in Ω, ${| \nabla u |}^{p (x) - 2} \partial u / \partial ν = λ f (x, u)$ on ∂Ω. Our technical approach is based on the three critical points theorem due to Ricceri.

CM-Selectors for pairs of oppositely semicontinuous multivalued maps with $_{p}$ -decomposable values

Hôǹg Thái Nguyêñ, Maciej Juniewicz, Jolanta Ziemińska (2001)

Studia Mathematica

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We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if F is an H-upper semicontinuous multivalued map on a separable metric space X, G is a lower semicontinuous multivalued map on X, both F and G take nonconvex $L_{p} (T, E)$ -decomposable closed values, the measure space T with a σ-finite measure μ is nonatomic, 1 ≤ p < ∞, $L_{p} (T, E)$ is the Bochner-Lebesgue space of functions defined on T with values in a Banach space E, F(x)...

Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions

Hông Thái Nguyêñ (2004)

Studia Mathematica

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Let Ω be a measure space, and E, F be separable Banach spaces. Given a multifunction $f : Ω \times E \to 2^{F}$ , denote by $N_{f} (x)$ the set of all measurable selections of the multifunction $f (\cdot, x (\cdot)) : Ω \to 2^{F}$ , s ↦ f(s,x(s)), for a function x: Ω → E. First, we obtain new theorems on H-upper/H-lower/lower semicontinuity (without assuming any conditions on the growth of the generating multifunction f(s,u) with respect to u) for the multivalued (Nemytskiĭ) superposition operator $N_{f}$ mapping some open domain G ⊂ X into $2^{Y}$ , where X and Y are...

On the Rockafellar theorem for $Φ^{γ (\cdot, \cdot)}$ -monotone multifunctions

S. Rolewicz (2006)

Studia Mathematica

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Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let $Γ : X \to 2^{Φ}$ be a cyclic $Φ^{γ (\cdot, \cdot)}$ -monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the $Φ^{γ (\cdot, \cdot)}$ -subdifferential of f, $Γ (x) \subset \partial_{Φ}^{γ (\cdot, \cdot)} {f |}_{x}$ .

Entire solutions to a class of fully nonlinear elliptic equations

Ovidiu Savin (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We study nonlinear elliptic equations of the form $F (D^{2} u) = f (u)$ where the main assumption on $F$ and $f$ is that there exists a one dimensional solution which solves the equation in all the directions $ξ \in ℝ^{n}$ . We show that entire monotone solutions $u$ are one dimensional if their $0$ level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.

On the Existence of Solutions for Abstract Nonlinear Operator Equations

Marek Galewski (2007)

Bollettino dell'Unione Matematica Italiana

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We provide a duality theory and existence results for a operator equation $\nabla T(x)=\nabla N(x)$ where $T$ is not necessarily a monotone operator. We use the abstract version of the so called dual variational method. The solution is obtained as a limit of a minimizng sequence whose existence and convergence is proved.

Minimax theorems without changeless proportion

Liang-Ju Chu, Chi-Nan Tsai (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: $i n f_{y \in Y} s u p_{x \in X} f (x, y) \leq s u p_{x \in X} i n f_{y \in Y} g (x, y)$ . We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: $s u p_{x \in X} f (x, y) \leq s u p_{x \in X} g (x, y)$ , ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some...

Some Remarks on Nonlinear Composition Operators in Spaces of Differentiable Functions

J. Appell, Z. Jesús, O. Mejía (2011)

Bollettino dell'Unione Matematica Italiana

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In this note we study the nonlinear composition operator $f\mapsto g\circ f$ in various spaces of differentiable functions over an interval. It turns out that this operator is always bounded in the corresponding norm, whenever it maps such a space into itself, but continuous only in exceptional cases.

On the principal eigencurve of the p-Laplacian related to the Sobolev trace embedding

Abdelouahed El Khalil, Mohammed Ouanan (2005)

Applicationes Mathematicae

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We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ $Δ ₚ u = {| u |}^{p - 2} u$ in Ω, ⎨ ⎩ ${| \nabla u |}^{p - 2} \partial u / \partial ν = {λ ϱ (x) | u |}^{p - 2} u + μ {| u |}^{p - 2} u$ on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.

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 $\pm 10^{- 7}$ .

$α$ -continuous multivalued maps

J. Eweret (1989)

Matematički Vesnik

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Periodic boundary value problem of a fourth order differential inclusion

Marko Švec (1997)

Archivum Mathematicum

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The paper deals with the periodic boundary value problem (1) $L_{4} x (t) + a (t) x (t) \in F (t, x (t))$ , $t \in J = [a, b]$ , (2) $L_{i} x (a) = L_{i} x (b)$ , $i = 0, 1, 2, 3$ , where $L_{0} x (t) = a_{0} x (t)$ , $L_{i} x (t) = a_{i} (t) L_{i - 1} x (t)$ , $i = 1, 2, 3, 4$ , $a_{0} (t) = a_{4} (t) = 1$ , $a_{i} (t)$ , $i = 1, 2, 3$ and $a (t)$ are continuous on $J$ , $a (t) \geq 0$ , $a_{i} (t) > 0$ , $i = 1, 2$ , $a_{1} (t) = a_{3} (t) \cdot F (t, x) : J \times R \to$ {nonempty convex compact subsets of $R$ }, $R = (- \infty, \infty)$ . The existence of such periodic solution is proven via Ky Fan’s fixed point theorem.