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Monotone operators. A survey directed to applications to differential equations

Jan Franců (1990)

Aplikace matematiky

The paper deals with the existence of solutions of the form A u = b with operators monotone in a broader sense, including pseudomonotone operators and operators satisfying conditions S and M . The first part of the paper which has a methodical character is concluded by the proof of an existence theorem for the equation on a reflexive separable Banach space with a bounded demicontinuous coercive operator satisfying condition ( M ) 0 . The second part which has a character of a survey compares various types of...

Monotone operators in divergence form with x -dependent multivalued graphs

Gilles Francfort, François Murat, Luc Tartar (2004)

Bollettino dell'Unione Matematica Italiana

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 ∘ rotation, for every fixed x ; in other words, the graph d ∈ 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 , ⋅ ) is pointwise monotone...

Monotonic solutions for quadratic integral equations

Mieczysław Cichoń, Mohamed M.A. Metwali (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Using the Darbo fixed point theorem associated with the measure of noncompactness, we establish the existence of monotonic integrable solution on a half-line ℝ₊ for a nonlinear quadratic functional integral equation.

Multifunctions of two variables: examples and counterexamples

Jürgen Appell (1996)

Banach Center Publications

A brief account of the connections between Carathéodory multifunctions, Scorza-Dragoni multifunctions, product-measurable multifunctions, and superpositionally measurable multifunctions of two variables is given.

Multilinear almost diagonal estimates and applications

Árpád Bényi, Nikolaos Tzirakis (2004)

Studia Mathematica

We prove that an almost diagonal condition on the (m + 1)-linear tensor associated to an m-linear operator implies boundedness of the operator on products of classical function spaces. We then provide applications to the study of certain singular integral operators.

Multilinear operators on C ( K , X ) spaces

Ignacio Villanueva (2004)

Czechoslovak Mathematical Journal

Given Banach spaces  X , Y and a compact Hausdorff space  K , we use polymeasures to give necessary conditions for a multilinear operator from C ( K , X ) into  Y to be completely continuous (resp.  unconditionally converging). We deduce necessary and sufficient conditions for  X to have the Schur property (resp.  to contain no copy of  c 0 ), and for  K to be scattered. This extends results concerning linear operators.

Multiple positive solutions for a second order delay boundary value problem on the half-line

K. G. Mavridis, Ch. G. Philos, P. Ch. Tsamatos (2006)

Annales Polonici Mathematici

Second order nonlinear delay differential equations are considered, and Krasnosel'skiĭ's fixed point theorem is used to establish a result on the existence of positive solutions of a boundary value problem on the half-line. This result can be used to guarantee the existence of multiple positive solutions. A specification of the result obtained to the case of second order nonlinear ordinary differential equations as well as to a particular case of second order nonlinear delay differential equations...

Multiple summing operators on l p spaces

Dumitru Popa (2014)

Studia Mathematica

We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of l p spaces. This characterization is used to show that multiple s-summing operators on a product of l p spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators T : l 4 / 3 × l 4 / 3 → l ₂ such that none of the associated linear operators is s-summing...

Multiplicity results for sign-changing solutions of an operator equation in ordered Banach space

Xian Xu, Bingjin Wang (2009)

Mathematica Bohemica

In this paper, we prove some multiplicity results for sign-changing solutions of an operator equation in an ordered Banach space. The methods to show the main results of the paper are to associate a fixed point index with a strict upper or lower solution. The results can be applied to a wide variety of boundary value problems to obtain multiplicity results for sign-changing solutions.

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