Operators Of Monotone Type Research Articles

Solvability of semilinear equations with compact perturbations of operators of monotone type

The solvability of the equation A u − T u + C u = f Au - Tu + Cu = f is studied under various assumptions of monotonicity and compactness on the operators A, T, and C, which map subsets of a reflexive Banach space X into its dual space. It is nowhere assumed that X possesses a Schauder basis or that the operator T is positive definite and selfadjoint. The results extend and/or improve recent results obtained by Chen, Kartsatos and Mabry, and Kesavan.

Ranges of Operators of Monotone Type in Banach Space

Let X be a reflexive Banach space with both X and X* locally uniformly convex. Let G ⊂ X be open, bounded, and convex, and T: Ḡ → X* an operator of monotone type. By degree theory of operators of type ( S +), certain range properties of T are established. The results are applied to the study of existence of a zero of T under various boundary conditions.

G-convergence of quasi-linear ordinary differential operators of monotone type

We study the notion ofG-convergence for systems of second order ordinary differential equations, when the differential operators are in a suitable class of monotone and coercive non linear maps. The main result of the paper is theG-compactness of this class and the characterization of theG-limit in terms of a suitable convergence of the inverse maps.

Nonlinear random operator equations and inequalities in Banach spaces

In this paper we give some new existence theorems for nonlinear random equations and inequalities involving operators of monotone type in Banach spaces. A random Hammerstein integral equation is also studied. In order to obtain random solutions we use some results from the existing deterministic theory as well as from the theory of measurable multifunctions and, in particular, the measurable selection theorems of Kuratowski/Ryll‐Nardzewski and of Saint‐Beuve.

Nonlinear random equations with noncoercive operators in Banach spaces

admits random solutions. In [ 121 Itoh proved the existence of solutions of nonlinear random equations with monotone operators. His method of proving the measurability of solutions was based on the selection theorem of Kuratowski and Ryll-Nardzewski [16]. In [13, 141 Itoh’s method was extended to obtain existence theorems for nonlinear random equations and inequalities with single-valued or multivalued operators of monotone type. The above results were derived under the basic assumption that the random operators are coercive. It is the purpose of this paper to give some new existence results concerning the solvability of nonlinear random equations with noncoercive operators. As in the coercive case, the measurability of solutions depends mainly on the selection theorem proved in [ 161. In Section 3, we prove the existence of solutions of nonlinear random equations with multivalued operators satisfying a Leray-Schauder-type boundary condition, generalizing thus some results in [ 151. In Section 4, we study the solvability of the random equation

Perturbations of maximal monotone random operators

Let X be a Banach space, X ∗ its dual, and Ω a measurable space. We study the solvability of nonlinear random equations involving operators of the form L + T , where L is a maximal monotone random operator from Ω × X into X ∗ and T : Ω × X → X ∗ a random operator of monotone type.

Nonlinear random equations involving operators of monotone type

The systematic study of random equations has been initiated by SpaEek [19] and Hans [9]. In recent years Kannan and Salehi [13] and Itoh [ 111 have treated nonlinear random equations with monotone operators. In [ 14, 151 the author studied nonlinear random equations and inequalities with singlevalued or multivalued operators of monotone type. It is the purpose of this paper to give some new existence theorems for nonlinear random equations with operators of monotone type. More precisely, in Section 3, we present an existence and perturbation theory for solutions of nonlinear random equations involving multivalued maximal monotone operators. In Section 4 we study a random Hammerstein equation in a Hilbert space involving a closed linear maximal monotone random operator and a random operator of type (M). In Section 5 we consider random equations with noncoercive pseudomonotone operators. Instead of coercivity a Leray-Schauder-type boundary condition is assumed.

Nonlinear random equations with maximal monotone operators in Banach spaces

Let X be a real Banach space, X* its dual space and ω a measurable space. Let D be a subset of X, L: Ω × D → X* a random operator and η:Ω →X* a measurable mapping. The random equation corresponding to the double [L, η] asks for a measurable mapping ξ: Ω → D such thatRandom equations with operators of monotone type have been studied recentely by Kannan and Salehi [7], Itoh [6] and Kravvarits [8].

A Degree for Operators of Monotone Type

Mathematische NachrichtenVolume 114, Issue 1 p. 53-62 Article A Degree for Operators of Monotone Type Eckehard Krauss, Eckehard Krauss Akademie der Wissenschaften Institut für Mathematik DDR - 1086 Berlin Mohrenstraße 39Search for more papers by this author Eckehard Krauss, Eckehard Krauss Akademie der Wissenschaften Institut für Mathematik DDR - 1086 Berlin Mohrenstraße 39Search for more papers by this author First published: 1983 https://doi.org/10.1002/mana.19831140104Citations: 5AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume114, Issue11983Pages 53-62 RelatedInformation

Nonlinear random operators of monotone type in Banach spaces

Nonlinear random operators of monotone type in Banach spaces

On the constructive solution of nonlinear functional equations

In the recent development of the theory of nonlinear operators in Banach spaces, a number of general existence theorems have been established for various classes of mappings by the use of compactness, convexity, or topological arguments that do not give a constructive procedure for the generation of the solutions thus proved to exist. Even in cases when the solutions are unique and one obtains them as limits of a precisely determined sequence of approximants, they often fail to satisfy an important and useful principle of constructivity in that the procedures have no effective control provided for the error at each state of the approximation. This is particularly the case for the Galerkin approximations used in the theory of operators of monotone type and for various fixed-point methods used in the theory of nonlinear accretive operators. About a decade ago, the writer developed existence results which satisfied these principles of constructivity for the case of a broad class of continuous monotone mappings in Hilbert spaces, and more generally, for continuous accretive mappings in Banach spaces X whose conjugate spaces X* are uniformly convex. These results were stated in the rather inaccessible paper [l] and developed in detail in the middle of a lengthy treatment of accretive operators in the writer’s paper [2]. It is our object here to develop these results explicitly and in detail in connection with the problem of constructivity. Our renewed interest in this question was stimulated by the recent paper by Bruck [3], who has developed an iteration procedure to obtain the solution of the equation (I + T)(u) = 0 for a continuous, bounded monotone operator T on a Hilbert space H with explicit control of the error. Bruck’s result has the curious feature that it depends on the assumption of the prior existence of the solution. Although the procedure we give is not an iteration procedure it corresponds to a simpler intuitive picture of the situation in the general context of accretive operators and the error control does not depend upon the assumption of the existence of a solution.

Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations

Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations

Nonlinear mappings of monotone type in Banach spaces

A new class of (nonlinear) operators of monotone type is introduced: a mapping T from the real Banach space X into the set 2 X∗ of all subsets of the conjugate space X ∗ is said to be generalized pseudo-monotone if for any sequence { u j } in the effective domain of T with u j⤳u in X and any corresponding sequence w j⤳w in X ∗, w j ϵ Tu j , for which lim sup( w j − w, u j − u) ⩽ 0, it follows that w ϵ Tu and ( w j , u j ) → ( w, u). It is shown that most of the known mappings of monotone type are generalized pseudo-monotone, as, for example, maximal monotone and pseudo-monotone mappings. The class of generalized pseudo-monotone mappings is then investigated in detail. An important part of that study is devoted to the discussion of the range of a generalized pseudo-monotone mapping.

An iteration method and operators of monotone type

An iteration method and operators of monotone type