scholarly journals Semilinear systems with a multi-valued nonlinear term

2017 ◽  
Vol 15 (1) ◽  
pp. 628-644
Author(s):  
In-Sook Kim ◽  
Suk-Joon Hong
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Nonlinear Term ◽  
Degree Theory ◽  
Existence Results ◽  
Topological Degree Theory ◽  
Dirichlet Problems ◽  
Semilinear Systems ◽  
Densely Defined ◽  
Monotone Type

Abstract Introducing a topological degree theory, we first establish some existence results for the inclusion h ∈ Lu − Nu in the nonresonance and resonance cases, where L is a closed densely defined linear operator on a Hilbert space with a compact resolvent and N is a nonlinear multi-valued operator of monotone type. Using the nonresonance result, we next show that abstract semilinear system has a solution under certain conditions on N = (N1, N2), provided that L = (L1, L2) satisfies dim Ker L1 = ∞ and dim Ker L2 < ∞. As an application, periodic Dirichlet problems for the system involving the wave operator and a discontinuous nonlinear term are discussed.

scholarly journals On the degree theory for densely defined mappings of class(S+)L

1999 ◽  
Vol 4 (3) ◽  
pp. 141-152 ◽  
Author(s):  
Juha Berkovits
Keyword(s):  
Topological Degree ◽  
Degree Theory ◽  
Elliptic Problems ◽  
Existence Results ◽  
Basic Properties ◽  
New Construction ◽  
Densely Defined ◽  
Monotone Type ◽  
Abstract Existence

We introduce a new construction of topological degree for densely defined mappings of monotone type. We also study the structure of the classes of mappings involved. Using the basic properties of the degree, we prove some abstract existence results that can be applied to elliptic problems.

Formally Normal Operators Having no Normal Extensions

1965 ◽  
Vol 17 ◽  
pp. 1030-1040 ◽  
Author(s):  
Earl A. Coddington
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Null Space ◽  
Normal Operator ◽  
Closed Linear Operator ◽  
Normal Operators ◽  
The Given ◽  
Densely Defined ◽  
Made In

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

scholarly journals Dirac structures on Hilbert spaces

1999 ◽  
Vol 22 (1) ◽  
pp. 97-108 ◽  
Author(s):  
A. Parsian ◽  
A. Shafei Deh Abad
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Smooth Manifold ◽  
Real Hilbert Space ◽  
Dirac Structure ◽  
Dirac Structures ◽  
Basic Properties ◽  
Hilbert Subspace ◽  
Densely Defined

For a real Hilbert space(H,〈,〉), a subspaceL⊂H⊕His said to be a Dirac structure onHif it is maximally isotropic with respect to the pairing〈(x,y),(x′,y′)〉+=(1/2)(〈x,y′〉+〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures onHare in one-to-one correspondence with isometries onH, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structureLonH, everyz∈His uniquely decomposed asz=p1(l)+p2(l)for somel∈L, wherep1andp2are projections. Whenp1(L)is closed, for any Hilbert subspaceW⊂H, an induced Dirac structure onWis introduced. The latter concept has also been generalized.

scholarly journals Rotational periodic solutions for fractional iterative systems

2021 ◽  
Vol 6 (10) ◽  
pp. 11233-11245
Author(s):  
Rui Wu ◽  
◽  
Yi Cheng ◽  
Ravi P. Agarwal ◽  
◽  
...  
Keyword(s):  
Existence And Uniqueness ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Degree Theory ◽  
Periodic Problem ◽  
Network System ◽  
Well Posedness ◽  
Topological Degree Theory ◽  
Neural Network System ◽  
Iterative Systems

<abstract><p>In this paper, we devoted to deal with the rotational periodic problem of some fractional iterative systems in the sense of Caputo fractional derivative. Under one sided-Lipschtiz condition on nonlinear term, the existence and uniqueness of solution for a fractional iterative equation is proved by applying the Leray-Schauder fixed point theorem and topological degree theory. Furthermore, the well posedness for a nonlinear control system with iteration term and a multivalued disturbance is established by using set-valued theory. The existence of solutions for a iterative neural network system is demonstrated at the end.</p></abstract>

scholarly journals Existence problems for homoclinic solutions

2002 ◽  
Vol 7 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Cezar Avramescu
Keyword(s):  
Fixed Point ◽  
Topological Degree ◽  
Degree Theory ◽  
Existence Results ◽  
Fixed Point Method ◽  
Point Method ◽  
Homoclinic Solutions ◽  
Topological Degree Theory

The problemx˙=f(t,x), x(−∞)=x(+∞), wherex(±∞):=limt→±∞x(t)∈ℝn, is considered. Some existence results for this problem are established using the fixed point method and topological degree theory.

scholarly journals Closed linear operators with domain containing their range

1984 ◽  
Vol 27 (2) ◽  
pp. 229-233 ◽  
Author(s):  
Schôichi Ôta
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Linear Operators ◽  
Closed Linear Operator ◽  
Unbounded Operators ◽  
Densely Defined ◽  
Closed Linear Operators

In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T* and is globally invariant under T and T*,then T is bounded. In the case of a Banach space (in particular, a C*-algebra) weshowed in [6] that a densely defined closed derivation in a C*-algebra with domaincontaining its range is automatically bounded (see the references in [6] and [7] for thetheory of derivations in C*-algebras).

scholarly journals Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian

2018 ◽  
Vol 20 (03) ◽  
pp. 1750032 ◽  
Author(s):  
Alexander Quaas ◽  
Aliang Xia
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Degree Theory ◽  
Elliptic Systems ◽  
Existence Results ◽  
Nonlinear Elliptic Problem ◽  
Topological Degree Theory ◽  
Scaling Method ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic

In this paper, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: [Formula: see text] where [Formula: see text] denotes the fractional Laplacian and [Formula: see text] is a smooth bounded domain in [Formula: see text]. It shown that under some assumptions on [Formula: see text] and [Formula: see text], the problem has at least one positive solution [Formula: see text]. Our proof is based on the classical scaling method of Gidas and Spruck and topological degree theory.

scholarly journals Nontrivial Solutions for the 2 n th Lidstone Boundary Value Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Yaohong Li ◽  
Jiafa Xu ◽  
Yongli Zan
Keyword(s):  
Linear Operator ◽  
Boundary Value Problem ◽  
Topological Degree ◽  
Boundary Value ◽  
Degree Theory ◽  
Topological Degree Theory ◽  
Nontrivial Solutions

In this paper, we study the existence of nontrivial solutions for the 2 n th Lidstone boundary value problem with a sign-changing nonlinearity. Under some conditions involving the eigenvalues of a linear operator, we use the topological degree theory to obtain our main results.

Existence results for nonlinear periodic boundary-value problems

2009 ◽  
Vol 52 (1) ◽  
pp. 79-95 ◽  
Author(s):  
John R. Graef ◽  
Lingju Kong
Keyword(s):  
Topological Degree ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Finite Interval ◽  
Nonlinear Differential Equations ◽  
Degree Theory ◽  
Periodic Boundary Conditions ◽  
Existence Results ◽  
Topological Degree Theory ◽  
Periodic Boundary Value Problems

AbstractWe study a class of second-order nonlinear differential equations on a finite interval with periodic boundary conditions. The nonlinearity in the equations can take negative values and may be unbounded from below. Criteria are established for the existence of non-trivial solutions, positive solutions and negative solutions of the problems under consideration. Applications of our results to related eigenvalue problems are also discussed. Examples are included to illustrate some of the results. Our analysis relies mainly on topological degree theory.

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