Eigenvalue Problems for Fredholm Operators with Set-Valued Perturbations

AbstractBy means of a suitable degree theory, we prove persistence of eigenvalues and eigenvectors for set-valued perturbations of a Fredholm linear operator. As a consequence, we prove existence of a bifurcation point for a non-linear inclusion problem in abstract Banach spaces. Finally, we provide applications to differential inclusions.

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SOLUTION BRANCHES OF NONLINEAR EIGENVALUE PROBLEMS ON RESTRICTED DOMAINS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972720000234 ◽

2020 ◽

Vol 102 (3) ◽

pp. 490-497

Author(s):

SHANE ARORA

Keyword(s):

Banach Spaces ◽

Bifurcation Point ◽

Eigenvalue Problems ◽

Accumulation Point ◽

Nonlinear Eigenvalue Problems ◽

Restricted Domains ◽

Nonlinear Eigenvalue

We extend bifurcation results of nonlinear eigenvalue problems from real Banach spaces to any neighbourhood of a given point. For points of odd multiplicity on these restricted domains, we establish that the component of solutions through the bifurcation point either is unbounded, admits an accumulation point on the boundary, or contains an even number of odd-multiplicity points. In the simple-multiplicity case, we show that branches of solutions in the directions of corresponding eigenvectors satisfy similar conditions on such restricted domains.

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Existence and bifurcation of solutions for Fredholm operators with nonlinear perturbations

Nagoya Mathematical Journal ◽

10.1017/s0027763000019887 ◽

1982 ◽

Vol 86 ◽

pp. 249-271 ◽

Cited By ~ 1

Author(s):

Yasuo Niikura

Keyword(s):

Banach Space ◽

Linear Operator ◽

Nonlinear Operator ◽

Real Banach Space ◽

Eigenvalue Problems ◽

Nonlinear Perturbations ◽

Fredholm Operators ◽

Nonlinear Eigenvalue Problems ◽

Bifurcation Of Solutions ◽

Nonlinear Eigenvalue

In this paper we shall discuss nonlinear eigenvalue problems for the equations of the formwhere L is a linear operator on a real Banach space X with non-zero kernel, K(-) is a linear or nonlinear operator on X and M(·, ·) is an operator from X X R into X. Equations of the form (1) arise in various fields of physics and engineering.

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Existence of solutions for functional boundary value problems at resonance on the half-line

Boundary Value Problems ◽

10.1186/s13661-020-01459-5 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Bingzhi Sun ◽

Weihua Jiang

Keyword(s):

Boundary Value Problems ◽

Banach Spaces ◽

Existence Of Solutions ◽

Boundary Value ◽

Coincidence Degree ◽

Degree Theory ◽

Half Line ◽

Projection Scheme ◽

At Resonance ◽

Functional Boundary

Abstract By defining the Banach spaces endowed with the appropriate norm, constructing a suitable projection scheme, and using the coincidence degree theory due to Mawhin, we study the existence of solutions for functional boundary value problems at resonance on the half-line with $\operatorname{dim}\operatorname{Ker}L = 1$ dim Ker L = 1 . And an example is given to show that our result here is valid.

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Projections on Tree-Like banach Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1985-049-2 ◽

1985 ◽

Vol 37 (5) ◽

pp. 908-920

Author(s):

A. D. Andrew

Keyword(s):

Banach Space ◽

Linear Operator ◽

Banach Spaces ◽

Bounded Linear Operator ◽

Unconditional Basis ◽

Reflexive Banach Space ◽

Separable Space ◽

Bounded Linear ◽

Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

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Existence Results for Fractional Differential Inclusions with Multivalued Term Depending on Lower-Order Derivative

Abstract and Applied Analysis ◽

10.1155/2012/423796 ◽

2012 ◽

Vol 2012 ◽

pp. 1-24 ◽

Cited By ~ 8

Author(s):

Xiaoyou Liu ◽

Zhenhai Liu

Keyword(s):

Boundary Conditions ◽

Lower Order ◽

Differential Inclusions ◽

Integral Boundary Conditions ◽

Existence Results ◽

Inclusion Problem ◽

Fractional Differential Inclusions ◽

Integral Boundary ◽

Separated Boundary Conditions ◽

Fractional Differential

This paper is concerned with a class of fractional differential inclusions whose multivalued term depends on lower-order fractional derivative with fractional (non)separated boundary conditions. The cases of convex-valued and non-convex-valued right-hand sides are considered. Some existence results are obtained by using standard fixed point theorems. A possible generalization for the inclusion problem with integral boundary conditions is also discussed. Examples are given to illustrate the results.

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A note on best approximation and invertibility of operators on uniformly convex Banach spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171291000832 ◽

1991 ◽

Vol 14 (3) ◽

pp. 611-614 ◽

Cited By ~ 1

Author(s):

James R. Holub

Keyword(s):

Banach Space ◽

Linear Operator ◽

Banach Spaces ◽

Best Approximation ◽

Bounded Linear Operator ◽

Convex Banach Space ◽

Sufficient Condition ◽

Uniformly Convex ◽

Bounded Linear ◽

Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

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Degree theory for mappings in non-reflexive banach spaces

Applied Mathematics and Computation ◽

10.1016/j.amc.2008.02.001 ◽

2008 ◽

Vol 202 (1) ◽

pp. 229-232 ◽

Cited By ~ 4

Author(s):

Fulong Wang ◽

Yuqing Chen ◽

Donal O’Regan

Keyword(s):

Banach Spaces ◽

Degree Theory ◽

Reflexive Banach Spaces

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Stability of essential approximate point spectrum and essential defect spectrum of linear operator

Filomat ◽

10.2298/fil1509983a ◽

2015 ◽

Vol 29 (9) ◽

pp. 1983-1994

Author(s):

Aymen Ammar ◽

Mohammed Dhahri ◽

Aref Jeribi

Keyword(s):

Linear Operator ◽

Measure Of Noncompactness ◽

Point Spectrum ◽

Fredholm Operators ◽

Approximate Point Spectrum ◽

Densely Defined

In the present paper, we use the notion of measure of noncompactness to give some results on Fredholm operators and we establish a fine description of the essential approximate point spectrum and the essential defect spectrum of a closed densely defined linear operator.

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