scholarly journals A New Method for Proving Existence Theorems for Abstract Hammerstein Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
C. E. Chidume ◽  
C. O. Chidume ◽  
Ma’aruf Shehu Minjibir
Keyword(s):  
Hilbert Space ◽  
Existence Theorems ◽  
Monotone Operators ◽  
New Method ◽  
Existence Of A Solution ◽  
Linear Map ◽  
Hammerstein Equation ◽  
Variational Techniques

An abstract Hammerstein equation is an equation of the formu+KFu=0. Anew methodis introduced to prove the existence of a solution of this equation whereKandFare nonlinear accretive (monotone) operators. The method does not involve the complicated technique of factorizing a linear map via a Hilbert space and does not involve the use of deep variational techniques.

scholarly journals Existence theorems for a second orderm-point boundary value problem at resonance

1995 ◽  
Vol 18 (4) ◽  
pp. 705-710 ◽  
Author(s):  
Chaitan P. Gupta
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence Theorems ◽  
Boundary Value ◽  
Continuation Theorem ◽  
Point Boundary ◽  
Existence Of A Solution ◽  
At Resonance ◽  
Problem At Resonance

Letf:[0,1]×R2→Rbe function satisfying Caratheodory's conditions ande(t)∈L1[0,1]. Letη∈(0,1),ξi∈(0,1),ai≥0,i=1,2,…,m−2, with∑i=1m−2ai=1,0<ξ1<ξ2<…<ξm−2<1be given. This paper is concerned with the problem of existence of a solution for the following boundary value problemsx″(t)=f(t,x(t),x′(t))+e(t),0<t<1,x′(0)=0,x(1)=x(η),x″(t)=f(t,x(t),x′(t))+e(t),0<t<1,x′(0)=0,x(1)=∑i=1m−2aix(ξi).Conditions for the existence of a solution for the above boundary value problems are given using Leray Schauder Continuation theorem.

scholarly journals Zeros of Nonlinear Monotone Operators in Hilbert Space*

1978 ◽  
Vol 21 (2) ◽  
pp. 213-219 ◽  
Author(s):  
R. Schöneberg
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Monotone Operators ◽  
Russian Mathematician ◽  
Image Position

Around 1960, the Russian mathematician Kachurovski [1] introduced the notion of monotone operators in Hilbert spaces: Let E be a Hilbert space and X ⊂ E. An operator T:X→E is said to be monotone, iff.

Existence for Second Order Differential Inclusions on ℝ+ Governed by Monotone Operators

2014 ◽  
Vol 14 (3) ◽  
Author(s):  
Gheorghe Moroşanu
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Differential Inclusions ◽  
Real Hilbert Space ◽  
Monotone Operators ◽  
Second Order

AbstractConsider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u″(t) + q(t)u′(t) ∈ Au(t) + f (t) for a.a. t ∈ ℝ

scholarly journals A new method for interpolating in a convex subset of a Hilbert space

2017 ◽  
Vol 68 (1) ◽  
pp. 95-120
Author(s):  
Xavier Bay ◽  
Laurence Grammont ◽  
Hassan Maatouk
Keyword(s):  
Hilbert Space ◽  
Convex Subset ◽  
New Method

scholarly journals Remarks on an operator Wielandt Inequality

2015 ◽  
Vol 30 ◽  
pp. 577-584
Author(s):  
Pingping Zhang
Keyword(s):  
Hilbert Space ◽  
Recent Result ◽  
Positive Operator ◽  
Upper Bounds ◽  
Linear Map ◽  
Wielandt Inequality

Let A be a positive operator on a Hilbert space H with 0 < m ≤ A ≤ M, and let X and Y be isometries on H such that X*Y = 0, p > 0, and Φ be a 2-positive unital linear map. Define Γ = (Φ(X*AY )Φ(Y*AY )^(−1)Φ(Y*AX)^p Φ(X*AX)^(−p). Several upper bounds for (1/2) |Γ + Γ*| are established. These bounds complement a recent result on the operator Wielandt inequality.

scholarly journals Operator Jensen's Inequality for Operator Superquadratic Functions

2019 ◽  
Author(s):  
Mohammad W. Alomari
Keyword(s):  
Hilbert Space ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Linear Map ◽  
Superquadratic Function ◽  
Hilbert Space Operators ◽  
Superquadratic Functions

In this work, an operator superquadratic function (in operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator convexity are pointed out. Equivalent statements of a non-commutative version of Jensen's inequality for operator superquadratic function are established. A generalization of the main result to any positive unital linear map is also provided.

scholarly journals Boundary value problems for nonlinear systems with generalized second-order differences

2011 ◽  
Vol 27 (1) ◽  
pp. 95-104
Author(s):  
RODICA LUCA ◽  
Keyword(s):  
Boundary Condition ◽  
Hilbert Space ◽  
Nonlinear Systems ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Real Hilbert Space ◽  
Boundary Value ◽  
Monotone Operators ◽  
Second Order ◽  
Differential Systems

In a real Hilbert space, we investigate the existence and uniqueness of the solutions for two classes of infinite nonlinear systems with generalized second-order differences, one of them subject to a boundary condition. Some applications to nonlinear differential systems with monotone operators are also presented.

scholarly journals VARIANTS OF ANDO–HIAI TYPE INEQUALITIES FOR DEFORMED MEANS AND APPLICATIONS

2020 ◽  
pp. 1-18 ◽  
Author(s):  
MOHSEN KIAN ◽  
MOHAMMAD SAL MOSLEHIAN ◽  
YUKI SEO
Keyword(s):  
Hilbert Space ◽  
Norm Inequality ◽  
Power Mean ◽  
Variable Operator ◽  
Linear Map ◽  
Power Means ◽  
Operator Mean ◽  
Positive Linear Map ◽  
Euclidean Mean

Abstract For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

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