Properties and Applications of the Resolvent Operator to a Volterra Integral Equation in Hilbert Space

1980 ◽  
Vol 11 (1) ◽  
pp. 82-91
Author(s):  
T. Kiffe ◽  
M. Stecher
Keyword(s):  
Hilbert Space ◽  
Integral Equation ◽  
Volterra Integral Equation ◽  
Resolvent Operator ◽  
The Resolvent Operator

scholarly journals Resolvent operator and spectrum of new type boundary value problems

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1671-1680 ◽  
Author(s):  
Oktay Mukhtarov ◽  
Hayati Olğar ◽  
Kadriye Aydemir
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Boundary Value Problems ◽  
Resolvent Operator ◽  
Boundary Value ◽  
Lebesgue Spaces ◽  
Direct Sum ◽  
New Type ◽  
Type Boundary ◽  
The Resolvent Operator

The aim of this study is to investigate a new type boundary value problems which consist of the equation -y''(x) + (By)(x) = ?y(x) on two disjoint intervals (-1,0) and (0,1) together with transmission conditions at the point of interaction x = 0 and with eigenparameter dependent boundary conditions, where B is an abstract linear operator, unbounded in general, in the direct sum of Lebesgue spaces L2(-1,0)( L2(0,1). By suggesting an own approaches we introduce modified Hilbert space and linear operator in it such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. We establish such properties as isomorphism and coerciveness with respect to spectral parameter, maximal decreasing of the resolvent operator and discreteness of the spectrum. Further we examine asymptotic behaviour of the eigenvalues.

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

Controllability for some nonlinear impulsive partial functional integrodifferential systems with infinite delay in Banach spaces

2020 ◽  
Vol 4 (2) ◽  
pp. 104-115
Author(s):  
Khalil Ezzinbi ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Resolvent Operator ◽  
Measure Of Noncompactness ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Mönch Fixed Point Theorem ◽  
The Resolvent Operator

This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of the work of K. Balachandran and R. Sakthivel (Journal of Mathematical Analysis and Applications, 255, 447-457, (2001)) and a host of important results in the literature, without assuming the compactness of the resolvent operator. An example is given for illustration.

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