scholarly journals New Type of Sturm-Liouville Problems in Associated Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
O. Sh. Mukhtarov ◽  
K. Aydemir
Keyword(s):  
Linear Operator ◽  
Asymptotic Behaviour ◽  
Spectral Parameter ◽  
Hilbert Spaces ◽  
Resolvent Operator ◽  
Sufficient Conditions ◽  
Transmission Conditions ◽  
Continuous Case ◽  
New Type ◽  
Sturm Liouville

We introduce a new type of discontinuous Sturm-Liouville problems, involving an abstract linear operator in equation. By suggesting own approaches we define some new Hilbert spaces to establish such properties as isomorphism, coerciveness, and maximal decreasing of resolvent operator with respect to spectral parameter. Then we find sufficient conditions for discreteness of the spectrum and examine asymptotic behaviour of eigenvalues. Obtained results are new even for continuous case, that is, without transmission conditions.

Approximate controllability of semilinear impulsive functional differential systems with non-local conditions

2020 ◽  
Vol 37 (4) ◽  
pp. 1070-1088 ◽  
Author(s):  
Sumit Arora ◽  
Soniya Singh ◽  
Jaydev Dabas ◽  
Manil T Mohan
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Resolvent Operator ◽  
Approximate Controllability ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Local Conditions ◽  
Functional Differential Systems ◽  
Non Local ◽  
Functional Differential

Abstract This paper is concerned with the approximate controllability of semilinear impulsive functional differential systems in Hilbert spaces with non-local conditions. We establish sufficient conditions for approximate controllability of such systems via resolvent operator and Schauder’s fixed point theorem. An application involving the impulse effect associated with delay and non-local conditions is presented to verify our claimed results.

scholarly journals Inverse problems for the Sturm–Liouville equation with a spectral parameter in the boundary condition

2017 ◽  
Author(s):  
Namig J. Guliyev
Keyword(s):  
Boundary Condition ◽  
Inverse Problems ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Eigenvalue Parameter ◽  
Sturm Liouville ◽  
Necessary And Sufficient

Inverse problems of recovering the coefficients of Sturm--Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

scholarly journals An inverse problem for the second-order integro-differential pencil

2019 ◽  
Vol 50 (3) ◽  
pp. 223-231 ◽  
Author(s):  
Natalia P. Bondarenko
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Spectral Parameter ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Uniqueness Of Solution ◽  
Convolution Kernel ◽  
Sturm Liouville ◽  
Necessary And Sufficient

We consider the second-order (Sturm-Liouville) integro-differential pencil with polynomial dependence on the spectral parameter in a boundary condition. The inverse problem is solved, which consists in reconstruction of the convolution kernel and one of the polynomials in the boundary condition by using the eigenvalues and the two other polynomials. We prove uniqueness of solution, develop a constructive algorithm for solving the inverse problem, and obtain necessary and sufficient conditions for its solvability.

scholarly journals ON THE EXPANSION FORMULA FOR A SINGULAR STURM-LIOUVILLE OPERATOR

2021 ◽  
Vol 21 (1) ◽  
pp. 67-76
Author(s):  
ULVIYE DEMIRBILEK ◽  
KHANLAR R. MAMEDOV
Keyword(s):  
Boundary Condition ◽  
Green Function ◽  
Spectral Parameter ◽  
Liouville Operator ◽  
Resolvent Operator ◽  
Liouville Problem ◽  
Scattering Data ◽  
Expansion Formula ◽  
Sturm Liouville

In this study, on the semi-axis, Sturm - Liouville problem under boundary condition depending on spectral parameter is considered. In what follows scattering data is defined and its properties are given for the problem. The kernel of resolvent operator which is Green function is constructed. Using Titchmarsh method, expansion is obtained according to eigenfunctions and expansion formula is expressed with the scattering data.

Stability of fractional neutral stochastic partial integro-differential equations

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Liping Xu ◽  
Zhi Li
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Hilbert Spaces ◽  
Stochastic Analysis ◽  
Resolvent Operator ◽  
Linear Part ◽  
Sufficient Conditions ◽  
Integral Inequalities

AbstractIn this paper, we are concerned with a class of fractional partial neutral stochastic integro-differential equations in Hilbert spaces. We assume that the linear part of this equation generates an α-resolvent operator and transform it into an integral equation. By the stochastic analysis and fractional calculus technique, and combining some integral inequalities, we obtain some sufficient conditions ensuring the exponential

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.

scholarly journals Input-to-State Stability of Lur’e Hyperbolic Distributed Complex-Valued Parameter Control Systems: LOI Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Zhixin Tai ◽  
Xingcheng Wang ◽  
Yan Shi ◽  
Hamid Reza Karimi
Keyword(s):  
Linear Operator ◽  
Control Systems ◽  
Comparison Principle ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Parameter Control ◽  
Operator Inequalities ◽  
Work Input ◽  
Delay Dependent ◽  
Complex Valued

In this work, input-to-state stability of Lur’e hyperbolic distributedcomplex-valuedparameter control systems has been addressed. Using comparison principle, delay-dependent sufficient conditions for the input-to-state stability in complex Hilbert spaces are established in terms of linear operator inequalities. Finally, numerical computation illustrates our result.

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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