Abstract
The aim of this paper is to investigate some problems related to perturbed integral equations. It is well known that many physical systems are governed by such equations, a fact which justifies the increasing interest in their study.
The results established can be considered as particular cases of the following scheme. Let us consider the equation(e) x = Ax + fx, whereA is a linear operator andf is in general a nonlinear operator. We suppose that the equationx = Ax + f has a unique solutionx belonging to a determinate function spaceD wheneverf belongs to a spaceB. Moreover, it is assumed that the mappingf →x is continuous fromB toD. Then, equation (e) has a unique solution inD iffx is an operator fromD toB satisfying a Lipschitz condition with a small enough constant. A similar statement holds iffx is a completely continuous operator fromD toB.
Various results are derived by special choices of the operatorsA andf and of the spacesB andD. Generally, we assume thatB = D and thatA is an integral operator of Volterra type or of convolution type.
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Corduneanu, C. Some perturbation problems in the theory of integral equations. Math. Systems Theory 1, 143–155 (1967). https://doi.org/10.1007/BF01705524
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DOI: https://doi.org/10.1007/BF01705524