Initial Value Problems in Linear Integral Operator Equations

  • L. P. Castro
  • M. M. Rodrigues
  • S. SaitohEmail author
Part of the Springer Optimization and Its Applications book series (SOIA, volume 94)


For some general linear integral operator equations, we investigate consequent initial value problems by using the theory of reproducing kernels. A new method is proposed which—in particular—generates a new field among initial value problems, linear integral operators, eigenfunctions and values, integral transforms and reproducing kernels. In particular, examples are worked out for the integral equations of Lalesco–Picard, Dixon, and Tricomi types.


Integral transform Reproducing kernel Isometric mapping Inversion formula Initial value problem Eigenfunction Eigenvalue Fourier integral transform Inverse problem Lalesco–Picard equation Dixon equation Tricomi equation 

Mathematics Subject Classification (2010):

Primary 45C05 Secondary 32A30 42A38 45A05 45D05 45E05 45P05 46E22 47A05 



This work was supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014. The third author is supported in part by the Grant-in-Aid for the Scientific Research (C)(2)(No. 24540113).


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mathematics, CIDMA—Center for Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal
  2. 2.Institute of Reproducing KernelsKiryuJapan

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