In recent years, the
reproducing kernel Hilbert space method has been used for obtaining approximate solutions in a wide class of ordinary differential, partial differential and integral equations.
The space [H.sub.w] is a
reproducing kernel Hilbert space whose reproducing kernel is
A Hilbert space H of functions on a nonempty abstract set E is called a
reproducing kernel Hilbert space if there exists a reproducing kernel K of H.
Particularly, the analytical approximate solutions for first-order up to higher-order IDEs have been obtained by the numerical integration techniques such as Runge-Kutta methods [13], Euler-Chebyshev methods [14], Wavelet-Galerkin method [15], and Taylor polynomials method [16] and by semianalytical-numerical techniques such as Adomian decomposition method [17],
reproducing kernel Hilbert space (RKHS) method [18-22], homotopy analysis method [23], and variational iteration method [24].
The hypothesis space H is defined as a
reproducing kernel Hilbert space (RKHS) whose elements are real continuous functions defined on X with a kernel K.
The
reproducing kernel Hilbert space (RKHS) property of localization space [P.sub.[psi]] will be described in the next theorem.
which shows that [H.sub.D](R) is a
reproducing kernel Hilbert space (see also [10]).
It is well known that [F.sup.2] is a
reproducing kernel Hilbert space with inner product