Convergence of a Projection-Difference Method for the Approximate Solution of a Parabolic Equation with a Weighted Integral Condition on the Solution

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In a Hilbert space, we consider an abstract linear parabolic equation defined on an interval with a nonlocal weighted integral condition imposed on the solution. This problem is solved approximately by a projection-difference method with the use of the implicit Euler method in the time variable. The approximation to the problem in the spatial variables is developed with the finite element method in mind. An estimate of the approximate solution is obtained, the convergence of the approximate solutions to the exact solution is proved, and the error estimates, as well as the orders of the rate of convergence, are established.