Abstract
In this paper, we consider a nonhomogeneous differential operator equation of first order u ′ t + A u t = f t . The coefficient operator A is linear unbounded and self-adjoint in a Hilbert space. We assume that the operator does not have a fixed sign. We associate to this equation the initial or final conditions u 0 = Φ or u T = Φ . We note that the Cauchy problem is severely ill-posed in the sense that the solution if it exists does not depend continuously on the given data. Using a quasi-boundary value method, we obtain an approximate nonlocal problem depending on a small parameter. We show that regularized problem is well-posed and has a strongly solution. Finally, some convergence results are provided.
Highlights
As is known, the nonhomogeneous problem is severely ill-posed in the sense of Hadamard, i.e., solutions do not always exist, and in the case of existence, these do not depend continuously on the given data
We mention that the same problem for the homogeneous equation is treated by Yurchuk and Ababneh [8] and by Bessila [9] by introducing different nonlocal conditions
We introduce into unhomogeneous differential operator equation a nonlocal boundary condition depending on a small parameter ε ∈ ]0, 1[ as follows:
Summary
A little more than fifty years of studying problems of this kind have shown that a great number of problems from various branches of classical mathematics (computational algebra, differential and integral equations, partial differential equations, and functional analysis) can be classified as inverse or ill-posed, and they are among the most complicated ones (since they are unstable and usually nonlinear). Inverse and ill-posed problems began to be studied and applied systematically in physics, geophysics, medicine, astronomy, and all other areas of knowledge where mathematical methods are used. The nonhomogeneous problem is severely ill-posed in the sense of Hadamard, i.e., solutions do not always exist, and in the case of existence, these do not depend continuously on the given data. We mention that the same problem for the homogeneous equation is treated by Yurchuk and Ababneh [8] and by Bessila [9] by introducing different nonlocal conditions
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