An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems

<jats:p>In this paper, we consider a nonhomogeneous differential operator equation of first order <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <msup>
                           <mrow>
                              <mi>u</mi>
                           </mrow>
                           <mrow>
                              <mo>′</mo>
                           </mrow>
                        </msup>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </mfenced>
                        <mo>+</mo>
                        <mi>A</mi>
                        <mi>u</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </mfenced>
                        <mo>=</mo>
                        <mi>f</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>t</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>. The coefficient operator <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mi>A</mi>
                     </math>
                  </jats:inline-formula> 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 <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>u</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mn>0</mn>
                           </mrow>
                        </mfenced>
                        <mo>=</mo>
                        <mi mathvariant="normal">Φ</mi>
                        <mtext> or </mtext>
                        <mi>u</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>T</mi>
                           </mrow>
                        </mfenced>
                        <mo>=</mo>
                        <mi mathvariant="normal">Φ</mi>
                     </math>
                  </jats:inline-formula>. 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.</jats:p>


Introduction
e terms "inverse problems" and "ill-posed problems" have been steadily and surely gaining popularity in modern science since the middle of the 20th century. 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). At the same time, 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. e reason is that solutions to inverse problems describe important properties of media under study, such as density and velocity of wave propagation, elasticity parameters, conductivity, dielectric permittivity and magnetic permeability, and properties and location of inhomogeneities in inaccessible areas. roughout this paper, H will denote a Hilbert space endowed with the inner product 〈., .〉 and the norm ‖.‖. A is a linear, unbounded self-adjoint operator which has a continuous spectrum with nonfixed sign on H. We assume that A admits a bounded inverse A − 1 within H and f is a given function in C([0, T], H). Let T be a positive real number and Φ is an element in H. We consider the problem of finding a function u: 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. In fact, from small noise contaminated physical measurements, the corresponding solutions have large errors. It becomes difficult to do numerical calculations. Hence, a regularization is in order.
ere have been many studies on the homogeneous case with the initial condition u(0) � Φ or Cauchy problems with final condition u(T) � Φ corresponding to A being of constant sign, using different approaches such as [1][2][3][4][5][6]. e unhomogeneous has been treated by many authors (see [7]).
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.
In this paper, we introduce into unhomogeneous differential operator equation a nonlocal boundary condition depending on a small parameter ε ∈ ]0, 1[ as follows: roughout this work, we denote by v n n≥1 the eigenvectors corresponding to positive eigenvalues λ n n≥1 and by w n n≥1 the eigenvectors corresponding to negative eigenvalues μ n n≥1 of the operator A.
e eigenvectors of A v n n≥1 , w n n≥1 form an orthogonal system in H with ‖v n ‖ � 1 and ‖w n ‖ � 1, ∀n ≥ 1.

The Approximate Problem
We approximate problem (1) by the following problem: where A is as above and f ∈ C([0, T], H) is represented by where e vector Φ ∈ H can be represented as follows: where ξ n � 〈v n Φ〉 and δ n � 〈w n , Φ〉. e classical solution of problem (2) is the function as follows: In the base v n , w n , it can be written that

Notation. If we denote
then the function u ε (t) can be written as follows: (3) is well-posed, and its unique solution is u ε (t) given by (8). Furthermore, and by using relations (12) and (13), we obtain 2 Journal of Mathematics en, and we deduce the convergence of series (10)
Proof. (i) For all ε ∈ ]0, 1[, the following value: is negative and increasing with ε since its derivative with respect to ε is as follows: and it takes its lowest value when ε � (1/2), and then we deduce (41). (ii) We have also and then On the other hand, the following function: is positive and decreasing with ε since the derivative with respect to ε is as follows: and it takes its greatest value for ε � 0, and then we obtain (43). (iii) Furthermore, we know that for all ε ∈ ]0, 1[, and then relation (44) is obtained.
and then we deduce (45) Proof. We have and then by using (12) and (41)-(44), we obtain Now, we prove that where M is a particular set dense in H. We choose Φ and f on M having the form: Journal of Mathematics ∀N < ∞. en, εe μ n T +(1 − ε) 〈w n , Φ〉 + (1 − ε)e μ n T εe μ n T +(1 − ε) I T h n 2 . (62) Using inequalities (45)-(48) in Lemma 2 and relation (12), we obtain that Finally, according to the Banach-Steinhaus theorem, (57) results from (59) and (63). □ Conclusion 1. Note that, in this work, using a quasiboundary value method, we study the nonhomogeneous differential-operator equation introducing nonlocal conditions. e results given in this paper generalized the results of the work given by Yurchuk and Ababneh [8] where they considered the homogeneous case. [10][11][12][13][14][15][16] Data Availability e content of our article is devoid of any particular data and therefore does not require any provision in this direction.

Conflicts of Interest
e authors declare that they have no conflicts of interest.