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

Authors: Nihed Teniou; Salah Djezzar
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we consider a nonhomogeneous differential operator equation of first orderu′t+Aut=ft. 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 u0=ΦoruT=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.


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## Body

## 1. Introduction

The 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. The 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.Throughout 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 C0,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:0,T⟶H, such that(1)u′t+Aut=ft,t∈0,T,u0=ΦoruT=Φ.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.There have been many studies on the homogeneous case with the initial conditionu0=Φ or Cauchy problems with final condition uT=Φ corresponding to A being of constant sign, using different approaches such as [1–6]. The 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 parameterε∈0,1 as follows:(2)u′t+Aut=ft,t∈0,T,εu0+1−εuT=Φ,Φ∈H.Throughout this work, we denote byvnn≥1 the eigenvectors corresponding to positive eigenvalues λnn≥1 and by wnn≥1 the eigenvectors corresponding to negative eigenvalues μnn≥1 of the operator A. The eigenvectors of Avnn≥1,wnn≥1 form an orthogonal system in H with vn=1 and wn=1,∀n≥1.


## 2. The Approximate Problem

We approximate problem (1) by the following problem:(3)u′t+Aut=ft,0≤t≤T,εu0+1−εuT=Φ,Φ∈H,where A is as above and f∈C0,T,H is represented by(4)ft=∑n≥1gntvn+∑n≥1hntwn,where(5)gnt=ft,vn,hnt=ft,wn,∀n≥1.The vectorΦ∈H can be represented as follows:(6)Φ=∑n≥1ξntvn+∑n≥1δntwn,where ξn=vnΦandδn=wn,Φ.The classical solution of problem (2) is the function as follows:(7)uεt=Φ−1−ε∫0Tfse−AT−sdse−ATε+1−εe−AT+∫0tfse−At−sds,for Φ∈H,ε∈0,1,t∈0,T,f∈C0,T,H.In the basevn,wn, it can be written that(8)uεt=∑n≥1e−λntε+1−εe−λnTξn−1−ε∫0Tgnse−λnT−sds+∫0tgnse−λnt−sdsvn+∑n≥1eμnT−tεeμnT+1−εδn−1−ε∫0Thnse−μnT−sds+∫0thnse−μnt−sdswn.


### 2.1. Notation

If we denote(9)Mn,ε=ε+1−εe−λnT−1,Nn,ε=εeμnT+1−ε−1,Itgn=∫0tgnse−λnt−sds,Ithn=∫0thnse−μnt−sds,then the function uεt can be written as follows:(10)uεt=∑n≥1e−λntMn,εξn−1−εITgn+Itgnvn+∑n≥1eμnT−tNn,εδn−1−εIThn+Ithnwn.Theorem 1.
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Problem (3) is well-posed, and its unique solution is uεt given by (8). Furthermore,(11)sup0≤t≤Tuεt2≤max2ε2,21−ε2Φ2+2T2f2.Proof.
For each0≤t≤T, we have(12)Itgn≤Tgn,Ithn≤Thn,(13)e−λntε+1−εe−λnT≤1ε,eμnT−tεeμnT+1−ε≤11−ε,and by using relations (12) and (13), we obtain(14)sup0≤t≤T∑n=kk+pe−λntε+1−εe−λnTvn,Φ−1−ε∫0Tgnse−λnT−sds+∫0tgnse−λnt−sdsvn2+sup0≤t≤T∑n=kk+peμnT−tεeμnT+1−εwn,Φ−1−ε∫0Thnse−μnT−sds+∫0thnse−μnt−sdswn2≤21ε2∑n=kk+pvn,Φ2+11−ε2∑n=kk+pwn,Φ2+T2∑n=kk+pgn2+T2∑n=kk+phn2.
Then,(15)sup0≤t≤Tuεt2≤max2ε2,21−ε2Φ2+2T2f2,and we deduce the convergence of series (10) which is the strong solution of problem (3).


## 2.1. Notation

If we denote(9)Mn,ε=ε+1−εe−λnT−1,Nn,ε=εeμnT+1−ε−1,Itgn=∫0tgnse−λnt−sds,Ithn=∫0thnse−μnt−sds,then the function uεt can be written as follows:(10)uεt=∑n≥1e−λntMn,εξn−1−εITgn+Itgnvn+∑n≥1eμnT−tNn,εδn−1−εIThn+Ithnwn.Theorem 1.
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Problem (3) is well-posed, and its unique solution is uεt given by (8). Furthermore,(11)sup0≤t≤Tuεt2≤max2ε2,21−ε2Φ2+2T2f2.Proof.
For each0≤t≤T, we have(12)Itgn≤Tgn,Ithn≤Thn,(13)e−λntε+1−εe−λnT≤1ε,eμnT−tεeμnT+1−ε≤11−ε,and by using relations (12) and (13), we obtain(14)sup0≤t≤T∑n=kk+pe−λntε+1−εe−λnTvn,Φ−1−ε∫0Tgnse−λnT−sds+∫0tgnse−λnt−sdsvn2+sup0≤t≤T∑n=kk+peμnT−tεeμnT+1−εwn,Φ−1−ε∫0Thnse−μnT−sds+∫0thnse−μnt−sdswn2≤21ε2∑n=kk+pvn,Φ2+11−ε2∑n=kk+pwn,Φ2+T2∑n=kk+pgn2+T2∑n=kk+phn2.
Then,(15)sup0≤t≤Tuεt2≤max2ε2,21−ε2Φ2+2T2f2,and we deduce the convergence of series (10) which is the strong solution of problem (3).


## 3. The Convergence Results

Lemma 1.
For allε∈0,1, we have the following estimates:(16)0≤ε1−e−λnTε+1−εe−λnT≤1−e−λnT≤1,(17)εε+1−εe−λnT≤1,(18)−1≤eμnT−1eμnT+1≤εeμnT−1εeμnT+1−ε≤0,(19)εeμnTεeμnT+1−ε≤1,(20)ε21−e−λnT2ε+1−εe−λnT2≤ε2e2λNT1−ε2,(21)ε2eμnT−12εeμnT+1−ε2≤ε21−ε2,(22)ε2e2μnTεeμnT+1−ε2≤ε21−ε2,where λN=maxn≥1λn.Proof.
(i)
The valueε1−e−λnT/ε+1−εe−λnT is positive and increasing with ε, and its derivative with respect to ε is as follows:(23)e−λnT1−e−λnTε+1−εe−λnT2≥0.It takes its greatest value whenε⟶1, i.e.,(24)0≤ε1−e−λnTε+1−εe−λnT≤1−e−λnT≤1.(ii)
We know that for allε∈0,1, we have(25)0≤1−εe−λnT≤1,and then(26)εε+1−εe−λnT≤1.(iii)
On the other hand, the following value:(27)εeμnT−1εeμnT+1−ε,is negative and decreasing withε as long as its derivative with respect to ε is as follows:(28)eμnT−1εeμnT+1−ε2≤0,and it takes its lowest value atε=1/2, i.e.,(29)−1≤eμnT−1eμnT+1≤εeμnT−1εeμnT+1−ε≤0.(iv)
We have also(30)0≤εeμnT≤εeμnT+1−ε,and then we obtain relation (19).(v)
For estimate (20), we have(31)ε21−e−λnT2≤ε2,1−εe−λnT≤ε+1−εe−λnT,and then(32)ε21−e−λnT2ε+1−εe−λnT2≤ε2e2λnT1−ε2≤ε2e2λNT1−ε2.(vi)
Furthermore, we have(33)ε2eμnT−12≤ε2,1εeμnT+1−ε2≤11−ε2,εeμnTεeμnT+1−ε≤εeμnT1−ε≤ε1−ε,and then we deduce (21) and (22).Theorem 2.
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For allΦ∈H, we have(34)limε⟶0Φ−uεT=0.Proof.
By definitions (6) and (8), we have(35)Φ−uεT2=∑n≥1ε1−e−λnTε+1−εe−λnTvn,Φ−εε+1−εe−λnTITgn2+∑n≥1εeμnT−1εeμnT+1−εwn,Φ−εeμnTεeμnT+1−εIThn2.
Using inequalities (12) and (16)–(19) (see Lemma 1), we obtain(36)Φ−uεT2≤2Φ2+T2f2.
Now, we demostrate that(37)limε⟶0Φ−uεT=0,∀Φ∈M,where M is some special set dense in the Hilbert space H. We take all Φ and f on M of the form as follows:(38)Φ=∑n=1Nξnvn+∑n=1+∞δnwn,f=∑n=1Ngnvn+∑n=1+∞hnwn,∀N<∞. Then, by virtue of the Banach–Steinhaus theorem, we obtain(39)Φ−uεT2≤2∑n=1Nε21−e−λnT2ε+1−εe−λnT2vn,Φ2+2∑n=1Nε2ε+1−εe−λnT2IT2gn+2∑n=1+∞ε2eμnT−12εeμnT+1−ε2wn,Φ2+2∑n=1+∞ε2e2μnTεeμnT+1−ε2IT2hn.
Combining relations (12), (17), and (20)–(22) (see lemma 1), we obtain(40)Φ−uεT2≤2ε2e2λNT1−ε2∑n=1Nvn,Φ2+T2∑n=1Ngn2+2ε21−ε2∑n=1+∞wn,Φ2+T2∑n=1+∞hn2≤2ε2e2λNT1−ε2Φ2+T2f2,which gives that limε⟶0Φ−uεT2=0.Lemma 2.
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For allε∈0,1, we have the following inequalities:(41)−1≤e−λnT−1e−λnT+1≤ε−11−e−λnTε+1−εe−λnT≤0,(42)−1≤1−ε1+ε≤1−εε+1−εe−λnT,(43)0≤1−ε1−eμnTεeμnT+1−ε≤1−eμnT≤1,(44)1−εeμnTεeμnT+1−ε≤1,(45)1−εε+1−εe−λnT≤1−εε,(46)ε−121−e−λnT2ε+1−εe−λnT2≤1−ε2ε2,(47)1−ε1−eμnTεeμnT+1−ε≤1−εεeμNT,(48)1−εeμnTεeμnT+1−ε≤1−εεeμNT,where μN=minn≥1μn.Proof.
(i)
For allε∈0,1, the following value:(49)ε−11−e−λnTε+1−εe−λnT,is negative and increasing withε since its derivative with respect to ε is as follows:(50)1−e−λnTε+1−εe−λnT2≥0,and it takes its lowest value whenε=1/2, and then we deduce (41).(ii)
We have also(51)11+ε≤1ε+1−εe−λnT≤1ε,and then(52)−1≤1−ε1+ε≤1ε+1−εe−λnT.On the other hand, the following function:(53)1−ε1−eμnTεeμnT+1−ε,is positive and decreasing withε since the derivative with respect to ε is as follows:(54)eμnTeμnT−1εeμnT+1−ε2≤0,and it takes its greatest value forε=0, and then we obtain (43).(iii)
Furthermore, we know that for allε∈0,1,(55)1εeμnT+1−ε≤11−ε,and then relation (44) is obtained.(iv)
We have(56)0≤1−εe−λnT≤1−ε,ε≤1−εe−λnT≤1+ε,and then we deduce (45) and (46).(v)
Finally, with simple increases, we can easily have inequalities (47) and (48).Theorem 3.
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For allΦ∈H, we have(57)limε⟶1Φ−uε0=0.Proof.
We have(58)Φ−uε0=∑n≥1ε−11−e−λnTε+1−εe−λnTvn,Φ+1−εε+1−εe−λnTITgnvn+∑n≥11−ε1−eμnTεeμnT+1−εwn,Φ+1−εeμnTεeμnT+1−εIThnwn,and then by using (12) and (41)–(44), we obtain(59)Φ−uε02≤2Φ2+T2f2.
Now, we prove that(60)limε⟶1Φ−uε0=0,∀Φ∈M,where M is a particular set dense in H. We choose Φ and f on M having the form:(61)Φ=∑n=1+∞ξnvn+∑n=1Nδnwn,f=∑n=1+∞gnvn+∑n=1Nhnwn,∀N<∞. Then,(62)Φ−uε02=∑n=1+∞ε−11−e−λnTε+1−εe−λnTvn,Φ+1−εε+1−εe−λnTITgn2+∑n=1N1−ε1−eμnTεeμnT+1−εwn,Φ+1−εeμnTεeμnT+1−εIThn2.
Using inequalities (45)–(48) in Lemma 2 and relation (12), we obtain that(63)Φ−uε02≤21−ε2ε2∑n=1+∞vn,Φ2+T2∑n=1+∞gn2+21−ε2ε2e−2μnT∑n=1Nwn,Φ2+T2∑n=1Nhn2≤21−ε2ε2e−2μnTΦ2+T2f2.
Finally, according to the Banach–Steinhaus theorem, (57) results from (59) and (63).Conclusion 1.
Note that, in this work, using a quasi-boundary value method, we study the nonhomogeneous differential-operator equation introducing nonlocal conditions. The results given in this paper generalized the results of the work given by Yurchuk and Ababneh [8] where they considered the homogeneous case. [10–16]



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