Existence and uniqueness solution of an inverse problems for degenerate differential equations

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1 Existence and uniqueness solution of an inverse problems for degenerate differential equations Mahmoud M. El-borai & Osama L. Mostafa & Hoda A. Fouad m m elborai@yahoo.com & moustafa labib@yahoo.com & hoda rg@yahoo.com Faculty of Science, Alexandria University, Alexandria, Egypt Khadug S. Sharnana & khadog 20000@yahoo.com Faculty of Science, Almerqeb University, Alkhomes, Libya Abstract In this paper we concerned with study existence and uniqueness of solutions for a class of inverse problems of degenerate differential equations.the main tool perturbation theory for linear operators. consider the inverse problems for degenerate differential equations of the form with the initial condition dbu(t) and the overdetermination condition = Au(t) + Bγ(t)f(t), u(0) = u 0 (u(t), v) = w(t) where A and B are closed linear operators in a Hilbert space H,f is a given abstract function with values in H,v is a given element in H,u 0 is an initial value,and {u, γ} are the unknown functions. Key words: Perturbation Theory of Linear Operators; Linear a c 0 Semigroup ; degenerate differential equations. Introduction The term inverse problems have been steadily and surely gaining popularity in moderm science since the middle of the 20th contury.alittle 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,functional analysis) can be classified as inverse.and they are among the most complicated ones ( since they are unstable and usually nonlinear).

2 2 Abstract inverse problem We establish existence and uniqueness of solutions for a class of inverse problems of degenerate differential equations.the main tool perturbation theory for linear operators. consider the inverse problems for degenerate differential equations of the form with the initial condition dbu(t) and the overdetermination condition = Au(t) + Bγ(t)f(t), () u(0) = u 0 (2) (u(t), v) = w(t) (3) where A and B are closed linear operators in a Hilbert space H,f is a given abstract function with values in H,v is a given element in H,u 0 is an initial value,and {u, γ} are the unknown functions. We introduce some facts a bout the generator of a c 0 -semigroup (see [4]). we denote by X a Banach space with norm. and A : D(A) X is the infinitesimal generator of a c 0 -semigroup of bounded linear operator T(t),t > 0,on X. It is well known that A is closed and its domain D(A) equipped with the graph norm x A = x + Ax becomes a Banach space,which we shall denote by X A. Theorem 2. Let A be a linear operator on X such that A β I is maximal dissipative with some real number β and I is a Identity operator, i.e A satisfies with the rang condition then ρ(a) (β, ) and A satisfies Re(γ, u) X β u 2 X for all γ Au (4) R(λ 0 I A) = X for some λ 0 > β. (5) (λ I A) X (λ β), λ > β. 2

3 By virtue of theorem (2.), if A is a linear operator on X with a maximal dissipative A β I, β R, a semigroup T(t) is generated by A on the whole space X. We will be interested in the case when [A (B ) ] = B A where A, B are the adjoint operators of A and B. The following perturbation result for linear operator will be helpful in the sequel ([3],[2]). Theorem 2.2 Let X be a Banach space, and let M be the infinitesimal generator of a c 0 - semigroup T(t) on X. If L : X M X M is a continuous linear operator,then M + L is the infinitesimal generator of a c 0 -semigroup on X. For more details a bout perturbation theory one can (see [5]). Consider the identification problem (), (2) and (3) where A and B are densely linear operators in the Hilbert space X, such that there adjoint operator satisfy D(A ) D(B ), f X, u 0 is an initial value and {u, γ} are the unknown functions. We assume Re(A s B s) X β s 2 X, for some S D(A ). (6) λ 0 ρ B (A ), for some λ 0 > β (7) Let T = A (B ), if h T u, then h = A (B ) u = A s and B s = u for some s D(A ), so that (h, u) X = (A s, B s) X On the other hand, for any h X, we have h = (λ 0 B A )s for some s D(A ). If we put u = B s, then and u B (D(A )) = D(T ) h (λ 0 I T )u, 3

4 According to theorem (2.), this proves that T β I = A (B ) β I is maximal dissipative in X. As a consequence, its adjoint [A (B ) ] β I is also maximal dissipative, so that, [A (B ) ] is the generator of a c 0 -semigroup on X. On the other hand, clearly (), (2) and (3) is written in the form du(t) = Mu(t) + γ(t)f(t), 0 t T (8) with the initial condition u(0) = u 0 H (9) and the overdetermination condition (u(t), v) = w(t) (0) Theorem 2.3 Let M and L be densely defined linear operators in the Hilbert space X such that D(L ) D(M ), f X, if (6), (7) are satisfied and M is a bounded linear operator on X with ρ M (L ) ρ M (L) 0. Then, for any u 0 D(L) such that Lu 0 R(M), the inverse problem (), (2) and (3) possesses a unique solution {u, γ}. Proof: with a coefficient operator M = B A. Moreover, we are interested in the case when [A (B ) ] and B A are coincide. If B is a bounded linear operator on X and ρ B (A ) ρ B (A) 0, then [A (B ) ] = B A and so B A is the generator of a c 0 -semigroup on X. Multiplying both sides of (8) by v scalarly in H, we obtain the relation d(u(t), v) = (Mu(t), v) + γ(t)(f(t), v) and using (0),we get then dw(t) = (Mu(t), v) + γ(t)(f(t), v) w (t) = (Mu(t), v) + γ(t)(f(t), v) γ(t) = substituting () in (8),we obtain (f(t), v) (w (t) (Mu(t), v)) () u (t) = Mu(t) + (f(t), v) (w (t) (Mu(t), v))f(t) (2) 4

5 common practice involves the operator L = (Mu(t), v)f(t) (3) (f(t), v) and equation (2) becomes u (t) = (M + L) u(t) + (f(t), v) w (t)f(t) (4) Plain calculation show that L is bounded in X M, L M = sup L M = sup (Mu(t), v)f(t) (f(t), v) sup (f(t), v) Mu v f. Theorem (2.2) now implies that M +L is the infinitesimal generator of a semigroup S(t), t 0. Since u 0 D(M),then the problem (4) with condition (9) has a unique solution u(t) u(t) = S(t)u 0 + t S(t s)w (s)f(s)ds (5) (f(t), v) 0 By () and (5) γ(t) is uniquely determined and the reduced problem (8), (9) and (0) possesses a unique solution (u, γ), see [6],[7],[8],[9]. References [] Awawdeh, F. and Jaradat, H. M., On a class of inverse problem for degenerate differential equations, World Academy of Science, Engineering and Technology 7, 200. [2] Desh, W. and Schappacher, W., On Relatively Perturbations of Liner a c 0 Semigroup, Annali della Scuola Normal Superiore di Pisa-Classe di Scienze, Sér.4, (2) (984) [3] Desh, W. and Schappacher, W., Some Perturbation Results for Analytic Semigroup.Math.Ann.28 (988) [4] Kato, T., Perturbation Theory of Linear Operators, Springer:New York-Berlin- Heidelberg, (976). [5] Lorenzi, A., An Introduction to Identification Problems via Functional Analysis.Inverse and Ill-Posed Problems Series.VSP, Utrecht (200). [6] El-Borai, M.M., Some probability densities and fundamental solution of fractional evolution equations, Chaos, Solitons and Fractals 4, 2002, [7] El-Borai, M.M., On some fractional evolution equations with non local conditions, Intrnational-J.of Pure-and Appl. Math., Vol.24, No.3, 2005,

6 [8] El-Borai, M.M., On the solvability of an inverse fractional abstract Cauchy problem, IJRRAS 4(4), 200. [9] El-Borai, M.M.,El-Nadi, K.E. and El-Akabawy, E.G., On some fractional evolution equations, Article in press, Computers and Mathematics with Applications,

On the solvability of an inverse fractional abstract Cauchy problem

$On the solvability of an inverse fractional abstract Cauchy problem$ On the solvability of an inverse fractional abstract Cauchy problem Mahmoud M. El-borai m ml elborai @ yahoo.com Faculty of Science, Alexandria University, Alexandria, Egypt. Abstract This note is devolved