EXPLICIT SOLUTIONS FOR NON HOMOGENEOUS STURM MOUVILLE OPERATOR PROBLEMS

EXPLICIT SOLUTIONS FOR NON HOMOGENEOUS STURM MOUVILLE OPERATOR PROBLEMS


. Introduction
Throught this paper H denotes a complex Hilbert space and L(H) denotes the algebra of all bounded linear operators on H.If T lies in L(H), its spectrum u(T) is the set of all complex numbers z such zI -T is not invertible in L(H).
Second order operator differential equations with constant operator coefficients appear in the theory of oscillatory and vibrating systems, [6,10,12] .Sturm-Liouville operator problems have been studied by several authors and with several techniques ( [13,14,15,16]) .For the scalar case these problems are completely studied [1,7] .In a recent paper [9] we studied the homogeneous Sturm-Liouville operator problem.X (2) (t) -~XQX(t) = 0 E,X(o) + E2X(')(0) = 0 F,X(a) + F2X(1)(a) = 0 0 < t < a = 1, 2, are in L(H).where A is a complex parameter and X(t) , Q, E¡, Fi, for i The method proposed in [9] is based on the existence of square roots for the operator AQ.The existence of square roots of matrices is treated in [3] and for the infinite-dimensional case a recent paper [2] studies this problem.In [4] and [8] methods for obtaining solutions of more general equations of polynomial type are given.
L .JÓDAR By differentiation in (2 .9)and considering (2 .8)one gets Thus we have and X(t) given by (2.3) is a solution of (2.2) if C(t) and D(t) satisfy (2.5).
Considering the equality by integration in (2.5) and taking into account (2.6) one gets From (2 .3)and (2.9) the operators C(0) and D(0) must verify (2.12) Taking into account (2.11) and solving (2.12) we have Hence, the result is proved .Theorem 1 of [9] gives a sufficient condition for the existence of one unique solution of problem (1 .1), the trivial one X(t) = 0 for all t E [0, a], when \ :~0 and Q is an invertible operator such that Next result shows that the same condition ensures the existence of only one solutionl for the non-homogeneous problem (1.2).Also a sufficient condition for the existence of solutions for (1 .2) and explicit expressions of them in terms of data are given.Theorem 2. Let Q an invertible operator such that condition (2 .13) is satisfied and le¡ Xo = exp(log«(AQ)/2) .Let us consider the operators P, Q and Y defined by ¡he expressions 1 P = -(_) j exp(sXo)X 0 1F(s)ds ; Q= ( 1 ) 2) has only one solution given by the operator function X (t) defined by (2.3) where C(t), D(t) are determined by (2.4) and Proof From theorem 1, the general solution of the operator differential equation (2 .1) is given by (2.3)-(2 .4).Note that from the proof of th .1, one gets (2 .18)X (1 )(t) = Xo exp(tXo )C(t) -Xo exp(-tXo )D(t) Thus, we have the following relationship .between Co = X(0), C1 = X(1)(0) and the operators C(0), D(0), or In order to find solutions of (1 .2),we are going to impose to the operators C(0), D(0), that X(t) = exp(tXo )C(t)+exp(-tXo)D(t), satisfies the boundary value conditions of (1 .2) .From th .1, we have Remark 1 .Note that given the operators C(0), D(0) such that X(t) = exp(tXo)C(t) + exp(-tXo)D(t) satisfies (1.2), the explicit expression of the operator functions C(t) and D(t) are given by (2.4), and from (2.19), it is equivalent to solve the Cauchy problem (2.2) with the initial conditions Co = C(0) + D(0) and Cl = Xo (C(0) -D(0)) .In order to compute the solution of problem (1 .2), it is necessary to solve (2.16) .For the finite dimensional case it is an easy matter ; for the infinite-dimensional case, and under the invertibility hypothesis of S, an explicit expression of S-1 is given in Lemma 1 of [9] .
Theorem 2 provides a sufficient condition for the existence of only one solution of problem (1 .2) in terms of the invertibility of the operator matrix S given by (2.14).In order to obtain a more concrete condition, in terms of data and a square root of AQ, the following corollary is an easy consequence of the, above theorem 2 and lemma 1 of [9] .Next example provides a lot of cases where the uniqueness property and an explicit expression of the solution of problem (1 .2) are available.

. The case \ = 0
We begin this section with an analogous result to theorem 1 of section 2, corresponding to the case \ = 0, It may be considered as a variation of the parameters method for the operator differential equation X(2)(t) = F(t).
Next result concems with the boundary value problem (3.8) Hence parts (i) and (ii) are established.
Next corollary provides sufiicient conditions in terms of data, in order to obtain the uniqueness of solution for Problem (3.8), as well as an explicit expression of the solution.Under this hypothesis, ¡he unique solution is given by where C(0) and D(0) are given by (3.18) (afl + F2 ) -1 F l W-1 E 2(F1 + aF2)-1 + IR Proof: It is an easy consequence of Lemma 1 of [9] and of the previous th.5 (1957) .

Corollary 6 .
Let us consider Problem(3 .8)and let R be the operator in L(H) deftined by (3 .9)(i) If El is invertible in L(H), then Problem (3.8) has only one solution, if the operator V = (aFl + F2) -Fl Ei 1 E2 is invertible in L(H), Under this hypothesis the unique solution of ¡he problem is given by where (3.17)C(0) = -El 1 E 2 V-1 R D(0) = V-' R(ii) If the operator aFl + F2 is invertible, then Problem (3.8) has only one solution if the operator W = El -E2(aFl + F2)-1 F1 is invertible in L(H) .