ON A CLASS OF NONCLASSICAL HYPERBOLIC EQUATIONS WITH NONLOCAL CONDITIONS

This paper proves the existence, uniqueness and continuous dependence of a solution of a class of nonclassical hyperbolic equations with nonlocal boundary and initial conditions. Results are obtained by using a functional analysis method based on an a priori estimate and on the density of the range of the linear operator corresponding to the abstract formulation of the considered problem. Nonclassical Hyperbolic Equations, Nonlocal Boundary Con

The paper is organized as follows.In Section 2, we state three assumptions on the functions involved in problem (1.1)-(1.4)and we reduce the posed problem to one with homogeneous boundary conditions.In addition, we present an abstract formulation of the considered problem and we define the strong solution of the problem.In Section 3, the uniqueness and continuous dependence of the solution are established.Finally, in Section 4 we prove the existence of the strong solution and offer remarks on its generalizations.

Preliminaries
First, we begin with the following assumptions on function : + Assumption A1: In Assumptions A1-A2, we assume that are positive constants.We also, -Ð3 oe !ß á ß *Ñ 3 assume that the functions and satisfy the following: F G Functions and satisfy the compatibility conditions: , respectively and Let us reduce problem (1.1)-(1.3)and (1.4 [respectively (1.4 ] with nonhomogeneous +Ñ ,Ñ boundary conditions (1.3), (1.4 [respectively (1.4 ] to an equivalent problem with +Ñ ,Ñ homogeneous conditions by assuming that we are able to find a function ?oe ?ÐBß > ß > Ñ " # defined as follows: where ; respectively Consequently, we have to find a function , which is a solution of the where We recall that is a scalar product on for which is not complete. We denote by a completion of for a scalar product defined by (2.5).The having finite norms and satisfying conditions (2.3), (2.4 [(2.4 ), respectively], and is a Hilbert space of +Ñ , J vector-valued functions with the finite norms Ð0 ß ß Ñ : < Finally, we give a definition of a strong solution.Let be the closure of the operator P P  with the domain .

Uniqueness and Continuous Dependence of the Solution
In this section we will prove an a priori estimate.The uniqueness and continuous dependence of the solution upon the data presented here are direct consequences of the a priori estimate.
satisfies the following a priori estimate: where is a constant independent of .- !?Proof: Taking the scalar product, in , of equation (2.1) and , we obtain , where and , we have It is easy to see that From (3.3)-(3.5)and from Assumption A1, we get where To finish the proof of Theorem 1, we will need the following result: Lemma 2: If and are nonnegative functions on the rectangle , is integrable on , and is nondecreasing in with respect to each of its variables separately, then " # The proof of the above lemma is analogous to the proof of Lemma 1 in Bouziani [6].
Continuing the proof of Theorem 1, we apply Lemma 2 to (3.6).For this purpose, we denote the left-hand side of (3.6) by , and the sum of three first integrals on the 0 Ð ß Ñ " " # 7 7 right-hand side of (3.6) by .This procedure eliminates the last integral of the right-0 Ð ß Ñ # " # 7 7 hand side of (3.6) and yields: According to Lemma 1, we bound below the first and the third terms on the left-hand side of (3.7) and we bound above the third and the fifth terms on the right-hand side of (3.7).Consequently, we obtain Since the right-hand side of inequality (3.8) does not depend on and , then, in the left-7 7 " # hand side of (3.8) we can take the supremum with respect to from to .P F J The proof of the above proposition is similar to the proof of Proposition 1 in Bouziani [4].Theorem 1 can be extended to cover strong solutions by passing to the limit.Corollary 1: Under the assumptions of Theorem , there exists a positive constant such " that where does not depend on .

Existence of the Solution
In this section we concentrate on the existence of the strong solution of problem (2.1)-(2.3),Ð#Þ%+Ñ Ð#Þ%,ÑÓ VÐPÑ [respectively, . The main idea is to demonstrate that the range is dense in J .
Theorem 2: Suppose that Assumptions A and A are satisfied.Then, for arbitrary To prove that the function , defined by (4.4), belongs to , we apply the following result: Lemma 3: If the assumptions of Proposition are satisfied, then the function , # ?defined by , has derivatives of the form and belonging to . See Lemma 2 in Bouziani [3].Proof: Relation (4.4) implies that (4.1) can be written in the form Integrating by parts, each term of (4.5), we obtain According to Assumptions A1 and A2, we get Observe that Adding inequalities (4.10)-(4.15)and applying (4.5), we get where .

Proposition 1 :
oe "ß#Ñ Therefore, we obtain (3.1) with .The proof of Theorem 1 is complete.The operator from to has a closure.