Mixed Problemwith an Integral Two-Space-Variables Condition for a Class of Hyperbolic Equations

e integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water �ow, and population dynamics. Cannon was the �rst who drew attention to these problems with an integral one-space-variable condition [1], and their importance has been pointed out by Samarskii [2]. e existence and uniqueness of the classical solution of mixed problem combining a Dirichlet and integral condition for the equation of heat demonstrated by cannon [1] using the potential method. Always using the potential method, Kamynin established in [3] the existence and uniqueness of the classical solution of a similar problem with a more general representation. Subsequently, more works related to these problems with an integral one-space-variable have been published, among them, we cite the work of Benouar and Yurchuk [4], Cannon and Van Der Hoek [5, 6], Cannon-Esteva-Van Der Hoek [7], Ionkin [8], Jumarhon and McKee [9], Kartynnik [10], Lin [11], Shi [12] and Yurchuk [13]. In these works, mixed problems related to one-dimensional parabolic equations of second order combining a local condition and an integral condition was discussed. Also, by referring to the articles of Bouziani [14–16] and Bouziani and Benouar [17–19], the authors have studied mixed problems with integral conditions for some partial differential equations, specially hyperbolic equation with integral condition which has been investigated in Bouziani [20]. e present paper is devoted to the study of problems with a boundary integral two-space-variables condition for second-order hyperbolic equation.


Introduction
e integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water �ow, and population dynamics.
Cannon was the �rst who drew attention to these problems with an integral one-space-variable condition [1], and their importance has been pointed out by Samarskii [2].e existence and uniqueness of the classical solution of mixed problem combining a Dirichlet and integral condition for the equation of heat demonstrated by cannon [1] using the potential method.
Always using the potential method, Kamynin established in [3] the existence and uniqueness of the classical solution of a similar problem with a more general representation.
e present paper is devoted to the study of problems with a boundary integral two-space-variables condition for second-order hyperbolic equation.

Setting of the Problem
In the rectangle Ω = (0, 1) × (0, ), with   , we consider the hyperbolic equation: where the coefficient (, ) is a real-valued function belonging to  1 (Ω) such that 0   0 ≤  (, ) ≤  1 ,  (, )  ≤  2 . ( in the rest of the paper, ,   ,  = 1,  , 12, denote strictly positive constants.we adjoin to (1) the initial conditions ℓ 1  =  (, 0) =  () ,   (0, 1) , where  and  are known functions.We will assume that the function  and  satisfy a compatibility conditions with (5), that is, e presence of integral terms in boundary conditions can, in general, greatly complicate the application of standard functional or numerical techniques specially the integral twospace-variables condition.In order to avoid this difficulty, we introduce a technique to transfer this problem to another classically less complicated one which does not contain integral conditions.For that, we establish the following lemma.
Proof.Let (, ) be a solution of (1)-( 5), we prove that So, by integrating (1) with respect to  over (0, ) and (, 1) and taking into account (6) Let now (, ) be a solution of (), then we are bound to prove that So, by integrating (1) with respect to  over (0, ) and taking into account that we obtain and by integrating (1) with respect to  over (, 1) and taking in consideration we obtain By combining the two preceding (15) and (17) and taking into account (7), we get

A Priori Estimate and Its Consequences
In this paper, we prove the existence and the uniqueness of the solution of the problem (1)-( 5) and of the operator equation where  = (ℒ, ℓ 1 , ℓ 2 ) with domain of di�nition  consisting of functions    2 (Ω) such that , ,  2  2   2 (Ω), and  satis�es conditions (3) and ( 4); the operator  is considered from  to , where  is the Banach space consisting of all functions (, ) having a �nite norm and  is the Hilbert space consisting of all elements ℱ = (   for which the norm is �nite.
eorem 2. For any function   , we have the inequality where  is a positive constant independent of .
Proof.Multiplying the (1) by the following : and integrating over Ω  , where Ω  = (0 1 × (0 , (1) by integrating over Ω   = Ω  = (0  × (0 .Consequently, Employing integration by parts in (24), and taking into account the boundary conditions in (, we obtain where By virtue of Lemma 7.1 in [21] and by using it twice, we �nd where where e right-hand side of (49) is independent of ; hence replacing the le-hand side by its upper bound with respect to  from 0 to , we obtain the desired inequality, where  = ( 12 ) 1/2 .Corollary 3. A solution of the problem (1)-( 5) is unique if it exists and depends continuously on ℱ ∈ .

Solvability of the Problem
To show the existence of solutions, we prove that () is dense in  for all  ∈  and for arbitrary ℱ = (  ) ∈ .eorem 4. Suppose the conditions of eorem 2 are satis�ed.en, the problem (1)-( 5) admits a unique solution  =  −1 ℱ.
We return to the proof of eorem 4. We have already noted that it is sufficient to prove that the set () dense in .

Proposition 5 .
Let the conditions of eorem 4 be satis�ed.�ffor  ∈  2 (Ω) and for all  ∈  0 (), we have In terms of the given function  and from the equality (55), we give the function  in terms of  as follows:Integrating by parts the le-hand side of (61) with respect to , and by taking the conditions of the function , we obtain and  satis�es the same conditions of the function .Ω    2   ≤ 0