On a Nonhomogeneous Timoshenko System with Nonlocal Constraints

Our main concern in this paper is to prove the well posedness of a nonhomogeneous Timoshenko system with two damping terms. The system is supplemented by some initial and nonlocal boundary conditions of integral type. The uniqueness and continuous dependence of the solution on the given data follow from some established a priori bounds, and the proof of the existence of the solution is based on some density arguments.


Introduction
Timoshenko [1] was the first who introduced a model describing the transverse vibration of a beam. More precisely, his research work concerns with the correction for shear of a differential equation for transverse vibrations of prismatic bars. This model was given by a system of two coupled hyperbolic partial differential equations complemented with some boundary conditions.
where L is the length of the beam in its equilibrium configuration. The function u models the transverse displacement of the beam, and v models the rotation angle of its filament. The coefficients ρ 1 ,ρ 2 ,κ, and κ * are, respectively, the density, the polar moment of inertia of a cross section, the shear modulus, and the Young's modulus of elasticity. In [2], the authors considered and proved some exponential decay results for a linear homogeneous Timoshenko system with a memory term of the form where ðx, tÞ ∈ ð0, LÞ × ð0,∞Þ. The same problem (2) was considered in [3] where the authors discussed the decay properties of the semigroup generated by a linear Timoshenko system with fading memory. In paper [4], the authors studied the exponential stability for the following Timoshenko system with two weak dampings.
In [5], the authors investigated the effect of both frictional and viscoelastic dampings. They considered in the domain ð0, LÞ × ð0,∞Þ the following system and proved some exponential and polynomial decay results. For more results concerning Timoshenko systems, we refer the reader to [6][7][8][9][10][11][12][13][14][15]. Motivated by the above systems, we consider a nonlocal initial boundary value problem for a nonhomogeneous Timoshenko system with memory term of type (2), complemented with boundary integral boundary conditions. The study of mixed problems with nonlocal conditions such as integral conditions goes back to the year 1963, when Cannon [16] used the potential method to investigate the existence and uniqueness of the solution of the heat equation subject to the specification of energy (integral constraint). This type of conditions arises mainly when the data cannot be measured directly on the boundary, but only their averages (weighted averages) are known. Due to their importance, physical significance (mean, total flux, total energy, etc.), and numerous applications in different fields of science and engineering, several authors extensively studied this type of problems, and we can cite, for example, [17][18][19][20][21][22][23][24]. Some recent new results on this direction were obtained (see [25,26]). In this work, a functional analysis method based on some a priori bounds and on the density of the range of the unbounded operator corresponding to the abstract formulation of the given problem is used to prove the well posedness of the posed problem. This work can be considered as a contribution to the development of the energy inequality method used to prove the well posedness of mixed problems with nonlocal conditions such as integral boundary conditions (see, for example, [17,18,[27][28][29][30][31]).

Formulation of the Problem and Function Spaces
In the bounded domain Q T = ð0, LÞ × ð0, TÞ, we consider the initial boundary value problem for a nonhomogeneous Timoshenko system with a viscoelastic term of the form ρ 1 , ρ 2 , κ 1 , and κ 2 are positive constants, f , g, φ, ψ, F, and G are given functions, and h :ℝ + → ℝ + is a twice differentiable function such that The convolution term represents the memory effect with a real valued function h of class C 2 : System (5) is supplemented with the initial conditions the boundary integral conditions This system of coupled hyperbolic equations represents a Timoshenko model for a thick beam of length L,where u is the transverse displacement of the beam and v is the rotation angle of the filament of the beam. The coefficients ρ 1 , ρ 2 , κ 1 , and κ 2 are, respectively, the density, the polar moment of inertia of a cross section, the shear modulus, and the Young modulus of elasticity. The integral conditions represent the averages (weighted averages) of the total transverse displacement of the beam and the rotation angle of the filament of the beam.
Our aim is to study the well posedness of the solution of problems (5), (9), and (10). That is, on the basis of some a priori bounds and on the density of the range of the operator generated by the problem under consideration, we prove the existence, uniqueness, and continuous dependence of the solution on the given data of problems (5), (9), and (10). We now introduce some function spaces needed throughout the sequel. Let L 2 ðQ T Þ be the Hilbert space of square integrable functions on Q T = ð0, 1Þ × ð0, TÞ,T < ∞, with scalar product and norm, respectively.
We also use the space L 2 ðð0, 1ÞÞ on the interval ð0, 1Þ, whose definition is analogous to the space on Q: Let B 1 2 ð0, L Þ be the space obtained by completion of the space C 0 ð0, LÞ of real continuous functions with compact support in the interval ð0, LÞ with respect to the inner product where I x θ = Ð x 0 θðζÞdζ for every fixed x ∈ ð0, LÞ: The associated norm is kθk 2 and Cð J ; B 1 2 ð0, LÞÞ the set of functions θð:,tÞ: To obtain a priori estimates for the solution, we write down our problems (5), (9), and (10) in its operator form: AU = H with U = ðu, vÞ, AU = ðL 1 ðu, vÞ, L 2 ðu, vÞÞ, and H = ðH 1 , H 2 Þ, where The operator A is an unbounded operator of domain of definition DðAÞ consisting of elements ðu, vÞ ∈ ðL 2 ð J ; L 2 ð0, LÞÞ 2 such that u x , v x , u t , v t , u tt , v tt , u xx , v xx belong to L 2 ð J ; L 2 ð0, LÞÞ and verify initial and boundary conditions (9) and (10). The operator A is acting from the Banach space B into the Hilbert space E, where B is the Banach space obtained by completing DðAÞ with respect to the norm and E = ½L 2 ðQ T Þ × ðL 2 ð0, LÞÞ 2 × ½L 2 ðQ T Þ × ðL 2 ð0, LÞÞ 2 is the Hilbert space consisting of vector-valued functions H = ðfF, φ, ψg, fG, f , ggÞ for which the norm

A Priori Estimate and Its Consequences
In this section, we establish an energy inequality from which we deduce the uniqueness and continuous dependence of solution of problems (5), (9), and (10) on the given data.

Theorem 1.
For any function U = ðu, vÞ ∈ DðAÞ, the following a priori estimate holds where C =De DT with D is a positive constant independent of U ≪ ðu, vÞ given by equation (41) below.
Proof. Define the integrodifferential operators and consider the identity where Q τ = ð0, LÞ × ð0, τÞ: The standard integration by parts of each term in (21) and conditions (9) and (10) give

Journal of Function Spaces
and in the same manner, we have Substituting equalities (22)- (30) into (21), we obtain By using the Cauchy ε -inequality, the last six terms in the right-hand side of (31) can be estimated as follows: Journal of Function Spaces If we let we then infer from (31)-(37) that By discarding the last three terms on the left-hand side of (39) and using the poincare ′ type inequality [27], we have ∥u :,τ ð Þ∥ 2 where 5

Journal of Function Spaces
Application of Gronwall's lemma (see [28]) to (40) gives The right-hand side of inequality (42) does not depend on τ. By taking the supremum with respect to τ over ½0, T, we get Then, the a priori estimate (19) follows with C = De DT . At the moment, we do not have any information about the range RðAÞ of the operator A except that RðAÞ ⊂ E, and we must extend A so that inequality (43) holds for the extension and its range is the whole space E. In this regard, we prove the following.

Proposition 2. The unbounded operator A : B → E admits a closure
A with domain of definition Dð AÞ: Proof. Let U n = ðu n , v n Þ ∈ DðAÞ be a sequence such that where L 1 u n , v n ð Þ= L 1 u n , v n ð Þ, ℓ 1 u n , ℓ 2 u n f g , Then, we must show that H 1 = f0g and H 2 = f0g: That is, F = 0,φ = 0,ψ = 0,G = 0,f = 0, and g = 0: Equality ((44)) implies that where D′ðQ T Þ is the space of distributions on Q T : By the continuity of derivation of in D ′ ðQ T Þ × D ′ ðQ T Þ: Then, from (45) it follows that in L 2 ðQ T Þ × L 2 ðQ T Þ. Therefore, in D′ðQ T Þ × D′ðQ T Þ. By virtue of the uniqueness of the limit in D′ðQ T Þ, the identities (48) and (50) lead to F = 0, and G = 0: Similarly, we have from (45) We observe from (44) and the obvious inequalities We conclude from (51) and (53) and the uniqueness of the limit in L 2 ð0, LÞ that φ = 0, and f = 0: In the same manner, we show that ψ = 0 and g = 0: The energy inequality (19) can be extended to The previous a priori bound shows that the operator A is injective and that A −1 is continuous from the range Rð AÞ onto B from which we assert that if a strong solution of problems (5), (9), and (10) exists, it is unique and depends continuously on the initial data ðφ, ψÞ, ðf , gÞ and the free terms F and G:

Solvability of the Posed Problem
Here is the main result of the paper.
Moreover, the solution U = ðu, vÞ and its time derivative Proof. It follows from Corollary 4 that in order to prove the existence of the strongly generalized solution of problems (5), (9), and (10), it is sufficient to show that the range RðA Þof the operator A is everywhere dense in the space E; that is, the operator A is injective. To this end, we first prove the density in the following special case.
Proof. Since relation (57) holds for any element of D 0 ðAÞ, we take an element U = ðu, vÞ with special form given by ð58Þ and consider the system It follows from the above relations that Lemma 7. The function W = ðω 1 , ω 2 Þ, defined by (60), belongs to ðL 2 ðQ T ÞÞ 2 : Proof (of Lemma). We use the t − averaging operator ρ ε introduced in [32]. By applying the operators ρ ε and ∂/∂t to the first equation in (59), we obtain then ∂ ∂t

Journal of Function Spaces
Using the t − averaging operator ρ ε properties, we infer from the above inequality that Since ρ ε u → u as ε → 0 in L 2 ðQ T Þ and the norm of ∂/∂t ðI 2 x ðu tt ÞÞ in L 2 ðQ T Þ is bounded, we conclude that ω 1 ∈ L 2 ð Q T Þ: Similarly, applying ∂/∂tρ ε to the second equation in (59), we conclude that ω 2 ∈ L 2 ðQ T Þ: Consequently, W = ðω 1 , ω 2 Þ ∈ ðL 2 ðQ T ÞÞ 2 : We now continue the proof of Theorem 6. Replacing the functions ω 1 and ω 2 given by (60) in (57), we obtain Invoking the boundary integral conditions and carrying out appropriate integrations by parts of each term, we have By using the Cauchy ε -inequality, we estimate each term of the right-hand side of the previous relations to get By combining equality (74) and inequalities (75)-(79) and taking we obtain By discarding the terms κ 2 /2∥I x v t ðx, TÞ∥ 2 L 2 ð0,LÞ , ∥vðx, TÞ ∥ 2 L 2 ð0,LÞ from the left-hand side of (81) and using the Frederick's inequality for the norm of v obtained from the norm of v t , it follows that ∥I x u tt :,s ð Þ∥ 2 where and M * is the Frederick's constant. Inequality (82) is important and fundamental in the proof; to use it, we introduce the new functions ξ, μ defined by Then, μðx, sÞ = v t ðx, TÞ, v t ðx, tÞ = μðx, sÞ − μðx, tÞ and ξð x, sÞ = u t ðx, TÞ,u t ðx, tÞ = ξðx, sÞ − ξðx, tÞ, and we have Consequently, inequality (82) reduces to ∥I x u tt :,s ð Þ∥ 2 Choose s 0 ≥ 0 such that s ∈ ½T − s 0 , T: and 2C * ðT − s 0 Þ = 1l2; then, inequality (86) implies It then follows from inequality (90) that ΛðTÞ = 0, and hence, W = ðω 1 , ω 2 Þ = 0 almost everywhere in Q T−s 0 . Proceeding in this way step by step along a rectangle of side s 0 , we prove that W ≡ 0 almost everywhere in Q T .
We now consider the general case for density.

Conclusion
In this article, we proved the well posedness of a nonhomogeneous Timoshenko system with a viscoelastic damping term. The coupled two hyperbolic equations were associated with initial conditions and nonlocal boundary conditions. The proofs of the results are mainly based on some energy and a priori estimates and on some density arguments. The method uses functional analysis tools such as operator theory and density arguments. It is found that the method is efficient and powerful for solving initial boundary value problems with nonlocal constraints. The a priori estimate for the solution can be provided by constructing suitable multiplicators and from which it is also possible to establish the solvability of the stated problem. We note, here, that no previous works were done for Timoshenko systems with nonlocal conditions of integral type.

Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

Conflicts of Interest
The authors declare that they have no competing interests.