On the well posedness of a mathematical model for a singular nonlinear fractional pseudo-hyperbolic system with nonlocal boundary conditions and frictional dampings

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Introduction
In the bounded domain Q = Ω × (0, T ) = {(x, t) : 0 < x < b, 0 < t < T },we are concerned with the well posedness of a nonlinear fractional system with frictional damping.More precisely, the model problem we have in mind is presented in the form  The functions f, g are L 2 (0, T ; L 2 ρ (Ω) given Lipschitzian functions, that is there exist two positive constants δ 1 , δ 2 such that for all (x, t) ∈ Q.The functions ϕ 1 , ψ 1 , ϕ 2 and ψ 2 are in H 1 ρ (Ω), and z 1 , z 2 are positive constants.The operator C ∂ β 0t denotes the left Caputo fractional derivative, defined in the second section, where 1 < β, γ < 2.
The paper is organized as follows: In Section 2, we introduce the needed function spaces, and state some important inequalities, and fractional calculus relations that will be used in the rest of the sequel.In Section 3, we reformulate the fractional linear system associated to the nonlinear problem (1.1) in its operator form.Then in Section 4, we prove the uniqueness of the solution of the fractional linear system, and we present the consequences of the obtained energy estimate (4.1) of the solution.In Section 5, we show the solvability of the associated linear problem.Finally, in Section 6, on the basis of the results obtained in Sections 4 and 5, and on the use of an iterative process, we prove the existence and uniqueness of the solution of the fractional nonlinear system (1.1).

Preliminaries and functions spaces 2.1 Functional spaces
Let L 2 (0, T ; L 2 ρ (Ω)) be the space consisting of all measurable functions Q : [0, T ] → L 2 ρ (Ω) with scalar product and with the associated finite norm and we denote by L 2 (0, T ; H 1 ρ (Ω)) the space of functions which are square integrable in the Bochner sense, with the inner product and the associated norm is We also introduce the fractional functional space W λ (Q T ) having the inner product and with norm We denote by C(0, T ; L 2 (Ω)) the set of all continuous functions V * (., t) : [0, T ] → L 2 (Ω) with the norm We recall some definitions of fractional derivatives and fractional integral [52,53].Let Γ(•) denote the Gamma function.For any positive integer n where: n − 1 < α < n, the Caputo derivative, and fractional integral of order α are respectively defined by The left Caputo derivative The right Caputo derivative (2.9) and the fractional integral (2.10) Lemma 2.1 [54].Let a nonnegative absolutely continuous function P(t) satisfy the inequality for almost all t ∈ [0, T ], where C is positive and k(t) is an integrable nonnegative function on [0, T ].
Lemma 2.2 [54].For any absolutely continuous function v(t) on [0, T ], the following inequality holds We use the following Gronwall-Bellman lemma Lemma 2.3 [55] Let R(s) be nonnegative and absolutely continuous on [0, T ], and suppose that for almost all s ∈ [0, T ], the function R satisfies the inequality where the functions J(s) and I(s) are summable and nonnegative on [0, T ].Then (2.12) We also use the following inequality [54] the Cauchyε-inequality where a and b are positive numbers.and the Poincare type inequalities [56] J where

Reformulation of the linear problem
We consider a fractional coupled system of the form supplemented by the initial conditions and the Neumann and integral boundary conditions We assume that there exists a solution (u, v) ∈ (C 2,2 (Q)) 2 consisting of the set of functions together with their partial derivatives of order 2 in x and t, which are continuous on Q.The solution of system (3.1)-(3.3)can be regarded as the solution of the operator equation X W = F , where W, X W and F are respectively the pairs and The operator X is considered from a space B into a space H, where B is a Banach space consisting of all functions (u, v) ∈ L 2 (0, T ; L 2 ρ (Ω)) 2 satisfying conditions (3.3) and having the finite norm and 4 is the Hilbert space consisting of vector-valued functions S = ({f, ϕ 1 , ψ 1 }, {g, ϕ 2 , ψ 2 }) with norm Let D(X ), be the domain of definition of the operator X , defined by:

Uniqueness of the solution
In this section, we prove the uniqueness result for the fractional system (3.1)-(3.3), that is we establish an energy inequality for the operator X and we give some of its consequences.
Theorem 4.1 For any (u, v) ∈ D(X ), f, g ∈ L 2 (0, T ; L 2 ρ (Ω)), and ϕ 1 , ψ 1 , ϕ 2 , ψ 2 ∈ H 1 ρ (Ω), the solution of the problem (3.1)-(3.3)verifies the a priori bound where M = Y * * e T Y * * is a positve constant with Proof.The fractional partial differential equations in (3.1), and the following fractional integrodifferential operators x (ξu t ), and Using boundary conditions (3.3), we evaluate the following terms on the LHS of (4.3) as follows . (4.15) In the same fashion, we have the equations (4.4)-(4.15)with β replaced by γ, and u replaced by v. Since 0 < β − 1 < 1, then by using Lemma 2.2, we have ) Combination of (4.2)-(4.18)yields By inserting (4.20)-(4.37)into (4.19), and taking where Replacing t by τ and integrating both sides of (4.38) with respect to τ over [0, t], we obtain Boundary integral conditions allow us to use the Poincre inequalities to get rid of the fourth integral term on the right-hand side of (4.40), and in the mean time, we use Poincare type inequality (2.15), we then have where If we leave only the last two terms on the left-hand side in inequality (4.42), and use the Gronwall-Bellman lemma 2.3 [55], with we obtain . Now by keeping only the fifth and sixth terms on the left-hand side of (4.42), and by taking into account the inequality (4.44), we have we see from (4.45), that where Owing to the inequalities we deduce from inequalities (4.42), (4.44), and (4.48) that where By virtue of poincare inequalities (4.41), and equivalence of norms, the inequality (4.51) takes the form where Now by adding the quantity u 2 to both sides of (4.53), we have Application of Gronwall's Lemma to (4.55) gives the inequality The independence of the right-hand side on t in (4.57), gives where M = Y * * e T Y * * .
It can be proved in a standard way that the operator X : B → H is closable.Let X be its closure.
Proposition 4.1 The operator X : B → H has a closure.
Proof: The proof can be established in a similar way as in [57].These are some consequences of Theorem 4.1.
Corollary 4.1 There exists a positive constant C such that where: C = √ C 7 .The inequality (4.59) means that inequality (4.1) can be extended to strong solutions after passing to limit.
We can deduce from inequality (4.59) that a strong solution of the system (3.1)-(3.3) is unique and depends continuously on F = (F 1 , F 2 ) ∈ H, where F 1 = {f, ϕ 1 , ϕ 2 } and F 2 = {g, ψ 1 , ψ 2 }, and that the image R(X ) of X is closed in H and R(X ) = R(X ).So in order to prove that the system (3.1)-(3.3) has a strong solution for arbitrary (F 1 , F 2 ) ∈ H, it is sufficient to prove that the range of 5 Existence of the solution of the linear system Proposition 5.1 If for some function: , and for all then Y * vanishes a.e in the domain Q.
Proof: We first set where We suppose that the functions p i (x, t) satisfy conditions (3.3) and such that Now by replacing (5.2) and (5.3) in the relation (5.1), we obtain L 2 ρ (Ω) , i = 1, 2, then, using conditions (3.3), and computation of each term of (5.4), gives x (5.12) Combination of (5.5)-(5.12)and (5.4), yields where After integration, we entail from (5.13) that (5.14) If we drop the last four terms on the left-hand side of (5.14), apply Lemma 2.1, and use inequality (2.13) , we have Application of inequality (2.13), reduces (5.15) to where (5.17) We infer from inequalities (5.16) and (5.14) that where If we now discard the last four terms in the left-hand side of (5.18), and apply Lemma 2.1, we get Hence, we deduce that Y * (x, t) = (y * 1 , y * 2 ) = (0, 0) almost everywhere in the domain Q.
for a positive constant C, independent of W .

The nonlinear system
We are now in a position to solve the nonlinear system (1.1).Relying on the results obtained previously, we apply an iterative process to establish the existence and uniqueness of the weak solution of the nonlinear system (1.1).If (u, v) is a solution of system (1.1) and (ψ, φ) is a solution of the homogeneous system The functions F and G are Lipschitzian functions for all (x, t) ∈ Q.According to Theorem 5.1, system (6.1) has a unique solution that depends continuously on (ϕ We must prove that the system (6.2) admits a unique solution.Suppose that w, U, V ∈ C 2 (Q), such that Consider the identity In light of the above assumptions, we obtain Using the symmetry in the system, and inserting equations (6.7)-(6.12)into (6.6),yields We write (6.13) in the form where A (w, U, V ) denotes the left-hand side of (6.13).
Theorem 6.2 If hypotheses (6.3) and (6.4) are satisfied, then the system (6.2) has only one solution.

Conclusion.
A Caputo fractional nonlinear pseudohyperbolic system supplemented by a classical and a nonlocal boundary condition of integral type is investigated.more precisely, in this research work, we search a function u(x, t) verifying (1.1).The associated fractional linear problem is reformulated, and the uniqueness and existence of the strong solutions are proved in a fractional Sobolev space.A priori bound for the solution is obtained from which the uniqueness of the solution follows.By using some density arguments, the solvability of the linear problem is established.To takle the well posedness of the fractional nonlinear problem, we relied on the obtained results for the linear fractional system, by applying a certain iterative process.Our study improves and develops some few existence results for the fractional initial boundary value problems when using the method of functional analysis, the so called energy inequality method.We would like to mention that the application of the used method is a little complicated while dealing with the posed problem in the presence of the nonlinear source terms, the fractional terms, the appearence of the singularity and the nonlocal integral conditions.