Contribution À L’étude De Quelques Problèmes Fractionnaires Àvec Des Conditions Non Locales

Résumé: In this work we are interested in nonlocal problems for hyperbolic integro- differential equations, with boundary conditions of the integral type. We have developed the method of energy inequalities for very complex problems, or, we have applied the method for a fractional problem (in the Caputo sense) two-dimensional, the conditions of the problem are integral (non-local) and classical conditions (Neumann and Dirichlet), We have studied all the above problems in terms of existence and uniqueness of the solution in each case. The thesis begins with an introduction which presents the history of the non-classical problems studied, the importance and the purpose of the topic. Chapter II is devoted to reminders of some fundamental preliminary concepts and basic tools in this work concerning operators, density in Hilbert spaces, functional spaces, fractional calculus, and certain technical lemmas. Chapter III is as follows : we give the statement of the fractional problem and the necessary function spaces. Then, we prove an a priori estimate from which we deduce the uniqueness of a strong solution of the problem posed. Also, we establish the existence of the solution of our problem, by proving that the closure of the value of the operator L generated by the problem, is dense in the space of Hilbert Y. The method used in classical problems (the a priori estimation), have been developed at the fractional level and with success. (This is the subject of our article). We establish the existence and the uniqueness of the strong solution for the associated linear problem. On the basis of the results obtained for the linear problem, we apply an iterative process in order to establish the good posed of the nonlinear problem (in chapter IV). A general conclusion. A comparison with other studies and methods car- ried out in the same field and the difficulties, a perspective is brought at the end of the thesis opening the way to future research and applications.

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