Boletín de la Sociedad Matemática Mexicana https://doi.org/10.1007/s40590-019-00266-y ORIGINAL ARTICLE On the existence of a local solution for an integro-differential equation with an integral boundary condition Nouri Boumaza 1 · Billel Gheraibia 2 Received: 18 June 2019 / Accepted: 26 October 2019 © Sociedad Matemática Mexicana 2019 Abstract In this paper, we consider a nonlinear hyperbolic equation with a nonlocal boundary condition. We apply the Faedo–Galerkin’s method to establish the local existence and uniqueness of a weak solution. Keywords Nonlinear hyperbolic equation · Faedo–Galerkin’s method · Integro-differential equation · Integral boundary condition · Local existence Mathematics Subject Classification 35L70 · 35R09 · 35A01 1 Introduction Boundary value problems with integral conditions are an interesting and important class of problems; this is due to the importance of nonlocal conditions appearing in the mathematical modeling of various phenomena of physics, ecology, biology, etc. The starting work on the use of nonlocal boundary conditions has been done by Cannon [4]; the presence of an integral term in boundary conditions can complicate the application of classical methods; therefore, several methods have been proposed for overcoming the difficulties arising from nonlocal conditions as functional methods, approximation methods (see [1,6,7,11]). Pulkina [17] has dealt with a hyperbolic problem with two integral conditions and has established the existence and uniqueness of generalized solutions using the fixed point arguments. The importance of approximation methods B Nouri Boumaza nouri.boumaza@univ-tebessa.dz Billel Gheraibia gheraibia.billel@univ-oeb.dz 1 Department of Mathematics and computer Science, Larbi Tebessi University, Tebessa, Algeria 2 Department of Mathematics and Computer Science, Larbi Ben M’Hidi University, 04000 Oum El-Bouaghi, Algeria