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On the existence of a local solution for an integro-differential equation with an integral boundary condition

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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.

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References

  1. Chen, Baili: Existence of solutions for quasilinear parabolic equations with nonlocal boundary conditions. Electron. J. Diff. Equ. 2011(18), 1–9 (2011)

    MathSciNet  Google Scholar 

  2. Balachandran, K., Park, J.Y.: Existence of a mild solution of a functional integro-differential equation with nonlocal condition. Bull. Korean Math. Soc. 38(1), 175–182 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Belin, S.A.: Existence of solutions for one dimensional wave equations with nonlocal conditions. Electron. J. Diff. Equ. 76, 1–8 (2001)

    MathSciNet  Google Scholar 

  4. Cannon, J.R.: The solution of the heat equation subject to the specification of energy. Quart. Appl. Math. 21, 155–160 (1963)

    Article  MathSciNet  Google Scholar 

  5. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill Book Company, New York (1955)

    MATH  Google Scholar 

  6. Dabas, J., Bahuguna, D.: An integro-differential equation with an integral boundary condition. Math. Comput. Model. 50, 123–131 (2009)

    Article  Google Scholar 

  7. Guezane-Lakoud, A., Dabas, J., Bahuguna, D.: Existence and uniqueness of generalized solutions to a telegraph equation with an integral boundary condition via Galerkin’s method. Int. J. Math. Math. Sci. 2011, 451492 (2011)

    Article  MathSciNet  Google Scholar 

  8. Guezane-Lakoud, A., Chaoui, A.: Rothe method applied to semilinear hyperbolic integro-differential equation with integral conditions. Int. J. Open Problems Complex Anal. 4, 1–14 (2011)

    MathSciNet  Google Scholar 

  9. Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities, vol. 1. Academic Press, New York (1969)

    MATH  Google Scholar 

  10. Lions, J.L.: Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires. Dunod/Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  11. Long, N.T., Ngoc, L.T.P.: On a nonlinear wave equation with boundary conditions of two-point type. J. Math. Anal. Appl. 385(2), 1070–1093 (2012)

    Article  MathSciNet  Google Scholar 

  12. Merad, A., Ozel, C., Kiliçman, A.: On solvability of the integro-differential hyperbolic equation with purely nonlocal conditions. Acta Math. Sci. 35B(2), 1–9 (2015)

    MATH  Google Scholar 

  13. Merad, A., Martín-Vaquero, J.: A Galerkin method for two-dimensional hyperbolic integro-differential equation with purely integral conditions. Appl. Math. Comput. 291, 386–394 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Metaxas, A.C., Meredith, R.J.: Industrial Microwave. Heating, Peter Peregrinus, London (1993)

    Google Scholar 

  15. Ngoc, L.T.P., Hang, L.N.K., Long, N.T.: On a nonlinear wave equation associated with the boundary conditions involving convolution. Nonlinear Anal. TMA 70(11), 3943–3965 (2009)

    Article  MathSciNet  Google Scholar 

  16. Phuong Ngoc, L.T., Triet, N.A., Long, N.T.: existence and exponential decay estimates for an N-dimensional nonlinear wave equation with a nonlocal boundary condition. Boundary Value Problems 2016, 20 (2016)

    Article  MathSciNet  Google Scholar 

  17. Pulkina, L.S.: A nonlocal problem with integral conditions for hyperbolic equations. Electron. J. Diff. Eqn. 45, 1–6 (1999)

    Google Scholar 

  18. Pulkina, L.S.: A nonlocal problem with an integral condition of the first kind for a multidimensional hyperbolic equation. Ross. Akad. Nauk 416(5), 597–599 (2007)

    MathSciNet  Google Scholar 

  19. Truong, L.X., Ngoc, L.T.P., Dinh, A.P.N., Long, N.T.: The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions. NA Ser. B Real World Appl. 11, 1289–1303 (2010)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and computer Science, Larbi Tebessi University, Tebessa, Algeria

    Nouri Boumaza

  2. Department of Mathematics and Computer Science, Larbi Ben M’Hidi University, 04000, Oum El-Bouaghi, Algeria

    Billel Gheraibia

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  1. Nouri Boumaza
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  2. Billel Gheraibia
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Correspondence to Nouri Boumaza.

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Boumaza, N., Gheraibia, B. On the existence of a local solution for an integro-differential equation with an integral boundary condition. Bol. Soc. Mat. Mex. 26, 521–534 (2020). https://doi.org/10.1007/s40590-019-00266-y

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  • DOI: https://doi.org/10.1007/s40590-019-00266-y

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