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Existence theorems for systems of nonlinear integro-differential equations. (English) Zbl 1185.45010

The solvability of systems of integro-differential equations of the form \[ \begin{cases} c_i (t) =\alpha_ i + \beta_i t + \int_0^t (\int_0^{\tau} (Gc)_i (s)ds) d\tau,\\ (Gc)i (t) = F_{1i} (t, c(t), c'(t)) + \int_0^t k(t - s)F_{2i} (c(s))ds,\end{cases}\tag{\(*\)} \]
under certain conditions is discussed, where \(c(t)=(c_1(t),c_2(t),\dots,c_m(t))\), \(0\leq t\leq T\), \(1\leq i\leq m\), \(F_{1i}:[0,T]\times\mathbb R^{2m}\to\mathbb R\), \(F_{2i}:\mathbb R^m\to\mathbb R\) and \(k:\mathbb R\to\mathbb R\) are given functions and \(\alpha_i\) and \(\beta_i\) are constants. It is shown that equations (\(*\)) are obtained from the consideration of certain initial-boundary value problems for second order partial differential equations. Further, the results are extended to equations which are more general than (\(*\)).

MSC:

45G15 Systems of nonlinear integral equations
45J05 Integro-ordinary differential equations
47G20 Integro-differential operators
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[1] Brezis H.: Analyse Fonctionnelle: Théorie et Applications. Masson, Paris (1983)
[2] Drábek P., Milota J.: Lectures on Nonlinear Analysis. Plzeň, Praha (2004) · Zbl 1078.47001
[3] Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1988)
[4] Le U.V.: The well-posedness of a semilinear wave equation associated with a linear integral equation at the boundary. Mem. Differ. Equ. Math. Phys. 44, 69–88 (2008) · Zbl 1161.35447
[5] Le U.V.: A contracted procedure for the unique solvability of a semilinear wave equation associated with a linear integral equation at the boundary. JP J. Fixed Point Theory Appl. 3(1), 49–61 (2008) · Zbl 1162.35422
[6] Le, U.V., Pascali, E.: A contracted procedure for a semi-linear wave equation associated with a full nonlinear damping source and a linear integral equation at the boundary. Mem. Differential Equations Math. Phys. (in press)
[7] Nguyen L.T., Le U.V., Nguyen T.T.T.: A shock problem involving a linear viscoelastic bar. Nonlinear Anal. 63, 198–224 (2005) · Zbl 1082.35108 · doi:10.1016/j.na.2005.05.007
[8] Rudin W.: Principles of Mathematical Analysis, 2nd edn. McGraw-Hill Book Company, New York (1964) · Zbl 0148.02903
[9] Rudin W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Company, New York (1986) · Zbl 0954.26001
[10] Rudin W.: Functional Analysis. McGraw-Hill Book Company, New York (1973) · Zbl 0253.46001
[11] Raviart P, A., Thomas J.M.: Introduction Àl’analyse Numérique des Equation aux Dérivées Partielles. Masson, Paris (1983)
[12] Roubíček T.: Nonlinear Partial Differential Equations with Applications. Birkhauser Verlag, Boston (2005) · Zbl 1087.35002
[13] Salsa S.: Partial Differential Equations in Action: from Modelling to Theory. Springer, Milano (2008) · Zbl 1146.35001
[14] Teschl, G.: Nonlinear Functional Analysis. Lecture Notes in Math. Vienna Univ., Austria (2001)
[15] Yosida K.: Functional Analysis. Springer Verlag, Berlin (1965) · Zbl 0126.11504
[16] Zeidler E.: Nonlinear Functional Analysis and its Applications, vol. II. Springer-Verlag, Leipzig (1989)
[17] Zheng S.M.: Nonlinear Evolution Equations. CRC Press, Florida (2004) · Zbl 1085.47058
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