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**Uniform exponential stability and applications to bounded solutions of integro-differential equations in Banach spaces.**
*(English)*
Zbl 06979944

Summary: Let \(\mathbb{X}\) be a Banach space. Let \(A\) be the generator of an immediately norm continuous \(C_0\)-semigroup defined on \(\mathbb{X}\). We study the uniform exponential stability of solutions of the Volterra equation
\[
u'(t) = Au(t)+\int _0^t a(t-s)Au(s)\,ds,\quad t\geq 0,\;u(0)=x,
\]
where \(a\) is a suitable kernel and \(x\in \mathbb {X}\). Using a matrix operator, we obtain some spectral conditions on \(A\) that ensure the existence of constants \(C,\omega >0\) such that \(\|u(t)\|\leq Ce^{-\omega t}\|x\|\), for each \(x\in D(A)\) and all \(t\geq 0\). With these results, we prove the existence of a uniformly exponential stable resolvent family to an integro-differential equation with infinite delay. Finally, sufficient conditions are established for the existence and uniqueness of bounded mild solutions to this equation.

### MSC:

47D06 | One-parameter semigroups and linear evolution equations |

45D05 | Volterra integral equations |

45N05 | Abstract integral equations, integral equations in abstract spaces |

47J35 | Nonlinear evolution equations |

45M10 | Stability theory for integral equations |

### Keywords:

exponential stability; \(C_0\)-semigroups; mild solutions; almost periodic; Volterra equations; heat equation with memory
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\textit{Y.-K. Chang} and \textit{R. Ponce}, J. Integral Equations Appl. 30, No. 3, 347--369 (2018; Zbl 06979944)

### References:

[1] | E. Álvarez and C. Lizama, Weighted pseudo almost automorphic mild solutions for two-term fractional order differential equations, Appl. Math. Comp. 271 (2015), 154–167. |

[2] | ——–, Weighted pseudo almost periodic solutions to a class of semilinear integro-differential equations in Banach spaces, Adv. Differ. Eqs. 2015 (2015). · Zbl 1283.65121 |

[3] | D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlin. Anal. 69 (2008), 3692–3705. · Zbl 1166.34033 |

[4] | I. Area, J. Losada and J.J. Nieto, On fractional derivatives and primitives of periodic functions, Abstr. Appl. Anal. 2014, art. ID 392598. · Zbl 1337.26008 |

[5] | I. Area, J. Losada and J.J. Nieto, On quasi-periodicity properties of fractional integrals and fractional derivatives of periodic functions, Int. Transf. Spec. Funct. 27 (2016), 1–16. · Zbl 1337.26008 |

[6] | Y-T. Bian, Y-K. Chang and J. Nieto, Weighted asymptotic behavior of solutions to semilinear integro-differential equations in Banach spaces, Electr. J. Differ. Eqs. 2014, (2014). · Zbl 1304.34128 |

[7] | Y-K. Chang, X-Y. Wei and G.M. N’Guérékata, Some new results on bounded solutions to a semilinear integro-differential equation in Banach spaces, J. Int. Eqs. Appl. 27 (2015), 153–178. · Zbl 1325.45012 |

[8] | Y-K. Chang, R. Zhang and G. N’Guérékata, Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations, Comp. Math. Appl. 64 (2012), 3160–3170. · Zbl 1268.34010 |

[9] | J. Chen, J. Liang and T. Xiao, Stability of solutions to integro-differential equations in Hilbert spaces, Bull. Belgium Math. Soc. 18 (2011), 781–792. · Zbl 1232.34106 |

[10] | J. Chen, T. Xiao and J. Liang, Uniform exponential stability of solutions to abstract Volterra equations, J. Evol. Eqs. 4 (2009), 661–674. · Zbl 1239.34065 |

[11] | B.D. Coleman and M.E. Gurtin, Equipresence and constitutive equation for rigid heat conductors, Z. Angew. Math. Phys. 18 (1967), 199–208. |

[12] | E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discr. Cont. Dynam. Syst. 2007, 277–285, suppl. · Zbl 1163.45306 |

[13] | K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer, New York, 2000. · Zbl 0952.47036 |

[14] | E. Fašangová and J. Prüss, Asymptotic behaviour of a semilinear viscoelastic beam model, Arch. Math. (Basel) 77 (2001), 488–497. |

[15] | M. Gurtin and A Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal. 31 (1968), 113–126. · Zbl 0164.12901 |

[16] | B. Haak, B. Jacob, J. Partington and S. Pott, Admissibility and controllability of diagonal Volterra equations with scalar inputs, J. Diff. Eqs. 246 (2009), 4423–4440. · Zbl 1161.93005 |

[17] | V. Kavitha, S. Abbas and R. Murugesu, Asymptotically almost automorphic solutions of fractional order neutral integro-differential equations, Bull. Malaysian Math. Sci. Soc. 39 (2016), 1075–1088. · Zbl 1347.34114 |

[18] | V. Keyantuo, C. Lizama and M. Warma, Asymptotic behavior of fractional-order semilinear evolution equations, Differ. Int. Eqs. 26 (2013), 757–780. · Zbl 1299.35309 |

[19] | A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations, North-Holland Math. Stud. 204 (2006). · Zbl 1092.45003 |

[20] | C. Lizama and G.M. N’Guérékata, Bounded mild solutions for semilinear integro-differential equations in Banach spaces, Int. Eqs. Oper. Th. 68 (2010), 207–227. · Zbl 1209.45007 |

[21] | C. Lizama and R. Ponce, Bounded solutions to a class of semilinear integro-differential equations in Banach spaces, Nonlin. Anal. 74 (2011), 3397–3406. · Zbl 1220.45012 |

[22] | L. Mahto and S. Abbas, PC-almost automorphic solution of impulsive fractional differential equations, Mediterr. J. Math. 12 (2015), 771–790. · Zbl 1325.34014 |

[23] | M.A. Meyers and K.K. Chawla, Mechanical behavior of materials, Cambridge University Press, Cambridge, 2009. · Zbl 1280.74002 |

[24] | K. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993. · Zbl 0789.26002 |

[25] | J.W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187–304. · Zbl 0227.73011 |

[26] | R. Ponce, Bounded mild solutions to fractional integro-differential equations in Banach spaces, Semigroup Forum 87 (2013), 377–392. · Zbl 1285.34071 |

[27] | ——–, Hölder continuous solutions for fractional differential equations and maximal regularity, J. Differ. Eqs. 255 (2013), 3284–3304. · Zbl 1321.34104 |

[28] | J. Prüss, Evolutionary integral equations and applications, Monogr. Math. 87 (1993). |

[29] | P. You, Characteristic conditions for a \(C_0\)-semigroup with continuity in the uniform operator topology for \(t > 0\) in Hilbert space, Proc. Amer. Math. Soc. 116 (1992), 991–997. · Zbl 0773.47023 |

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