Evolution equations in Fréchet spaces

Yıl 2018, Cilt: 1 Sayı: 1, 33 - 38, 27.05.2018

Said Abbas Amaria Arara Mouffak Benchohra Fatima Mesri

https://doi.org/10.33187/jmsm.419917

Öz

This paper deals with the existence of mild solutions for a class of evolution equations. The technique used is a generalization of the classical Darbo fixed point theorem for Fr\'{e}chet spaces associated with the concept of measure of noncompactness.

Anahtar Kelimeler

Densely defined operator, evolution system, fixed point, evolution equation, mild solution

Kaynakça

• [1] S. Abbas, W. Albarakati and M. Benchohra, Successive approximations for functional evolution equations and inclusions, J. Nonlinear Funct. Anal., Vol. 2017 (2017), Article ID 39, pp. 1-13.
• [2] S. Abbas and M. Benchohra, Advanced Functional Evolution Equations and Inclusions, Springer, Cham, 2015.
• [3] N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series, 246. Longman Scientific & Technical, Harlow; John Wiley & Sons, New York, 1991.
• [4] S. Baghli and M. Benchohra, Global uniqueness results for partial functional and neutral functional evolution equations with infinite delay, Differential Integral Equations, 23 (2010), 31–50.
• [5] S. Baghli and M. Benchohra, Multivalued evolution equations with infinite delay in Fr´echet spaces, Electron. J. Qual. Theo. Differ. Equ. 2008, No. 33, 24 pp.
• [6] A. Baliki and M. Benchohra, Global existence and asymptotic behaviour for functional evolution equations, J. Appl. Anal. Comput. 4 (2) (2014), 129–138.
• [7] A. Baliki and M. Benchohra, Global existence and stability for neutral functional evolution equations, Rev. Roumaine Math. Pures Appl. LX (1) (2015), 71-82.
• [8] M. Benchohra and I. Medjadj, Global existence results for second order neutral functional differential equation with state-dependent delay. Comment. Math. Univ. Carolin. 57 (2016), 169-183.
• [9] D. Bothe, Multivalued perturbation of m-accretive differential inclusions, Isr. J. Math. 108 (1998), 109-138.
• [10] S. Dudek, Fixed point theorems in Fr´echet Algebras and Fr´echet spaces and applications to nonlinear integral equations, Appl. Anal. Discrete Math., 11 (2017), 340-357.
• [11] S. Dudek and L. Olszowy, Continuous dependence of the solutions of nonlinear integral quadratic Volterra equation on the parameter, J. Funct. Spaces, V. 2015, Article ID 471235, 9 pages.
• [12] A. Freidman, Partial Differential Equations, Holt, Rinehat and Winston, New York, 1969.
• [13] M. Frigon and A. Granas, R´esultats de type Leray-Schauder pour des contractions sur des espaces de Fr´echet, Ann. Sci. Math. Qu´ebec 22 (2) (1998), 161-168.
• [14] S. Heikkila and V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations, Marcel Dekker Inc., New York, 1994.
• [15] H. M¨onch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4(1980), 985-999.
• [16] L. Olszowy and S. We¸drychowicz, Mild solutions of semilinear evolution equation on an unbounded interval and their applications, Nonlinear Anal. 72 (2010), 2119-2126.
• [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
• [18] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences 119, Springer-Verlag, New York, 1996. [19] K. Yosida, Functional Analysis, 6th edn. Springer-Verlag, Berlin, 1980.