J. Nonl. Evol. Equ. Appl. 2011 n° 6, pp. 91-100, published on October 24, 2011

Fractional Nonlinear Nonlocal Delay Evolution Equations With New Delay Resolvent Family

Amar Debbouche

Department of Mathematics, Faculty of Science, Guelma University, Guelma, Algeria

Received July 18, 2011
Accepted October 5, 2011

Communicated by Gaston N'Guérékata

Abstract.  In this note, we establish the existence and uniqueness of mild and classical solutions of a class of some nonlinear nonlocal delay evolution equations of fractional orders in Banach spaces. By introducing a new approach in the semigroup theory named delay resolvent family, the solution representation is formulated. Sufficient conditions set, fractional calculus and fixed point theorem will be needed. An example that provides the abstract results is also given.
Keywords: Fractional delay evolution equation, nonlocal condition, mild and classical solutions, resolvent family, fixed point theorem.
2010 Mathematics Subject Classification:   34K05, 34G20, 26A33, 35A05.

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