Journal of Prime Research in Mathematics

# The nonlinear time-fractional differential equations with integral conditions.

H. Merad $$^{a,*}$$, F. Merghadi $$^b$$, A. Merad $$^c$$

$$^a$$ Laboratory of Mathematics, Informatics and Systems (LAMIS), Larbi Tebessi University – Tebessa, Algeria
$$^b$$ Department of Mathematics and Informatics, Laboratory of Mathematics, Informatics and Systems (LAMIS), Larbi
TebessiUniversity – Tebessa, Algeria.
$$^c$$ Department of Mathematics and Informatics, Laboratory of Dynamical Systems and Control, Larbi Ben M’hidi University of Oum El Bouaghi, Algeria.
Correspondence should be addressed to: merahad@gmail.com

### Abstract

In this paper, we present a nonlinear equation modeling a time-fractional pseudoparabolic problem, involving fractional Caputo derivative where the fractional order is 0 < α < 1. We first started with the associated linear problem, we establish the energy inequalities to obtaine a priori estimate,and demonstrate the density of the operator’s range generated. Accordingly, the existence and uniqueness of the weak solutions are given, then we use the preceding results to handle the nonlinear case via an iterative process.

#### Keywords:

Energy inequality, fractional evolution equations, integral condition, priori estimate.