Summary of Research.
Our research has focused on boundary value problems for nonlinear ordinary differential equations and semi-linear partial differential equations. We deal with continuous and discontinuous nonlinearities. Our analysis is based on topological degree, topological transversality theory, upper and lower solutions methods, fixed points theory and variational methods.
1. Past and Recent Research.
As a Professor at the University of Tlemcen, I initiated the graduate programs in Mathematics starting with the Academic Year 1986-1987 (with options in Nonlinear Functional Analysis and Applications). As a result twenty-one of my students successfully graduated with a Master s degree and four graduated with a Ph.D degree.These four Doctors are actually very active in research.
a. Multipoint boundary value problems
We have made major contributions to the study of multipoint boundary value problems for
second order differential equations.We have dealt with the multiparameter bifurcation problems. To prove our results we have used several techniques from nonlinear analysis: bifurcation theory for Fredholm maps with negative index, topological degree, upper and lower solutions method, topological transversality theory.We have succeeded in generalizing several known results of Henderson, O Regan , Gupta, Ntouyas and Tsamatos. Our results, also, have implications to some results in chapter six of the book by Chow and Hale. We have, also, obtained extensions of the results of Erbe and Wang on the existence of positive solutions.
b. Discontinuous differential equations and free boundary problems
With my graduate students Bouguima and Lakmeche we have obtained several interesting results on the existence of solutions of differential equations with discontinuous right-hand sides. As a byproduct we were led to study free boundary problems for semi-linear elliptic partial differential equations. Our results generalize some of the work of C.A. Stuart.
c. Impulsive differential equations
This is a new area of investigation initiated quite recently, mainly by D.Bainov and co-workers.
With my graduate students Benchohra and Lakmeche , we have made several contributions to the study of initial and boundary value problems for such equations both in the scalar case and in the abstract case (in a Banach space setting). Moreover, we have been successfull in analyzing multivalued differential equations. Our results generalize in several directions, those of Bainov et al., Erbe, Liu, Frigon and O Regan.
d. Periodic boundary value problems
Two techniques were used to analyze second order periodic boundary value problems, method of truncature and the topological transversality theory. In the first case we obtained extensions of many known results of Nieto, Rachunkova; and in the the second case, we were able to adapt an integral monotonicity condition due to O Regan to the problem under consideration, which allowed us to use the topological transversality theorem. Also, with my graduate student N.Daoudi Merzagui , we considered nonautonomous periodic boundary value problems by variational methods. We obtained extensions of results of Ahmad, Zanolin, Willem and others.
2. Current Research
I am studying nonlocal initial value and boundary problems for differential equations, differential inclusions, evolution equations, both in the scalar case and the abstract case. In the latter case, the techniques of semi-group theory of linear operators play an important role in our analysis.
3. Future Research
I intend to consider Optimal Control Problems, especially the controllability of impulsive dynamical systems in Banach spaces, and the application of set-valued analysis to such problems.