چكيده به لاتين
In this thesis, we first apply Edelstein fixed point theorem to prove the existence of solutions for the linear and nonlinear Fredholm integral equations of the second kind and then we give some suitable conditions under which uniformly Lipschitzian mappings have a fixed point. We investigate the solvability of Fredholm integral equations of second kind by the obtained results. Also, we present a generalization of Darbo's fixed point theorem and using it we prove an epsilon- fixed point result. Furthermore, we present a generalization of Darbo and Sadovski'i fixed point theorem in uniformly convex Banach spaces.
We also consider the nonlinear eigenvalue problem L u = lambda f ( x , u ) , posed in a smooth bounded domain Omega subseteq R with Dirichlet boundary condition, where L is a uniformly elliptic second-order linear differential operator, and f is a suitable function. We present some upper and lower bounds for the extremal parameter and extremal solution and apply the results to the explosion problem in a flow. Furthermore, we consider the special case, i.e., the problem - Delta u = lambda f ( x , u ) and give pointwise lower bounds for the super-solutions under some appropriate conditions of f , and apply them to find upper and lower bounds for the extremal parameter and the extremal solution that improve and extend several results in the literature. Finally, we derive a priori bounds for positive super-solutions of problem - Delta-p = lambda rho ( x ) f ( u ) , where p > 1 and Delta-p is the p-Laplacian operator and give sharp upper and lower bounds for the extremal parameter. As a by-product of our results, we obtain a lower bound for the principal eigenvalue of the p-Laplacian.
