Most of the known cases of strong gravitational lensing involve multiple imaging of an active galactic nucleus. The properties of lensed active galactic nuclei make them promising systems for astrophysical applications of gravitational lensing. So we present a simple model for strong lensing in the gravitational lensed systems to calculate the age of four lensed galaxies, in the present work we take the freedman models with (k curvature index =0) Euclidian case, and the result show a good agreement with the other models.

Among a variety of approaches introduced in the literature to establish duality theory, Fenchel duality was of great importance in convex analysis and optimization. In this paper we establish some conditions to obtain classical strong Fenchel duality for evenly convex optimization problems defined in infinite dimensional spaces. The objective function of the primal problem is a family of (possible) infinite even convex functions. The strong duality conditions we present are based on the consideration of the epigraphs of the c-conjugate of the dual objective functions and the ε-c-subdifferential of the primal objective functions.

The aim of this article, we define new iterative methods called three-step type in which Jungck resolvent CR-iteration and resolvent Jungck SP-iteration are discussed and study rate convergence and strong convergence in Banach space to reach the fixed point which is differentially solve of nonlinear equations. The studies also expanded around it to find the best solution for nonlinear operator equations in addition to the varying inequalities in Hilbert spaces and Banach spaces, as well as the use of these iterative methods to approximate the difference between algorithms and their images, where we examined the necessary conditions that guarantee the unity and existence of the solid point. Finally, the results show that resolvent CR-iter