We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for quasi-linear elliptic operator:where Ω is assumed to be a polygonal bounded domain in R 2 , f ∈ L 2 (Ω), and α is a bounded function which satisfies the strictly monotone assumption. We estimated the actual error in the H 1 -norm by an indicator η which is composed of L 2 -norms of the element residual and the jump residual. The main result is divided into two parts; the upper bound and the lower bound for the error. Both of them are accompanied with the data oscillation and the α-approximation term emerged from nonlinearity. The design of the adaptive finite element algorithm were included accordingly.

In this paper, we prove a common fixed point theorem for multi-valued mappings in complete b-metric spaces. The conditions for existence of a common fixed point had been investigated. The main result can be regarded as a generalization of previous results in complete metric space.

This paper aims to extend the concept of cyclic single-valued mapping to the case of multi-valued mapping in dislocated quasi-b-metric spaces. The existence of fixed point in dislocated quasi-b-metric spaces.

In this paper, we introduce an analytical approximate solution of nonlinear fractional Volterra population growth model based on the Caputo fractional derivative and the Riemann fractional integral of the symmetry order. The residual power series method and Adomain decomposition method are implemented to find an approximate solution of this problem. The convergence analysis of the proposed technique has been proved. A numerical example is given to illustrate the method.

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