Fixed point theorems in partial metric spaces with an application

Abstract

Matthews [12] introduced a new distance P on a nonempty set X, which he called a partial metric. The purpose of this paper is to present some fixed point results for weakly contractive type mappings in ordered partial metric space. An application to nonlinear fractional boundary value problem is also presented.

On a new four-dimensional model of memristor-based chaotic circuit in the context of nonsingular Atangana–Baleanu–Caputo operators

“…1, the system started to bifurcate at q = 0.95, and then for q ∈ (0.95, 1] the system is chaotic. This observation is verified by the phase portrait of given model (21) depicted in Figs. 2 and 3 for q = 0.95 and q = 0.99, respectively.…”

“…Bifurcation analysis of system (21) related to fractional derivative order q and three of the parameters in the model β 1 , β 2 , and β 3 are performed. In addition, a Matlab pseudo-code for Lyapunov exponents of fractional systems called the Danca algorithm [56] is used to quantify the chaos by calculating Lyapunov exponents for different fractional orders of model (21). The values for parameters are a 1 = 0.3, a 2 = 0.8, ς = 1.4, β 1 = 10, β 2 = 13, and β 3 = 0.1.…”

“…The eigenvalues of the trivial EP of model (21) for β 3 = 0.4, (a 1 = 0.3, a 2 = 0.8, ς = 1.4, β 1 = 10, β 2 = 13) are given as λ 1 = 0, λ 2 = -1.3113 ± 2.7417i, and λ 3 = 2.2226, and hence the trivial EP of system ( 21) is unstable for β 3 = 0.4. Therefore the chance for system (21) in this particular case is to tend to a nontrivial EP as illustrated in Fig. 11.…”

“…The impact of applying different initial conditions on dynamic system ( 21) is addressed using different initial conditions and simulation results in this section. Since chaotic systems are famous to be sensitive to given initial conditions, it is better to investigate dynamic system (21) in terms of different initial conditions. The values of used parameters are q = 0.98, β 1 = 10, β 2 = 13, β 3 = 0.1, a 1 = 0.3, a 2 = 0.8, ς = 1.4.…”

A coincident point principle for two weakly compatible mappings in partial S-metric spaces

“…For example, see [1,2,3,4,5,6,8,11,13,15,17,18,19,20,21]. However, many researchers prove the existence and uniqueness of a coincident point and common fixed point for two self-mappings on different types of metric spaces.…”