Journal of Applied Mathematics and Computing

In this paper, we first introduce the concepts of dominating bipolar fuzzy graphs and double dominating bipolar fuzzy graphs. Then we present the notions of energy of dominating bipolar fuzzy graphs and the energy of double dominating bipolar fuzzy graphs. We discuss some properties and bounds for the energy of dominating bipolar fuzzy graphs and energy of double dominating bipolar fuzzy graphs.

Our contribution is the development of novel representations and investigations of main properties of the MPCEP inverse. Precisely, we present representations of the MPCEP inverse which involve appropriate Moore–Penrose inverses, projections and full-rank decompositions, as well as limit and integral representations. Determinantal representations for the MPCEP inverse are also established. We study perturbation formulae with perturbation bounds of MPCEP inverse. An equivalent condition for the continuity of the MPCEP inverse is proposed. The MPCEP inverse of a suitable upper block triangular matrix is given. The successive matrix squaring algorithm and splitting method for computing the MPCEP inverse are presented. Some appropriate constrained systems of linear equations are solved applying the MPCEP inverse.

A new smoothing function similar with the well known Fischer-Burmeister function is given. Based on this new function, a smoothing-type Newton method is proposed for solving second-order cone programming. At each iteration, the proposed algorithm solves only one system of linear equations and performs only one line search. This algorithm can start from an arbitrary point and it is Q-quadratically convergent under a mild assumption. Preliminary numerical results demonstrate the effectiveness of the method.

In this paper, we propose and investigate a predator–prey model with harvesting and reserve area for prey in the presence of toxicity. We research the boundedness of the solutions and the existence of the equilibria of this system. By analyzing the characteristic equations, the local asymptotically stability of feasible equilibria is discussed. By use of Lyapunov function method, we also obtain some sufficient conditions for globally stability of these equilibria. The optimal harvesting policy are discussed by using the Pantryagin’s maximum principle. Finally, numerical simulations are carried to verify the theoretical conclusions.

Give a graph G, we color its all edges. If any two adjacent edges gets the different colors, then we call this color a proper edge coloring of G. Give a proper edge coloring of G, if only two colors alternately appear on a cycle, then the cycle is called bichromatic. Acyclic edge coloring of a graph G means that there are no bichromatic cycles in G. The acyclic chromatic index of a graph G is the minimum number k such that G has an acyclic edge coloring using k colors. Denoted $${\chi ^{'}_a}(G)$$ as the acyclic chromatic index of G. A planar graph is a graph that can be embedded in the plane in such a way that no two edges intersect geometrically except at a vertex to which they are both incident. In this paper, we use the discharging method to prove that $${\chi ^{'}_a}(G)\le \varDelta (G)+ 2$$ if G is a planar graph and there is an integer $$k_v \in \{3, 4, 5\}$$ such that v is not contained in any $$k_v$$ -cycle for every vertex v.

In this paper we establish some new nonlinear Gronwall-Bellman-Type integral inequalities with two variables, which include a non-constant term outside the integrals. These inequalities generalize the results in Chen et al. (J. Inequal. Appl. 2009:258569, 15 p., 2009). We apply our result to a boundary value problem of a partial differential equation for boundedness, uniqueness and continuous dependence.

A modified BFGS algorithm for solving the unconstrained optimization, whose Hessian matrix at the minimum point of the convex function is of rank defects, is presented in this paper.The main idea of the algorithm is first to add a modified term to the convex function for obtain an equivalent model, then simply the model to get the modified BFGS algorithm. The superlinear convergence property of the algorithm is proved in this paper. To compared with the Tensor algorithms presented by R. B. Schnabel (seing [4],[5]), this method is more efficient for solving singular unconstrained optimization in computing amount and complication.

The temporal discretisation of a moderate semilinear parabolic problem in an abstract setting by the two-step backward differentiation formula with variable step sizes is analysed. Stability as well as optimal smooth data error estimates are derived if the ratios of adjacent step sizes are bounded from above by 1.91.

By using exponential dichotomy and the Banach fixed-point theory and combining Yang inequality, some sufficient conditions are derived ensuring existence and global exponential stability of almost periodic solution for CNNs with variable coefficients and delays. Without assuming the boundedness of signal function, these results obtained here can be expected to have highly important significance in designs and applications of the networks. We extend and improve previously known results.