We First introduce a three-step iterative algorithm for approximating the fixed points of the hemicontractive mappings in Banach spaces. Consequently, we prove the strong convergence of the proposed algorithm under some assumptions. Since three-step iterations include Ishikawa iterations as special cases, our result continue to hold for these problems. Our main results can be viewed as an important refinement of the previously known results.

A Korpelevich-like algorithm has been introduced for solving a generalized variational inequality. It is shown that the presented algorithm converges strongly to a special solution of the generalized variational inequality. MSC: 47H05; 47J25

Based on an epidemic model which Manvendra and Vinay [Mathematical model to simulate infections disease, VSRD-TNTJ3(2) (2012) 60–68] have proposed, we consider the dynamics and therapeutic strategy of a SEIS epidemic model with latent patients and active patients. First, the basic reproduction number is established by applying the method of the next generation matrix. By means of appropriate Lyapunov functions, it is proven that while the basic reproduction number 0 < R0 < 1, the disease-free equilibrium is globally asymptotically stable and the disease eliminates; and if the basic reproduction number R0 > 1, the endemic equilibrium is globally asymptotically stable and therefore the disease becomes endemic. Numerical investigations of their basin of attraction indicate that the locally stable equilibria are global attractors. Second, we consider the impact of treatment on epidemic disease and analytically determine the most effective therapeutic strategy. We conclude that the most effective therapeutic strategy consists of treating both the exposed and the infectious, while treating only the exposed is the least effective therapeutic strategy. Finally, numerical simulations are given to illustrate the effectiveness of the proposed results.

Let {tn}⊂(0,1) be such that tn → 1 as n → ∞, let α and β be two positive numbers such that α + β = 1, and let f be a contraction. If T be a continuous asymptotically pseudocontractive self‐mapping of a nonempty bounded closed convex subset K of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence {tn}, we show the existence of a sequence {xn} n satisfying the relation xn = (1 − tn/kn)f(xn) + (tn/kn)Tnxn and prove that {xn} converges strongly to the fixed point of T, which solves some variational inequality provided T is uniformly asymptotically regular. As an application, if T be an asymptotically nonexpansive self‐mapping of a nonempty bounded closed convex subset K of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by z0 ∈ K, zn+1 = (1 − tn/kn)f(zn) + (αtn/kn)Tnzn + (βtn/kn)zn converges strongly to the fixed point of T.

We first construct an implicit algorithm for solving the minimization problemminx∈Ω∥x∥, whereΩis the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping, and the solution set of some variational inequality. Further, we suggest an explicit algorithm by discretizing this implicit algorithm. We prove that the proposed implicit and explicit algorithms converge strongly to a solution of the above minimization problem.