An Adaptive Prediction-Correction Method for Solving Large-Scale Nonlinear Systems of Monotone Equations with Applications

and Applied Analysis 3 Lemma 3. Let {x k } and {z k } be any sequence generated by Algorithm 1. Suppose that g is monotone and that the solution set of (1) is not empty, then {x k } and {z k } are both bounded. Furthermore, it holds that lim k→∞ 󵄩󵄩󵄩󵄩xk − zk 󵄩󵄩󵄩󵄩 = 0, (13) lim k→∞ 󵄩󵄩󵄩󵄩xk+1 − xk 󵄩󵄩󵄩󵄩 = 0. (14) Proof. From (6), we have ⟨g (z k ) , x k − z k ⟩ = −α k ⟨g (z k ) , d k ⟩ ≥ σα 2 k 󵄩󵄩󵄩󵄩dk 󵄩󵄩󵄩󵄩 2 = σ 󵄩󵄩󵄩󵄩xk − zk 󵄩󵄩󵄩󵄩 2 . (15) Let x∗ be an arbitrary point such that g(x∗) = 0. Taking account of the monotonicity of g, we have ⟨g (z k ) , x k − x ∗ ⟩ = ⟨g (z k ) , x k − z k ⟩ + ⟨g (z k ) , z k − x ∗ ⟩ ≥ ⟨g (z k ) , x k − z k ⟩ + ⟨g (x ∗ ) , z k − x ∗ ⟩ = ⟨g (z k ) , x k − z k ⟩ . (16) From (7), (14), and (16), it follows that 󵄩󵄩󵄩󵄩xk+1 − x ∗󵄩󵄩󵄩󵄩 2 = 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 x k − ⟨g (z k ) , x k − z k ⟩ 󵄩󵄩󵄩󵄩g (zk) 󵄩󵄩󵄩󵄩 2 g (z k ) − x ∗ 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 2 = 󵄩󵄩󵄩󵄩xk − x ∗󵄩󵄩󵄩󵄩 2 − ⟨g (z k ) , x k − z k ⟩ 2 󵄩󵄩󵄩󵄩g (zk) 󵄩󵄩󵄩󵄩 2 ≤ 󵄩󵄩󵄩󵄩xk − x ∗󵄩󵄩󵄩󵄩 2 − σ 󵄩󵄩󵄩󵄩xk − zk 󵄩󵄩󵄩󵄩 4 󵄩󵄩󵄩󵄩g (zk) 󵄩󵄩󵄩󵄩 2 . (17) Hence the sequence {‖x k −x ∗ ‖} is decreasing and convergent; moreover, the sequence {‖x k ‖} is bounded. Since the g is continuous, there exists a constant C > 0 such that 󵄩󵄩󵄩󵄩g (zk) 󵄩󵄩󵄩󵄩 ≤ C. (18) By the Cauchy-Schwarz inequality, themonotonicity of g and (15), we have 󵄩󵄩󵄩󵄩g (xk) 󵄩󵄩󵄩󵄩 ≥ ⟨g (x k ) , x k − z k ⟩ 󵄩󵄩󵄩󵄩xk − zk 󵄩󵄩󵄩󵄩 ≥ ⟨g (z k ) , x k − z k ⟩ 󵄩󵄩󵄩󵄩xk − zk 󵄩󵄩󵄩󵄩 ≥ σ 󵄩󵄩󵄩󵄩xk − zk 󵄩󵄩󵄩󵄩 . (19) From (18) and (19), we obtain that {z k } is also bounded. It follows from (17) and (18) that