eSS implements an intensification mechanism already in the global phase, which exploits the promising directions defined by a pair of solutions in the RefSet.
Let and be a pair of solutions to (1.3) in a domain.
Let and be a pair of solutions to the nonhomogeneous -harmonic equation (1.1) in a domain.
Let and be a pair of solution to the conjugate -harmonic tensor in.
It is easy to see that ((u, v in C[0, 1]times C[0, 1]) is a pair of solution to the system (1) if and only if ((u, v)) is a pair of solution of the following nonlinear integral system: left { textstylebegin{array}l} u(t)=lambdaint_{0}^{1}G_{alpha}(t, s)f s, v s)),ds, v(t)=muint_{0}^{1}G_{beta}(t, s g s, u(s)),ds.
Step 4. Crossover Selecting a pairs of solutions in step 3 randomly to use them as the parents for crossover operations.
In the next step, a pair of parent solutions will be picked from the selected parents to generate a new solution with crossover operation; meanwhile, mutation(s) can be optionally applied to certain element(s) within a parent individual to generate a new one.
x, s ∈ Ω ¯ and 〈 x, s 〉 = 0 ; x, s ∈ Ω ¯ and x ∘ s = 0. Using Lemma 1.2, we can check (see Proposition 2.1 in [13]) that finding a pair of optimal solutions ( x, y, s ) of (P) and (D) is equivalent to solving the following Newton system: { A x = b, A T y + s = c, x ∘ s = 0, x, s ∈ Ω ¯, y ∈ R m. (4).
In Section 3, we show that in the presence of a pair of upper and lower solutions the problem (1.1 - 1.2 1.1 - 1.2east one solution.
It is shown that the sequence of iterations from a linear iteration process converges monotonically and quadratically to a unique solution in a sector between a pair of upper and lower solutions.
Let ( u, v ) be a pair of positive solutions of (1.4) in the critical case (1.5).
