An elementary proof that real roots nonexistence of nonlinear equations and its optimal solutions

In this paper,we give an elementary proof that the real roots nonexistence of nonlinear equation such as: x+y+z=n,where n =1,2,3,(x,y,z)∈ by contradiction. Thus, when n takes all positive integers between 4 and 1981, the value of function f= x+y+z based on the exact solution of the above equations is still real.Finally,numerical experiments presents a set of approximate numerical solutions by using standard genetic algorithm(SGA), adaptive particle swarm optimization algorithm(APSO) and artificial fish swarm algorithm(AFSA).


Introduction
We consider the nonlinear equations given by equations group (1) with the condition x,y and z belongs to . (1) Thus we can prove that the Eq.(1) has no real roots [1] and discuss the value of the function such as: f= x n +y n +z n , which is based on the exact solution of Eq. (1) and n∈ Z + .The main result is stated as follows: Theorem 1. Let x,y and z be real numbers, the Eq. (1) has no solutions in real number field. In the following section, we will prove Theorem 1 by contradiction [2,3]. We first assume that a solution, (x,y,z),to Eq.(1) exists, and then prove that it yields a contradiction. The paper is organized as follows: In Section 2,we give a completely elementary proof of the nonexistence result of Eq.(1) in Theorem 1.The numerical experiments and some bionic inspired algorithms are presented in Section 3. Section 4 summarizes the conclusions and points deserving further attention.
x y x z y z xy xz yz xyz x y z Substituting equalities(7),(9),and (3)   x y z x y z x y z x y xy z z x y Substituting equality (5) into equality (12),we have that x y z x y z x y xy z z x y xyz Combining equalities (3),(13),(5) and (14),we obtain the equality (9).

Derivation of elementary methods
In this section, we can obtain the value of function (2) based on the exact solution of Eq.(1) using elementary methods such as: when n=4, the value of function (2) is equal to 25/6, when n=5,the value of function (2) is equal to 6 and when n=6, the value of function (2) is equal to 103/12. Using equality (4),we obtain x y z x y z x y x z y z + + + + = Combining equalities (11) and (15),we obtain So the value of function (2) is shown with the parameter n=4.
Using equality (7) and (11),we obtain The equality (24) can be rewritten as x y x z y z xyz z x y y x z x y z The equality (4) can be rewritten as Combining equalities (9),(26),(27),(28) and (25),we obtain x y x z y z x y z x y z Substituting equalities (3) and (5) into the equality (29),we have Using equality (5),we obtain The equality (31) can be rewritten as x y x z y z Substituting equality (30) into the equality (32),we have So the value of function (2) is shown with the parameter n=6.What follows is using Matlab's solve function to obtain the exact solutions for a calculation when n takes all positive integers between 4 and 1981: Then, we obtain six groups complex roots of Eq.(1). Using these, we can get the value of function (2). Table 1 illustrates the value of function(2) when n can be given from 1 to 30. When n is equal to 1981,the value of function (2) is approximately 1.704774E308+0i.Numberic experiment shows that the value of function (2) is still real until n is equal to 1981.

The optimal approximate real solutions
Consider the general forms of nonlinear equations where fi(x1,x2, … ,xn)=Ai(i=1,2, … ,n) is a real-valued function on area D of the n-dimensional Euclidean space , and at least one of which is a nonlinear function, X=(x1,x2,…,xn) T is the vector variables and Ai(i=1,2,…,n) is the constant. In order to apply intelligent optimization algorithms for solving nonlinear equations, the problem of solving nonlinear equations is transformed into a nonlinear least squares problem,formulated as a function optimization problem:  Step 1:Initialize:the population size is fishnum, the biggest evolution generation is maxgen,the visual distance is visual,the crowded factor is δ (0< δ <1),the largest trying number is try_number and so on. Initialize the current position X[fishnum] for fishnum artificial fishes randomly. Calculate the food consistence Y[fishnum] for every artificial fishes. Mark the bulletin board.
Step 2:Each artificial fish in the population performs swarming behavior and following behavior with preying behavior if needed,to search for the position where the food resources are richer than the current position. If the position after finishing an iteration is better than the bulletin board, update bulletin board and go to step 3.
Step 4:The algorithm reaches an end. Output the best optimal solutions.
Herein we present the standard genetic algorithm that is implemented within a ga function from MATLAB TM Optimization Toolbox to perform the tests.As described in [7],the adaptive particle swarm optimization algorithm is given. Next, we now consider the fitness function such as: The parameters for SGA are as follows: the population size PopulationSize is equal to 300,the maximum iteration Generations is equal to 150,the tolerance TolFun is equal to 1e-10.
The parameters for APSO algorithm are as follows: the population size PopulationSize is equal to 300, the maximum iteration Generations is equal to 150,the maximum weight is equal to 0.9,the minimum weight is equal to 0.6.
The evolution implementation process of above three algorithms is given in Fig.1.The optimal coordinate movement trajectory and the fitness function image with AFSA algorithm is shown in Fig.2.

Conclusion and future work
In this note, we give a proof that is completely elementary to establish the real roots nonexistence of Eq.(1). Even more importantly, the value of the function(2) when n takes all positive integers between 4 and 1981 is still real, which is calculated in numeric experiment. We conjecture that when n takes all positive integers, the value of function (2) may be always a real number. To the best of our knowledge, however, no strict mathematical proof method has been found. It is expected that the strict mathematical proof method is shown in future work.