Conditions for Existence and Uniqueness for the Solution of Nonlinear Problems

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1 4 Indian Journal of Biomechanics: Special Issue (NCBM) 7-8 March Conditions for Existence and Uniqueness for the Solution of Nonlinear Problems R.P. Shama, A.K.Wadhwani EED, MITS, Gwalior, M.P., India Abstract New conditions are derived for existence and uniqueness for the solutions of nonlinear algebraic equations arising in non-linear resistive networks, power and biomechanical systems. It is proved that a continuous differentiable function f(x) = y, x Є R n and y Є R n has at least one solution for every finite y, if the Jacobian is positive or negative definite for all the values of appropriate large x. In addition to this if the Jacobian is nonsingular for all the values of x, then function f is a C' diffeomorphic. In this paper new conditions for existence and uniqueness for the solution of the nonlinear problems are introduced. The vector functional form and diffeomorphism is presented for the solution of the nonlinear resistive networks, power and biomechanical systems. Introduction The conditions for existence and uniqueness for the solutions of non-linear resistive and power networks problems were studied by R.S.Palais [], C.A.Desoer et al., [], T. Ohtuski and Watenbe [5], L.O.Chua [6], T. Fujisawa and E.S.Kuh [9], further Mock [] and another group of R.E.Bank and D.J.Rose [5]. They have presented the papers in this area to define the diffeomorphism with suitable power network or power and biomechanical system problems. Not to much work related with necessary and sufficient conditions for existence and uniqueness of solutions and diffeomorphism of a nonlinear vector function is reported. Thus there is a need to work on specific vector functional form of the non-linear equation for the study of existence, uniqueness and C' diffeomorphic solution of the problems.. Diffeomorphism: A diffeomorphism is a kind of isomorphism of smooth manifolds. It is invertible mathematical function that maps one differentiable manifold to another, such that both the function and its inverse are smooth. This can be used for the solution of power networks, biomechanical systems and various problems like existence, uniqueness of solution and other diffeomorphic studies [-7].. Definition: Mathematically when given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism, if both functions are: f: M N and its inverse f ˉ¹: N M are differentiable (if these functions are r- times continuously Differentiable, f is called a C' diffeomorphism) Consider an equation of the form: 4

2 4 f (x) = y ( ) Where f is a continuous differentiable function of x Є R n in to y Є R n. It is well known that this equation is of great interest in the analysis and study of nonlinear restive networks [6, 7] as well as power networks []. Given equation (), we wish to know if there is at least one x for every y. This is known as the existence problem. In some situations we wish to know, if every y has exactly one solution. This is known as the uniqueness problem. A close look at the literature reveals the following: (i) Not much work is reported on the problems associated with existence of solutions of the equation () in its general form. Several research papers have (ii) appeared on special forms of f [9]. The uniqueness problem is relatively well studied [, 6, 7]. But most of the conditions are sufficient but not necessary. Therefore there is a need for further study. In this paper the new answers will be provided for the existence and uniqueness problems, based on the concept of positive definiteness of the Jacobian matrix for some value of x only (unlike some conditions which should be checked for all the values of x). This is presented in the next section. In next section the results are to be compared with the results available in literature to bring out the differences. Existence and Uniqueness Problems. Definition: A matrix J (x) will be called a positive definite (P.D) matrix for a given, x if for every Z Є R n, there exists a real constant r > 0 such that Z T J(x) Z Є J Z () It will be said to be negative definite (N.D) if, () is - Z T J(x) Z Є J Z ( ) Replaced by ( ) in the above definition.. Theorem. (Existence Theorem): Referring to equation (), there is at least one solution for every finite y if there exists a real constant r > 0 such that J (x) f / is positive definite for all x r. Proof: Consider the following homotopy [8] h (x,λ) = λ f (x) + ( - λ) x, 0 λ. () Let S be an n-dimensional sphere of radius M, where M > r. Let S denote the boundary of this sphere. Pre-multiplying equation () with x T, we have x T h(x,λ) = x T f(x) + ( - λ) x ( ) Whenever the conditions of the theorem are satisfied, h (x,λ) 0 for all x Є S. To prove this we first note that this is true for λ = 0. For all other values of λ, it can be done as follows: For any x Є S, we can write x = ut, where u = x / x. Define H u ( t ) u T ( ut,λ ) [6]. Then 4

3 4 λ H u t ( t) h n n i = ( u j ) ui i= j= j = u T h u h Where h = ( x, λ), f = λ + ( λ) I f By assumption is either positive definite or, negative definite for all x Є S. Take f f (- λ ) I, if is positive definite and - ( - λ ) I if is negative definite. Then h is either P.D. for all x Є S, i.e., there exists a constant α > 0 such that the slope of H u (t) is greater than α for all t such that x Є S. This implies that (t) is nonzero for all t such that x Є S. Thus f(x) is homotopic to x [9]. The deg (f, S, o) =. The above argument also implies that H u (t) as t. H u But, H u (t) = u T h (u t,λ) l u T h (ut, λ) = h (ut, ) as u =. h ( u t, λ ) as t. Putting, λ =, we have f ( x ). as x. i.e., f ( x ) is norm coercive. Let y () and y () be any two finite points (i. e., none of their coordinates is arbitrarily large).then y () and y () have no solution on f(s) as f is norm coercive. From degree invariance [], deg ( f, s, y () ) = deg. (f, s, y () ) = deg. (f, s, 0) =. Hence the result.. Theorem. (Uniqueness Theorem): Let f be continuous and differentiable for all x. Then it is a C diffeomorphism of R n onto R n. If (i) det j 0 for all x in R n Where j = f /. (ii) There exists a constant M, or J is N.D. for all x satisfying x M. Proof: In theorem It is shown that f (x) is non coercive whenever the conditions of the theorem are satisfied. Thus the conditions of the theorem are satisfied. Thus the 4

4 44 conditions of the following theorem due to Palais are satisfied: f : R R is a diffeomorphism of R onto itself if the function (i) f (x) is continuously differentiable (ii) det. J o for all x in R (iii) f (x) as x. Hence the required result. This is generalization of a result due to Ohtsuki and Watanabe, [5]. Remarks and Examples. The conditions of Theorem are both necessary and sufficient conditions in one dimension. But in higher dimensions it is not so as the following example shows clearly.. y = x x + + x y = x x x x J = is not P.D for x = 0 and x = large. But it can be 0 x shown to be onto.. Note that theorem is for onto mapping. For, this consider y = e x. This is a homeomorphism. But it does not map onto R'. It is evident that this function does not satisfy the conditions of the Theorem and Theorem 4. Consider an equation of the form [0]. F (x) = f(x) + A x = y Where f (x) = [f (x ) f (x ) f n (x n ) ] T. Let A be a positive semi definite matrix. Let f i (x i ) be strictly monotonic mapping R onto R. Then F (x) is a homeomorphism as theorem is satisfied. Consider the following equations: F( x, x F( x, x ) ) x x + x = + 5x 0 x y = y Fujisawa and Kuh [9] showed than f(x) = y will have one and only one solution for every y if there exists a positive constant Є > 0 such that det J Є, det J /J Є,, det J n /det J n- Є for all x Є R n. Where J k denotes the first k rows and he first k columns of he Jacobian matrix J. Our conditions are in general but different because for finite values of x here it is only required that det J n 0. Vehovec [7] stated that f is c diffeomorphism 44

5 45 if (i) there exists a constant Є such that det f/ Є > 0 for all x (ii) the Jacobian matrix f/ is bounded for all x. our conditions are valid even if one or more elements are not bounded. For example consider y = e x + x. The Jacobians are not bounded. But det J for all x, thus satisfying the Theorem. Consider y = x + x And, y = - x + x. The positive definiteness (P.D) condition of Theorem is not satisfied. But it satisfies the ratio condition of Fujisawa and Kuh [9]. However, pre-multiplying f(x) = y with [ 0; - ], Theorem., is satisfied. Thus transformation of coordinate axes enhances the applicability of theorems and. The matrix A is not positive definite or class P. But the conditions of the Theorem are satisfied. Therefore, it is a homeomorphism. Thus this condition is different from that of Wilson [0] in which A is required to be a P 0 matrix. 4 Conclusions The work presented in this paper deals with the existence, uniqueness and determination of solutions of resistive networks, power and biomechanical systems. The systems are characterized by the specific functional form of the nonlinear equation no., which is used for the solutions of the resistive or power networks and biomechanical problems. The necessary and sufficient conditions are derived, defined and presented with illustration. The Jacobian form is used for mathematical formulation, calculations and theoretical study of the system diffeomorphism. References. R. S. Palais, Natural operation on differential forms, Trans. American Math. Soc.Vol.9, pp , July C. A. Desoer and J. Katzenelson, Non linear RLC Networks, Bell system. Tech. Journal, vol. 44 pp- 6-98, Jan J. Katzenlson, An algorithm for solving non-linear resistive networks, Bell system. Tech. Journal. Vol. 44 pp Oct E. S. Kuh, Representation of non-linear Networks, Proc. NEC- vol. pp , T. Ohtsuki and H. watanbe, State variable analysis of RLC net works containing non-linear coupling elements, IEEE Trans.on circuit Theory vol. ct-6 pp-6-8, Feb L. O. Chua, Introduction to non-linear network theory, McGraw Hill Pub M. Vechovec, Simple criterion for the global regularity of vector valued functions, Electronics letter, vol. 5, pp , Dec E.S.Kuh and I.N.Hajj, Nonlinear circuit theory: resistive networks, Proceeding IEEE, vol. 59, No., pp.40 55, March

6 46 9. T. Fujisawa and E. S. Kuh, Some results on existence and uniqueness of solutions of non- linear net- works, IEEE Trans. Circuit theory Vol.ct-8, pp , Sept A. N. Wilson, Some aspects of theory of non- linear net-works, Proc. IEEE, vol.6, pp.09-, August 97.. M.S. Mock, Analysis of mathematical model of semiconductor devices, Dublin, Ireland; Boole Press, ch., p.8, 97.. M.S. Berger, Nonlinearity and functional analysis, New York, N.Y; Academic Press, L. O. Chua and N. N. Wang, On the application of degree theory to analysis of resistive non-linear networks, Int. Journal of circuit Theory and Applications vol.5.pp M. S. Berger, Non-linearity and Functional Analysis, M.Y. Academic Press R. E. Bank and D. J. Rose, Global approximate Newton methods, Numerical Math. vol.7, No , Banyaga,Augustin The structure of classical diffeomorphism groups, Mathematics and its Applications, 400, Kluwer Academic, ISBN , Milnor, W. John, Collected Works Vol. III, Differential topology, American Mathematical Society, ISBN ,