PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume, Number, Pages S -9939(XX- ON THE DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS N. S. HOANG AND A. G. RAMM (Communicated by Yingfei Yi Abstract. A version of the Dynamical Systems Method (DSM for solving ill-posed nonlinear equations F (u = f with monotone operators F in a Hilbert space is studied in this paper under less restrictive assumptions on the nonlinear operators F than the assumptions used earlier. A new method of proof of the basic results is used. An a posteriori stopping rule, based on a discrepancytype principle, is proposed and justified mathematically under weaker assumptions on the nonlinear operator F, than the assumptions used earlier.. Introduction In this paper we study a version of the Dynamical Systems Method (DSM (see [] for solving the equation (. F (u = f, where F is a nonlinear Fréchet differentiable monotone operator in a real Hilbert space H, and equation (. is assumed solvable. Monotonicity means that (. F (u F (v, u v, u, v H. Here,, denotes the inner product in H. It is known (see, e.g., [], that the set N := {u : F (u = f} is closed and convex if F is monotone and continuous. A closed and convex set in a Hilbert space has a unique minimal-norm element. This element in N we denote y, F (y = f. The local existence of the solution to (.3 below can be proved under our assumptions. We assume that f is not known but f δ, the noisy data, are known, and f δ f δ. If F (u is not boundedly invertible, then solving for u, given noisy data f δ, is often (but not always an ill-posed problem. When F is a linear bounded operator many methods for stable solution of (. were proposed (see [7], [8], [9], [], [] and references therein. However, when F is nonlinear then the theory is less complete. The DSM for solving equation (. was studied extensively in [] [8], [4]-[6], where also numerical examples, illustrating efficiency of the algorithms, based on Mathematics Subject Classification. Primary 65R3; 47J5; 47J6; 47J35. Key words and phrases. Dynamical systems method (DSM, nonlinear operator equations, monotone operators, discrepancy principle. c XXXX American Mathematical Society
N. S. HOANG AND A. G. RAMM the DSM methods, were given. In [] the following version of the DSM for solving equation (. was studied: (.3 u δ = ( F (u δ + a(ti ( F (uδ + a(tu δ f δ, uδ ( = u. Here F is a monotone operator, and a(t > is a continuous function, defined for all t, strictly monotonically decaying, lim a(t =. These assumptions on a(t hold throughout the paper and are not repeated. Additional assumptions on a(t will appear in Theorem 3.. Convergence of the above DSM was proved in [] for any initial value u with an a priori choice of stopping tim, provided that a(t is suitably chosen. In this paper an a posteriori choice of t δ is formulated and justified rigorously. The theory of monotone operators is presented in many books, e.g., in [], [], []. Some of the results of the theory of monotone operators, used in this paper, can be found in []. In [9] methods for solving well-posed nonlinear equations in a finite-dimensional space are discussed. Methods for solving equation (. with monotone operators are quite important in many applications. It is proved in [] that solving any solvable linear operator equation Au = f with a closed densely defined linear operator A can be reduced to solving equation (. with a monotone operator. Equations (. with monotone operators arise often when the physical system is dissipative. In the earlier papers and in monograph [] it was assumed that F is locally twice Fréchet differentiable, and a nonlinear differential inequality ([], p.97 was used in a study of the behavior of the solution to the DSM (.3. The smoothness assumptions on F are weakened in this paper, the method of our proofs is new, and, as a result, the proofs are shorter and simpler than the earlier ones. The assumptions on the regularizing function a(t are also weakened. In this paper we propose and justify a stopping rule for solving ill-posed equation (. based on a discrepancy principle (DP for the DSM (.3. The main result of this paper is Theorem 3. in which a DP is formulated, the existence of the stopping tim is proved, and the convergence of the DSM (.3 with the proposed DP is justified under some natural assumptions for a wide class of nonlinear equations with monotone operators. Our result is novel because the convergence of the DSM is justified under less restrictive assumptions on F than in [], where twice Fréchet differentiability was assumed and the DP was not established for problem (.3. Moreover, the rate of decay of the function a(t as t can be arbitrary in the power scale, while in [] a(t was often assumed to satisfy the condition a(tdt = which implies the decay in the power scale not faster than O( t as t. These new theoretical results are useful practically. The auxiliary results in our paper are borrowed from [3] and their proofs are omitted. Let us consider the following equation. Auxiliary results (. F (V δ,a + av δ,a f δ =, a >, where a = const. It is known (see, e.g., [] that equation (. with monotone continuous operator F has a unique solution for any f δ H. Let us recall the following result from [, p.]:
ON THE DSM FOR SOLVING NOE 3 Lemma.. Assume that equation (. is solvable, y is its minimal-norm solution, and F is monotone and continuous. Then where V,a solves (. with δ =. lim V,a y =, a Lemma. (Lemma 3, [3]. If (. holds and F is continuous, then V δ,a = O( a as a, and (. lim a F (V δ,a f δ = F ( f δ. Let a = a(t, < a(t, and assume a C [,. Then the solution V δ (t := V δ,a(t of (. is a function of t. From the triangle inequality one gets: F (V δ ( f δ F ( f δ F (V δ ( F (. From Lemma. it follows that for large a( one has: ( F (V δ ( F ( M V δ ( = O. a( Therefore, if F ( f δ > Cδ, then F (V δ ( f δ (C ɛδ, where ɛ > is arbitrarily small, for sufficiently large a( >. Below the words decreasing and increasing mean strictly decreasing and strictly increasing. Lemma.3 (Lemma, [3]. Assume F ( f δ >. Let < a(t, and F be monotone. Denote φ(t := F (V δ (t f δ, ψ(t := V δ (t, where V δ (t solves (. with a = a(t. Then φ(t is decreasing, and ψ(t is increasing. Lemma.4 (cf. Lemma 4, [3]. Assume < a(t. Then the following inequality holds (.3 lim F (V δ (t f δ δ. Remark.5. Let V := V δ (t δ=, so F (V + a(tv f =. Let y be the minimalnorm solution to the equation F (u = f. We claim that (.4 V δ V δ a. Indeed, from (. one gets F (V δ F (V + a(v δ V = f δ f. Multiply this equality by V δ V and use (. to obtain δ V δ V f δ f, V δ V This implies (.4. Similarly, from the equation = F (V δ F (V + a(v δ V, V δ V a V δ V. F (V + av F (y =
4 N. S. HOANG AND A. G. RAMM one can derive that (.5 V y. From (.4 and (.5, one gets the following estimate: (.6 V δ V + δ a y + δ a. Lemma.6. Let a(t satisfy (3.. Then one has (.7 e e s ȧ(s Vδ (s ds a(t V δ(t, t. Proof. Let us check that ( (.8 e t d ȧ(t dt a(te t, t >. One has (.9 ( d dt a(te t = a(te t 4 Thus, inequality (.8 is equivalent to + ȧ(te t = a(te t 4 ȧ(t e t 3 (. ȧ(t a(t, t >. 4 Inequality (. holds because by our assumptions the function a(t satisfies (3.. Integrating both sides of (.8 from to t, one gets (. e s ȧ(s ds a(te t a(e < a(te t, t. Multiplying (. by e V δ (t, using the fact that V δ (t is increasing (see Lemma.3, one gets (.7. Lemma.6 is proved. Denote 3. Main result A := F (u δ (t, A a := A + ai, where I is the identity operator, and u δ (t solves the following Cauchy problem: (3. u δ = A a(t [F (u δ + a(tu δ f δ ], u δ ( = u, where u H. Assume (3. < a(t, 6 ȧ(t, t. a(t Assume that equation (. has a solution, possibly nonunique, and y is the minimal norm solution to this equation. Let f be unknown but f δ be given, f δ f δ. Theorem 3.. Let a(t satisfy (3.. Let C > and ζ (, ] be constants such that Cδ ζ > δ. Assume that F : H H is a Fréchet differentiable monotone operator, u is an element of H, satisfying the following inequalities (3.3 F (u + a(u f δ 4 a( V δ(, F (u f δ > Cδ ζ,.
ON THE DSM FOR SOLVING NOE 5 where V δ (t := V δ,a(t solves (. with a = a(t. Then the solution u δ (t to problem (3. exists on an interval [, T δ ], lim δ T δ =, and there exists a uniqu, t δ (, T δ, such that lim δ t δ = and (3.4 F (u δ (t δ f δ = Cδ ζ, F (u δ (t f δ > Cδ ζ, t [, t δ. If ζ (, and t δ satisfies (3.4, then (3.5 lim δ u δ (t δ y =. Remark 3.. In Theorem 3. the existence of t δ satisfying (3.4 is guaranteed for any ζ (, ]. However, we prove relation (3.5 for ζ (,. If ζ = it is possible to prove that u δ (t δ convergences to a solution to (., but it is not known whether this solution is the minimal-norm solution of (. if (. has more than one solution. Further results on the choices of ζ require extra assumptions on F and y. Since the minimal-norm solution y satisfies the relation F (y f δ = f f δ δ, it is natural to choose C > and ζ (, so that Cδ ζ be close to δ. One can choose u satisfying the first inequality in (3.3. Indeed, if u approximates V δ (, the solution to equation (., with a small error, then the first inequality in (3.3 is satisfied. The first inequality in (3.3 is a sufficient condition for (3.6, i.e., (3.6 e F (u + a(u f δ 4 a(t V δ(t, t, to hold. In our proof inequality (3.6 is used at t = t δ. The stopping tim is often sufficiently large for the quantity a(t δ to be large. This follows from the fact that lim e t a(t = (see (3.3 below. In this case inequality (3.6 with t = t δ is satisfied for a wide range of u. The second inequality in (3.3 is a natural assumption because if this inequality does not hold and u is not too large, then u can be considered as an approximate solution to (.. Proof of Theorem 3.. The uniqueness of t δ follows from (3.4. Indeed, if t δ and τ δ > t δ both satisfy (3.4, then the second inequality in (3.4 does not hold on the interval [, τ δ. Let us prove the existence of t δ. From (3., one obtains: d ( F (uδ + au δ f δ = Aa u δ + ȧu δ = ( F (u δ + au δ f δ + ȧuδ. dt This and (. imply: d [ (3.7 F (uδ F (V δ + a(u δ V δ ] = [ F (u δ F (V δ + a(u δ V δ ] + ȧu δ. dt Denote v := v(t := F (u δ (t F (V δ (t + a(t(u δ (t V δ (t, Multiply (3.7 by v and get (3.8 This implies h := h(t := v(t. hḣ = h + v, ȧ(u δ V δ + ȧ v, V δ h + h ȧ u δ V δ + ȧ h V δ. (3.9 ḣ h + ȧ u δ V δ + ȧ V δ.
6 N. S. HOANG AND A. G. RAMM Since F (u δ F (V δ, u δ V δ, one obtains from two equations and v, u δ V δ = F (u δ F (V δ + a(t(u δ V δ, u δ V δ, v, F (u δ F (V δ = F (u δ F (V δ + a(t u δ V δ, F (u δ F (V δ, the following two inequalities: (3. a u δ V δ v, u δ V δ u δ V δ h, and (3. F (u δ F (V δ v, F (u δ F (V δ h F (u δ F (V δ. Inequalities (3. and (3. imply: (3. a u δ V δ h, F (u δ F (V δ h. Inequalities (3.9 and (3. imply ( (3.3 ḣ h By the assumption, ȧ a ȧ a + ȧ V δ., so inequality (3.3 implies (3.4 ḣ h + ȧ V δ. Inequality (3.4 implies: (3.5 h(t h(e + e From (3.5 and (3., one gets (3.6 F (u δ (t F (V δ (t h(e + e e s ȧ(s Vδ (s ds. Hence, using the triangle inequality and (3.6, one gets: (3.7 F (u δ (t f δ F (V δ (t f δ + h(e + e e s ȧ Vδ ds. e s ȧ Vδ ds. Since a(s V δ (s = F (V δ (s f δ is decreasing, by Lemma.3, one obtains: (3.8 lim e The assumption lim ȧ(t a(t (3.9 lim e s ȧ Vδ ds lim e = implies: e s e s ȧ(s a(s a( V δ( ds e t ȧ(s a(s a( V δ( ds. Indeed, if I := e s ȧ(s a(s a( V δ( ds < then (3.9 is obvious. If I =, then (3.9 follows from L Hospital s rule. It follows from (3.7 (3.9 and Lemma.4 that =. (3. lim F (u δ (t f δ lim F (V δ (t f δ + lim e e s ȧ Vδ ds δ.
ON THE DSM FOR SOLVING NOE 7 The assumption F (u f δ > Cδ ζ > δ and inequality (3. imply the existence of a t δ > such that (3.4 holds because F (u δ (t f δ is a continuous function of t. We claim that (3. lim δ t δ =. Let us prove (3.. From the triangle inequality and (3.6 one gets (3. F (u δ (t f δ F (V δ (t f δ F (V δ (t F (u δ (t a(t V δ (t h(e e e s ȧ Vδ ds. Recall that a(t satisfies (3. by our assumptions. From (3. and Lemma.6 one obtains (3.3 From (3.3 we have a(t V δ(t e e s ȧ Vδ (s ds. (3.4 h(e 4 a( V δ( e, t. It follows from (3. that (3.5 e a( a(t. Specifically, inequality (3.5 is obviously true for t =, and ( ( a(te t = a(te t ȧ(t >, a(t t by (3.. Therefore, one gets from (3.5 and (3.4 the following inequality: (3.6 e t h( 4 a(t V δ( 4 a(t V δ(t, t, Here, we have used the inequality V δ (t V δ (t for t < t, established in Lemma.3 in Section. From (3.4 and (3. (3.6, one gets (3.7 Cδ ζ = F (u δ (t δ f δ 4 a(t δ V δ (t δ. From (.4 and the triangle inequality one derives (3.8 a(t V (t a(t V (t V δ (t +a(t V δ (t δ+a(t V δ (t, t. It follows from (3.7 and (3.8 that (3.9 lim δ a(t δ V (t δ lim δ ( δ + 4Cδ ζ =. Since V (t increases (see Lemma.3, the above formula implies lim δ a(t δ =. Since < a(t, it follows that lim δ t δ =, i.e., (3. holds. Let us prove that (3.3 lim e t a(t =. We claim that, for sufficiently large t >, the following inequality holds: t (3.3 > ln a (t.
8 N. S. HOANG AND A. G. RAMM By L Hospital s rule and (3., one obtains t (3.3 lim ln a (t = lim a (t ȧ(t a 3 (t a(t = lim 4 ȧ(t =. This implies that (3.3 holds for t > sufficiently large. From (3.3 one concludes (3.33 lim e t a(t lim e ln a (t a(t = lim a(t =. Thus, relation (3.3 is proved. From (3.7, (3.3, (3.6 and (.6 one gets (3.34 Cδ ζ a(t δ V δ (t δ ( + + 4 This and the relation lim δ δ δ ζ 7 4 ( a(t δ y + δ. =, for a fixed ζ (,, imply δ ζ (3.35 lim δ a(t δ 7 y 4C < y C. It follows from inequality (3.5 and the first inequality in (3. that (3.36 a(t u δ (t V δ (t h(e + e e s ȧ(s Vδ (s ds. From (3.35 and the first inequality in (.6 one gets, for sufficiently small δ, the following inequality (3.37 V δ (t y + δ y < y + a(t C, t t δ. Therefore, (3.38 lim δ Let us show that δ e s ȧ(s V δ (s ds a(t δ (3.39 lim δ ( y + y tδ lim C δ δ e s ȧ(s ds =. a(t δ e s ȧ(s ds. a(t δ The denominator of (3.39 tends to as δ by (3.3. Thus, if the numerator of (3.39 is bounded then (3.39 holds. Otherwise, relation (3. and L Hospital s rule yield (3.4 lim δ δ e s ȧ(s ds = lim a(t δ It follows from (3.38 and (3.39 that (3.4 lim δ From (3.4, (3.36, and (3., one gets δ e s ȧ(s V δ (s ds a(t δ e t ȧ(t e t a(t e t ȧ(t =. =. (3.4 lim δ u δ (t δ V δ (t δ = lim δ h(t δ a(t δ =. It is now easy to finish the proof of Theorem 3..
ON THE DSM FOR SOLVING NOE 9 From the triangle inequality and inequality (.4 one obtains (3.43 u δ (t δ y u δ (t δ V δ (t δ + V (t δ V δ (t δ + V (t δ y u δ (t δ V δ (t δ + δ a(t δ + V (t δ y, where V (t δ = V,a(tδ (see equation (.. From (3.4, (3., inequality (3.43, and Lemma. one obtains (3.5. Theorem 3. is proved. References. K. Deimling, Nonlinear functional analysis, Springer Verlag, Berlin, 985.. N.S. Hoang and A. G. Ramm, Dynamical systems method for solving linear finite-rank operator equations, Ann. Polon. Math., 95, N, (9, 77-93. 3. N.S. Hoang and A. G. Ramm, A discrepancy principle for equations with monotone continuous operators, Nonlinear Analysis Series A: Theory, Methods & Applications, 7, (9, 437-435. 4. N.S. Hoang and A. G. Ramm, An iterative scheme for solving nonlinear equations with monotone operators, BIT Numer. Math. 48, N4, (8, 75-74. 5. N.S. Hoang and A. G. Ramm, Dynamical Systems Gradient method for solving nonlinear equations with monotone operators, Acta Appl. Math., 6, (9, 473-499. 6. N.S. Hoang and A. G. Ramm, A new version of the Dynamical Systems Method (DSM for solving nonlinear equations with monotone operators, Diff. Eqns and Appl.,, N, (9, -5. 7. V. Ivanov, V. Tanana and V. Vasin, Theory of ill-posed problems, VSP, Utrecht,. 8. V. A. Morozov, Methods of solving incorrectly posed problems, Springer Verlag, New York, 984. 9. J. Ortega, W. Rheinboldt, Iterative solution of nonlinear equations in several variables, SIAM, Philadelphia,.. D. Pascali and S. Sburlan, Nonlinear mappings of monotone type, Noordhoff, Leyden, 978.. A. G. Ramm, Dynamical systems method for solving operator equations, Elsevier, Amsterdam, 7.. A. G. Ramm, Global convergence for ill-posed equations with monotone operators: the dynamical systems method, J. Phys A, 36, (3, L49-L54. 3. A. G. Ramm, Dynamical systems method for solving nonlinear operator equations, International Jour. of Applied Math. Sci.,, N, (4, 97-. 4. A. G. Ramm, Dynamical systems method for solving operator equations, Communic. in Nonlinear Sci. and Numer. Simulation, 9, N, (4, 383-4. 5. A. G. Ramm, DSM for ill-posed equations with monotone operators, Comm. in Nonlinear Sci. and Numer. Simulation,, N8, (5,935-94. 6. A. G. Ramm, Discrepancy principle for the dynamical systems method, Communic. in Nonlinear Sci. and Numer. Simulation,, N, (5, 95-7. A. G. Ramm, Dynamical systems method (DSM and nonlinear problems, in the book: Spectral Theory and Nonlinear Analysis, World Scientific Publishers, Singapore, 5, -8. (ed J. Lopez-Gomez. 8. A. G. Ramm, Dynamical systems method (DSM for unbounded operators, Proc.Amer. Math. Soc., 34, N4, (6, 59-63. 9. A. G. Ramm, Dynamical systems method for solving linear ill-posed problems, Ann. Polon. Math., 95, N3, (9, 53-7.. M.M. Vainberg, Variational methods and method of monotone operators in the theory of nonlinear equations, Wiley, London, 973. Mathematics Department, Kansas State University, Manhattan, KS 6656-6, USA E-mail address: nguyenhs@math.ksu.edu Mathematics Department, Kansas State University, Manhattan, KS 6656-6, USA E-mail address: ramm@math.ksu.edu