Nonlinear Analysis 71 2009 2744 2752 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A nonlinear inequality and applications N.S. Hoang A.G. Ramm Mathematics Department Kansas State University Manhattan KS 66506-2602 USA a r t i c l e i n f o a b s t r a c t Article history: Received 27 December 2008 Accepted 15 January 2009 MSC: 65J15 65J20 65N12 65R30 47J25 47J35 A nonlinear inequality is formulated in the paper. An estimate of the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can be applied to the study of global existence of solutions to nonlinear PDE. 2009 Elsevier Ltd. All rights reserved. Keywords: Nonlinear inequality Dynamical systems method DSM Stability 1. Introduction In this paper the following differential inequality ġt γ tgt + αtg p t + βt t p > 1 ġ = dg dt 1 where gt 0 is studied. In Eq. 1 αt βt and γ t are continuous functions defined on [ where 0 is a fixed number and αt 0 t. Under the assumptions of Theorem 1 see below we prove that there exist solutions to inequality 1 and all nonnegative solutions to inequality 1 are defined globally i.e. for all t. Estimates of the rate of decay of solutions to this inequality are obtained and formulated in 4 and 23. These new results can be used in a study of dynamical systems both continuous and discrete and large time behavior of solutions to nonlinear evolution equations. For example inequality 1 is used in Section 3 in a study of the Dynamical Systems Method DSM for solving nonlinear equations of the type Fu = f where F : H H is a monotone operator and H is a Hilbert space. The DSM we study in Section 3 is the continuous analog of the regularized Newton s method for solving equations with monotone operators. The local boundedness of the second Fréchet derivative of F was assumed earlier in a study of a similar method inequality 1 with p = 2 was used and an estimate for the decay of gt as t was derived with the use of a comparison lemma and a closed form solution to a special Riccati s equation see [1]. It is not possible to extend the argument from [1] to the case p 2. The estimate of solutions to inequality 1 with p = 2 was also used in [2] in a study of a DSM for solving ill-posed operator equations. In this paper sufficient conditions on α β and γ are found which yield the global existence and an estimate of the rate of decay of solutions to 1. The method of the proof of these results is different from that in [1]. It does not require the Corresponding author. E-mail addresses: nguyenhs@math.ksu.edu N.S. Hoang ramm@math.ksu.edu A.G. Ramm. 0362-546X/$ see front matter 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.112
N.S. Hoang A.G. Ramm / Nonlinear Analysis 71 2009 2744 2752 2745 knowledge of a closed form solution of a differential equation. Discrete analogs of the inequality 1 are also found see Theorems 4 and 6. These new results can be applied to the study of the global existence of solutions to nonlinear PDE. The inequality 1 is formulated in such a way that the function g 0 is assumed to be defined on the interval [. However the proof of the basic result Theorem 1 in Section 2 makes it clear that one may assume nothing a priori about the domain of definition of a solution g 0 to 1: in the proof of Theorem 1 it is established that any nonnegative solution g 0 to 1 is defined on [ provided that conditions 2 and 3 are satisfied. There is a solution on [ + δ to 1 with sign replaced by = sign as follows from the standard results on the existence of the local solution to ordinary differential equations. Therefore inequality 1 has a local solution. Under the assumptions 2 and 3 of Theorem 1 its proof in Section 2 shows that δ = so that every g 0 solving 1 locally is defined on [ and satisfies estimate 4. Theorem 4 of Section 2 is a discrete analog of Theorem 1. This Theorem is broadly applicable in a study of iterative methods for solving operator equations. The paper is organized as follows: In Section 2 the main results namely Theorems 1 2 4 6 and 7 are formulated and proved. An upper bound for gt is obtained under some conditions on α β γ. This upper bound gives a sufficient condition for the relation lim t gt = 0 to hold and also gives a rate of decay of gt as t. In Section 3 a version of the DSM is studied. The main result in this Section is Theorem 8. In its proof an application of Theorem 1 is essential. 2. Main results Theorem 1. Let αt βt and γ t be continuous functions on [ and αt 0 t. Suppose there exists a function µt > 0 µ C 1 [ such that αt µ p t + βt 1 [ γ t µt ]. 2 µt µt Let gt 0 be a solution to inequality 1 such that µ g < 1. Then gt exists globally and the following estimate holds: 3 0 gt < 1 µt t. Consequently if lim t µt = lim gt = 0. t 4 5 t γ sds Proof. Denote wt := gte 0. Then inequality 1 takes the form where ẇt atw p t + bt w = g := g 0 6 at := αte 1 p γ sds Denote ηt = e t γ sds 0. µt From inequality 3 and relation 8 one gets w = g < 1 µ = η. It follows from the inequalities 2 6 and 9 that 1 ẇ α µ p + β 1 µ t γ bt := βte sds 0. 7 [ γ µt ] 0 = d µ dt t γ sds e 0 µt From the inequalities 9 and 10 it follows that there exists δ > 0 such that wt < ηt t + δ. To continue the proof we need two Claims. 8 9 = η. 10 t= 11
2746 N.S. Hoang A.G. Ramm / Nonlinear Analysis 71 2009 2744 2752 Claim 1. If wt ηt t [ T] T > ẇt ηt t [ T]. 12 13 Proof of Claim 1. It follows from inequalities 2 6 and the inequality wt ηt that ẇt e 1 p γ sds αt ep e t γ sds [ 0 γ µt µt µt = d dt Claim 1 is proved. Denote e γ sds µt t=t γ sds µ p t ] t γ sds + βte 0 = ηt t [ T]. 14 T := sup{δ R + : wt < ηt t [ + δ]}. 15 Claim 2. One has T =. Claim 2 says that every nonnegative solution gt to inequality 1 satisfying assumption 3 is defined globally. Proof of Claim 2. Assume the contrary i.e. T <. From the definition of T and the continuity of w and η one gets wt ηt. It follows from inequality 16 and Claim 1 that ẇt ηt t [ T]. This implies 16 17 wt w = T ẇsds Since w < η by assumption 3 it follows from inequality 18 that wt < ηt. T ηsds = ηt η. 18 Inequality 19 and inequality 17 with t = T imply that there exists an ɛ > 0 such that wt < ηt T t T + ɛ. This contradicts the definition of T in 15 and the contradiction proves the desired conclusion T =. Claim 2 is proved. It follows from the definitions of ηt and wt and from the relation T = that gt = e γ sds wt < e γ sds ηt = 1 µt t >. 21 Theorem 1 is proved. Let us prove that the strict inequality sign in 3 can be replaced by sign and inequality 4 remains valid if the < sign is replaced in 4 by the sign. 19 20 Theorem 2. Let α β γ and g be as in Theorem 1 µt > 0 µ C 1 [ and let condition 2 hold. Assume also that g µ 1. Then the following inequality holds: gt 1 µt t. 22 23
N.S. Hoang A.G. Ramm / Nonlinear Analysis 71 2009 2744 2752 2747 Proof. By Theorem 1 the desired conclusion holds if g µ < 1. Let g := b < µ 1. Denote by G b t the solution Gt 0 to the differential equation which is obtained from inequality 1 by replacing sign by = sign. Fix an arbitrarily large T > 0. The solution G b t exists on the interval [ T] by Theorem 1 and again by Theorem 1 the estimate 0 G b t < µ 1 t holds on this interval. Let b µ 1. From the continuity of G b t with respect to b uniform with respect to t [ T] T < and from the estimate 0 G b t < µ 1 t one concludes that Gt := lim b µ 1 G bt exists solves the differential equation 1 with sign replaced by = sign and 0 Gt µ 1 t. Since T > is arbitrarily large the conclusion of Theorem 2 is proved for the solution Gt of the differential equation 1. If g 0 solves differential inequality 1 gt Gt by the known comparison lemma from [1] p.99. Thus Theorem 2 is proved. Corollary 3. Let αt βt and γ t be continuous functions on [ and αt 0 t. Suppose there exists a function µt > 0 µ C 1 [ such that [ 0 αt θµ p 1 γ µt ] µt βt 1 θ µ [ γ µt µt ]. Let gt 0 be a solution to inequality 1 such that µ g 1. Then gt exists globally and the following estimate holds: u := du θ = const 0 1 24 dt 25 26 0 gt 1 t 0. µt Consequently if lim t µt = lim gt = 0. t 27 28 Let us consider a discrete analog of Theorem 1. We wish to study the following inequality: g n+1 g n h n γ n g n + α n g p n + β n h n > 0 0 < h n γ n < 1 p > 1 and the inequality: g n+1 1 γ n g n + α n g p n + β n n 0 0 < γ n < 1 p > 1 where g n β n γ n and α n are positive sequences of real numbers. Under suitable assumptions on α n β n and γ n we obtain an upper bound for g n as n. In particular we give sufficient conditions for lim n g n = 0 and estimate the rate of decay of g n as n. This result can be used in a study of evolution problems in a study of iterative processes and in a study of nonlinear PDE. Theorem 4. Let α n γ n and g n be nonnegative sequences of numbers and the following inequality holds: g n+1 g n or equivalently h n γ n g n + α n g p n + β n h n > 0 0 < h n γ n < 1 29 g n+1 g n 1 h n γ n + α n h n g p n + h nβ n h n > 0 0 < h n γ n < 1. 30 If there is a monotonically growing sequence of positive numbers n=1 such that the following conditions hold: α n µ p n + β n 1 γ n +1 h n g 0 1 µ 0 31 32 g n 1 n 0. 33 Therefore if lim n = lim n g n = 0.
2748 N.S. Hoang A.G. Ramm / Nonlinear Analysis 71 2009 2744 2752 Proof. Let us prove 33 by induction. Inequality 33 holds for n = 0 by assumption 32. Suppose that 33 holds for all n m. From inequalities 29 31 and from the induction hypothesis g n 1 n m one gets g m+1 g m 1 h m γ m + α m h m g p m + h mβ m 1 µ m 1 h m γ m + h m α m 1 µ m 1 h m γ m + h m µ m = 1 µ m µ m+1 µ m µ 2 m + h m β m γ m µ m+1 µ m µ m h m µ p m = 1 µ m+1 µ m+1 µ m 1 µ 2 m 1 µ m µ m+1 = 1 µ m+1 µ m 2 µ m+1 µ 2µ 1. 34 n m+1 µ m+1 Therefore inequality 33 holds for n = m+1. Thus inequality 33 holds for all n 0 by induction. Theorem 4 is proved. Corollary 5. Let α n γ n and g n be nonnegative sequences of numbers and the following inequality holds: g n+1 g n 1 h n γ n + α n h n g p n + h nβ n h n > 0 0 < h n γ n < 1. 35 If there is a monotonically growing sequence of positive numbers n=1 such that the following conditions hold: α n θµ p 1 n β n 1 θ γ n +1 h n γ n +1 h n θ = const 0 1 36 37 g 0 1 µ 0 38 g n 1 n 0. 39 Therefore if lim n = lim n g n = 0. Setting h n = 1 in Theorem 4 and in Corollary 5 one obtains the following results: Theorem 6. Let α β γ n and g n be sequences of nonnegative numbers and g n+1 g n 1 γ n + α n g p n + β n 0 < γ n < 1. 40 If there is a monotonically growing sequence n=1 > 0 such that the following conditions hold g 0 1 µ 0 α n µ p n + β n 1 γ n +1 h n n 0 41 g n 1 n 0. 42 Theorem 7. Let α β γ n and g n be sequences of nonnegative numbers and g n+1 g n 1 γ n + α n g p n + β n 0 < γ n < 1. 43 If there is a monotonically growing sequence n=1 > 0 such that the following conditions hold α n θµ p 1 n γ n µ n+1 θ = const 0 1 44
β n 1 θ γ n µ n+1 N.S. Hoang A.G. Ramm / Nonlinear Analysis 71 2009 2744 2752 2749 45 g 0 1 µ 0 46 g n 1 n 0. 47 3. Applications Let F : H H be a Fréchet-differentiable map in a real Hilbert space H. Assume that sup F u M = Mu 0 R u Bu 0 R 48 where M is a constant Bu 0 R := {u : u u 0 R} u 0 H is some element R > 0 and there is no restriction on the growth of MR as R i.e. arbitrary fast growing nonlinearities F are admissible. Consider the equation: Fv = f 49 in H. Assume that F 0 that is F is monotone: Fu Fv u v 0 u v H. 50 Let Eq. 49 have a solution possibly non-unique and denote by y the unique minimal-norm solution to 49. If F is monotone and continuous N f := {u : Fu = f } is a closed convex set in H see e.g. [1]. Such a set in a Hilbert space has a unique minimal-norm element. So the solution y is well defined. Let us assume that Fv Fu F uu v M p R u v p u v B0 R 51 where 1 < p < 2. For brevity let us denote M p := M p R. The main result of this Section is the following Theorem: Theorem 8. Assume that F is a monotone operator satisfying conditions 48 and 51 that Eq. 49 has a solution and y is its minimal-norm solution. Let u 0 be an arbitrary element in H. Assume that at = d c+t where 0 < b p 1 c max 1 2b b p 1 and d > 0 is sufficiently large so that condition 61 holds see below. Let ut be the solution to the following DSM: u = A 1 at [Fu + atu f ] u0 = u 0 52 where A := F ut and A a := A + ai. Then lim ut y = 0. t 53 Let us recall the following result see [1] p. 112: Lemma 9. Assume that Eq. 49 is solvable y is its minimal-norm solution and the operator F is monotone and continuous. Then lim V a y = 0 a 0 where V a solves the equation FV a + av a f = 0 54 and a 0 is a parameter.
2750 N.S. Hoang A.G. Ramm / Nonlinear Analysis 71 2009 2744 2752 Lemma 10. Let M p c 1 and g 0 be nonnegative constants and p 1 2. Then there exist a positive constant > 0 and a monotonically decaying function at > 0 at 0 such that the following conditions hold: p 1 [ M p at 1 2 a q t ȧt c 1 at a q [ t 2 g 0 a0 < 1. 1 q ȧt at ] 1 q ȧt at Proof. Choose the function at and positive constants b c and d such that at = d ] qp 1 = 1 55 c max 1 2bq 0 < b p 1 58 c + t b where the constant d > 0 will be specified later. Then q ȧt at = qb c + t qb c 1 t 0. 2 Thus inequality 55 holds if 4M p p 1 where the relation qp 1 = 1 was used. Choose 4M p q. Then inequality 55 is satisfied for any d > 0. Choose d max g 0 c b + 1 4c 1 b p 1. Then inequality 57 is satisfied. From the relations 58 and inequalities 61 and 59 one gets ȧt c 1 a q+1 t = c 1 b d q c + t c 1b c 1b 1 bq d q 4c 1 b = 1 4 1 [ 1 q ȧt ] 62 2 at where inequality 59 was used. This implies inequality 56. Lemma 10 is proved. Remark 11. One can choose d and so that the quantity a0 is uniformly bounded as M p. Indeed using inequality 60 one can choose 56 57 59 60 61 = 4M p q. 63 Using inequality 61 one can choose d = max g 0 c b + 1 4c 1 b p 1. 64 It follows from 63 and 64 and the assumption p 1 2 that the quantity a0 = d is bounded as M c b p. Indeed if M p because M p = 1 4 p 1 and p > 1. Moreover p 1 < as because p 1 < 1. Thus d = g 0 c b + 1 as so d < g c b 0 + 1 as. Proof Proof of Theorem 8. Denote wt := ut Vt gt := wt 65 where Vt solves Eq. 54 with a = at. Eq. 52 can be rewritten as ẇ = V A 1 at [Fu FV + atw]. From inequality 51 one gets 66 Fu FV + aw = A a uw + K K M p w p. 67 Multiplying 66 by w and using 67 one obtains gġ g 2 + M p A 1 at g 1+p + V g. 68
N.S. Hoang A.G. Ramm / Nonlinear Analysis 71 2009 2744 2752 2751 Multiply Eq. 54 with a = at by V y and use the monotonicity of F to get 0 = FV + atv Fy V y atv V y. 69 It follows from this inequality that Vt y t 0. 70 Differentiating Eq. 54 with a = at with respect to t one gets F V V + at V + ȧtv = 0. 71 This inequality and inequality 70 imply V = ȧta 1 ȧt at V at y 72 where the estimate A 1 1 a was used. This estimate holds because of the assumption A 0. From inequalities 68 and a 72 and the relation gt 0 one gets ġ gt + c 0 at g p + ȧt at c 1 c 0 := M p c 1 := y. 73 This inequality is inequality 1 with γ = 1 αt = c 0 at βt = c ȧt 1 at. 74 Let us now apply Corollary 3 with µt = a q t > 0 qp 1 = 1 θ = 1 2 75 where and at satisfy conditions 55 57. From inequalities 55 57 and Corollary 3 one concludes that gt < at. From the triangle inequality one gets ut y ut Vt + Vt y. 77 76 One has: ut Vt = gt at Using 70 one gets a0. Vt y Vt + y 2 y. 78 79 Inequalities 77 79 imply the following estimates: ut y at a0 + Vt y + 2 y =: R. 80 By Remark 11 one can choose d and such that the quantity a0 is uniformly bounded as M p. Thus one concludes that R can be chosen independently of M p = M p R. Inequality 80 implies that the trajectory of ut stays for all t 0 inside a ball By R where R is a sufficiently large fixed number. Since at 0 as t it follows from the first inequality in 80 and Lemma 9 that lim ut y = 0 t where we took into account that Vt = V at. Theorem 8 is proved. In conclusion let us outline a general approach to a study of the large time behavior of solutions to evolution equations based on Theorem 1 of this paper. A wide class of dynamical systems is described by the evolution equation: u = Atu + ht u + f u0 = u 0 81 82
2752 N.S. Hoang A.G. Ramm / Nonlinear Analysis 71 2009 2744 2752 where u = ut H H is a Hilbert space At is a linear operator function in H u ht u is a nonlinear operator in H and f = f t H. Without specifying here all the assumptions on A h and f let us assume that Re Au u γ t u 2 Re ht u u αt u 1+p f t βt p > 1. 83 Multiplying the evolution equation by u denoting ut := gt and using our assumptions 83 one gets ġ γ tgt + αtg p t + βt p > 1. 84 Theorem 1 is directly applicable and under the assumption of Theorem 1 one obtains the global existence of the solution ut and an estimate 4 of its large time behavior. An interesting novel result is an approach to studying by this method nonlinear parabolic-type problems with elliptic operator A which degenerates as t. Specifically if A is an elliptic operator with coefficients depending on t and γ t 0 as t one obtains inequality 1 for gt = ut and γ t 0. Choosing µt properly one can satisfy assumptions 2 and 3 and apply Theorem 1. For example let γ t = M 1 1 + t 1 αt = M 2 1 + t a βt = M 3 1 + t b µt = 1 + t ν. It is not difficult to choose a b ν and so that assumptions 2 and 3 hold. References [1] A.G. Ramm Dynamical Systems Method for Solving Operator Equations Elsevier Amsterdam 2007. [2] N.S. Hoang A.G. Ramm Dynamical systems gradient method for solving nonlinear operator equations with monotone operators Acta Appl. Math. in press doi:10.1007/s10440-008-9308-1.