GENERALIZATION OF GRONWALL S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS

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1 Communications in Applied Analysis 19 (215), GENERALIZATION OF GRONWALL S INEQUALITY AND ITS APPLICATIONS IN FUNCTIONAL DIFFERENTIAL EQUATIONS TINGXIU WANG Department of Mathematics, Texas A&M University-Commerce Commerce, TX USA tingxiu.wang@tamuc.edu ABSTRACT. In this paper, we briefly review the recent development of research on Gronwall s inequality. Then obtain a result for the following nonlinear integral inequality: m ( w(u(t)) K + (t ) αi(t) f i (s) H ij (u(s))g ij max α i j=1 s h ξ s u(ξ) ) ds. As an application, we study the abstract functional differential equation, du dt = f(t, u t ) with Lyapunov s second method. Then, we obtain an estimate of solutions of functional differential equations, u = F(t, u t ) with conditions like: i) W 1 ( u(t) X ) V (t, u t ) W 2 ( u t CX ) ii) V (3.1) (t, u t). AMS (MOS) Subject Classification. 26D1, 34A4, 34D2, 34K INTRODUCTION In 1919, Gronwall [7] gave the following lemma when he studied a system of differential equations with a parameter. Theorem 1.1 (Gronwall s Original Inequality). Let α, a, b and h be nonnegative constants, and u : [α, α + h] [, ) be continuous. If u(t) then α [bu(s) + a]ds, α t α + h, u(t) ahe bh, α t α + h. His lemma stayed basically quietly until Bellman generalized it in 1943, which is now commonly known as Gronwall s Inequality, or Gronwall-Bellman s Inequality. This version of Gronwall s inequality can be found in many references, for example [1, 5, 12]. Received December 15, $15. c Dynamic Publishers, Inc.

2 68 T. WANG Theorem 1.2 (Gronwall s Inequality). Let α, β and c be nonnegative constants, and u, f : [α, β] [, ) continuous. If then u(t) c + α f(s)u(s)ds, α t β, u(t) ce R t α f(s)ds, α t β. In 1958, Bellman [3] generalized Theorem 1.2, his own result, by allowing that c is a nonnegative and nondecreasing function, which is stated as following, and can be founded in many references, for example [8, 12]. Theorem 1.3 (Gronwall-Bellman s Inequality). If u(t) and α(t) are real valued continuous functions on [a, b], α(t) is nondecreasing, and β(t) is integrable on [a, b] with then u(t) α(t) + a β(s)u(s)ds for a t b, u(t) α(t)e R t a β(r)dr, for a t b. Today, Gronwall s inequality has been found very useful in research of boundedness and stability of differential equations; research of inequalities of the Gronwall type has exploded; and generalization of Gronwall s inequality has gone into many different directions. For example, the integral in Gronwall s inequality is generalized to iterated integrals or a sum of integrals; the function u(t) is generalized to a format of W(u(t)) and from a one-variable function to a two-variable function with double integrals; the upper limit of the integral is generalized to a function, instead of a simple variable t; and the integral inequality is generalized to a difference inequality. Applications of this type of inequalities are also expanded from ordinary differential equations to functional differential equations, fractional differential equations and difference equations. In 1998, Pachpatte published a book [12] summarizing the development of inequalities of the Gronwall type up to Here are some examples of new generalizations of the Gronwall type in the past a few years. In 27, Pachpatte [13] investigated the following two types of inequalities of the Gronwall type involving double integrals and summations: u(x, y) c + + x s p(s, y)u(s, y)ds + x y g(σ, τ)u(σ, τ)dτdσ + f(s, t)[u(s, t) a b h(σ, τ)u(σ, τ)dτdσ]dtds,

3 NONLINEAR INTEGRAL INEQUALITIES 681 and n 1 n 1 u(n, m) c + p(s, m)u(s, m) + s= s 1 t 1 + g(σ, τ)u(σ, τ) + σ= τ= s= α m 1 t= σ= τ= f(s, t)[u(s, t) β h(σ, τ)u(σ, τ)]. In 212, Bohner, Hristova and Stefanova [4] studied the following inequality: ψ(u(t)) k + + α i (t ) βj (t) m j=1 β j (t ) f i (s)u p (s)ω i (u(s))ds ( ) g j (s)u p (s) ω j max u(ξ) ds. (1.1) ξ [s h,s] In 213, Lin [11] gave the following result of the Gronwall type when he considered fractional differential equations: Let β i > be constants; b i (t) defined on [, T) be bounded, continuous and monotonically increasing; i = 1, 2,..., n, a(t) and u(t) be continuous. If u(t) a(t) + b(t) (t s) β 1 u(s)ds, t [, T), (1.2) then k u(t) a(t) + [b i (t)γ(β i ( )] k ) k=1 i =1 Γ β i (t s) P k β i 1 a(s)ds. Meanwhile, by reversing the direction of Inequality (1.2), Lin also gave the following result: Let β i > be constants; b i (t) defined on [, T) be bounded, continuous, nonnegative and monotonically increasing; i = 1, 2,..., n; u(t) be continuous. If u(t) b i (t) (t s) β i 1 u(s)ds, t [, T), then u(t), t [, T). In 214, Wang, Zheng and Guo [14] studied the following nonlinear delay sumdifference inequality: u(m, n) a(m, n) + k α i (m 1) s=α i () β i (n 1) s=β i () f i (m, n, s, t)w i (u(s, t)), m, n N.

4 682 T. WANG 2. A NONLINEAR INTEGRAL INEQUALITY In this section, we generalize Inequality (1.1) to the following inequality and simplify the proof of [4]: w(u(t)) K + α i (t ) j=1 m ( ) f i (s) H ij (u(s))g ij max u(ξ) ds, t t < T. s h ξ s α(t ) (2.1) Notations and conditions for this inequalty are as following: Let h >, K >, t, and T are constants, t < T, R + = [, ). (A1) α i C 1 ([t, T), R + ) are nondecreasing and α i (t) t, t [t, T), i = 1, 2,..., n; (A2) f i C([, T), R + ) for i = 1, 2,..., n; (A3) H ij, G ij C(R +, R + ) are nondecreasing, and H ij (x) >, G ij (x) > if x > ; (A4) w C(R +, R + ) is increasing, w() =, and lim t w(t) = ; (A5) u C([ h, T), R + ). Theorem 2.1. Let u(t) satisfy (2.1) with conditions (A1) through (A5). Then [ ( )] α(t) u(t) w 1 K + W 1 f i (s)ds, t t < T where W(r) = r 1 H m (w 1 (K + s))g m (w 1 (K + s)) ds, r < H(r) = max {H ij(r)}, G(r) = max {G ij(r)}. i n, j m i n, j m Proof. Define Z(t) = K + α i (t ) j=1 m ( ) f i (s) H ij (u(s))g ij max u(ξ) ds, t t < T. s h ξ s Then Z(t) is nondecreasing, and w(u(t)) Z(t), t t < T. By (A4), w 1 exists and has the same properties as those of w. Therefore, In addition, u(t) w 1 (Z(t)), t t < T. max u(ξ) max s h ξ s s h ξ s w 1 (Z(ξ)) = w 1 (Z(s)), t s < T.

5 NONLINEAR INTEGRAL INEQUALITIES 683 Let That is Therefore, Z(t) K + K + = K + R(t) = α i (t ) α i (t ) α i (t ) α i (t ) f i (s) f i (s) m j=1 ( ) H(u(s))G max s h ξ s ds m H(w 1 (Z(s)))G(w 1 (Z(s)))ds j=1 f i (s)h m (w 1 (Z(s)))G m (w 1 (Z(s)))ds. f i (s)h m (w 1 (Z(s)))G m (w 1 (Z(s)))ds. Clearly, Z(t) K + R(t). Then R (t) = f i (α i (t))h m (w 1 (Z(α i (t))))g m (w 1 (Z(α i (t))))α i (t) f i (α i (t))h m (w 1 (Z(t)))G m (w 1 (Z(t)))α i (t) f i (α i (t))h m (w 1 (K + R(t)))G m (w 1 (K + R(t)))α i (t) = H m (w 1 (K + R(t)))G m (w 1 (K + R(t))) R (t) H m (w 1 (K + R(t)))G m (w 1 (K + R(t))) d(w(r(t))) dt f i (α i (t))α i (t). f i (α i (t))α i (t). f i (α i (t))α i (t). Integrate the above inequality from t to t, W(R(t)) W(R(t )) f i (α i (s))α i (s)ds = t W(R(t)) α(t) α(t ) R(t) W 1 ( α(t) α(t ) f i (s)ds ) f i (s)ds Z(t) K + W 1 ( α(t) u(t) w 1 [ α(t ) K + W 1 ( α(t) α(t ) ) f i (s)ds α(t) α(t ) )] f i (s)ds f i (s)ds.

6 684 T. WANG Inequality (2.1) generalizes Inequality (1.1). Let us consider a very simple case of Inequality (2.1) for applications. Corollary 2.2. Given or w(u(t)) K + w(u(t)) K + t f(s)h(u(s))ds, t t < T, (2.2) t f(s)h( max s h ξ s u(ξ))ds, t t < T, (2.3) where f, H, w and u satisfy (A2), (A3), (A4) and (A5), respectively. Then [ ( )] u(t) w 1 K + W 1 f(s)ds, t where W(r) = r 1 ds, r <. H(w 1 (K+s)) 3. APPLICATIONS IN CONJUNCTION WITH LYAPUNOV S SECOND METHOD Let us consider the general abstract functional differential equations with finite delay du dt = F(t, u t). (3.1) First, let us set forth some notation and terminology. Suppose that (X, X ) and (Y, Y ) are Banach spaces and X Y with u Y N u X for some constant N > and all u X Y. Denote C X = C([ h, ],X), where h > is a constant and u CX = sup{ u(s) X : h s } for u C X. Thus (C X, CX ) is also a Banach space. If u : [t h, β) X for some β > t, define u t C X by u t (s) = u(t + s) for s [ h, ], where t [t, β). For a positive constant H, by C XH we denote the subset of C X for which φ CX < H for each φ C X. F : R + C X Y is a function. Here R + = [, ). du dt is the strong derivative [9] of u at t in (X, X ). Definition 3.1. A function u(t, φ) is said to be a solution of Eq. (3.1) with initial function φ C X at t = t and having value u(t, t, φ) if there is a β > such that u : [t h, t + β) X Y with u t C X Y for t t < t + β, u t = φ, and u(t, t, φ) satisfies Eq. (3.1) on (t, t + β) in (Y, Y ). In this section, a wedge, denoted by W i, is a continuous and strictly increasing function from R + R + with W i () =, which is related to properties of continuous

7 NONLINEAR INTEGRAL INEQUALITIES 685 scalar functionals (called Lyapunov functionals) V : R + C XH R + which are differentiated along the solutions of Eq. (3.1) by the relation V (3.1)(t, φ) = sup lim δ + sup[v (t + δ, u t+δ(t, φ)) V (t, φ)]/δ and V (t, ) = for all t R +, where u(t, φ) is a solution of Eq. (3.1) satisfying u t = φ and the first sup runs over such solutions, since the solution of Eq. (3.1) may not be unique. Detailed consequences of this derivative are discussed in [6], [8], [9]. Lyapunov s second method has generated numerous results on stability and boundedness. For example, please refer to Burton [6], Hale and Lunel [8], Ladas, Lakshmikantham and Leela [9, 1], Wang [2, 21], and Wu [22]. The following are two typical theorems [6] about uniform stability and uniformly asymptotical stability, which are stated for a system of ordinary functional differential equations: X = F(t, X t ), X R n. (3.2) Theorem 3.1. Let D > be a constant, V (t, ϕ t ) be a scalar continuous function in (t, ϕ t ) and locally Lipschitz in ϕ t when t t < and ϕ t C(t) with ϕ t < D. Suppose that F(t, ) =, V (t, ) =, and i) W 1 ( ϕ(t) ) V (t, ϕ t ) < W 2 ( ϕ t ), ii) V (3.2) (t, ϕ t). Then X = is uniformly stable. Theorem 3.2. Let D > be a constant, V (t, ϕ t ) be a scalar continuous function in (t, ϕ t ) and locally Lipschitz in ϕ t when t t < and ϕ t C(t) with ϕ t < D. Suppose that F(t, ) =, V (t, ) =, and i) W 1 ( ϕ(t) ) V (t, ϕ t ) < W 2 ( ϕ(t) ) + W 3 ( t h ϕ(s) 2 ds), ii) V (3.2) (t, ϕ t) W 4 ( ϕ(t) ). Then X = is uniformly asymptotically stable. It has been an interest for a long time if we can use conditions similar to (i) and (ii) in the above theorems to obtain an inequality about solutions of Equation (3.2). Wang has published a few papers [15, 16, 17, 18, 19] on this type of research. Here is a typical one. The proof is done by application of Theorem 1.3 (Gronwall-Bellman s Inequality). Theorem 3.3 (Wang, [17]). Let V : R + C XH R + be continuous and D : R + C XH R + be continuous along the solutions of Eq. (3.1), and η, L, and P : R + R + be integrable. Suppose the following conditions hold: i) W 1 ( u(t) X ) V (t, u t ) W 2 (D(t, u t )) + t h L(s)W 1( u(s) X )ds, ii) V (3.1) (t, u t) η(t)w 2 (D(t, u t )) + P(t).

8 686 T. WANG Then the solutions of Eq. (3.1), u(t) = u(t, t, φ), satisfy the following inequality: [ ] R s t W 1 ( u(t) X ) K + P(s)e η(r)dr R R t s+h t [ η(s)+l(s)(e s ds e η(r)dr 1)]ds, t t, t where K = V (t, φ) + [e R t +h t η(r)dr 1] h L(s + t )W 1 ( φ(s) X )ds. (3.3) Now, as an application of Theorem 2.1, we prove another inequality with similar conditions to those in Theorem 3.1. Theorem 3.4. Let V : R + C XH R + be continuous and D : R + C XH R + be continuous along the solutions of Eq. (1). Suppose the following conditions hold: i) W 1 ( u(t) X ) V (t, u t ) W 2 ( u t CX ) ii) V (3.1) (t, u t). Then where [ u(t) X W1 1 V (t, u t ) + W 1 (t t ) ], t t, W(r) = r 1 W 2 (W1 1 ds. (V (t, u t )e t + s)) Proof. For convenience, let us denote V (t) = V (t, u t ). With Condition (ii), V (3.1)(t) W 2 ( u t CX ) + W 2 ( u t CX ) V (t) + W 2 ( u t CX ) Thus, V (t) + V (t) W 2 ( u t CX ). Multiplying e t, we get That is d(v (t)et ) dt Thus, With Condition (i), V (t)e t + V (t)e t W 2 ( u t CX )e t. W 2 ( u t CX )e t. Integrating from t to t, we get V (t)e t V (t )e t Apply Corollary 2.2, we get t W 2 ( u s CX )e s ds e t V (t) V (t )e t+t + V (t ) + t W 2 ( u s CX )ds t W 2 ( u s CX )ds. W 1 ( u(t) X ) V (t, u t ) V (t ) + u(t) X W 1 1 t W 2 ( u s CX )ds. t W 2 ( u s CX )ds. [ V (t ) + W 1 (t t ) ].

9 NONLINEAR INTEGRAL INEQUALITIES 687 ACKNOWLEDGMENTS This research was supported by a Texas A&M University-Commerce Internal Research Grant. REFERENCES [1] R. Bellman, The stability of solutions of linear differential equations, Duke Math. J. 1 (1943) no. 4, [2] R. Bellman, Stability Theory of Differential Equations, McGraw-Hill Book Company, Inc., New York, [3] R. Bellman, Asymptotic series for the solutions of linear differential-difference equations, Rendiconti del Circolo Matematico di Palermo 7 (1958) no. 2, [4] M. Bohner, S. Hristova, K. Stefanova, Nonlinear integral inequalities involving maxima of the unknown scalar functions, Math. Inequal. Appl. 15 (212), no. 4, [5] T. A. Burton, Volterra Integral and Differential Equations. Second Edition, in Mathematics in Science and Engineering, 22, Elsevier, New York, 25. [6] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-Differential Equations, in Mathematics in Science and Engineering, 178, Academic Press, Orlando, Florida, [7] T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. 2 (1919), no. 4, [8] J. Hale and S. Lunel, Introduction to Functional-Differential Equations, in Applied Matheamtics Science, 99, Springer, New York, [9] G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, in Mathematics in Science and engineering, Vol. 85, Academic Press, New York-London, [1] V. Lakshmikantham, and S. Leela, Differential and Integral Inequalities, Vol. I and II, Academic Press, New York, [11] Shi-you Lin, Generalized Gronwall inequalities and their applications to fractional differential equations, J. Inequal. Appl. 213 (213), 9 pp. [12] B. G. Pachpatte, Inequalities for Differential and Integral Equations, in Mathematics in Science and Engineering, 197, Academic Press, San Diego, [13] B. G. Pachpatte, On a new integral inequality applicable to certain partial integrodifferential equations. Inequality Theory and Applications. Vol. 5, , Nova Sci. Publ., New York, 27. [14] H. Wang, K. Zheng, C. Guo, Chun-xiang, Nonlinear discrete inequality in two variables with delay and its application, Acta Math. Appl. Sin. Engl. Ser. 3 (214), no. 2, [15] T. Wang, Lower and upper bounds of solutions of functional differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Analy. 2 (213) no. 1, [16] T. Wang, Inequalities of Solutions of Volterra Integral and Differential Equations, Electron. J. Qual. Theory Differ. Equ. 29, Special Edition I, No. 28, 1 pp. [17] T. Wang, Exponential stability and inequalities of abstract functional differential equations, J. Math. Anal. Appl. 342 (26), no.2, [18] T. Wang, Inequalities and stability in a linear scalar functional differential equation, J. Math. Anal. Appl. 298 (24), no. 1,

10 688 T. WANG [19] T. Wang, Wazewski s inequality in a linear Volterra integrodifferential equation, Volterra Equations and Applications, edited by C. Corduneanu and I. W. Sandberg, Vol. 1: Stability and Control: Theory, Methods and Applications, Gordon and Breach, Amsterdam, 2. [2] T. Wang, Stability in abstract functional-differential equations I: General theorems, J. Math. Anal. Appl. 186 (1994), [21] T. Wang, Stability in abstract functional-differential equations II: Applications, J. Math. Anal. Appl. 186 (1994), [22] J. Wu, Theory and Applications of Partial Functional Differential Equations, in Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996.