Journal of Inequalities in Pure and Applied Mathematics

Transcription

1 Journal of Inequalities in Pure and Applied Mathematics SOME NEW DISCRETE NONLINEAR DELAY INEQUALITIES AND APPLICATION TO DISCRETE DELAY EQUATIONS WING-SUM CHEUNG AND SHIOJENN TSENG Department of Mathematics University of Hong Kong Hong Kong Department of Mathematics Tamkang University Tamsui, Taiwan volume 7, issue 4, article 122, Received 07 September, 2005; accepted 27 January, Communicated by: S.S. Dragomir Abstract Home Page c 2000 Victoria University ISSN electronic):

2 Abstract In this paper, some new discrete Gronwall-Bellman-Ou-Iang-type inequalities are established. These on the one hand generalize some existing results and on the other hand provide a handy tool for the study of qualitative as well as quantitative properties of solutions of difference equations Mathematics Subject Classification: 26D10, 26D15, 39A10, 39A70. Key words: Gronwall-Bellman-Ou-Iang-type Inequalities, Discrete inequalities, Difference equations. 1 Introduction Discrete Inequalities with Delay Immediate Consequences Application References Page 2 of 36

3 1. Introduction It is widely recognized that integral inequalities in general provide an effective tool for the study of qualitative as well as quantitative properties of solutions of integral and differential equations. While most integral inequalities only give the global behavior of the unknown functions in the sense that bounds are only obtained for integrals of certain functions of the unknown functions), the Gronwall-Bellman type see, e.g. [3] [8], [10] [12], [15] [18]) is particularly useful as they provide explicit pointwise bounds of the unknown functions. A specific branch of this type of inequalities is originated by Ou-Iang. In his paper [13], in order to study the boundedness behavior of the solutions of some 2nd order differential equations, Ou-Iang established the following beautiful inequality. Theorem 1.1 Ou-Iang [13]). If u and f are non-negative functions on [0, ) satisfying u 2 x) c for some constant c 0, then ux) c + x 0 x 0 fs)us)ds, fs)ds, x [0, ), x [0, ). While Ou-Iang s inequality is interesting in its own right, it also has numerous important applications in the study of differential equations see, e.g., [2, 3, 9, 11, 12]). Over the years, various extensions of Ou-Iang s inequality have been established. These include, among others, works of Agarwal [1], Page 3 of 36

4 Ma-Yang [10], Pachpatte [14] [18], Tsamatos-Ntouyas [19], and Yang [20]. Among such extensions, the discretization is of particular interest because analogous to the continuous case, discrete versions of integral inequalities should, in our opinion, play an important role in the study of qualitative as well as quantitative properties of solutions of difference equations. It is the purpose of this paper to establish some new discrete Gronwall- Bellman-Ou-Iang-type inequalities giving explicit bounds to unknown discrete functions. These on the one hand generalize some existing results in the literature and on the other hand give a handy tool to the study of difference equations. An application to a discrete delay equation is given at the end of the paper. Page 4 of 36

5 2. Discrete Inequalities with Delay Throughout this paper, R + = 0, ) R, Z + = R + Z, and for any a, b R, R a = [a, ), Z a = R a Z, Z [a,b] = Z [a, b]. If X and Y are sets, the collection of functions of X into Y, the collection of continuous functions of X into Y, and that of continuously differentiable functions of X into Y are denoted by FX, Y ), CX, Y ), and C 1 X, Y ), respectively. As usual, if u is a real-valued function on Z [a,b], the difference operator on u is defined as un) = un + 1) un), n Z [a,b 1]. In the sequel, summations over empty sets are, as usual, defined to be zero. The basic assumptions and initial conditions used in this paper are the following: Assumptions A1) f, g, h, k, p FZ 0, R 0 ) with p non-decreasing; A2) w CR 0, R 0 ) is non-decreasing with wr) > 0 for r > 0; A3) σ FZ 0, Z) with σs) s for all s Z 0 and < a := inf{σs) : s Z 0 0; A4) ψ FZ [a,0], R 0 ); and A5) φ C 1 R 0, R 0 ) with φ non-decreasing and φ r) > 0 for r > 0. Initial Conditions I1) xs) = ψs) for all s Z [a,0] ; I2) ψ σs)) φ 1 ps)) for all s Z 0 with σs) 0. Page 5 of 36

6 Theorem 2.1. Under Assumptions A1) A5), if x FZ a, R 0 ) is a function satisfying the nonlinear delay inequality 2.1) φ xn)) pn) + φ x σs))) {fs) + gs)x σs)) + hs)w x σs))) for all n Z 0 with initial conditions I1) I2), then )] 2.2) xn) Φ {Φ 1 exp gs) φ 1 pn)) + fs) for all n Z [0,], where Φ CR 0, R) is defined by Φr) := r 1 ds ws), r > 0, n 1 + exp gs) ) n 1 and 0 is chosen such that the RHS of 2.2) is well-defined, that is, )] Φ exp gs) φ 1 pn)) + fs) for all n Z [0,]. + exp n 1 gs) ) n 1 ht) ht) I m Φ Page 6 of 36

7 Proof. Fix ε > 0 and N Z [0,]. Define u : Z [0,N] R 0 by 2.3) un) := φ {ε 1 + pn) + φ x σt))) [ft) + gt)x σt)) + ht)w x σt)))]. By A5), u is non-decreasing on Z [0,N]. For any n Z [0,N], by A5) again, 2.4) un) φ 1 ε + pn)) > 0. As φ un)) > φ xn)), we have 2.5) un) > xn). Next, observe that if σn) 0, then by A3), σn) Z [0,N] and so x σn)) < u σn)) un). On the other hand, if σn) 0, then by A3) again, σn) Z [a,0] and so by I1), I2), A1), A5) and 2.4), x σn)) = ψ σn)) φ 1 pn)) φ 1 pn)) φ 1 pn) + ε) un). Hence we always have 2.6) x σn)) un) for all n Z [0,N]. Page 7 of 36

8 Therefore, for any s Z [0,N 1], by 2.3) and 2.6), φ u)s) = φ us + 1)) φ us)) = φ x σs))) {fs) + gs)x σs)) + hs)w x σs))) φ us)) {fs) + gs)us) + hs)w us)). On the other hand, by the Mean Value Theorem, we obtain φ u)s) = φ us + 1)) φ us)) = φ ξ) us) for some ξ [us), us + 1)]. Observe that by 2.4) and A5), φ ξ) > 0. Thus by the monotonicity of φ, for any s Z [0,N 1], Summing up, we have us) φ us)) {fs) + gs)us) + hs)w us)) φ ξ) fs) + gs)us) + hs)w us)). un) u0) = us) n 1 fs) + hs)w us)) + gs)us), Page 8 of 36

9 or [ ] un) φ 1 ε + pn)) + fs) + hs)w us)) + gs)us) for all n Z [0,N]. Hence by the discrete version of the Gronwall-Bellman inequality see, e.g., [16, Corollary 1.2.5]), [ ] un) φ 1 ε + pn)) + fs) + hs)w us)) exp gs) 2.7) [ ] N 1 φ 1 ε + pn)) + fs) + hs)w us)) gs) N 1 exp for all n Z [0,N]. Denote by vn) the RHS of 2.7). Then v is non-decreasing and for all n Z [0,N], 2.8) un) vn). Therefore, for any t Z [0,N 1], vt) = vt + 1) vt) N 1 = ht)w ut)) exp gs) N 1 ht)w vt)) exp gs). Page 9 of 36

10 On the other hand, by the Mean Value Theorem, we have Φ v)t) = Φ vt + 1)) Φ vt)) = Φ η) vt) = 1 wη) vt) for some η [vt), vt + 1)]. Observe that by 2.4), 2.8), and A2), wη) > 0. Therefore, as w is non-decreasing, Φ v)t) 1 N 1 wη) ht)w vt)) exp gs) N 1 ht) exp gs) for all t Z [0,N 1]. Summing up, we have On the other hand, N 1 Φ v)t) ht) exp gs). Φ v)t) = Φ vn)) Φ v0)) Page 10 of 36

11 therefore, = Φ vn)) Φ Φ vn)) Φ exp N 1 exp gs) N 1 gs) φ 1 ε + pn)) + φ 1 ε + pn)) + for all n Z [0,N]. In particular, taking n = N we have Φ vn)) Φ exp N 1 Since N Z [0,] is arbitrary, gs) φ 1 ε + pn)) + N 1 fs) N 1 )] fs) )] N 1 + ht) exp gs) N 1 fs) N 1 + exp gs) )] Φ vn)) Φ exp gs) φ 1 ε + pn)) + fs) n 1 + exp gs) )] ) N 1, ht). ) n 1 ht) Page 11 of 36

12 for all n Z [0,]. Hence )] vn) Φ {Φ 1 exp gs) φ 1 ε + pn)) + fs) n 1 + exp gs) and so by 2.5) and 2.8), ) n 1 xn) < un) vn) )] Φ {Φ 1 exp gs) φ 1 ε + pn)) + fs) n 1 + exp gs) for all n Z [0,]. Finally, letting ε 0 +, we conclude that ht) ) n 1 )] xn) Φ {Φ 1 exp gs) φ 1 pn)) + fs) + exp gs) for all n Z [0,]. ht) ) n 1 ht) Page 12 of 36

13 Remark 1. In many cases the non-decreasing function w satisfies 1 ds = ws). For example, w = constant > 0, ws) = s, etc., are such functions. In such cases Φ ) = and so we may take, that is, 2.2) is valid for all n Z 0. Theorem 2.2. Under Assumptions A1) A5), if x FZ a, R 0 ) is a function satisfying the nonlinear delay inequality { φ xn)) pn) + φ x σs))) fs) + gs)x σs)) s 1 + hs) kt)w x σt))) for all n Z 0 with initial conditions I1) I2), then )] 2.9) xn) Φ {Φ 1 exp gs) φ 1 pn)) + fs) + exp gs) ) n 1 s 1 hs)kt) for all n Z [0,β], where Φ CR 0, R) is as defined in Theorem 2.1, and β 0 is chosen such that the RHS of 2.9) is well-defined, that is, )] Φ exp gs) φ 1 pn)) + fs) Page 13 of 36

14 + exp gs) ) n 1 s 1 hs)kt) I m Φ for all n Z [0,β]. Proof. Fix ε > 0 and M Z [0,β]. Define u : Z [0,M] R 0 by [ 2.10) un) := φ {ε 1 + pm) + φ x σδ))) fδ) + gδ)x σδ)) δ=0 ] δ 1 +hδ) kt)w x σt))). By A5), u is non-decreasing on Z [0,M]. For any n Z [0,M], by A5) again, 2.11) un) φ 1 ε + pm)) > 0. As φ un)) > φ xn)), we have 2.12) un) > xn). Using the same arguments as in the derivation of 2.6) in the proof of Theorem 2.1, we have 2.13) x σn)) un) for all n Z [0,M]. Page 14 of 36

15 Hence for any s Z [0,M 1], by 2.10) and 2.13), φ u)s) = φ us + 1)) φ us)) { s 1 = φ x σs))) fs) + gs)x σs)) + hs) kt)w x σt))) { s 1 φ us)) fs) + gs)us) + hs) kt)w ut)). On the other hand, by the Mean Value Theorem, φ u)s) = φ us + 1)) φ us)) = φ ξ) us) for some ξ [us), us + 1)]. Observe that by 2.12) and A5), φ ξ) > 0. Thus by the monotonicity of φ, for any s Z [0,M 1], { us) φ us)) s 1 fs) + gs)us) + hs) kt)w ut)) φ ξ) Summing up, we have s 1 fs) + gs)us) + hs) kt)w ut)). un) u0) = us) Page 15 of 36

16 s 1 fs) + hs) kt)w ut)) + gs)us), or [ ] s 1 un) φ 1 ε + pm)) + fs) + hs) kt)w ut)) + gs)us) for all n Z [0,M]. Hence by the discrete version of the Gronwall-Bellman inequality see, e.g., [16, Corollary 1.2.5]), [ un) φ 1 ε + pm)) + fs) 2.14) [ ] s 1 + hs) kt)w ut)) exp gs) φ 1 ε + pm)) + M 1 fs) ] s 1 + hs) kt)w ut)) gs) M 1 exp for all n Z [0,M]. Denote by vn) the RHS of 2.14). Then v is non-decreasing and for all n Z [0,M], 2.15) un) vn). Page 16 of 36

17 Therefore, for any δ Z [0,M 1], vδ) = vδ + 1) vδ) δ 1 ) = hδ) kt)w ut)) hδ) δ 1 ) kt)w vt)) hδ)w vδ)) δ 1 ) kt) On the other hand, by the Mean Value Theorem, gs) M 1 exp gs) M 1 exp gs). M 1 exp Φ v)δ) = Φ vδ + 1)) Φ vδ)) = Φ η) vδ) = 1 wη) vδ) for some η [vδ), vδ + 1)]. Observe that by 2.11), 2.14), and A2), wη) > 0. Therefore, as w is non-decreasing, Φ v)δ) 1 hδ)w vδ)) wη) δ 1 ) hδ) kt) δ 1 ) kt) gs) M 1 exp gs) M 1 exp Page 17 of 36

18 for all δ Z [0,M 1]. Summing up, we have or Φ v)δ) hδ) δ=0 δ=0 Φ vn)) Φ v0)) + hδ) δ=0 δ 1 ) kt) δ 1 ) kt) ) M 1 = Φ φ 1 ε + pm)) + fs) + hδ) δ=0 δ 1 ) kt) gs), M 1 exp gs) M 1 exp M 1 exp for all n Z [0,M]. In particular, taking n = M this yields Φ vm)) Φ φ 1 ε + pm)) + M 1 fs) M 1 + hδ) δ=0 ) ] gs) M 1 exp gs) M 1 exp δ 1 ] gs) ) kt) gs). M 1 exp Page 18 of 36

19 Since M Z [0,β] is arbitrary, ) ] Φ vn)) Φ φ 1 ε + pn)) + fs) exp gs) + hδ) δ 1 ) kt) exp gs) δ=0 for all n Z [0,β]. Hence ) ] vn) Φ {Φ 1 φ 1 ε + pn)) + fs) exp gs) + hδ) δ 1 ) kt) exp gs) δ=0 and so by 2.12) and 2.15), xn) < un) vn) ) ] Φ {Φ 1 φ 1 ε + pn)) + fs) exp gs) + hδ) δ 1 ) kt) exp gs) δ=0 Page 19 of 36

20 for all n Z [0,β]. Finally, letting ε 0 +, we conclude that )] xn) Φ {Φ 1 exp gs) φ 1 pn)) + fs) for all n Z [0,β]. + exp gs) ) n 1 δ 1 hδ)kt) Remark 2. Similar to the previous remark, in case Φ ) =, 2.9) holds for all n Z 0. Theorem 2.3. Under Assumptions A1), A3) and A4), if x FZ a, R 0 ) is a function satisfying the nonlinear delay inequality x r n) c r + x r σs)) {fs) + gs)x r σs)), n Z 0, with initial conditions I1) and δ=0 I3) ψ σs)) c for all s Z 0 with σs) 0, where r, c > 0 are constants, then [ n ) xn) c r 1 fs)) n s=1 n 1 gs) t=s 1 ft)) ] 1 r for all n Z [0,γ], where γ 0 is chosen such that the RHS of 2.16) is welldefined. Page 20 of 36

21 Proof. Define u FZ 0, R 0 ) by 2.17) u r n) := c r + x r σs)) {fs) + gs)x r σs)), n Z 0. Clearly, u 0 is non-decreasing and 2.18) xn) un) for all n Z 0. Similar to the derivation of 2.6) in the proof of Theorem 2.1, we easily establish By 2.17), for any n Z 0, x σn)) un) for all n Z 0. u r n) = u r n + 1) u r n) = x r σn)) {fn) + gn)x r σn)) u r n) {fn) + gn)u r n) u r n + 1) {fn) + gn)u r n). As u0) = c, by elementary analysis, we infer from 2.17) that 2.19) un) yn) for all n Z [0,ρ] where Z [0,ρ] is the maximal lattice on which the unique solution yn) to the discrete Bernoulli equation y r n) = y r n + 1) {fn) + gn)y r n), n Z ) y0) = c Page 21 of 36

22 is defined. Now the unique solution for 2.20) is see, e.g., [1]) 2.21) yn) = [ 1 fs)) c r n 1 n s=1 n 1 gs) t=s 1 ft)) ] 1 r for all n Z [0,γ]. The assertion now follows from 2.18), 2.19) and 2.21). Page 22 of 36

23 3. Immediate Consequences Direct application of the results in Section 2 yields the following consequences immediately. Corollary 3.1. Under Assumptions A1) A4), if x FZ a, R 0 ) is a function satisfying the nonlinear delay inequality 3.1) x n) pn) + x 1 σs)) {fs) + gs)x σs)) + hs)w x σs))) for all n Z 0 with initial conditions I1) and I4) ψ σs)) p 1 s) for all s Z0 with σs) 0, where 1 is a constant, then )] 3.2) xn) Φ {Φ 1 exp 1 g) p 1 1 n) + fs) ) + exp 1 1 g) ht) for all n Z [0,µ], where µ 0 is chosen such that the RHS of 3.2) is welldefined for all n Z [0,µ], and Φ is defined as in Theorem 2.1. Page 23 of 36

24 Proof. Let φ : R 0 R 0 be defined by φr) = r, r R 0. Then φ satisfies Assumption A5). By 3.1) we have φ xn)) pn)+ { fs) φ x σs))) Furthermore, it is easy to see that + gs) hs) x σs)) + w x σs))). φ xs)) p 1 s) = φ 1 ps)) for all s Z 0 with σs) 0. Thus Theorem 2.1 applies and the assertion follows. Remark 3. i) In Corollary 3.1, if we set = 2, pn) c 2, gn) 0, we have x 2 n) c 2 + x σs)) {fs) + hs)w x σs))), n Z 0 implies [ ] xn) Φ {Φ 1 c + 1 fs) + 1 hs), n Z [0,µ]. 2 2 This is the discrete analogue of a result of Pachpatte in [14]. Furthermore, if σ = id, this reduces to a result of Pachpatte in [18]. ii) In case Φ ) =, 3.2) holds for all n Z 0. Page 24 of 36

25 Corollary 3.2. Under Assumptions A1) A4) with p FZ 0, R + ), if x FZ a, R 1 ) satisfies the nonlinear delay inequality 3.3) x n) pn) + x σs)) {fs) + gs) ln x σs)) + hs)w ln x σs))) for all n Z 0 with initial conditions I1) and I5) ψ σs)) 1 ln ps)) for all s Z 0 with σs) 0, where > 0 is a constant, then [ ) 3.4) xn) exp {Φ 1 Φ exp 1 gs) )) 1 ln pn) + 1 fs) ) ] + exp 1 1 gs) ht) for all n Z [0,ν], where ν 0 is chosen such that the RHS of 3.4) is welldefined for all n Z [0,ν], and Φ is defined as in Theorem 2.1. Page 25 of 36

26 Proof. Letting yn) = ln xn), 3.3) becomes 3.5) exp yn)) pn) + exp y σs))) {fs) + gs)y σs)) + hs)w y σs))). Let φ : R 0 R 0 be defined by φr) = expr), r R 0. Then φ satisfies Assumption A5). Hence from 3.5), we have φ yn)) pn)+ { fs) φ y σs))) Furthermore, it is easy to see that + gs) hs) y σs)) + w y σs))). ψ σs)) 1 ln ps)) = φ 1 ps)) for all s Z 0 with σs) 0. Thus Theorem 2.1 applies and we have )] yn) Φ {Φ 1 exp 1 1 gs) ln pn) + 1 fs) ) + exp 1 1 gs) ht) for all n Z [0,ν], and from this the assertion follows. Page 26 of 36

27 Remark 4. In case Φ ) =, 3.4) holds for all n Z 0. Corollary 3.3. Under Assumptions A1) A4), if x FZ a, R 0 ) satisfies the nonlinear delay inequality { 3.6) x n) pn) + x 1 σs)) fs) + gs)x σs)) s 1 + hs) kt)w x σt))) for all n Z 0 with initial conditions I1) and I4), where 1 is a constant, then )] 3.7) xn) Φ {Φ 1 exp 1 gs) p 1 1 n) + fs) + exp 1 gs) 1 n 1 ) s 1 hs) kt) for all n Z [0,η], where η 0 is chosen such that the RHS of 3.7) is welldefined for all n Z [0,η], and Φ is defined as in Theorem 2.1. Proof. Let φ : R 0 R 0 be defined by φr) = r, r R 0. Then φ satisfies Page 27 of 36

28 Assumption A5). By 3.6), { φ xn)) pn) + φ fs) x σs))) + gs) x σs)) for all n Z 0. Furthermore, it is easy to see that + hs) s 1 kt)w x σt))) ψ σs)) p 1 s) = φ 1 ps)) for all s Z 0 with σs) 0. Thus Theorem 2.2 applies and we have )] xn) Φ {Φ 1 exp 1 gs) p 1 1 n) + fs) ) + exp 1 gs) 1 s 1 hs)kt) for all n Z [0,η]. Remark 5. i) In Corollary 3.3, if we put = 2, pn) c 2, gn) 0, we have { s 1 x 2 n) c 2 + x σs)) fs) + hs) kt)w x σt))), n Z 0 Page 28 of 36

29 implies [ ] xn) Φ {Φ 1 c + 1 fs) + 1 s 1 hs) kt), n Z [0,η]. 2 2 This is the discrete analogue of a result of Pachpatte in [14]. Furthermore, if σ = id and w = id, this reduces to a result of Pachpatte in [18]. ii) In case Φ ) =, 3.7) holds for all n Z 0. Corollary 3.4. Under Assumptions A1) A4) with p FZ 0, R + ), if x FZ a, R 1 ) satisfies the nonlinear delay inequality { 3.8) x n) pn) + x σs)) fs) + gs) ln x σs)) for all n Z 0 with initial conditions I1) and s 1 + hs) kt)w ln x σt))) I6) ψ σs)) 1 ln ps)) for all s Z 0 with σs) 0, where > 0 is any constant, then [ ) 3.9) xn) exp {Φ 1 Φ exp 1 gs) Page 29 of 36

30 1 ln pn) + 1 fs) ) + exp 1 gs) )) 1 ] s 1 hs) kt) for all n Z [0,λ], where λ 0 is chosen such that the RHS of 3.9) is welldefined for all n Z [0,λ], and Φ is defined as in Theorem 2.1. Proof. Letting yn) = ln xn), 3.8) becomes { 3.10) exp yn)) pn) + exp y σs))) fs) + gs)y σs)) s 1 + hs) kt)w y σt))) for all n Z 0. Let φ : R 0 R 0 be defined by φr) = expr), r R 0. Then φ satisfies Assumption A5). Hence from 3.10), we have φ yn)) pn) + φ y σs))) { fs) + gs) hs) s 1 y σs)) + kt)w y σt))) Page 30 of 36

31 for all n Z 0. Furthermore, it is easy to check that ψ σs)) 1 ln ps)) = φ 1 ps)) for all s Z 0 with σs) 0. Thus Theorem 2.2 applies and we have )] yn) Φ {Φ 1 exp 1 1 gs) ln pn) + 1 fs) ) 1 + exp gs) 1 s 1 hs)kt) for all n Z [0,λ], and from this the assertion follows. Remark 6. i) In Corollary 3.4, if we set = 2, pn) c 2, gn) 0, then { s 1 x 2 n) c 2 + x 2 σs)) fs) + hs) kt)w ln x σt))), n Z 0 implies [ ) ] xn) exp {Φ 1 1 Φ 2 ln pn) + 1 fs) + 1 s 1 hs) kt) 2 2 n Z [0,λ]. This is the discrete version of a result of Pachpatte in [14]. ii) In case Φ ) =, 3.9) holds for all n Z 0. Page 31 of 36

32 4. Application Consider the discrete delay equation 4.1) x n) = F n, x σn)), ) G n, s, x σs))), n Z 0 with initial conditions I1) and I4), where 1 is a constant, σ, ψ satisfy Assumptions A3), A4), x FZ a, R), F CZ 0 R 2, R), and G CZ 2 0 R, R). If F, G satisfy F n, u, v) pn) + K v, n Z 0, u, v R, Gn, s, v) [fs) + gs) v + hs)w v )] v 1, n, s Z 0, v R, for some p, f, g, h, w satisfying A1) and A2), and some constant K > 0, then every solution of 4.1) satisfies xn) = F n, x σn)), G n, s, x σs)))) pn) + K G n, s, x σs))) n 1 pn) + K G n, s, x σs))) n 1 pn) + K [fs) + gs) x σs)) + hs)w x σs)) )] x σs)) 1 Page 32 of 36

33 for all n Jx) := the maximal existence lattice on which x is defined. Applying Corollary 3.1, this yields xn) Φ 1 {Φ exp K )] g) p 1 K n) + fs) ) + exp K K g) ht) for all n Jx) Z [0,µ]. This gives the boundedness of solutions of 4.1). Page 33 of 36

34 References [1] R.P. AGARWAL, Difference and Inequalities, Marcel Dekker, New York, [2] D. BAINOV AND P. SIMEONOV, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, [3] E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer-Verlag, New York, [4] R. BELLMAN, The stability of solutions of linear differential equations, Duke Math. J., ), [5] I. BIHARI, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar., ), [6] W.S. CHEUNG, On some new integrodifferential inequalities of the Gronwall and Wendroff type, J. Math. Anal. Appl., ), [7] W.S. CHEUNG AND Q.H. MA, Nonlinear retarded integral inequalities for functions in two variables, to appear in J. Concrete Appl. Math. [8] T.H. GRONWALL, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., ), [9] H. HARAUX, Nonlinear Evolution Equation: Global Behavior of Solutions, Lecture Notes in Mathematics, v.841, Springer-Verlag, Berlin, Page 34 of 36

35 [10] Q.M. MA AND E.H. YANG, On some new nonlinear delay integral inequalities, J. Math. Anal. Appl., ), [11] D.S. MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, New York, [12] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, [13] L. OU-IANG, The boundedness of solutions of linear differential equations y + At)y = 0, Shuxue Jinzhan, ), [14] B.G. PACHPATTE, A note on certain integral inequalities with delay, Period. Math. Hungar., ), [15] B.G. PACHPATTE, Explicit bounds on certain integral inequalities, J. Math. Anal. Appl., ), [16] B.G. PACHPATTE, Inequalities for Finite Difference, Marcel Dekker, New York, [17] B.G. PACHPATTE, On some new inequalities related to a certain inequality arising in the theory of differential equations, J. Math. Anal. Appl., ), [18] B.G. PACHPATTE, On some new inequalities related to certain inequalities in the theory of differential equations, J. Math. Anal. Appl., ), Page 35 of 36

36 [19] P. Ch. TSAMATOS AND S.K. NTOUYAS, On a Bellman-Bihari type inequality with delay, Period. Math. Hungar., ), [20] E.H. YANG, Generalizations of Pachpatte s integral and discrete inequalities, Ann. Differential, ), Page 36 of 36