Nonlinear Gronwall Bellman Type Inequalities and Their Applications

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1 Article Noliear Growall Bellma Tye Iequalities ad Their Alicatios Weimi Wag 1, Yuqiag Feg 2, ad Yuayua Wag 1 1 School of Sciece, Wuha Uiversity of Sciece ad Techology, Wuha 4365, Chia; wugogre@163.com (W.W.); wagyuayua@wust.edu.c (Y.W.) 2 Hubei Provice Key Laboratory of Systems Sciece i Metallurgical Process, Wuha 4365, Chia * Corresodece: yqfeg6@126.com; Tel.: Academic Editor: Hari M. Srivastava eceived: 2 February 217; Acceted: 23 May 217; Published: 31 May 217 Abstract: I this aer, some oliear Growall Bellma tye iequalities are established. The, the obtaied results are alied to study the Hyers Ulam stability of a fractioal differetial equatio ad the boudedess of solutios to a itegral equatio, resectively. Keywords: oliear Growall Bellma iequalities; differetial equatio; itegral equatio; Hyers Ulam stability; boudedess 1. Itroductio The study of Growall Bellma iequalities has bee aid much attetio ad develoed at a high rate i the last three decades. These iequalities lay a imortat role i may fields. They are alied to the ivestigatio of the stability, boudedess, global existece, uiqueess, ad cotiuous deedece o the iitial or boudary value ad arameters of solutios to differetial equatios, itegral equatios, as well as differece equatios [1 11]. They are also used to study the regularized family of models for homogeeous icomressible two-hase flows [12], the state of the high oliear circuit [13], the Cousi roblems ad the emergece of the Sheaf Cocet [14]. ecetly, Willett [15] discussed the liear iequality: u(t) w (t) w i (t) v i (s)u(s)ds (t I) (1) ad Li [16] exteded the study to the liear iequality as follows: u(t) a(t) b i (t) (t s) βi 1 u(s)ds (2) Several geeralizatios of the Growall iequality were established ad the alied to rove the uiqueess of solutios for fractioal differetial equatios with various derivatives. I this aer, we are cocered with the followig oliear Growall Bellma-tye iequality: u (x) a(x) ω i (x) h i (t)g i (t, u(t) (x t) β i 1 f i (t, u(t) (3) Mathematics 217, 5, 31; doi:1.339/math5231

2 Mathematics 217, 5, 31 2 of 11 Some ew results are obtaied ad the alied to ivestigate the qualitative roerties of differetial ad itegral equatios. This aer is orgaized as follows: I Sectio 2, we itroduce some defiitios ad otatios. Some oliear Growall Bellma-tye iequalities are reseted i Sectio 3. I Sectio 4, the obtaied results are alied to rove the Hyers Ulam stability of a fractioal differetial equatio ad the boudedess of the solutios to a itegral equatio. 2. Prelimiaries I this sectio, we recall some stadard defiitios ad otatios. Lemma 1. (see [17]). Assume that a, q with =. The, for ay b >, we have: a q q b q a q b q (4) Lemma 2. (see [15]) Suose that: u(t) w (t) w i (t) v i (s)u(s)ds (t I) (5) where v i w j (i = 1, 2, ; j =, 1,, ) ad v i u(i = 1, 2,, ) are locally itegrable o I, all fuctios are assumed o-egative. The: u E w (6) where E i (i =, 1,, ) is defied iductively as the comositio of i 1 fuctioal oerators; i.e., E i = D i D i 1 D, where: 3. Mai esults D w = w D j w = w (E j 1 w j )(ex (7) v j E j 1 w j ) v j wds (j = 1, 2,, ) (8) I this sectio, we will establish some oliear Growall Bellma-tye iequalities. The first result is the followig: Theorem 1. Suose that 1, β i > are costats, f i C(, ) with f i (x, u) f i (x, v) l i (u v) for u v, where l i is the Lischitz costats, ad all the fuctios are oegative ad cotiuous o [,T), with b i (i = 1, 2,, ) are bouded ad odecreasig fuctios. If the followig iequality is satisfied: the: u (x) a(x) (x t) βi 1 f i (t, u(t) (9) u(x) (ã(x) ( (x t) k β i 1 ã(t)) 1 ( 1 b 1 ) k k [l i b i (x)γ(β i )] Γ( k β i ) (1) for ay b > ad t [, T], where: ã(x) = a(x) (x t) βi 1 f i (t, 1 b 1 (11)

3 Mathematics 217, 5, 31 3 of 11 Proof. Deote u (x) = v(x), the we have u(x) = v 1 (x) : Accordig to Lemma 1: v(x) a(x) (x t) βi 1 f i (t, v 1 (t) (12) v(x) a(x) = a(x) f i (t, 1 a(x) 1 b 1 = ã(x) 1 b 1 (x t) βi 1 f i (t, 1 b 1 (x t) βi 1 [ f i (t, 1 b 1 b 1 ) f i (t, 1 b 1 )]dt l i (x t) β i 1 v(t (x t) βi 1 f i (t, 1 b 1 l i (x t) βi 1 v(t v(t) 1 b 1 v(t) 1 b 1 ) (13) Now let: Bϕ(x) = 1 b 1 x [, T), for locally itegrable fuctios ϕ. The: By mathematical iductio method, we have: l i (x t) βi 1 ϕ(t (14) v(x) ã(x) Bv(x). (15) v(x) ã(x) B(ã(x) Bv(x)) = ã(x) Bã(x) B 2 v(x) ã(x) Bã(x) B 2 (ã(x) Bv(x)) = ã(x) Bã(x) B 2 ã(x) B 3 v(x) k 1 B m ã(x) B k v(x). m= (16) Let r = 1 b 1. We assert that: B k v(x) r k k [l i b i (x)γ(β i )] 1,2, k =1 Γ( k β i ) (x t) k β i 1 v(t (17) for ay k N ad B k v(x) as k for each x i x < T.

4 Mathematics 217, 5, 31 4 of 11 I fact, (i) If k = 1, the: Bv(x) = 1 =1 r [l i b i (x)γ(β i )] Γ(β i ) rl i b i (x) 1 =1 = Bv(x) (x t) β i 1 v(t (x t) β i 1 v(t (18) So we kow the iequality (17) holds for k = 1. (ii) Assume that the iequality (17) holds for k = j; that is: B j v(x) (iii) For k = j 1, we have: r j j [l i b i (x)γ(β i )] 1,2, j =1 Γ( j β i ) (x t) j β i 1 v(t (19) B j1 v(x) = B(B j v(x)) rl i (x t) β i 1 ( r j j [l i b i (t)γ(β i )] 1,2, j =1 Γ( j β i ) (t s) j β i 1 v(s)ds (2) Owig to the mootoicity of b i (i = 1, 2,, ), we get: B j1 v(x) rl i (x t) β i 1 1,2, j =1 r j j [l i b i (t)γ(β i )] Γ( j β i ) (t s) j β i 1 v(s)dsdt (21) Iterchagig the order of itegratio, we have: B j1 v(x) r j1 l i 1,2, j =1 j [l i b i (x)γ(β i )] Γ( j β i ) Γ(β i )Γ( j β i ) Γ(β i j β i ) (x s) j β i βi 1 v(s)ds = r j1 1,2, j =1 (x s) j β i β i 1 v(s)ds [l i Γ(β i )] j [l i b i (x)γ(β i )] Γ(β i j β i ) (22) Let i = (j 1), the:

5 Mathematics 217, 5, 31 5 of 11 B j1 v(x) r j1 (j1) =1 1,2, j =1 (x s) j β i β (j1) 1 v(s)ds = r j1 j1 [l i b i (x)γ(β i )] 1,2, (j1) =1 Γ( j1 β i ) [l (j1) b (j1) (x)γ(β (j1) )] j [b i (x)γ(β i )] Γ(β (j1) j β i ) (x s) j1 β i 1 v(s)ds (23) This imlies that the iequality (17) holds for k = j 1. Hece, it holds for ay k N. Sice b i (i = 1, 2,, ) are bouded, b i < M i (M i > )(i = 1, 2,, ): B k v(x) 1,2, k =1 [l 1 b 1 (t)γ(β 1 )][b 2 (x)γ(β 2 )] [b k (x)γ(β k )] Γ(β 1 β 2 β k ) (x t) k β i 1 v(t 1,2, k =1 (l 1 l 2 l k )(M 1 M 2 M k ) (Γ(β 1 )Γ(β 2 ) Γ(β k )) Γ(β 1 β 2 β k ) (x t) k β i 1 v(t (24) ad accordig to the roerty of the Gamma fuctio, B k v(x) as for x [, T), the: v(x) ã(x) ( (x t) k β i 1 ã(t). ( 1 b 1 ) k k [l i b i (x)γ(β i )] Γ( k β i ) (25) Hece: u(x) (ã(x) ( (x t) k β i 1 ã(t)) 1 ( 1 b 1 ) k k [l i b i (x)γ(β i )] Γ( k β i ) (26) This comletes the roof. By Theorem 1, the mai result i [16] is a secial case of Theorem 1 for = 1, f i (t, u(t)) = u(t). Corollary 1. (see [16]). For ay t [, T): u(t) a(t) b i (t) (t s) βi 1 u(s)ds (27) where all the fuctios are o-egative ad cotiuous. The costats β i >, b i (i = 1, 2,, ) are the bouded ad mootoic icreasig fuctios o [,T). The: u(t) a(t) ( k [b i (t)γ(β i )] Γ( k β i ) (t s) k β i 1 a(s)ds) (28)

6 Mathematics 217, 5, 31 6 of 11 Corollary 2. Uder the hyothesis of Theorem 1, if a(x) is icreasig o [.T], the: u(x) (ã(t) ( k= 1,2, k =1 k [l i b i (x)γ(β i )] Γ( k β i 1) ( 1 b 1 ) k T k β i )) 1 (29) Proof. Sice a(x) is icreasig, ã(x) is also icreasig: u(x) (ã(x) ( (x t) k β i 1 ã(t)) 1 ã 1 (x)(1 = (ã(x) (ã(t) ( (x t) k β i 1 dt)) 1 ( k= 1,2, k =1 ( k= 1,2, k =1 ( 1 b 1 ) k k [l i b i (x)γ(β i )] Γ( k β i ) ( 1 b 1 ) k k [l i b i (x)γ(β i )] Γ( k β i ) ( 1 b 1 ) k k [l i b i (x)γ(β i )] Γ( k β i 1) ( 1 b 1 ) k k [l i b i (x)γ(β i )] Γ( k β i 1) t k β i )) 1 T k β i )) 1 (3) The roof is comleted. Theorem 2. Uder the coditios of Corollary 2, if ω i (x)(i = 1, 2,, ) are bouded ad mootoic icreasig. g i C(, ) with g i (x, u) g i (x, v) T i (u v) for u v, where T i is the Lischitz costat. If the followig iequality is satisfied: u (x) a(x) ω i (x) h i (t)g i (t, u(t) (x t) β i 1 f i (t, u(t) (31) the: u(x) (E ã(x) ( k= 1,2, k =1 k [l i b i (x)γ(β i )] Γ( k β i 1) ( 1 b 1 ) k t k β i )) 1 (32) where E i (i =, 1,, ) is defied iductively as the comositio of i 1 fuctioal oerators; i.e., E i = D i D i 1 D, where: D w = w D j w = w (E j 1 w j )(ex (33) v j E j 1 w j ) v j wds (j = 1, 2,, ) (34)

7 Mathematics 217, 5, 31 7 of 11 Proof. Deote u (x) = v(x), u(x) = v 1 (x). The: where: v(x) a(x) a(x) a(x) ω i (x) (x t) βi 1 f i (t, v 1 (t) ω i (x) f i (t, 1 h i (t)g i (t, v 1 (t) (x t) βi 1 f i (t, 1 b 1 v(t) 1 b 1 h i (t)g i (t, 1 b 1 v(t) 1 b 1 (x t) βi 1 [ f i (t, 1 b 1 v(t) 1 b 1 ) b 1 x )]dt ω i (x) h i (t) b 1 ) f i (t, 1 [g i (t, 1 b 1 v(t) 1 b 1 ) g i (t, 1 b 1 ) g i (t, 1 b 1 )]dt a(x) 1 b 1 1 b 1 l i (x t) β i 1 v(t (x t) βi 1 f i (t, 1 b 1 T i ω i (x) = ã(x) 1 b 1 1 b 1 l i T i ω i (x) ã(x) = a(x) ω i (x) h i (t)v(t ω i (x) (x t) β i 1 v(t h i (t)v(t (x t) βi 1 f i (t, 1 b 1 h i (t)g i (t, 1 b 1 h i (t)g i (t, 1 b 1 (35) (36) Let: The: z(x) = ã(x) 1 b 1 v(x) z(x) 1 b 1 T i ω i (x) h i (t)v(t (37) l i (x t) βi 1 v(t (38) By Iequality (37), we derive that z(x) is oegative ad icreasig. Accordig to Corollary 1, we have: v(x) z(x) ( ( 1 b 1 ) k k [l i b i (x)γ(β i )] Γ( k β i 1) x k β i ) (39) k= 1,2, k =1

8 Mathematics 217, 5, 31 8 of 11 Combig (39) with (37): z(x) ã(x) 1 b 1 (z(t) ( k= 1,2, k =1 ã(x) 1 b 1 T i ω i (x) h i (t) ( 1 b 1 ) k k [l i b i (t)γ(β i )] Γ( k β i 1) T i ω i (x) h i (t)z(t t k β i ) (4) where: h i (x) = h i (t) By Lemma 2, we obtai: ( k= 1,2, k =1 ( 1 b 1 ) k k [l i b i (t)γ(β i )] Γ( k β i 1) t k β i ) (41) z(x) E ã(x) (42) By iequalities (39) ad (42), we obtai (32). The roof is comleted. Corollary 3. Uder the hyothesis of Theorem 2, if = 1, the: 4. Alicatios u(x) E ã(x) ( k= 1,2, k =1 k [l i b i (x)γ(β i )] Γ( k β i 1) I this sectio, we reset two examles as alicatios of our results. t k β i ) (43) Examle 1. The followig iitial value roblems of fractioal differetial equatio was cosidered i [16]: D β i u(t) = f (t, u(t)), D 1 β i u(t) t= = δ, (44) where < β 1 < β 2 < < β < 1, t [, T), f satisfies f (t, u(t)) f (t, v(t)) l u(t) v(t) ; l is the Lischitz costat. D β ad Iβ deote the iema Liouville fractioal derivative ad fractioal itegral oerators, resectively. The uiqueess of solutios was roved by Growall Bellma iequality. I this sectio, we study the Hyers Ulam stability of this iitial value roblem. As we kow, if u(t) is a solutio of the differetial Equatio (44), the u(t) satisfies the followig itegral equatio: u(t) = I β Theorem 3. If u ε (t) C[, T] satisfies: f (t, u(t)) 1 I β β i u(t) δ tβ 1 Γ(β ) D β i u ε(t) f ε (t, u ε (t)) ε, the there exists a solutio of (44) such that u(t) u ε (t) kε for all t [, T]. Where: k = Γ(β ) Γ(β 1) (1 k=1 1,2, k =1 k [b i (t)γ(β i )] Γ( k β i 1) (45) T k β i (46)

9 Mathematics 217, 5, 31 9 of 11 b i (t) = 1 Γ(β β i ). Proof. Accordig to (45), we kow that u ε (t) satisfies: I β ε u ε(t) [I β 1 f ε(t, u ε (t)) I β β i u ε (t) δ tβ 1 Γ(β ) ] Iβ ε. (47) The: u(t) u ε (t) I β f (t, u(t)) 1 I β 1 f ε(t, u ε (t)) I β 1 ε 1 Γ(β β i ) I β β i u(t) I β β i u ε (t) (t s) β β i 1 u(s) u ε (s) ds (48) By Theorem 1, we have: t β u(t) u ε (t) ε Γ(β 1) k=1 1,2, k =1 (t s) k β i 1) I β εds Γ(β ) ε Γ(β 1) (1 k=1 1,2, k =1 k [b i (t)γ(β i )] Γ( k β i ) k [b i (t)γ(β i )] Γ( k β i 1) T k β i ) (49) 1 where b i (t) =. This comletes the roof. Γ(β β i ) Examle 2. Cosider the itegral equatio as follows: u(x) = u() ω i (x) h i (t)g i (t, u(t) (x t) β i 1 f i (t, u(t) (5) This icludes the iteger ad fractioal itegral arts. We assert the solutio of this itegral equatio is bouded o [, T], rovided ω i, g i, f i ( i = 1, 2,, ) satisfy the assumtios of Corollary 3. Theorem 4. Let u be a solutio of (5) o [, T]. If ω i, g i, f i (i = 1, 2,, ) satisfy the assumtios of Corollary 7, the for x [, T]: u(x) E u() ( k= 1,2, k =1 Proof. If u is a solutio of (5), the by Corollary 3: k [l i b i (x)γ(β i )] Γ( k β i 1) T k β i ) (51)

10 Mathematics 217, 5, 31 1 of 11 u(x) u() E u() E u() ω i (x) h i (t) g i (t, u(t)) dt (x t) β i 1 f i (t, u(t)) dt ( k= 1,2, k =1 ( k= 1,2, k =1 k [l i b i (x)γ(β i )] Γ( k β i 1) k [l i b i (x)γ(β i )] Γ( k β i 1) t k β i ) T k β i ) (52) emark 1. The result of Corollary 3 also ca be used to rove the uiqueess of solutios to fractioal differetial equatios. Ackowledgmets: This research is suorted by the Natural Sciece Foudatio of Chia ( ) ad the Doctoral Fud of Educatio Miistry of Chia ( ). Author Cotributios: Yuqiag Feg itroduced the roblem ad gave eough suggestios; Weimi Wag ad Yuayua Wag ivestigated the roblem; Weimi Wag wrote the aer. Coflicts of Iterest: The authors declare o coflict of iterest. efereces 1. Bihari, I. A geeralizatio of a lemma of Bellma ad its alicatio to uiqueess roblem of differetial equatio. Acta Math. Acad. Sci. Hug. 1956, 7, Che, C.; Cheg, J.; Zhao, D. Growall Bellma-Tye itegral iequalities ad alicatios to BVPs. J. Iequal. Al. 29, 29, Daa, F. Itegral iequalities of Growall Bellma Bihari tye ad asymtotic behavior of certai secod order oliear differetial equatios. J. Math. Aal. Al. 1985, 18, Feg, Q.; Zheg, B. Geeralized Growall Bellma-tye delay dyamic iequalities o time scales ad their alicatios. Al. Math. Comut. 212, 218, Feg, Q.; Meg, F.; Zheg, B. Growall Bellma tye oliear delay itegral iequalities o time scales. J. Math. Aal. Al. 211, 382, Ferreira,.A.C.; Torres, D.F.M. Geeralized retarded itegral iequalities. Al. Math. Lett. 29, 22, Jug, S.-M. Hyers Ulam assias Stability of Fuctioal Equatios i Noliear Aalysis; Sriger Otimizatio ad Its Alicatios; Sriger: New York, NY, USA, 211; Volume Kim, Y. Growall, Bellma ad Pachatte tye itegral iequalities with alicatio. Noliear Aal. 29, 27, e2641 e Li, L.; Meg, F.; Ju, P. Some ew itegral iequalities ad their alicatios i studyig the stability of olieari tegro-differetial equatios with time delay. J. Math. Aal. Al. 21, 377, Liu, F.; Zhao, X. O the Hyers Ulam stability of a iohomogeeous liear differetial equatios of third-order. Math. Pract. Theory 213, 43, Agarwal,.; Deg, S.; Zhag, W. Geeralizatio of a retarded Growall-like iequality ad its alicatios. Al. Math. Comut. 25, 165, Cal, C.G.; Medjo, T.T. O a regularized family of models for homogeeous icomressiable two-hase flows. J. Noliear Sci. 214, 24, Dig, W.; Feg, P. The study of uiqueess of the steady state of the high degrees oliear oautoomous circuits by Growall s Lemma. J. Huaiyi Ist. Techol. 22, 11, (I Chiese)

11 Mathematics 217, 5, of Chorlay,. From roblems to structures: The cousi roblems ad the emergece of the sheaf cocet. Arch. Hist. Exact. Sci. 21, 64, Willett, D. A liear geeralizatio of Growall s iequality. Proc. Am. Math. Soc. 1965, 16, Li, S. Geeralized Growall iequalitilies ad their alicatios to fractioal differetial equatios. J. Iequal. Al. 213, 213, Jiag, F.; Meg, F. Exlicit bouds o some ew oliear itegral iequality with delay. J. Comut. Al. Math. 27, 25, c 217 by the authors. Licesee MDPI, Basel, Switzerlad. This article is a oe access article distributed uder the terms ad coditios of the Creative Commos Attributio (CC BY) licese (htt://creativecommos.org/liceses/by/4./).