On Gronwall s Type Integral Inequalities with Singular Kernels

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1 Filoma 31:4 (217), DOI /FIL17441A Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp:// On Gronwall s Type Inegral Inequaliies wih Singular Kernels Maksa Ashyraliyev a a Deparmen of Sofware Engineering, Bahcesehir Universiy, Isanbul, Turkey Absrac. In his paper, he generalizaions of Gronwall s ype inegral inequaliies wih singular kernels are esablished. In applicaions, heorems on sabiliy esimaes for he soluions of he nonliner inegral equaion and he inegral-differenial equaion of he parabolic ype are presened. Moreover, hese inequaliies can be used in he heory of fracional differenial equaions. 1. Inroducion Inegral inequaliies play an imporan role in he heory of differenial equaions. They are useful o invesigae properies of soluions of differenial equaions, such as exisence, uniqueness, and sabiliy, see for insance [1 4]. Gronwall s inegral inequaliy [5] is one of he mos widely applied resuls in he heory of inegral inequaliies. Due o various moivaions, several generalizaions and applicaions of Gronwall s ype inegral inequaliy and heir discree analogues have been obained and used exensively, see for insance [1, 6 9]. In [1], he following generalizaion of Gronwall s inegral inequaliy wih wo dependen limis is obained. Theorem 1.1. Assume ha v() is a coninuous funcion on [ 1, 1] and he inequaliies v() C + L s n() v(s)ds, 1 1 hold, where C = cons, L = cons. Then for v() he following inequaliies are saisfied. v() C exp (2L ), 1 1 In he presen paper, wo generalizaions of Gronwall s ype inegral inequaliies wih singular kernels are presened. In applicaions, heorems on sabiliy esimaes for he soluions of he nonliner inegral equaion and he inegral-differenial equaion of he parabolic ype are presened. We noe ha hese resuls may have an applicaion in he heory of fracional differenial equaions [1]. 21 Mahemaics Subjec Classificaion. Primary 26D1; Secondary 45K5 Keywords. Gronwall s Inegral Inequaliy, Inegral Inequaliies wih Singular Kernels, Inegral-Differenial Equaions Received: 31 December 215; Revised; 4 April 216; Acceped: 9 April 216 Communicaed by Eberhard Malkowsky address: maksa.ashyralyyev@eng.bahcesehir.edu.r (Maksa Ashyraliyev)

2 2. Inegral Inequaliies wih Singular Kernels M. Ashyraliyev / Filoma 31:4 (217), We consider he generalizaions of Gronwall s ype inegral inequaliies wih he singular kernel and wo dependen limis. Theorem 2.1. Assume ha v() is a coninuous funcion on [ 1, 1], a() is an inegrable funcion on [ 1, 1] and he inequaliies v() a() + L s n() hold, where L, β. Then for v() he inequaliies n=1 ( s ) β 1 v(s)ds, 1 1 (1) v() a() + 2 s n() ( s ) nβ 1 a(s)ds, 1 1 (2) are saisfied, where Γ(β) is he Gamma funcion. Proof. We denoe Bv() = L s n() ( s ) β 1 v(s)ds, 1 1. (3) Using (1), for nonnegaive funcions a and v we ge v() B k a() + B n v(), 1 1, (4) k= where n N. We will now show ha B n v() as n, which will prove (2). For his we firs prove ha B n v() = 2 s n() ( s ) nβ 1 v(s)ds, 1 1 (5) holds for any n N. Noe ha (5) follows direcly from (3) when n = 1. Assume ha (5) holds for some n N. Then for 1 we have B n+1 v() = L = L 2 = L 2 + τ ( s ) β 1 B n v(s)ds = L 2 ( s ) β 1 s n(s) ( s τ ) nβ 1 dτds ( s) β 1 τ ( + s) β 1 ( τ) nβ 1 dsdτ + (s τ ) nβ 1 dτds + ( s) β 1 (s τ) nβ 1 dsdτ + ( + s) β 1 τ s ( τ ) nβ 1 dτds ( s) β 1 (s + τ) nβ 1 dsdτ τ ( + s) β 1 ( + τ) nβ 1 dsdτ

3 = L 2 n = L 2 n = L 2 n So, we have M. Ashyraliyev / Filoma 31:4 (217), τ +τ z β 1 ( τ z) nβ 1 dzdτ + z β 1 ( + τ z) nβ 1 dzdτ 1 ( τ) (n+1)β 1 dτ + ( + τ) (n+1)β 1 dτ ρ β 1 (1 ρ) nβ 1 dρ Γ(β) ( τ ) (n+1)β 1 dτ. Γ(nβ + β) B n+1 v() = 2 n +1 ( s ) (n+1)β 1 v(s)ds, Γ((n + 1)β) 1. (6) In he similar way, for 1 < we have B n+1 v() = L ( s ) β 1 B n v(s)ds = L 2 = L 2 = L 2 ( s) β 1 ( s ) β 1 s n(s) s ( s τ ) nβ 1 dτds (s τ ) nβ 1 dτds + ( + s) β 1 ( τ ) nβ 1 dτds ( s) β 1 (s τ) nβ 1 dsdτ + τ + ( + s) β 1 ( τ) nβ 1 dsdτ + = L 2 n = L 2 n = L 2 n So, we have τ τ z β 1 ( τ z) nβ 1 dzdτ + ( τ) (n+1)β 1 dτ + Γ(β) ( τ ) (n+1)β 1 dτ. Γ(nβ + β) ( s) β 1 (s + τ) nβ 1 dsdτ τ τ ( + s) β 1 ( + τ) nβ 1 dsdτ +τ ( + τ) (n+1)β 1 dτ z β 1 ( + τ z) nβ 1 dzdτ 1 ρ β 1 (1 ρ) nβ 1 dρ B n+1 v() = 2 n +1 ( s ) (n+1)β 1 v(s)ds, Γ((n + 1)β) 1 <. (7)

4 M. Ashyraliyev / Filoma 31:4 (217), Combining (6) and (7), we prove by inducion ha (5) holds for any n N. Since B n v() for any n N and [ 1, 1], we have B n+1 v() B n v() = 2LΓ(β) Γ((n + 1)β) ( s ) (n+1)β 1 v(s)ds ( s ) nβ 1 v(s)ds 2LΓ(β) Γ((n + 1)β) as n. Therefore, lim n B n v() =. Then, by leing n in (4) and using (5), we obain he inequaliies (2). Theorem 2.1 is proved. Noe ha by puing a() cons in he Theorem 2.1, we obain he following resul. Corollary 2.2. Assume ha v() is a coninuous funcion on [ 1, 1] and he inequaliies v() C + L s n() ( s ) β 1 v(s)ds, 1 1 hold, where C, L, β. Then for v() he inequaliies v() C are saisfied. ( ) n 2L Γ(β) n= Γ(nβ + 1), 1 1 Noe ha by puing β = 1 in he Corollary 2.2, we obain he Theorem 1.1. Theorem 2.3. Assume ha v() is a coninuous funcion on [ 1, 1], a() is an inegrable funcion on [ 1, 1] and he inequaliies v() a() + L s n() s α 1 v(s)ds, 1 1 (8) hold, where L, 1 α >. Then for v() he following inequaliies hold v() a() + Proof. We denoe n=1 Bv() = L s n() 2 L n s n() s α 1 ( α s α ) a(s)ds, 1 1. (9) s α 1 v(s)ds, 1 1. (1) Using (8), for nonnegaive funcions a and v we ge v() B k a() + B n v(), 1 1, (11) k=

5 where n N. Le us prove ha B n v() = 2 L n s n() M. Ashyraliyev / Filoma 31:4 (217), s α 1 ( α s α ) v(s)ds, 1 1 (12) holds for any n N. Noe ha (12) follows direcly from (1) when n = 1. Assume ha (12) holds for some n N. Then for 1 we have B n+1 v() = L = 2 L n+1 = 2 L n+1 + = 2n L n+1 α n n! So, we have s α 1 B n v(s)ds = 2 L n+1 s α 1 s n(s) τ α 1 ( s α τ α ) dτds τ τ α 1 B n+1 v() = 2n L n+1 α n n! s α 1 τ α 1 τ α 1 (s α τ α ) dτds + τ s α 1 (s α τ α ) dsdτ + () α 1 (() α τ α ) dsdτ + () α 1 ( τ) α 1 τ α 1 ( α τ α ) n dτ + ( τ) α 1 ( α ( τ) α ) n dτ. In he similar way, for 1 < we have s τ α 1 (() α τ α ) dτds τ s α 1 (s α ( τ) α ) dsdτ τ ( τ) α 1 () α 1 (() α ( τ) α ) dsdτ s α 1 ( α s α ) n v(s)ds, 1. (13) B n+1 v() = L s α 1 B n v(s)ds = 2 L n+1 s α 1 s n(s) τ α 1 ( s α τ α ) dτds = 2 L n+1 = 2 L n+1 s α 1 τ α 1 τ α 1 (s α τ α ) dτds + τ s α 1 (s α τ α ) dsdτ + τ + τ α 1 () α 1 (() α τ α ) dsdτ + = 2n L n+1 α n n! () α 1 ( τ) α 1 s τ α 1 (() α τ α ) dτds τ s α 1 (s α ( τ) α ) dsdτ τ ( τ) α 1 () α 1 (() α ( τ) α ) dsdτ τ α 1 (() α τ α ) n dτ + ( τ) α 1 (() α ( τ) α ) n dτ.

6 M. Ashyraliyev / Filoma 31:4 (217), So, we have B n+1 v() = 2n L n+1 α n n! s α 1 (() α s α ) n v(s)ds, 1 <. (14) Combining (13) and (14), by inducion we prove ha (12) holds for any n N. Since B n v() for any n N and B n+1 v() B n v() = 2L nα s α 1 ( α s α ) n v(s)ds s α 1 ( α s α ) v(s)ds 2L as n, nα we ge lim n B n v() =. Then, by leing n in (11) and using (12), we obain he inequaliies (9). Theorem 2.3 is proved. Noe ha by puing a() cons in he Theorem 2.3, we obain he following resul. Corollary 2.4. Assume ha v() is a coninuous funcion on [ 1, 1] and he inequaliies v() C + L s n() s α 1 v(s)ds, 1 1 hold, where C, L, 1 α >. Then for v() he following inequaliies hold ( 2L α ) v() C exp, 1 1. α Noe ha by puing α = 1 in he Corollary 2.4, we obain he Theorem Applicaions Firs, we consider he nonliner inegral equaion x() = () + f (, s; x(s)) ds, 1 1. (15) Assume ha he funcion () is coninuous on [ 1, 1]. Suppose he kernel f of he equaion (15) is coninuous on [ 1, 1] [ 1, 1] (, ) and f saisfies on [ 1, 1] [ 1, 1] (, ) he following condiion f (, s; x(s)) L( s ) β 1 x(s), < s < 1 (16) for L and β. Theorem 3.1. Suppose ha he assumpion (16) holds. Then, for he soluion of equaion (15) he following sabiliy esimae x() max () ( ) n 2L Γ(β) 1 1 Γ(nβ + 1) n= is saisfied for any [ 1, 1].

7 M. Ashyraliyev / Filoma 31:4 (217), The proof of he Theorem 3.1 is based on he Theorem 2.1. Second, we consider he inegral-differenial equaion (see [11]) du() d + s n()au() = B(s)u(s)ds + f (), 1 1 (17) in an arbirary Banach space E wih unbounded linear operaors A and B() in E wih dense domain D(A) D(B()) and B()A 1 E E M 1, 1 α >, [ 1, ) (, 1]. (18) 1 α A funcion u() is called a soluion of he equaion (17) if he following condiions are saisfied: i) u() is coninuously differeniable on [ 1, 1]. The derivaives a he endpoins are undersood as he appropriae unilaeral derivaives. ii) The elemen u() belongs o D(A) for all [ 1, 1], and he funcions Au() and B()u() are coninuous on [ 1, 1]. iii) u() saisfies he equaion (17). A soluion of he equaion (17) defined in his manner will from now on be referred o as a soluion of he equaion (17) in he space C(E) = C([ 1, 1], E) of all coninuous funcions ϕ() defined on [ 1, 1] wih values in E equipped wih he norm ϕ C(E) = max 1 1 ϕ() E. We consider (17) under he assumpion ha he operaor A generaes an analyic semigroup exp{a} ( ), i.e. he following esimaes hold: e A E E M, Ae A E E M, 1. (19) Theorem 3.2. Suppose ha assumpions (18) and (19) for he operaors A and B() hold. Assume ha f () is a coninuously differeniable on [ 1, 1] funcion. Then here is a unique soluion of he equaion (17) and he following sabiliy inequaliies du() d, Au() E M E f () E f (s) E ds hold for any [ 1, 1], where M does no depend on f () and. The proof of he Theorem 3.2 is based on he following formula [11] u() = s n()a 1 f () s n()e A A 1 f () s n() + s n() and he Theorem 2.3. e ( s )A A 1 f (s)ds [ I e ( s )A ] A 1 B(s)u(s)ds, 1 1

8 We noe ha he inequaliy du() max 1 1 d + max Au() E M max f () E E M. Ashyraliyev / Filoma 31:4 (217), does no hold in general in he arbirary Banach space E and for he general srong posiive operaor A (see [12]). Neverheless, we can esablish he following heorem. Theorem 3.3. Suppose ha he esimaes (19) for he operaor A hold and B()A 1 Eλ E λ M 1, 1 α >, [ 1, ) (, 1]. 1 α Assume ha f () is a coninuous funcion on [ 1, 1]. Then here is a unique soluion of he equaion (17) and sabiliy inequaliies du() d, Au() Eλ M (λ) max f () E λ Eλ 1 1 hold for any [ 1, 1], where M (λ) does no depend on f () and. Here he fracional spaces E λ = E λ (E, A) ( < λ < 1), consising of all v E for which he following norms are finie: v Eλ = sup z 1 λ Aexp{ za}v E. <z Finally, we noe ha his approach allows us o exend our discussion o he sudy of he iniial-value problem d 2 u() + Au() = d 2 u() = u, u () = u B(ρ)u(ρ)dρ + f (), 1 1, for inegral-differenial equaion in an arbirary Banach space E wih unbounded linear operaors A and B() in E wih dense domain D(A) D(B()) saisfying he assumpion (18). The sabiliy esimaes for he soluion of he equaion (2) can be obained in he similar way. (2) 4. Conclusions In his paper, wo generalizaions of Gronwall s ype inegral inequaliies wih singular kernels are presened. In applicaions, heorems on sabiliy esimaes for he soluions of he nonliner inegral equaion and he inegral-differenial equaion of he parabolic ype are presened. Moreover, applying he resuls of he paper [1], he fracional differenial equaion D α ±v() = f (, v()), < α < 1, 1 1 can be invesigaed in he similar way, where D α ±v() = 1 Γ(1 α) s n() ( s ) α v (s)ds. The sabiliy esimaes for he soluion of his fracional differenial equaion can be obained in he similar way.

9 M. Ashyraliyev / Filoma 31:4 (217), References [1] M. Ashyraliyev, A Noe on he Sabiliy of he Inegral-Differenial Equaion of he Hyperbolic Type in a Hilber Space, Numerical Funcional Analysis and Opimizaion 29 (28) [2] S. G. Krein, Linear Differenial Equaions in a Banach Space, Nauka, Moscow, [3] E. F. Beckenbach, R. Bellman, Inequaliies, Springer, Berlin, [4] S. Ashirov, Y. D. Mamedov, A Volerra-ype inegral equaions, Ukr. Ma. Journal 4:4 (1988) [5] T. H. Gronwall, Noe on he derivaives wih respec o a parameer of he soluions of a sysem of differenial equaions, Annals of Mahemaics 2:4 (1919) [6] T. Nurimov, D. Filaov, Inegral Inequaliies, FAN, Tashken, [7] A. Corduneanu, A noe on he Gronwall inequaliy in wo independen variables, Journal of Inegral Equaions 4:3 (1982) [8] Z. Direk, M. Ashyraliyev, FDM for he inegral-differenial equaion of he hyperbolic ype, Advances in Difference Equaions 214:132 (214) 1 8. [9] A. Ashyralyev, E. Misirli, O. Mogol, A noe on he inegral inequaliies wih wo dependen limis, Journal of Inequaliies and Applicaions 21 (21) [1] A. Ashyralyev, A noe on fracional derivaives and fracional powers of operaors, Journal of Mahemaical Analysis and Applicaions 357 (29) [11] M. Ashyraliyev, A noe on he sabiliy of he inegral-differenial equaion of he parabolic ype in a Banach space, Absrac and Applied Analysis 212 (212) [12] A. Ashyralyev, P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equaions, Birkhauser, Basel, 1994.