On Inequality for the Non-Local Fractional Differential Equation

Transcription

1 Globl Journl of Pure nd Applied Mthemtics. ISSN Volume 13, Number ), pp Reserch Indi Publictions On Inequlity for the Non-Locl Frctionl Differentil Eqution Trchnd L. Holmbe Deprtment of Mthemtics, Ki Shnkrro Gutte ACS College, Dhrmpuri, Beed, Mhrshtr, Indi. E Mohmmed Mzhr-Ul-Hque 1 Dr. B.A.M. University, Aurngbd, Mhrshtr, Indi. Abstrct In the pper the nonlocl frctionl differentil eqution with boundry conditions will be treted nd will form the inequlities like Lypunov inequlity, Hrtmn nd Wintner inequlity for the solution of nonlocl frctionl differentil eqution with the help of the greens function involved in the solution of nonlocl frctionl differentil eqution with boundry conditions. AMS subject clssifiction: 2A33, 34A08, 34K37, 34A08, 34A40, 34C10. Keywords: nonlocl problem, Lypunov inequlity, Hrtmn nd Wintner inequlity, Frctionl differentil eqution, differentil eqution inequlities. 1. Introduction In mny engineering nd scientific disciplines such s physics, chemistry, erodynmics, electrodynmics of complex medium, polymer rheology, economics, control theory, signl nd imge processing, biophysics, blood flow phenomen, etc the frctionl differentil nd integrl equtions represents the processes in more effective mnner thn 1 Corresponding uthor.

2 982 Trchnd L. Holmbe nd Mohmmed Mzhr-Ul-Hque by integer order. Becuse of this the subject of frctionl order differentil nd integrl equtions becme the interest of mthemticins nd reserchers. Due to vst ppliction in vrious fields inequlity of differentil eqution involving differentil opertors with frctionl order is lso one of the interesting topic for reserchers nd mthemticins. As the inequlity provides the distnce between consecutive zeros of the solution so it is very useful in pplictions e.g. in oscilltion nd Sturmin theory, symptotic theory, disconjugcy, eigenvlue problems e.t.c the Lypunov inequlity nd some of its generliztions gives successful results [, 7, 9, 10, 11, s Lypunov inequlity ps) ds > 4 1.1) b studied by Russin mthemticin A. M. Lipunov [4 for nontrivil solution of x t) + pt)xt) = 0,<t<b with x)= xb) = 0 1.2) where p :[,b R is continuous function, provides lower bound for the distnce between consecutive zeros of xt). Most of the properties of Lipunov inequlity nd its generliztion studied by [12, 13, 14, 15, 1, 17, 18, 19 nd the references therein. The number of generliztion re present, one of them is b s)s )p + s)ds > b,p + s) = mx s b {ps), 0} 1.3) studied by Hrtmn nd Wintner [5 nd lso mny other extensions to frctionl order differentil eqution with boundry conditions were obtined by [20, 21, 22, 23, 24, 25, 2. In the present pper we will find such type of inequlities for the non-locl frctionl differentil equtions with boundry conditions c D α ut) = qt)f t, ut),c D η ut)) u) = u) = u ) = 0,u b) = γ uτ) 1.4) where c D α,c D η denotes the Cputo frctionl derivtive of order α nd η, < t, τ < b, 3 <α,η 4 nd q :[,b Ris continuous function. 2. Auxiliry Results We need the following definitions,lemms nd theorems in the sequel. The extensively studied frctionl derivtive nd integrl is Riemnn-Liouville frctionl derivtive nd integrl defined by [1, 2, 3, 27, 28, 29, 30. Definition 2.1. The Riemnn-Liouville frctionl integrl of order α of rel vlued function f is defined by I α fz)= 1 z z t) α1 ft)dt 2.5) Ɣα

3 On Inequlity for the Non-Locl Frctionl Differentil Eqution 983 nd when = 0 this becomes I α fz)= 1 Ɣα z 0 z t) α1 ft)dt 2.) The property studied by T.L. Holmbe nd Mohmmed Mzhr-Ul-Hque [8 is, Let α>0,β >0 nd f be ny function. Then I α k I β k fz))= I β k I α k Ik α I β k fz))= I β k I k α fz)) 2.7) α+β fz))= Ik fz)= for the kgenerlized frctionl integrl. 1 kɣ k α + β) z 0 z t) α+β k 1 ft)dt 2.8) Definition 2.2. The Riemnn-Liouville frctionl derivtive of order α of rel vlued function f is defined by where n =[lph+1. ) D α fz)= 1 d n z z t) nα1 ft)dt 2.9) Ɣn α) dz Definition 2.3. The Cputo frctionl derivtive of order α of rel vlued function f is defined by c D α fz)= 1 z z t) nα1 f n t)dt 2.10) Ɣn α) where n =[α+1. Definition 2.4. A rel vlued function f from I R is clled Crtheodry function if 1. f is mesurble on R. 2. f is continuous on I. 3. There exist Lebesgue function h on I such tht h is n Upper bound of f on I. Theorem 2.5. Kolmogorov compctness criterion [19) Let L p 0,T),1 p< if i) is bounded in L p 0,T)nd ii) u h u s h 0 uniformly with respect to u, then is reltively compct in L p 0,T)where u h t) = 1 h +h t us)ds.

4 984 Trchnd L. Holmbe nd Mohmmed Mzhr-Ul-Hque The following lemm will be very useful Lemm 2.. Assume tht f C,b) L 1, b) with frctionl derivtive of order α>0 tht belongs to C,b) L 1, b) Then I α c D α f )t) = c 0+c 1 t)+c 2 t) 2 +c 3 t) 3 + +c n1 t) n1 +c n t) n for t [,b,c i R nd n =[α Min Results 2.11) Theorem 3.1. For ny 4 <α 5nd u C[,b the nonlocl frctionl boundry vlue problem c D α ut) = qt)f t, ut),c D η ut)) u) = u) = u ) = 0,u 3.12) b) = γ uτ) hs unique solution given by ut) = Gt, s)qs)f s, us), c D η us))ds t )3 γ [ γτ ) 3 Gτ, s)qs)f s, us), c D η us))ds 3.13) where α 1)α 2)α 3)t ) Gt, s) = 1 3 b s) α4 ts) α1, s t b Ɣα) α 1)α 2)α 3)t ) 3 b s) α4, t s b Proof. For the given non-locl frctionl boundry vlue problem the solution by lemm 2. is ut) = c 0 + c 1 t ) + c 2 t ) 2 + c 3 t ) t s) α1 qs)f s, us), c D η Ɣα) us))ds 3.14) u) = u ) = u ) = 0 implies c 0 = c 1 = c 2 = 0 ut) = c 3 t ) t s) α1 qs)f s, us), c D η Ɣα) us))ds u t) = 3c 3 t ) α 1)t s) α2 qs)f s, us), c D η Ɣα) us))ds u t) = c 3 t ) + 1 α 1)α 2)t s) α3 qs)f s, us), c D η Ɣα) us))ds u t) = c α 1)α 2)α 3)t s) α4 qs)f s, us), c D η Ɣα) us))ds 3.15)

5 On Inequlity for the Non-Locl Frctionl Differentil Eqution 985 u b) = γ uτ) c Ɣα) = γ α 1)α 2)α 3)b s) α4 qs)f s, us), c D η us))ds c 3 τ ) ) τ s) α1 qs)f s, us), c D η Ɣα) us))ds c 3 [ γτ ) 3 = 1 Ɣα) c 3 = γ Ɣα) τ s) α1 qs)f s, us), c D η us))ds α 1)α 2)α 3)b s) α4 qs)f s, us), c D η us))ds γ Ɣα)[ γτ ) 3 α 1)α 2)α 3) Ɣα)[ γτ ) 3 τ s) α1 qs)f s, us), c D η us))ds b s) α4 qs)f s, us), c D η us))ds 3.1) from ut) = γt ) 3 Ɣα)[ γτ ) 3 τ s) α1 qs)f s, us), c D η us))ds α 1)α 2)α 3)t )3 b Ɣα)[ γτ ) 3 b s) α4 qs)f s, us), c D η us))ds + 1 t s) α1 qs)f s, us), c D η Ɣα) us))ds 3.17) 1 [ γτ ) 3 = 1 [ [ γτ ) 3 [ γτ ) 3 + γτ ) 3 = 1 [ γτ ) 3 = 1 γτ )3 [1 + [ γτ ) 3 γt ) 3 ut) = Ɣα)[ γτ ) 3 τ s) α1 qs)f s, us), c D η us))ds α 1)α 2)α 3)t )3 γτ )3 [1 + Ɣα) [ γτ ) Ɣα) b s) α4 qs)f s, us), c D η us))ds t s) α1 qs)f s, us), c D η us))ds 3.18)

6 98 Trchnd L. Holmbe nd Mohmmed Mzhr-Ul-Hque thus we hve ut) = γt ) 3 Ɣα)[ γτ ) 3 α 1)α 2)α 3)t )3 Ɣα) α 1)α 2)α 3)t )3 Ɣα) τ s) α1 qs)f s, us), c D η us))ds α 1)α 2)α 3)t )3 γτ ) 3 Ɣα)[ γτ ) Ɣα) b s) α4 qs)f s, us), c D η us))ds t b s) α4 qs)f s, us), c D η us))ds b s) α4 qs)f s, us), c D η us))ds t s) α1 qs)f s, us), c D η us))ds 3.19) ut) = γt ) 3 Ɣα)[γτ ) 3 α 1)α 2)α 3)t )3 Ɣα) α 1)α 2)α 3)t )3 Ɣα) τ s) α1 qs)f s, us), c D η us))ds α 1)α 2)α 3)t )3 γτ ) 3 Ɣα)[ γτ ) 3 b s) α4 qs)f s, us), c D η us))ds α 1)α 2)α 3)t )3 γτ ) 3 Ɣα)[ γτ ) 3 τ + 1 Ɣα) b s) α4 qs)f s, us), c D η us))ds t b s) α4 qs)f s, us), c D η us))ds b s) α4 qs)f s, us), c D η us))ds t s) α1 qs)f s, us), c D η us))ds 3.20)

7 On Inequlity for the Non-Locl Frctionl Differentil Eqution 987 ut) = 1 [ α 1)α 2)α 3)t ) 3 b s) α4 t s) α1 Ɣα) qs)f s, us), c D η us))ds 1 α 1)α 2)α 3)t ) 3 b s) α4 Ɣα) t qs)f s, us), c D η us))ds t ) 3 γ Ɣα)[ γτ ) 3 [ α 1)α 2)α 3)τ ) 3 b s) α4 τ s) α1 qs)f s, us), c D η us))ds t ) 3 γ b [ α 1)α 2)α 3)τ ) 3 b s) α4 Ɣα)[ γτ ) 3 τ qs)f s, us), c D η us))ds 3.21) where ut) = Gt, s)qs)f s, us), c D η us))ds Gt, s) = 1 Ɣα) t )3 γ [ γτ ) 3 Gτ, s)qs)f s, us), c D η us))ds 3.22) α 1)α 2)α 3)t ) 3 b s) α4 α 1)α 2)α 3)t ) 3 b s) α4 ts) α1, s t b, t s b Theorem 3.2. Greens function from theorem 3.1 stisfies the following conditions 1. Gt, s) is non-negtive for t,sɛ[,b. 2. Gt, s) is non-decresing with respect to the first vrible. 3. Gt, s) Gb, s) for t,sɛ[,b. Proof. 1. Since α 4 so for t s b Gt, s) = α 1)α 2)α 3)t )3 b s) α4 Ɣα) 0

8 988 Trchnd L. Holmbe nd Mohmmed Mzhr-Ul-Hque nd for s t b Gt, s) = α 1)α 2)α 3)t )3 b s) α4 t s)α1 Ɣα) Ɣα) α 1)α 2)α 3)t s)3 t s) α4 t s)α1 Ɣα) Ɣα) { } t s)α1 α 1)α 2)α 3) = 0 Ɣα) 2. Agin for t s b Gt, s) = α 1)α 2)α 3)t )3 b s) α4 Ɣα) Gt, s) t nd for s t b = α 1)α 2)α 3)t )2 b s) α4 2Ɣα) Gt, s) = α 1)α 2)α 3)t )3 b s) α4 t s)α1 Ɣα) Ɣα) Gt, s) = α 1)α 2)α 3)t )2 b s) α4 α 1)t s)α2 t 2Ɣα) Ɣα) α 1)α 2)α 3)t s)2 t s) α4 α 1)t s)α2 2Ɣα) Ɣα) { } t s)α2 α 1)α 2)α 3) 2α 1) = 0 Ɣα) 2 this shows tht the Greens function is non decresing nd immeditely stisfies. 3. Gt, s) Gb, s) for t,sɛ[,b. Theorem 3.3. Assume tht ft,x,y) c 1 x m +c 2 x n,t [,b,c 1,c 2 > 0,m,n b, Suppose the non-locl frctionl boundry vlue problem c D α ut) = qt)f t, ut),c D η ut)) u) = u) = u ) = 0,u b) = γ uτ) hs nontrivil continuous solution then Ɣα) 1 + b ) 1 )3 γ [ γτ ) 3 { α 1)α 2)α 3)b ) 3 b s) α4 b s) α1} qs) ds )

9 On Inequlity for the Non-Locl Frctionl Differentil Eqution 989 Proof. Here we re tking Bnch spce C[,b ={u :[,b R : u is continuous} with the stndrd supremum norm u = mx {ut) : t b} we hve the solution for the given non-locl frctionl boundry vlue problem now for t [,b ut) ut) ut) = Gt, s)qs)f s, us), c D η us))ds t )3 γ b [ γτ ) 3 Gτ, s)qs)f s, us), c D η us))ds Gt, s) qs) f s, us), c D η us)) ds + b )3 γ [ γτ ) 3 Gb, s) qs) f s, us), c D η us)) ds Gτ, s) qs) f s, us), c D η us)) ds + b )3 γ [ γτ ) 3 Gb, s) qs) f s, us), c D η us)) ds ut) 1 + b ) )3 γ b [ γτ ) 3 Gb, s) qs) { c 1 u m + c 2 c D η u } ds ut) 1 + b ) )3 γ b [ γτ ) 3 Gb, s) qs) { c 1 u +c 2 c D η u } ds for U R such tht mx { u,c 1 u +c 2 c D η u } U implies b ) )3 γ b [ γτ ) 3 Gb, s) qs) ds 3.24) 1 + b ) 1 )3 γ b [ γτ ) 3 Gb, s) qs) ds 3.25) now from the obtined Greens function 1 + b ) 1 )3 γ [ γτ ) 3 1 Ɣα) { α 1)α 2)α 3)b ) 3 b s) α4 b s) α1} qs) ds

10 990 Trchnd L. Holmbe nd Mohmmed Mzhr-Ul-Hque ) 1 Ɣα) 1 + b )3 γ [ γτ ) 3 { α 1)α 2)α 3)b ) 3 b s) α4 b s) α1} qs) ds Theorem 3.4. Let the non-locl frctionl boundry vlue problem hs nontrivil continuous solution then Ɣα 3) b ) α c D α ut) = qt)f t, ut),c D η ut)) u) = u) = u ) = 0,u b) = γ uτ) 1 + b ) 1 )3 γ [ γτ ) 3 Proof. Since { α 1)α 2)α 3)b ) 3 b s) α4 b s) α1} so the inequlity 3.2 becomes 3.2) qs) ds 3.27) { α 1)α 2)α 3)b ) 3 b s) α4} 3.28) Ɣα) α 1)α 2)α 3)b ) b ) 1 )3 γ [ γτ ) 3 Ɣα 3) b ) b ) 1 )3 γ [ γτ ) 3 b s) α4 qs) ds b s) α4 qs) ds by tking ψs) = b s) α4 where s [,b we hve ψ s) =α 4)b s) α5 0 now it is obvious tht ψs) = b s) α4 b ) α4 so Ɣα 3) b ) 3 Ɣα 3) b ) 3 Ɣα 3) b ) α 1 + b ) 1 )3 γ [ γτ ) b )3 γ [ γτ ) 3 ) b )3 γ [ γτ ) 3 ) 1 b ) α4 qs) ds b ) α3 qs) ds qs) ds 3.29)

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13 On Inequlity for the Non-Locl Frctionl Differentil Eqution 993 [33 Trchnd L. Holmbe, Mohmmed Mzhr - Ul-Hque, Govind P. Kmble, Approximtions to the Solution of Cuchy Type Weighted Nonlocl Frctionl Differentil Eqution, Nonliner Anlysis nd Differentil Equtions, Vol. 4, 201, no. 15, [34 Mohmmed Mzhr - Ul-Hque, Trchnd L. Holmbe, Govind P. Kmble, Solution to Weighted non-locl frctionl differentil eqution, Interntionl Journl of Pure nd Applied Mthemtics, Vol. 108, No. 1, 201, pp [35 Mohmmed Mzhr-ul-Hque, T.L. Holmbe, Positive Solutions of Qudrtic Frctionl Integrl Eqution With Generlized Mittg Leffler Function, Interntionl Journl of Mthemtics And its Applictions Volume 5, Issue 1-A 2017), ISSN: