Existence of solution for Hilfer fractional differential problem with nonlocal boundary condition
Hanan A. Wahash, Mohammed S. Abdo, Satish K. Panchal, Sandeep P. Bhairat
aa r X i v : . [ m a t h . G M ] S e p Stud. Univ. Babe¸s-Bolyai Math. ??(??), No. ??, 1–4
Existence of solution for Hilfer fractional dif-ferential problem with nonlocal boundarycondition
Hanan A. Wahash, Mohammed S. Abdo, Satish K.Panchal and Sandeep P. Bhairat
Abstract.
This paper is devoted to study the existence of a solution toHilfer fractional differential equation with nonlocal boundary condition.We use the equivalent integral equation to study the considered Hilferdifferential problem with nonlocal boundary condition. The M¨onch typefixed point theorem and the measure of the noncompactness techniqueare the main tools in this study. We demonstrate the existence of asolution with a suitable illustrative example.
Mathematics Subject Classification (2010):
Keywords: fractional differential equations, Hilfer fractional derivatives,Existence, Fixed point theorem.
1. Introduction
The calculus of arbitrary order has been extensively studied in the last fourdecades. It has been proved to be an adequate tool in almost all branchesof science and engineering. Because of its widespread applications, fractionalcalculus is becoming an integral part of applied mathematics research. Indeed,fractional differential equations have been found useful to describe abundantphenomena in physics and engineering, and the modest amount of work inthis direction has taken place, see [1, 6, 14] and references therein. For basicdevelopment and theoretical applications of fractional differential equations,see [20, 22].In the past two decades, the fractional differential equations are ex-tensively studied for existence, uniqueness, continuous dependence and sta-bility of the solution. For some fundamental results in existence theory
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H.A. Wahash, M.S. Abdo, S.K. Panchal and S.P. Bhairat ∗ of various fractional differential problems with initial and boundary con-ditions, see survey papers [1, 6], the monograph [22], the research papers[2, 3, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 21, 25, 26] and references therein.In the year 2018, Thabet et al. [24] investigated the existence of a solu-tion to BVP for Hilfer FDEs: D µ,νa + z ( t ) = f ( t, z ( t ) , Sz ( t )) , < µ < , ≤ ν ≤ , t ∈ ( a, b ] , (1.1) I − γa + (cid:2) uz ( a + ) + vz ( b − ) (cid:3) = w, µ ≤ γ = µ + ν (1 − µ ) , u, v, w ∈ R , (1.2)by using the M¨onch fixed point theorem.Recently, in [5], Abdo et al. obtained the existence of the solutions ofBVP for the class of Hilfer FDEs: D µ,νa + z ( t ) = f ( t, z ( t )) , p − < µ < p, ≤ ν ≤ I − γa + (cid:2) cz ( a + ) + dz ( b − ) (cid:3) = e, µ ≤ γ = µ + ν (1 − µ ) , (1.4)by using the Schauder, Schaefer and Krasnosel’skii’s fixed point theorems.Motivated by works cited above, in this paper, we consider the nonlocalboundary value problem for a class of Hilfer fractional differential equations(HNBVP): D µ,νa + z ( t ) = f ( t, z ( t )) , < µ < , ≤ ν ≤ , t ∈ ( a, b ] , (1.5) I − γa + cz ( a + )+ I − γa + dz ( b − ) = m X k =1 λ k z ( τ k ) , τ k ∈ ( a, b ] , µ ≤ γ = µ + ν − µν, (1.6)where D µ,νa + is the generalized Hilfer fractional derivative of order µ andtype ν , I − γa + is the Riemann-Liouville fractional integral of order 1 − γ , f : ( a, b ] × R → R be a function such that f ( t, z ) ∈ C − γ [ a, b ] for any z ∈ C − γ [ a, b ] and c, d ∈ R , for k = 1 , , · · · , m .The measure of noncompactness technique and a fixed point theorem ofMonch type are the main tools in this analysis.The paper is organized as follows: Some preliminary concepts relatedto our problem are listed in Section 2 which will be useful in the sequel. InSection 3, we first establish an equivalent integral equation of BVP and thenwe present the existence of its solution. An illustrative example is providedin the last section.
2. Preliminaries
In this section, we present some definitions, lemmas and weighted spaceswhich are useful in further development of this paper.Let J = [ a, b ] and J = ( a, b ] ∞ < a < b < + ∞ . Let C ( J , E ) ,AC ( J , E ) and C n ( J , E ) be the Banach spaces of all continuous, absolutelycontinuous, p − times continuous and continuously differentiable functions onn Hilfer Fractional BVP with nonlocal boundary conditions 3 J , respectively. Here L p ( J , E ) , p > , is the Banach space of measurablefunctions on J with the L p norm where k p k L p = Z ba | p ( s ) | p ds ! p < ∞ . Let L ∞ ( J , E ) be the Banach space of measurable functions z : J −→ E which are bounded and equipped with the norm k z k L ∞ = inf { e > k z k ≤ e, a.e t ∈ J } . Moreover, for a given set V of functions v : J −→ E let us denoteby V ( t ) = { v ( t ) : v ∈ V ; t ∈ J } , V ( J ) = { v ( t ) : v ∈ V ; t ∈ J } . Definition 2.1. [22]
Let g : [ a, ∞ ) → R is a real valued continuous function.The left sided Riemann-Liouville fractional integral of g of order µ > isdefined by I µa + g ( t ) = 1Γ( µ ) Z ta ( t − s ) µ − g ( s ) ds, t > a, (2.1) where Γ( · ) is the Euler’s Gamma function and a ∈ R . provided the right handside is pointwise defined on ( a, ∞ ) . Definition 2.2. [22]
Let g : [ a, ∞ ) → R is a real valued continuous function.The left sided Riemann-Liouville fractional derivative of g of order µ > isdefined by D µa + g ( t ) = 1Γ( p − µ ) d n dt n Z ta ( t − s ) n − µ − g ( s ) ds, (2.2) where n = [ µ ] + 1 , and [ µ ] denotes the integer part of µ. Definition 2.3. [20]
The left sided Hilfer fractional derivative of function g ∈ L ( a, b ) of order < µ < and type ≤ ν ≤ is denoted as D µ,νa + anddefined by D µ,νa + g ( t ) = I ν (1 − µ ) a + D p I (1 − ν )(1 − µ ) a + g ( t ) , D n = d n dt n . (2.3) where I µa + and D µa + are Riemann-Liouville fractional integral and derivativedefined by (2.1) and (2.2) , respectively. Remark 2.4.
From Definition 2.3, we observe that:(i) The operator D µ,νa + can be written as D µ,νa + = I ν (1 − µ ) a + D p I (1 − γ ) a + = I ν (1 − µ ) a + D γ , γ = µ + ν − µν .(ii) The Hilfer fractional derivative can be regarded as an interpolator be-tween the Riemann-Liouville derivative ( ν = 0) and Caputo derivative( ν = 1) as D µ,νa + = ( DI (1 − µ ) a + = D µa + , if ν = 0; I (1 − µ ) a + D = c D µa + , if ν = 1 . H.A. Wahash, M.S. Abdo, S.K. Panchal and S.P. Bhairat ∗ (iii) In particular, if γ = µ + ν − µν, then( D µ,νa + g )( t ) = (cid:16) I ν (1 − µ ) a + (cid:16) D γa + g (cid:17)(cid:17) ( t ) , where (cid:16) D γa + g (cid:17) ( t ) = ddt (cid:16) I (1 − ν )(1 − µ ) a + g (cid:17) ( t ) . Definition 2.5. [22]
Let ≤ γ < . The weighted spaces C γ [ a, b ] and C n − γ [ a, b ] are defined by C γ [ a, b ] = { g : ( a, b ] → R : ( t − a ) γ g ( t ) ∈ C [ a, b ] } , and C nγ [ a, b ] = { g : ( a, b ] → R , g ∈ C n − [ a, b ] : g ( n ) ( t ) ∈ C γ [ a, b ] } , n ∈ N with the norms k g k C γ = k ( t − a ) γ g k C = max {| ( t − a ) γ g ( t ) | : t ∈ [ a, b ] } , and k g k C n − γ = n − X k =0 k g ( k ) k C + k g ( n ) k C − γ , (2.4) respectively. Furthermore we recall following weighted spaces C µ,ν − γ [ a, b ] = (cid:8) g ∈ C − γ [ a, b ] : D µ,νa + g ∈ C − γ [ a, b ] (cid:9) , γ = µ + ν (1 − µ ) (2.5) and C γ − γ [ a, b ] = (cid:8) g ∈ C − γ [ a, b ] : D γa + g ∈ C − γ [ a, b ] (cid:9) , γ = µ + ν (1 − µ ) . where Let < µ < , ≤ ν ≤ and γ = µ + ν − µν . Clearly, D µ,νa + g = I ν (1 − µ ) a + D γa + g and C γ − γ [ a, b ] ⊂ C µ,ν − γ [ a, b ] . Lemma 2.6. [14] If µ > and ν > , and g ∈ L ( a, b ) for t ∈ [ a, b ] , then thefollowing properties hold: (cid:16) I µa + I νa + g (cid:17) ( t ) = (cid:16) I µ + νa + g (cid:17) ( t ) and (cid:16) D µa + I νa + g (cid:17) ( t ) = g ( t ) . In particular, if f ∈ C γ [ a, b ] or f ∈ C [ a, b ] , then the above properties hold foreach t ∈ ( a, b ] or t ∈ [ a, b ] respectively. Lemma 2.7. [22]
For t > a, we have (i). I µa + ( t − a ) δ − = Γ( δ )Γ( δ + µ ) ( t − a ) δ + µ − , µ ≥ , δ > . (ii). D µa + ( t − a ) µ − = 0 , µ ∈ (0 , . Lemma 2.8. [20]
Let µ > , ν > and γ = µ + ν − µν. If g ∈ C γ − γ [ a, b ] , then I γa + D γa + g = I µa + D µ,νa + g, D γa + I µa + g = D ν (1 − µ ) a + g. n Hilfer Fractional BVP with nonlocal boundary conditions 5 Lemma 2.9. [20]
Let < µ < , ≤ ν ≤ and g ∈ C − γ [ a, b ] . Then I µa + D µ,νa + g ( t ) = g ( t ) − I (1 − ν )(1 − µ ) a + g ( a )Γ( µ + ν (1 − µ )) ( t − a ) µ + ν (1 − µ ) − , for all t ∈ ( a, b ] , Moreover, if γ = µ + ν − µν, g ∈ C − γ [ a, b ] and I − γa + g ∈ C n − γ [ a, b ] , then I γa + D γa + g ( t ) = g ( t ) − I − γa + g ( a )Γ( γ ) ( t − a ) γ − , for all t ∈ ( a, b ] . Lemma 2.10. [21] If ≤ γ < and g ∈ C γ [ a, b ] , then ( I µa + g )( a ) = lim t → a + I µa + g ( t ) = 0 , < µ ≤ γ. Lemma 2.11. [23]
Let E be a Banach space and let Υ E be the bounded subsetsof E . The Kuratowski measure of noncompactness is the map α : Υ E −→ [0 , ∞ ) defined by α ( S ) = inf { ε > S ⊂ ∪ mi =1 S i and the diam ( S i ) ≤ ε } ; S ⊂ Υ E . Lemma 2.12. [18]
For all nonempty subsets S , S ⊂ E . The Kuratowskimeasure of noncompactness α ( · ) satisfies the following properties: α ( S ) = 0 ⇐⇒ S is compact ( S is relatively compact);2. α ( S ) = α ( S ) = α ( conv S ) , where where S and conv S denote the closureand convex hull of the bounded set S respectively;3. S ⊂ S = ⇒ α ( S ) ≤ α ( S );4. α ( S + S ) ≤ α ( S ) + α ( S ) , where S + S = { s + s : s ∈ S , s ∈ S } ;5. α ( κ S ) = | κ | α ( S ) , κ ∈ R ;For more details, see [5, 8, 19]. Lemma 2.13. [23]
Let B be a bounded, closed and convex subset of a Banachspace E such that ∈ B ; and let T be a continuous mapping of B into itself.If for every subset V of B V = co T ( V ) or V = T ( V ) ∪ { } = ⇒ α ( V )=0 holds. Then T has a fixed point. Lemma 2.14. [27]
Let B be a bounded, closed and convex subset of a Banachspace C ( J , E ) ; F is a continuous function on J × J ; and a function f : J × E −→ E satisfying the Carath´eodory conditions, and assume there exists ρ ∈ L P ( J , R + ) such that, for each t ∈ J and each bounded set B ∗ ⊂ E ; onehas lim r −→ + α ( f ( J t,r × B ∗ )) ≤ ρ ( t ) α ( B ∗ ) , where J t,r ∈ [ t − r, t ] ∩ J . If V is an equicontinuous subset of B ; then α (cid:18)(cid:26) Z J F ( t, s ) f ( s, z ( s )) ds : z ∈ V (cid:27)(cid:19) ≤ Z J k F ( t, s ) k ρ ( s ) α ( V ( s )) ds. H.A. Wahash, M.S. Abdo, S.K. Panchal and S.P. Bhairat ∗ Lemma 2.15. [12]
Let γ = µ + ν − µν where < µ < and ≤ ν ≤ . Let f : J × E → E be a function such that f ( t, z ) ∈ C − γ ( J , E ) for any z ∈ C − γ ( J , E ) . If z ∈ C γ − γ ( J , E ) , then z satisfies IVP (1.3) - (1.4) if andonly if z satisfies the Volterra integral equation z ( t ) = z a Γ( γ ) ( t − a ) γ − + 1Γ( µ ) Z ta ( t − s ) µ − f ( s, z ( s )) ds, t > a. (2.6)
3. Main results
Now we prove the existence of solution of HNBVP (1.5)-(1.6) in C γ − γ ( J , E ) ⊂ C µ,ν − γ ( J , E ) . Definition 3.1.
A function z ∈ C γ − γ ( J , E ) is said to be a solution of HNBVP (1.5) - (1.6) if z satisfies the differential equation D µ,νa + z ( t ) = f ( t, z ( t )) on ( a, b ] ,and the nonlocal condition I − γa + (cid:2) cz ( a + ) + dz ( b − ) (cid:3) = m X k =1 λ k z ( τ k ) . In the beginning, we need the following axiom lemma:
Lemma 3.2.
Let < µ < , ≤ ν ≤ where γ = µ + ν − µν , and f : J × R → R be a function such that f ( t, z ) ∈ C − γ ( J , E ) for any z ∈ C − γ ( J , E ) . If z ∈ C γ − γ ( J , E ) , then z satisfies HNBVP (1.5) - (1.6) if and only if z satisfiesthe following integral equation z ( t ) = ( t − a ) γ − Γ( γ ) 1( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − f ( s, z ( s )) ds − ( t − a ) γ − Γ( γ ) d ( c + d − A ) 1Γ(1 − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds + 1Γ( µ ) Z ta ( t − s ) µ − f ( s, z ( s )) ds, (3.1) where A = m X k =1 λ k ( τ k − a ) γ − Γ( γ ) , and c + d = A . Proof: In view of Lemma 2.15, the solution of (1.5) can be written as z ( t ) = I − γa + z ( a + )Γ( γ ) ( t − a ) γ − + 1Γ( µ ) Z ta ( t − s ) µ − f ( s, z ( s )) ds, t > a. (3.2)Applying I − γa + on both sides of (3.2) and taking the limit t → b − , weobtain I − γa + z ( b − ) = I − γa + z ( a + ) + 1Γ(1 − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds. (3.3)n Hilfer Fractional BVP with nonlocal boundary conditions 7Now, we substitute t = τ k in (3.2) and multiply by λ k to obtain λ k z ( τ k ) = λ k " I − γa + z ( a + )Γ( γ ) ( τ k − a ) γ − + 1Γ( µ ) Z τ k a ( τ k − s ) µ − f ( s, z ( s )) ds . (3.4)Using the nonlocal boundary condition (1.6) with (3.3) and (3.4), we have I − γa + z ( a + ) = 1 c m X k =1 λ k z ( τ k ) − dc I − γa + z ( a + )+ dc Γ(1 − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds. Therefore, by (3.4), we have I − γa + z ( a + ) = 1 c m X k =1 λ k I − γa + z ( a + )Γ( γ ) ( τ k − a ) γ − + 1 c m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − f ( s, z ( s )) ds − dc I − γa + z ( a + ) − dc − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds. = 1( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − f ( s, z ( s )) ds − d ( c + d − A ) 1Γ(1 − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds, (3.5)Submitting (3.5) into (3.2), we obtain z ( t ) = ( t − a ) γ − Γ( γ ) 1( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − f ( s, z ( s )) ds − ( t − a ) γ − Γ( γ ) d ( c + d − A ) 1Γ(1 − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds + 1Γ( µ ) Z ta ( t − s ) µ − f ( s, z ( s )) ds. (3.6) H.A. Wahash, M.S. Abdo, S.K. Panchal and S.P. Bhairat ∗ Conversely, applying I − γa + on both sides of (3.1), using Lemma 2.6 and2.7, some simple computations gives I − γa + (cid:0) cz ( a + ) + dz ( b − ) (cid:1) = c ( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − f ( s, z ( s )) ds − cd ( c + d − A ) 1Γ(1 − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds + d ( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − f ( s, z ( s )) ds − d ( c + d − A ) 1Γ(1 − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds + d Γ(1 − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds.I − γa + (cid:0) cz ( a + ) + dz ( b − ) (cid:1) = (cid:18) c ( c + d − A ) + d ( c + d − A ) (cid:19) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − f ( s, z ( s )) ds − (cid:18) d − cd ( c + d − A ) − d ( c + d − A ) (cid:19) Z ba ( b − s ) − γ + µ Γ(1 − γ + µ ) f ( s, z ( s )) ds = c + d ( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − f ( s, z ( s )) ds − Ad ( c + d − A ) 1Γ(1 − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds From (3.4) and (3.5), we conclude that I − γa + (cid:0) cz ( a + ) + dz ( b − ) (cid:1) = m X k =1 λ k z ( τ k ) , which shows that the boundary condition (1.6) is satisfied.Next, applying D γa + on both sides of (3.1) and using Lemma 2.7 and2.8, we have D γa + z ( t ) = D ν (1 − µ ) a + f (cid:0) t, z ( t ) (cid:1) . (3.7)Since z ∈ C γ − γ ( J , E ) and by definition of C γ − γ ( J , E ), we have D γa + z ∈ C − γ ( J , E ), therefore, D ν (1 − µ ) a + f = DI − ν (1 − µ ) a + f ∈ C − γ ( J , E ) . For f ∈ C − γ ( J , E ), it is clear that I − ν (1 − µ ) a + f ∈ C − γ ( J , E ). Hence f and I − ν (1 − µ ) a + f satisfy the hypothesis of Lemma 2.9.n Hilfer Fractional BVP with nonlocal boundary conditions 9Now, applying I ν (1 − µ ) a + on both sides of (3.7), we have I ν (1 − µ ) a + D γa + z ( t ) = I ν (1 − µ ) a + D ν (1 − µ ) a + f (cid:0) t, z ( t ) (cid:1) . Using Remark 2.4 (i), relation (3.7) and Lemma 2.9, we get I γa + D γa + z ( t ) = f (cid:0) t, z ( t ) (cid:1) − I − ν (1 − µ ) a + f (cid:0) a, z ( a ) (cid:1) Γ( ν (1 − µ )) ( t − a ) ν (1 − µ ) − , for all t ∈ J . By Lemma 2.10, we have I − ν (1 − µ ) a + f (cid:0) a, z ( a ) (cid:1) = 0. Therefore D µ,νa + z ( t ) = f (cid:0) t, z ( t ) (cid:1) . This completes the proof.To prove the existence of solutions for the problem at hand, let us makethe following hypotheses.(H1) The function f : J × E → E satisfies the Carath`eodory conditions.(H2) f : J × E → E is a function such that f ( · , z ( · )) ∈ C ν (1 − µ )1 − γ ( J , E ) forany z ∈ C − γ ( J , E ) and there exists ρ ∈ L p ( J , R + ) with p > µ and p > γ such that (cid:13)(cid:13) f (cid:0) t, z (cid:1)(cid:13)(cid:13) ≤ ρ ( t ) k z ( t ) k (cid:1) , for each t ∈ J , and all z ∈ E. (H3) The inequalities G : = (cid:0) γ ) (Λ q,µ,γ ) q ( c + d − A ) m X k =1 λ k Γ( µ ) ( τ k − a ) γ + µ − + (cid:1) γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) (∆ q,µ,γ ) q Γ(1 − γ + µ ) + (Λ q,µ,γ ) q Γ( µ ) (cid:1) ( b − a ) µ (cid:1) k ρ k L p < , and L ∗ : = (cid:0) m Γ( γ ) ( b − a ) γ − ( c + d − A ) m X k =1 λ k ( τ k − a ) µ Γ( µ + 1)+ (cid:0) γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) − γ + µ ) + 1Γ( µ + 1) (cid:1) ( b − a ) µ (cid:1) k ρ k L p < Theorem 3.3.
Assume that (H1)-(H3) are satisfied. Then HNBVP (1.5) - (1.6) has at least one solution in C γ − γ ( J , E ) ⊂ C µ,ν − γ ( J , E ) . ∗ Proof.
Transform the problem (1.5)-(1.6) into a fixed point problem. Definethe operator T : C − γ ( J , E ) −→ C − γ ( J , E ) as T z ( t ) = ( t − a ) γ − Γ( γ ) 1( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − f ( s, z ( s )) ds − ( t − a ) γ − Γ( γ ) d ( c + d − A ) 1Γ(1 − γ + µ ) Z ba ( b − s ) − γ + µ f ( s, z ( s )) ds + 1Γ( µ ) Z ta ( t − s ) µ − f ( s, z ( s )) ds. (3.8)Clearly, from Lemma 3.2, the fixed points of T are solutions to (1.5)-(1.6).Let B R = n z ∈ C − γ ( J , E ) : k z k C − γ ≤ R o . We shall show that T satisfiesthe conditions of M¨onch’s fixed point theorem. The proof will be given in thefollowing four steps:Step1: We show that T ( B R ) ⊂ B R . By definition of T , hypothesis ( H )and H¨older’s inequality, we have (cid:13)(cid:13) ( T z )( t )( t − a ) − γ (cid:13)(cid:13) = 1Γ( γ ) 1( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − k f ( s, z ( s )) k ds + 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) − γ + µ ) Z ba ( b − s ) − γ + µ k f ( s, z ( s )) k ds + ( t − a ) − γ Γ( µ ) Z ta ( t − s ) µ − k f ( s, z ( s )) k ds ≤ γ ) 1( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − ( s − a ) γ − ρ ( s ) k z k C − γ ds + 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) Z ba ( b − s ) − γ + µ Γ(1 − γ + µ ) ( s − a ) γ − ρ ( s ) k z k C − γ ds + ( t − a ) − γ Γ( µ ) Z ta ( t − s ) µ − ( s − a ) γ − ρ ( s ) k z k C − γ ds ≤ γ ) m X k =1 λ k Γ( µ ) (cid:18)Z τ k a ( τ k − s ) ( µ − q ( c + d − A ) ( s − a ) ( γ − q ds (cid:19) q k ρ k L p k z k C − γ + 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) Z ba ( b − s ) ( − γ + µ ) q Γ(1 − γ + µ ) ( s − a ) ( γ − q ds ! q × k ρ k L p k z k C − γ + ( t − a ) − γ Γ( µ ) × (cid:18)Z ta ( t − s ) ( µ − q ( s − a ) ( γ − q ds (cid:19) q k ρ k L p k z k C − γ . (3.9)n Hilfer Fractional BVP with nonlocal boundary conditions 11Since q > , p > µ and p + q = 1 , the change of variable s = a − u ( τ k − a )yields (cid:18)Z τ k a ( τ k − s ) ( µ − q ( s − a ) ( γ − q ds (cid:19) q ≤ (Λ q,µ,γ ) q ( τ k − a ) γ + µ − , (3.10)the change of variable s = a − u ( b − a ) gives Z ba ( b − s ) ( − γ + µ ) q ( s − a ) ( γ − q ds ! q ≤ (∆ q,µ,γ ) q ( b − a ) µ , (3.11)and the change of variable s = a − u ( t − a ) gives us (cid:18)Z ta ( t − s ) ( µ − q ( s − a ) ( γ − q ds (cid:19) q ≤ (Λ q,µ,γ ) q ( t − a ) γ + µ − , (3.12)where Λ q,µ,γ := Γ( q ( µ −
1) + 1)Γ( q ( γ −
1) + 1)Γ( q ( µ + γ −
2) + 2) , and ∆ q,µ,γ := Γ( q ( µ − γ ) + 1)Γ( q ( γ −
1) + 1)Γ( q ( µ −
1) + 2) . Substitution of (3.10),(3.11) and (3.12) into (3.9) leads (cid:13)(cid:13) ( T z )( t )( t − a ) − γ (cid:13)(cid:13) ≤ γ ) 1( c + d − A ) m X k =1 λ k Γ( µ ) (Λ q,µ,γ ) q ( τ k − a ) γ + µ − k ρ k L p k z k C − γ + 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) − γ + µ ) (∆ q,µ,γ ) q ( b − a ) µ k ρ k L p k z k C − γ + ( t − a ) − γ Γ( µ ) (Λ q,µ,γ ) q ( t − a ) γ + µ − k ρ k L p k z k C − γ . For any z ∈ B R , we obtain kT z k C − γ ≤ (cid:18) γ ) (Λ q,µ,γ ) q ( c + d − A ) m X k =1 λ k Γ( µ ) ( τ k − s ) γ + µ − (cid:19) + 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) (∆ q,µ,γ ) q Γ(1 − γ + µ ) + (Λ q,µ,γ ) q Γ( µ ) (cid:1) ( b − a ) µ (cid:1) k ρ k L p R. By (H3), we have kT z k C − γ ≤ G R ≤ R, that is, T ( B R ) ⊂ B R . Step 2. We shall prove that T is completely continuous.The operator T is continuous. Let { z n } n ∈ N is a sequence such that z n → z ∗ in B R . Then for each t ∈ J , we have (cid:13)(cid:13)(cid:0) ( T z n )( t ) − ( T z )( t ) (cid:1) ( t − a ) − γ (cid:13)(cid:13) = 1Γ( γ ) 1( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − k f ( s, z n ( s )) − f ( s, z ( s )) k ds + 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) Z ba ( b − s ) − γ + µ Γ(1 − γ + µ ) k f ( s, z n ( s )) − f ( s, z ( s )) k ds + ( t − a ) − γ Γ( µ ) Z ta ( t − s ) µ − k f ( s, z n ( s )) − f ( s, z ( s )) k ds ≤ γ ) 1( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − ( s − a ) γ − ds × (cid:13)(cid:13) f (cid:0) · , z n ( · ) (cid:1) − f (cid:0) · , z ( · ) (cid:1)(cid:13)(cid:13) C − γ + 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) − γ + µ ) Z ba ( b − s ) − γ + µ ( s − a ) γ − ds × (cid:13)(cid:13) f (cid:0) · , z n ( · ) (cid:1) − f (cid:0) · , z ( · ) (cid:1)(cid:13)(cid:13) C − γ + ( t − a ) − γ Γ( µ ) Z ta ( t − s ) µ − ( s − a ) γ − ds (cid:13)(cid:13) f (cid:0) · , z n ( · ) (cid:1) − f (cid:0) · , z ( · ) (cid:1)(cid:13)(cid:13) C − γ (cid:13)(cid:13)(cid:0) ( T z n )( t ) − ( T z )( t ) (cid:1) ( t − a ) − γ (cid:13)(cid:13) ≤ c + d − A ) B ( γ, µ )Γ( µ )Γ( γ ) m X k =1 λ k ( τ k − a ) γ − µ Γ( µ ) (cid:13)(cid:13) f (cid:0) · , z n ( · ) (cid:1) − f (cid:0) · , z ( · ) (cid:1)(cid:13)(cid:13) C − γ + (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) ( b − a ) µ Γ( µ + 1) (cid:13)(cid:13) f (cid:0) · , z n ( · ) (cid:1) − f (cid:0) · , z ( · ) (cid:1)(cid:13)(cid:13) C − γ + ( b − a ) µ Γ( µ ) B ( γ, µ )Γ( µ ) (cid:13)(cid:13) f (cid:0) · , z n ( · ) (cid:1) − f (cid:0) · , z ( · ) (cid:1)(cid:13)(cid:13) C − γ . By (H1) and the Lebesgue dominated convergence theorem, we have k ( T z n − T z ) k C − γ −→ as n −→ ∞ , which means that operator T is continuous on B R .Step 3. T ( B R ) is relatively compact.From Step 1, we have T ( B R ) ⊂ B R . It follows that T ( B R ) is uniformlybounded i.e. T maps B R into itself. Moreover, we show that operator T isn Hilfer Fractional BVP with nonlocal boundary conditions 13equicontinuous on B R . Indeed, for any a < t < t < b and z ∈ B R , we get (cid:13)(cid:13) ( t − a ) − γ (cid:0) T z (cid:1) ( t ) − ( t − a ) − γ (cid:0) T z (cid:1) ( t ) (cid:13)(cid:13) ≤ µ ) (cid:13)(cid:13)(cid:13)(cid:13) ( t − a ) − γ Z t a ( t − s ) µ − f (cid:0) s, z ( s ) (cid:1) ds − ( t − a ) − γ Z t a ( t − s ) µ − f (cid:0) s, z ( s ) (cid:1) ds (cid:13)(cid:13)(cid:13)(cid:13) ≤ k f k C − γ Γ( µ ) (cid:13)(cid:13)(cid:13)(cid:13) ( t − a ) − γ Z t a ( t − s ) µ − ( s − a ) γ − ds − ( t − a ) − γ Z t a ( t − s ) µ − ( s − a ) γ − ds (cid:13)(cid:13)(cid:13)(cid:13) ≤ k f k C − γ B ( γ, µ )Γ( µ ) k ( t − a ) µ − ( t − a ) µ k , which tends to zero as t → t , independent of z ∈ B R , where B ( · , · ) is aBeta function. Thus we conclude that T ( B R ) is equicontinuous on B r andtherefore is relatively compact. As a consequence of Steps 1 to 3 togetherwith Arzela-Ascoli theorem, we conclude that T : B R → B R is completelycontinuous operator.Step 4: The M¨onch condition is satisfied.Let V be a subset of B R such that V ⊂ co ( T ( V ) ∪ { } ) . V is bounded andequicontinuous, and therefore the function t −→ α ( V ( t )) is continuous on J . By (H2)-(H3), Lemma 2.6, and the properties of the measure α, for each t ∈ J α ( V ( t )) ≤ α ( T ( V )( t ) ∪ { } ) ≤ α ( T ( V )( t )) ≤ γ ) ( t − a ) γ − ( c + d − A ) m X k =1 λ k Γ( µ ) Z τ k a ( τ k − s ) µ − ρ ( s ) α ( V ( s )) ds + 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( t − a ) γ − ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) − γ + µ ) Z ba ( b − s ) − γ + µ ρ ( s ) α ( V ( s )) ds + 1Γ( µ ) Z ta ( t − s ) µ − ρ ( s ) α ( V ( s )) ds ≤ γ ) ( b − a ) γ − ( c + d − A ) m X k =1 λ k Γ( µ ) (cid:18)Z τ k a ( τ k − s ) ( µ − q ds (cid:19) q k ρ k L p mα ( V ( b ))+ 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( b − a ) γ − ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) − γ + µ ) Z ba ( b − s ) ( − γ + µ ) q ds ! q × k ρ k L p α ( V ( b )) + 1Γ( µ ) (cid:18)Z ta ( t − s ) ( µ − q ds (cid:19) q k ρ k L p α ( V ( b )) . where we have used the fact that1 q < ⇒ µ − q + 1 < µ , < µ < , ∗ and 1 q < ⇒ − γ + µ ) q + 1 < − γ + µ ) , < µ < γ < . Hence α ( V ( t )) ≤ (cid:0) m Γ( γ ) ( b − a ) γ − ( c + d − A ) m X k =1 λ k ( τ k − a ) µ Γ( µ + 1)+ 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) ( b − a ) µ Γ( − γ + µ ) + ( t − a ) µ Γ( µ + 1) (cid:1) k ρ k L p < . It follows that k α ( V ) k L ∞ (1 − L ∗ ) ≤ . This means k α ( V ) k L ∞ = 0 , i.e. α ( V ( t )) = 0 for all t ∈ J . Thus V ( t )is relatively compact in E . In view of Arzela-Ascoli theorem, V is relativelycompact in B R . An application of Theorem 2.13 shows that T has a fixedpoint which is a solution of HNBVP (1.5)-(1.6). The proof is complete. (cid:3)
4. An example
We consider the Hilfer fractional differential equation with nonlocal boundarycondition ( D µ,ν + z ( t ) = f (cid:0) t, z ( t ) (cid:1) , t ∈ (0 , , < µ < , ≤ ν ≤ ,I − γ + (cid:2) z (0 + ) + z (1 − ) (cid:3) = z ( ) , µ ≤ γ = µ + ν − µν, (4.1)where f (cid:0) t, z ( t ) (cid:1) = t sin | z ( t ) | , µ = , ν = , γ = , c = , d = , λ = and τ = . Let E = R + and J = (0 , . Clearly we can see that √ tf (cid:0) t, z ( t ) (cid:1) = √ t sin z ( t ) ∈ C ([0 , , R + ) , andhence f (cid:0) t, z ( t ) (cid:1) ∈ C ([0 , , R + ) . Also, observe that, for t ∈ (0 ,
1] and for any z ∈ C − ([0 , , R + ): (cid:13)(cid:13) f (cid:0) t, z ( t ) (cid:1)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) t sin | z ( t ) | (cid:13)(cid:13)(cid:13)(cid:13) ≤ t k z ( t ) k . Therefore, the conditions (H1) and (H2) is satisfied with ρ ( t ) = t. Select p = , we have k p k L = k p k L = Z (cid:12)(cid:12)(cid:12)(cid:12) s (cid:12)(cid:12)(cid:12)(cid:12) ds ! = 148 . It is easy to checkthat conditions in (H3) are satisfied too. Indeed, by some simple computationswith q = , we getΛ q,µ,γ := Γ( q ( µ −
1) + 1)Γ( q ( γ −
1) + 1)Γ( q ( µ + γ −
2) + 2) = Γ( )Γ( )Γ( ) , ∆ q,µ,γ := Γ( q ( µ − γ ) + 1)Γ( q ( γ −
1) + 1)Γ( q ( µ −
1) + 2) = Γ( )Γ( )Γ( ) , n Hilfer Fractional BVP with nonlocal boundary conditions 15 G := (cid:18) γ ) (Λ q,µ,γ ) q ( c + d − A ) λ Γ( µ ) ( τ − a ) γ + µ − (cid:19) + 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) × (∆ q,µ,γ ) q Γ(1 − γ + µ ) + (Λ q,µ,γ ) q Γ( µ ) (cid:0) ( b − a ) µ (cid:1) k ρ k L p ≃ . < L ∗ := (cid:0) m Γ( γ ) ( b − a ) γ − ( c + d − A ) m X k =1 λ k ( τ k − s ) µ Γ( µ + 1) + 1Γ( γ ) (cid:12)(cid:12)(cid:12)(cid:12) d ( c + d − A ) (cid:12)(cid:12)(cid:12)(cid:12) × ( b − a ) µ Γ( − γ + µ ) + ( b − a ) µ Γ( µ + 1) (cid:1) k ρ k L p ≃ . < , ( m = 1) . An application of Theorem 3.3 implies that problem (4.1) has a solution in C − ([0 , , R + ). References [1] Abbas, S., Benchohra, M., Lazreg, J.E. and Zhou, Y.
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Rend. Semin. Mat. U. Pad., (1986), 1–14. n Hilfer Fractional BVP with nonlocal boundary conditions 17 Hanan A. Wahash“Department of Mathematics”,Dr.Babasaheb Ambedkar Marathwada University,Aurangabad - 431001, (M.S) Indiae-mail: [email protected]
Mohammed S. Abdo“Department of Mathematics”,Dr.Babasaheb Ambedkar Marathwada University,Aurangabad - 431001, (M.S) Indiae-mail: [email protected]
Satish K. Panchal“Department of Mathematics”,Dr.Babasaheb Ambedkar Marathwada University,Aurangabad - 431001, (M.S) Indiae-mail: [email protected]
Sandeep P. Bhairat “Faculty of Engineering Mathematics”,Institute of Chemical Technology Mumbai,Marathwada Campus, Jalna - 431 203 (M.S) India.e-mail: [email protected]