# Nonlocal Fractional Hybrid Boundary Value Problems Involving Mixed Fractional Derivatives and Integrals via a Generalization of Darbo’s Theorem

Authors: Ayub Samadi; Sotiris K. Ntouyas; Jessada Tariboon
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this work, a new existence result is established for a nonlocal hybrid boundary value problem which contains one left Caputo and one right Riemann–Liouville fractional derivatives and integrals. The main result is proved by applying a new generalization of Darbo’s theorem associated with measures of noncompactness. Finally, an example to justify the theoretical result is also presented.


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## Body

## 1. Introduction

In the past years, fractional differential equations have attracted a lot of attention from many research studies as they have played a key role in many basic sciences such as chemistry, control theory, biology, and other arenas [1–3]. In addition, boundary conditions of differential models are the strongest tools to extend applications of those equations [4–6]. In fact, fractional differential equations can be extended by creating different types of boundary conditions. Newly, many authors have studied various types of boundary conditions to obtain new results of differential models.The following hybrid differential equation was studied by Dhage and Lakshmikantham [7]:(1)ddtxtht,xt=ωt,xt,a.e t∈J,xt0=x0∈ℝ,where h and ω are continuous functions from J×ℝ into ℝ∖0 and ℝ, respectively. Based on the above work, the Caputo hybrid boundary value problem of the form (2)DC0+pxtht,xt=ωt,xt,a.e t∈I≔0,L,a1x0h0,x0+a2xLhL,xL=d,was studied by Hilal and Kajouni [8] in which 0<p<1, h and ω are continuous functions from J×ℝ into ℝ∖0 and ℝ, respectively, and a1,a2,andd are real constants with a1+a2≠0. For some recent results on hybrid fractional differential equations, see [9–12].In [13], the authors proved the following integro-differential equation: (3)Dc1−α1DRL0+α2ut+θI−1pI0+qf1t,ut=ft,ut,t∈0,1,u0=uξ=0,u1=δuμ,0<ξ<μ<1,where Dc1−α1 and DRL0+α2 indicate right Caputo and left Riemann–Liouville fractional derivatives of orders α1∈1,2 and α2∈0,1, respectively, f1,f2:0,1×ℝ⟶ℝ are continuous functions, and the symbols I−1p and I0+q denote both right and left Riemann–Liouville fractional integrals of orders p,q>0, respectively. Ahmad et al. [13] applied Banach and Krasnosel’skiĭ fixed point theorems as well Leray–Schauder nonlinear alternative to obtain main results. We point out that fractional differential equations containing mixed fractional derivatives appear in the study of variational principles [14].For some recent results for boundary value problems involving left or/and right fractional derivatives, we refer to the papers [15–31] and references therein.In the present paper, we combine mixed fractional derivatives and hybrid fractional differential equations. More precisely, we investigate the existence of solutions for the following hybrid boundary value problem which contains both left Caputo and right Riemann–Liouville fractional derivatives and integrals and nonlocal hybrid conditions of the form:(4)Dc1−α1DRL1+α2utgt,ut+θI−1pI0+qf1t,ut=f2t,yt,t∈J≔0,1,u0g0,u0=uξgξ,uξ=0,u1g1,u1=δuμgμ,uμ,0<ξ<μ<1,where Dc1−α1 and DRL0+α2 are right Caputo and left Riemann–Liouville fractional derivatives of orders α1∈1,2 and α2∈0,1, respectively, and the symbols I−1p and I0+q denote both right and left Riemann–Liouville fractional integrals of orders p,q>0, respectively, f1,f2∈CJ×ℝ,ℝ, g∈CJ×ℝ,ℝ∖0, and δ,θ∈ℝ. An existence result is obtained via a new extension of Darbo’s theorem associated with measures of noncompactness.The structure of the paper has been organized as follows. Section2 presents some basic definitions and lemmas which will be applied in the future. In Section 3, we prove an existence result for problem (4). Finally, we present an example to illustrate the obtained result.


## 2. Preliminaries

Now, some basic notations are recalled from [2].Definition 1.
For an integrable functionϕ:0,∞⟶ℝ, we define the left and right Riemann–Liouville fractional integrals of order β>0, respectively, by (5)I0+βϕt=∫0tt−sβ−1Γβϕsds,I1−βϕt∫t1s−tβ−1Γβϕsds.Definition 2.
For the functionϕ:0,∞⟶ℝ in which ϕ∈Cn0,∞, we define the left Riemann–Liouville fractional derivative and the right Caputo fractional derivative of order β∈n−1,n, respectively, by(6)DRL0+βϕt=dndtn∫0tt−sn−β−1Γn−βϕsds,Dc1−βϕt=−1n∫t1s−tn−β−1Γn−βϕnsds.Lemma 1.
Ifp>0 and q>0, then the following relations hold almost everywhere on a,b:(7)I1−pI−1qfx=I1−p+qfx,I0+pI0+qfx=I0+p+qfx.As the technique of measure of noncompactness will be applied to obtain our main result, we recall some basic facts about the notion of measure of noncompactness.Assume thatZ is the real Banach space with the norm ⋅ and zero element θ. For a nonempty subset X of Z, the closure and the closed convex hull of X will be denoted by X¯ and ConvX, respectively. Also, MZ and NZ denote the family of all nonempty and bounded subsets of Z and its subfamily consisting of all relatively compact sets, respectively.Definition 3 (see [32]).
We say that a mappingh:MZ⟶0,∞ is a measure of noncompactness, if the following conditions hold true:(1)
The family Kerh=X∈MZ:hX=0 is nonempty and Ker h⊆NZ(2)
X1⊆Y1⇒hX1≤hY1(3)
hX¯=hX(4)
hConvX=hX(5)
hαX+1−αY≤αhX+1−αhY for α∈0,1(6)
For the sequenceXn of closed sets from MZ in which Xn+1⊆Xn for n=1,2,… and limn⟶∞hXn=0, we have ∩n=1∞Xn≠∅In [33], some generalizations of Darbo’s theorem have been proved by Samadi and Ghaemi. Also, in [34], Darbo’s theorem was extended, and the following result was presented which is basis for our main result.Theorem 1.
LetT be a continuous self-mapping operator on the set D, where D denotes a nonempty, bounded, closed, and convex subset of a Banach space Z. Assume that, for all nonempty subset X of D, we have(8)θ1hX+θ2hTX≤θ2hX,where h is an arbitrary measure of noncompactness defined in Z and θ1,θ2∈U. Then, T has a fixed point in D.In Theorem1, let U indicate the set of all pairs θ1,θ2 where the following conditions hold true:(U1) θ1tn↛0 for each strictly increasing sequence tn(U2) θ2 is strictly increasing function(U3) If αn be a sequence of positive numbers, then limn⟶∞αn=0⇔limn⟶∞θ2αn=−∞(U4) Let ln be a decreasing sequence in which ln⟶0 and θ1ln<θ2ln−θ2ln+1, then we have ∑n=1∞ln<∞Next, the definition of a measure of noncompactness in the spaceC0,1 is recalled which will be applied later. Fix Y∈MC0,1, and for ε>0 and y∈Y, we define(9)φy,ɛ=supyt−ys:t,s∈0,1,t−s≤ɛ,φY,ɛ=supφy,ɛ:y∈Y,φ0Y=limɛ⟶0φY,ɛ.Banas and Goebel [32] proved that φ0Y is a measure of noncompactness in the space C0,1.Lemma 2 (see [32]).
The measure of noncompactnessφ0 on CI satisfies the following condition:(10)φ0XY≤Xφ0Y+Yφ0X,for all X,Y⊆CI.


## 3. Main Existence Result

In this section, an existence result of problem (4) is investigated. In view of [13], Lemma 2, we present the following lemma which is an essential tool in our consideration.Lemma 3.
LetH1,F1∈C0,1∩L0,1, g∈C0,1,ℝ∖0, and Λ≠0. Then, the solution of the problem(11)Dc1−α1DRL0+βutgt,ut+λI−1pI0+qH1t=F1t,t∈0,1,u0g0,u0=uξgξ,uξ=0,u1g1,u1=δuμgμ,uμ,0<ξ<μ<1,has the form:(12)ut=gt,ut∫0tt−sα2−1Γα2I1−αF1s−λI1−α1+pI0+qH1sds+a1tδ∫0μμ−sα2−1Γα2I1−αF1s−λI1−α1+pI0+qHsds−∫011−sα2−1Γα2I1−α1F1s−λI1−α1+pI0+qH1sds+a2t∫0ξξ−sα2−1Γα2I1−α1F1s−λI1−α1+pI0+qH1sds,where(13)a1t=1Λξα2+1tα2−ξα2tα2+1,a2t=1Λtα21−δμα2+1−tα2+11−δμα2,Λ=ξα2+11−δμα2−ξα21−δμα2+1.Now, the hypotheses which will be applied to prove the main result of this section are presented.H1g:0,1×ℝ⟶ℝ∖0 is a continuous function, and there exists a positive real number d>0 provided that(14)gt,x1−gt,x2≤e−dx1−x2,wheret∈I and x1,x2∈ℝ. Moreover, assume that g¯=supgt,0;t∈0,1.H2f1,f2:0,1×ℝ⟶ℝ are continuous functions provided that(15)f1t,u≤M1,f1t,u−f1t,v≤k1u−v,f2t,u≤M2,f2t,u−f2t,v≤k2u−v,whereM1,M2,k1,k2≥0 and u,v∈ℝ.H3 The inequality(16)e−dr0+g¯M1Γα1+1+θM2Γα1+p+1Γq+1Δ≤r0,has a positive solutionr0. Also, assume that(17)M1Γα1+1+θM2Γα1+p+1Γq+1Δ<1,where(18)Δ=1Γα2+11+a1¯|δ|μα2+1+a2¯ξα2,a1¯=maxt∈0,1a1t,a2¯=maxt∈0,1a2t.Theorem 2.
Suppose that the hypothesesH1−H3 are true. Then, the hybrid boundary value problem (4) has at least one solution on 0,1.Proof.
Due to Lemma3, assume that the operator T has been defined on CI,I≔0,1 as follows:(19)T1ut=F1ut+F1¯utG1ut,where(20)G1ut=gt,ut,F1ut=∫0tt−sα2−1Γα2I1−α1f2s,us−θI1−α1+pI0+qf1s,usds,F1¯ut=a1tδ∫0μμ−sα1−1Γα2I1−α1f2s,us−θI1−α1+pI0+qf2s,usds−∫011−sα2−1Γα2I1−α1f2s,us−θI1−α1+pI0+qf1s,usds+a2t∫0ξξ−sα2−1Γα2I1−α1f2s,us−θI1−α1+pI0+qf1s,usds.
First, we show thatT1u∈CI in which u∈CI. In view of assumption H1, we conclude that G1u∈CI, u∈CI. Consequently, by proving F1u,F2¯u∈CI, the claim is obtained. Let ln be a sequence in 0,1 such that ln⟶l. Then, due to our assumptions, we get(21)F1uln−F1ul≤∫0lln−sα2−1−l−sα2−1Γα2I1−α1f2s,us−θI1−α1+pI0+qf1s,usds+∫llnln−sα2−1Γα2I1−α1f2s,us−θI1−α1+pI0+qf1s,usds≤M1Γα2∫0lnln−sα2−1−l−sα2−1I1−α11ds+M11Γα2∫llnln−sα2−1I1−α11ds+θM2Γα2∫0lln−sα2−1−l−sα2−1I1−α1+pI0+q1ds+θM2Γα2∫llnln−sα2−1I1−α1+pI0+q1ds≤M1Γα1+1Γα2+1+θM2Γq+1Γα2+1γα1+p+1×2ln−lα2+lnα2−lα2⟶0.
Hence,F1u∈CI. To obtain that F1¯u∈CI, by the definitions of a1 and a2, we have a1ln−a1l⟶0 and a2ln−a2l⟶0. Hence, we have F1¯uln−F1¯ul⟶0. Consequently, T1u∈CI for all x∈CI.
Now, we prove that the ballDr0=u∈CI:u≤r0 is mapped into itself by the operator T. Let us fix u∈CI. Hence, due to existence assumptions, for t∈I, we have(22)T1ut≤e−dut+g¯∫0tt−sα2−1Γα2I1−α1f2s,us−θI1−α1+pI0+qf1s,usds+a1tδ∫0μμ−sα2−1Γα2I1−α1f2s,us−θI1−α1+pI0+qf2s,usds−∫011−sα2−1Γα2I1−α1f2s,us−θI1−α1+pI0+qf1s,usds+a2t∫0ξξ−sα2−1Γα2I1−α1f2s,us−θI1−α1+pI0+qf1s,usds≤e−dr0+g¯M1Γα1+1+θM2Γα1+p+1Γq+1Δ.
Consequently, according to assumptionH3 we conclude that T maps the ball Dr0 into itself.
Now, the continuity property of the operatorT is considered on the ball Dr0. To do this, fix ɛ>0 and take u,v∈Dr0 such that u−v≤ɛ. Then, for t∈I, we have(23)T1ut−T1vt=F1utG1ut−F1vtG1vt+F1¯utG1ut−F1¯vtG1vt≤F1utG1ut−F1vtG1ut+F1vtG1ut−F1vtG1vt+F1¯utG1ut−F1¯vtG1ut+F1¯vtG1ut−F1¯vtG1vt≤gt,ut−gt,vtF1vt+gt,utF1ut−F1vt+gt,ut−gt,vtF1¯vt+gt,utF1¯ut−F1¯vt.
Then, we have(24)T1ut−T1vt≤e−dεF1vt+F1¯vt+gt,utε∫0tt−sα2−1Γα2k1I1−α11+k2|θ|I1−α1+pI0+q1ds+gt,utεa1tδ∫0μμ−sα2−1Γα2k1I1−α11+k2|θ|I1−α1+pI0+q1ds+∫011−sα2−1Γα2k1I1−α11+k2|θ|I1−α+pI0+q1ds+|a2t|∫0ξξ−sα2−1Γα2k1I1−α11+k2|θ|I1−α1+pI0+q1ds≤εk1Γα1+1+|θ|k2Γα1+p+1Γq+1e−dΔ+e−dr0+g¯Δ+1.
Consequently, the continuity property ofT is obtained on the ball Dr0.
To finish the proof, condition (8) of Theorem 1 is proved. Consider X as a nonempty subset of the ball Dr0 and assume that u∈X, ɛ>0 be arbitrarily constant. Choose l1,l2∈0,1 such that l1<l2 and l2−l1<ɛ. Taking into account our assumptions, we get(25)G1ul1−G1ul2=gl1,ul1−gl2,ul2≤gl1,ul1−gl1,ul2+gl1,ul2−gl2,ul2≤e−dφX,ɛ+φg,ɛ,where(26)φg,ɛ=supgl1,u−gl2,u;l1,l2∈I,l1−l2<ɛ,u∈−r0,r0.
Consequently,(27)φG1X,ɛ≤e−dφX,ɛ+φg,ɛ.
Asg is uniformly continuous on I×−r0,r0, we have φg,ɛ⟶0 as ɛ⟶0. Thus, from (27), we conclude that(28)φ0G1X≤e−dφ0X.
Next, we estimateφ0F1X and φ0F1¯X. In view of (21), since F1x is uniformly continuous on 0,1, then for fixed ε>0, there exists δ>0 such that, for l1,l2∈I with l2−11<δ≤ε, we have(29)φ0F1X≤ε.
Besides, sincea1 and a2 are uniformly continuous on 0,1, for l1,l2∈0,1 with l2−11<δ≤ε, we have a2l2−a2l1<ε and also a1l2−a1l1<ε. Consequently, we conclude that φ0F1¯X=0. Now, we estimate φ0T1X for X⊆Dr0. By applying (28) and (29) and Lemma 2 and using the fact that φ0F1¯X=0, we get(30)φ0T1X=φ0G1XF1X+F1¯X≤F1X+F1¯Xφ0G1X+G1Xφ0F1X+φ0F1¯X≤e−dφ0XM1Γα1+1+θM2Γα1+p+1Γq+1Δ.
Consequently, we derive that(31)d+lnφ0T1X≤lnφ0X.
Thus, we conclude the contractive condition in Theorem1 with θ1t=d and θ2t=lnt. Thus, by Theorem 1, at least one solution is obtained for the operator T in Dr0 which is a solution of problem (4) and the proof is completed.Now, the following example is investigated to show the applicability of the obtained result.Example 1.
Consider the following hybrid boundary value problem:(32)D1−3/2D0+1/2ute−d/1+t+|ut|+2I1−4/3I0+5/4e−t100cosut=e−t100sinut,u0e−d/1+|u0|=u2/3e−d/t+1+u2/3=0,u1e−d/2+|u1|=12u3/4e−d/1+3/4+u3/4=0.
By putting(33)gt,ut=e−d1+t+ut,f1t,ut=e−t100cosut,f2t,ut=e−t100sinut,in problem (4), we conclude the above hybrid boundary value problem as a special case of problem (4). Now, the conditions of Theorem 2 are checked. For all l∈0,1 and u1,u2∈ℝ, we have(34)gl,u1−gl,u2≤e−du1−u2.
Moreover, we haveg¯=sup|gl,0;l∈0,1=e−d. Besides, the functions f1 and f2 are continuous, and for all l∈0,1 and u,v∈ℝ, we have(35)f2t,u≤1100,f2t,u−f2t,v≤1100u−v,f1t,u≤1100,f1t,u−f1t,v≤1100u−v.
In this exampleα1=3/2, α2=1/2, θ=2, p=4/3, q=5/4, μ=3/43/4, δ=1/2, and ξ=2/3. Hence, Λ≈0.242, a1¯=1.121 and a2¯=1.168. Consequently, the existent inequality in condition H3 has the form:(36)e−dr0+11001/100Γ5/2+2/100Γ23/6Γ9/4Δ≤r0.
Obviously, the above inequality has the positive solutionr0, for example r0=1. Moreover, according to the obtained values we have(37)M1Γα1+1+θM2Γα1+p+1Γq+1Δ<1.
Thus, we conclude all conditions of Theorem2, and hence at least one solution of the mapping T1 is obtained on 0,1 which is a solution of problem (32).


## 4. Conclusion

We have studied a nonlocal hybrid boundary value problem which contains both left Caputo and right Riemann–Liouville fractional derivatives and integrals and nonlocal hybrid conditions. An existence result is proved by applying a new generalization of Darbo’s fixed point theorem associated with measures of noncompactness. The result obtained in this paper is new and significantly contributes to the existing literature on the topic.



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