# Existence Results for Generalized Bagley-Torvik Type Fractional Differential Inclusions with Nonlocal Initial Conditions

Authors: Lizhen Chen; Gang Li
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this article, we prove the existence of solutions for the generalized Bagley-Torvik type fractional order differential inclusions with nonlocal conditions. It allows applying the noncompactness measure of Hausdorff, fractional calculus theory, and the nonlinear alternative for Kakutani maps fixed point theorem to obtain the existence results under the assumptions that the nonlocal item is compact continuous and Lipschitz continuous and multifunction is compact and Lipschitz, respectively. Our results extend the existence theorems for the classical Bagley-Torvik inclusion and some related models.


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## Body

## 1. Introduction

In this article, we will consider the following generalized Bagley-Torvik type fractional differential inclusions:(1)Dcv1zt-χcDv2zt∈Gt,zt,t∈0,1=B,z0=hzwhere Dcv1 and Dcv2 are Caputo fractional derivatives with 0<ν1≤1 and 0<ν2<ν1, χ∈R is a constant, and G is a multifunction.By introducing nonlocal conditions into the initial-value problems, Byszewski and Lakshmikantham [1] provided a more accurate model for the nonlocal initial valued problem since more information was incorporated in the experiment. As a result, the negative impact of single initial value can be significantly reduced. Concerning the initial-value problems, the most recent developments can be referred to in [2, 3]. On the other hand, the fractional calculus and fractional differential equations have many real applications in biology, physics, and natural sciences and a number of results on this topic have emerged in the last decade [4–6]. In [7], EI-Sayed and Ibrahim initiated the research on fractional multivalued differential inclusions. After that, many authors were devoted to study the existence of solutions for fractional differential inclusions [8–11]. Very recently, Wang et al. [12, 13] studied the controllability and topological structure of the solution set for fractional impulsive differential inclusions.For the problem of fractional differential inclusions, multiterm fractional differential equations are a hot research direction owning to their wide use in practice and technique sciences, for example, physics, mechanics, and chemistry. An important result on multiterm fractional calculus is formulated by Bagley and Torvik in [14]. Here the authors deduced and tested a relation Ax′′(t)+BcD3/2x(t)+Cx(t)=g(t), where x(t) is a function describing the motion of thin plates in Newtonian fluids, A=m, the mass of thin rigid plate, B=2svρ, where s is area of the plate immersed in Newtonian fluid, v is viscosity, ρ is the fluid density, C=k, the stiffness of the spring, and g(t) is an external force. Later, the above equation was called Bagley-Torvik equation [15]. Based on this model, the nonlinear multiterm fractional differential equations were rediscovered and popularized by Kaufmann and Yao in [16]. As far as the author knows, there are few papers on the existence of the generalized Bagley-Torvik type fractional differential inclusion (1) except for Ibrahim, Dong, and Fan [17]. They studied the following equation: (2)Dcθvs-acDδvs+gs,vs=0;0<s<1,v0=v0;v1=v1;where Dcθ and Dcδ are the Caputo fractional derivatives, 1<θ≤2, 1≤δ<θ.In this article, we shall be concerned with the existence of the generalized Bagley-Torvik type fractional differential inclusions (1) by employing the noncompactness measure of Hausdorff, fractional calculus, and the nonlinear alternative for Kakutani maps fixed point theorem, when h is compact continuous and Lipschitz continuous and G is compact and Lipschitz, respectively. Our theory improves the results in [6, 8–10, 16, 17] and extends (generalizes) the corresponding results of Bagley-Torvik equation.The structure of this article is as follows: some preliminary knowledge is introduced in Section2; some existence criteria are derived from (1) in Section 3; in the end, we use an example to illustrate an application of the main result.


## 2. Preliminaries

LetY be a metric space and W a normed space; K(W)={Δ⊆W:nonempty}; Kb(W)={Δ⊆K(W):bounded}; Kbf(W)={Δ⊆K(W):closedandbounded}; Kpc(W)={Δ⊆K(W):compactandconvex}; Kfc(W)={Δ⊆K(W):closedandconvex}.LetE be a Banach space. Lp(B¯,E)≔{u:B¯→Eis  measurable  and∫01u(t)pdt<+∞}. For u∈Lp(B¯,E), 1<p<∞, the norm is defined as up=(∫01u(t)pdt)1/p, and p′ is the conjugate exponent; that is, 1/p+1/p′=1. Denote the Hausdorff measure of noncompactness (MNC) γ:Kb(E)→R+ as (3)γD=inf⁡ρ>0:there  exist  finite  pointx1,x2,…,xn∈EwithD⊂⋃i=1nBxi,ρ.Proposition 1 (see [18]).
The MNCγ enjoy the following properties:(1)
monotone, ifΔ1,Δ2∈Kb(E), Δ1⊆Δ2⇒γ(Δ1)≤γ(Δ2)(2)
nonsingular, ifγ({b}∪Δ)=γ(Δ), for all b∈E and Δ⊆Kb(E)(3)
regular, ifγ(Δ)=0⇔Δ is relative compact(4)
algebraically semiadditive, ifγ(Δ1+Δ2)≤γ(Δ1)+γ(Δ2), for Δ1,Δ2∈Kb(E)(5)
γ(ωΔ)=ωγ(Δ)We now denote the sequential MNCγ0 as follows: (4)γ0Δ=sup⁡γxm:m≥1:xmm=1∞⊆Δ.One can easily have that (5)γ0Δ≤γΔ≤2γ0Δ.Moreover, γ0(Δ)=γ(Δ), if E is separable.Proposition 2 (see [18]).
For every bounded setΔ⊆C(B¯,E), s∈B¯, the inequality (6)γΔs≤γcΔholds, where Δ(s)={δ(s):δ∈Δ}⊂E and γc is the MNC defined in C(B¯,E). Moreover, if Δ is equicontinuous, then γ(Δ(s)) is continuous. Moreover, (7)γcΔ=sup⁡γΔs:s∈B¯.Lemma 3 (see [19]).
Suppose{gm}m=1∞⊂L1(B¯,E), a∈L1(B¯,R+); the inequality gmt≤a(t) holds; for every t∈B¯, m≥1. Then γ({gm(t)}m=1∞)∈L1(B¯,R+), and (8)γ∫0tgmsds:m≥1≤2∫0tγgmsm=1∞ds.In what follows, we will recall some necessary definitions and lemmas of the fractional order differential and integra theory, which can be found in the literature [15].Definition 4.
Supposeg∈L1(B¯,E), ι>0. If ∫0s(s-η)ι-1/Γιg(η)dη<∞, then (9)I0+ιgs=∫0ss-ηι-1Γιgηdηis called ι order Riemann-Liouville fractional integral of g.Definition 5.
Supposeg∈L1(B¯,E), ι>0. we define (10)Dc0+ιgs=1Γn-ι∫0sgnηs-ηι-n+1dηas the ι order Caputo fractional derivative of g, where n=[ι]+1.Lemma 6.
Supposeι>0 and g∈L1(B¯,E). Consider the following differential equation: (11)Dc0+ιgt=0.Then there exists some constants dk∈E, k=0,1,2,…n-1 such that (12)gt=d0+d1t+d2t2+⋯+dn-1tn-1,where n=[ι]+1.Lemma 7.
Supposeg∈L1(B¯,E), and g(n)∈L1(B¯,E); then there exist dk∈E, k=0,1,2,…n-1 satisfying (13)I0+ιcD0+ιgt=gt+d0+d1t+d2t2+⋯+dn-1tn-1.For simplicity, we denoteDc0+ι and I0+ι by Dcι and Iι, respectively. And let ((p-1)/(ν1p-1))(p-1)/p=C, ν1-ν2=C1.Lemma 8.
Let0<ν1≤1, 0<ν2<ν1. Assume that g∈L1(B¯,E). Then the solution of the equation, (14)Dcv1zs-χcDv2zs=gs,a.e.s∈B,z0=z0,has the following form: (15)zs=1-χsC1ΓC1+1z0+χ∫0ss-ηC1-1ΓC1zηdη+∫0ss-ην1-1Γν1gηdη.Proof.
In light of0<ν1≤1, from Lemma 7, there exists d0 satisfying (16)Iν1cDν1zs=zs+d0,s∈B¯.Moreover, (17)Iν1cDν1zs=χIν1cDν2zs+Iν1gs.By Lemma 6, it follows that (18)Iν1cDν2zs=IC1Iν2cDν2zs=IC1zs+d0=IC1zs+d0sC1ΓC1+1.So we have (19)zs+d0=χIC1zs+χd0sC1ΓC1+1+Iν1gs.Since z(0)=z0, we obtain d0=-z0. Substituting the value of d0 into (19), the proof is complete.Givenz0∈E, g∈L1(B¯,E). Define the operator S by the formula (20)Sgs=1-χsC1ΓC1+1z0+χ∫0ss-ηC1-1ΓC1zηdη+∫0ss-ην1-1Γν1gηdη,∀s∈B¯;that is, Sg is the solution of the above system (14).For anyw belonging to C(B¯,E), set (21)Selw=g∈L1B¯,E:gs∈Gs,wsa.e.s∈B¯.For given gw∈Sel(w), we define Sh(w)gw as a set of solutions to the generalized Bagley-Torvik type fractional differential system (22)Dcv1zs-χcDv2zs=gws,a.es∈B,z0=hw.Before ending this section, we define the solution of the generalized Bagley-Torvik type fractional differential inclusions (1).Definition 9.
Ifz(·)∈C(B¯,E) and (23)zt=1-χtC1ΓC1+1gz+χ∫0tt-ηC1-1ΓC1zηdη+∫0tt-ην1-1Γν1gηdη,t∈B¯,where g∈Sel(z), then z is a solution of the generalized Bagley-Torvik type fractional differential inclusions (1).


## 3. Main Results

Here, we shall derive some existence criteria for the generalized Bagley-Torvik type differential system (1), when h is compact continuous and Lipschitz continuous and G is compact and Lipschitz, respectively. Our basic tools are noncompactness measure of Hausdorff, fractional calculus, and the nonlinear alternative for Kakutani maps fixed point theorem. Before proceeding, we assume that p>1/ν1 and h and G satisfy the following conditions:(h1)h:C(B¯,E)→E is compact and continuous mapping, and there exist positive constants c1, c2 satisfying h(z)≤c1z+c2( H G 1 )   ∀ z ∈ E, G(·,E):B¯→Kfc(E) admits a measurable selection( H G 2 )   G ( t , · ) : E → K f c ( E ) is upper semicontinuous, for almost every t∈B¯( H G 3 ) For every z∈E and t∈B¯, G(t,z)≔sup{ζ:ζ∈G(t,z)}≤l(t)Σ(z), where l∈Lp(B¯,R+), and Σ:R+→R+ is an increasing and continuous function( H G 4 ) For almost every t∈B¯, there exists b∈Lp(B¯,R+); the inequality γ(G(t,Δ))≤b(t)γ(Δ) holds, where Δ⊂E is a bounded set.In the sequel, we introduce some important lemmas which are crucial to derive existence results.Lemma 10 (see [20]).
Under assumptions(HG1)-(HG4), if there exists sequence {wm}m=1∞⊂C(B¯,E), {gm}m=1∞⊂L1(B¯,E), and gm∈Sel(wm), m≥1, such that wm→w and gm⇀g, then g∈Sel(w).Lemma 11.
Suppose that (h1) is satisfied; then, for any bounded setΔ⊂C(B¯,E), we obtain (24)γShwgwt:w∈Δ,gw∈Selw≤4χΓC1+1+4CbpΓν1γcΔ.Proof.
According to (5), for any ϵ>0, there exist {wm}m=1+∞⊂Δ and {gm}m=1+∞⊂L1(B¯,E) satisfying gm∈Sel(wm) for all m≥1 and (25)γShwgwt:w∈Δ,gw∈Selw≤2γShwmgmt:m≥1+ϵ.Since (26)Shwmgmt=1-χtC1ΓC1+1hwm+χ∫0tt-ηC1-1ΓC1wmηdη+∫0tt-ην1-1Γν1gmηdη,now by (h1) and Proposition 1, it follows that (27)γShwmgmt:m≥1=γ1-χtC1ΓC1+1hwm+χ∫0tt-ηC1-1ΓC1wmηdη+∫0tt-ην1-1Γν1gmηdη):m≥1=γχ∫0tt-ηC1-1ΓC1wmηdη+∫0tt-ην1-1Γν1gmηdη):m≥1.By (HG4), we obtain (28)γgmsm=1+∞≤γ(Gs,wmsm=1+∞≤bsγwmsm=1+∞≤bsγΔs.Applying Proposition 1 and Lemma 3, one can easily achieve (29)γShwmgmt:m≥1≤2χΓC1+1γcΔ+2Γν1∫0tt-ην1-1bηγΔηdη≤2χΓC1+1+2CbpΓν1γcΔ.It follows from (25) that (30)γShwgwt:w∈Δ,gw∈Selw≤4χΓC1+1+4CbpΓν1γcΔ+ϵ.Since ϵ>0 is arbitrary, we have (31)γShwgwt:w∈Δ,gw∈Selw≤4χΓC1+1+4CbpΓν1γcΔ.A nonlinear alternative for Kakutani maps, which is significant to develop our main results, is introduced as follows.Theorem 12 (see [21]).
LetΔ⊂E be a closed convex set and W⊆Δ be an open set with 0∈W. Suppose T:W¯→Kfc(Δ) is compact, upper semicontinuous. If there exist no v∈∂W and 0<μ<1 satisfying v∈μT(v), then T has at least one fixed point.Next, we shall prove the existence result whenh is compact continuous.Theorem 13.
Suppose(h1) and (HG1)-(HG4) are satisfied. If there exists L>0 such that (32)Γν1ΓC1+1-ΓC1+1+χc1-χLΓν1ΓC1+1+χc2+ΓC1+1ΣLlpC>1,and (33)4χΓC1+1+4CbpΓν1-1<0,then the generalized Bagley-Torvik type differential inclusion (1) has at least one solution in C(B¯,E).Proof.
From ([22]), according to conditions (HG1)-(HG3), Sel(w) is not empty, for every w∈C(B¯,E). Using (23), we define the following multivalued operator: (34)T:CB¯,E→K(CB¯,Eas (35)Tw=z∈CB¯,E:zt=1-χtC1ΓC1+1hw+χ∫0tt-ηC1-1ΓC1wηdη+∫0tt-ην1-1Γν1gηdη,g∈Selw.
Clearly, ifw is a fixed point of T, then w is a solution of (1). Let r>0; Br is a bounded ball, defined as Br={w∈C(B,E):w≤r}. Suppose Δ⊂Br is a bounded set and belongs to co¯({0}∪T(Δ)). In order to utilize the nonlinear alternative for Kakutani maps, we first need to show that γ(Δ)=0. Let s1,s2∈B¯, 0<s1<s2, z∈T(Δ); there exist w∈Δ and h∈Sel(w) such that (36)zt=1-χtC1ΓC1+1hw+χ∫0tt-ηC1-1ΓC1wηdη+∫0tt-ην1-1Γν1gηdη,t∈B¯.By Hölder inequality and (HG3), we have (37)∫0s2s2-ην1-1gηdη-∫0s1s1-ην1-1gηdη≤∫0s1s2-ην1-1-s1-ην1-1gηdη+∫s1s2s2-ην1-1gηdη≤ν1-1∫0s1s1-sν1-2s2-s1gηdη+lpΦrs2-s1ν1-1/pC≤ν1-1∫0s1s1-ην1-1p′dη2-ν1/1-ν1p′∫0s1s2-s1ν1-1p′dη1/ν1-1p′∫0s1lηpdη1/pΣr+lpΣrs2-s1ν1-1/pC≤ν1-1s1s2-s1∫0s1s1-ην1-1p′dη2-ν1/1-ν1p′lpΣr+lpΣrs2-s1ν1-1/pC.Taking into account the above inequality, one has (38)zs2-zs1≤χs2C1-s1C1ΓC1+1hw+χΓC1∫0s2s2-ηC1-1wηdη-∫0s1s1-ηC1-1wηdη+1Γν1∫0s2s2-ην1-1gηdη-∫0s1s1-ην1-1gηdη≤χs2C1-s1C1ΓC1+1c1r+c2+χrΓC1+1s2C1-s1C1+2s2-s1C1+1Γν1ν1-1s1s2-s1∫0s1s1-ην-1p′dη2-ν1/1-ν1p′lpΣr+lpΣrs2-s1ν1-1/pC.Applying the above inequality, zs2-zs1→0, as s1→s2. Hence, T(Δ) and Δ⊆co¯({0}∪T(Δ)) are equicontinuous.
Also because, for everyt∈B¯, (39)TΔt⊂Shwgwt:w∈Δ,gw∈Selw.It follows from Lemma 11 that (40)γTΔt≤γShwgwt:w∈Δ,gw∈Selw≤4χΓC1+1+4CbpΓν1γΔ.Thus, one can obtain (41)γco¯0∪TΔ=γTΔ≤4χΓC1+1+4CbpΓν1γΔ.Since Δ⊆co¯({0}∪T(Δ)) and by (33), we know that γ(Δ)=0.
Clearly, the multioperatorT has convex values. We still need to prove the multioperator T is closed on Δ. Suppose {wm}m=1∞⊂Δ with wm→w in C(B¯,E) and zm∈T(wm) with zm→z in C(B¯,E). Moreover, assume {gm}m=1+∞⊂L1(B¯,E) is a sequence satisfying gm∈Sel(wm) for any m≥1 and (42)zmt=1-χtC1ΓC1+1hwm+χ∫0tt-ηC1-1ΓC1wmηdη+∫0tt-ην1-1Γν1gmηdη.Then the set {gm}m=1∞ is integrally bounded. From (HG3), one can achieve that (43)gmη≤Gη,wmη)≤lηΣwm,a.e.η∈B¯,m=1,2,….
Sincewm→w in C(B¯,E), therefore {wm:m≥1} is bounded and belongs to C(B¯,E). Invoking (HG4), we infer that (44)γgmηm=1∞≤γGη,vmηm=1∞≤bηγwmηm=1∞=0.So the sequence {gm}m=1∞ is semicompact. Applying the proposition (see [22]), {gm}m=1∞ is weakly compact. Therefore, there is g∈L1(B¯,E) satisfying gm⇀g. Moreover, one has (45)∫0tt-ην1-1Γν1gmηdη→∫0tt-ην1-1Γν1gηdη,which, together with (42) and g being continuous, implies that (46)zmt→1-χtC1ΓC1+1hw+χ∫0tt-ηC1-1ΓC1wηdη+∫0tt-ην1-1Γν1gηdη.Then, one can obtain (47)zt=1-χtC1ΓC1+1gw+χ∫0tt-ηC1-1ΓC1wηdη+∫0tt-ην1-1Γν1gηdη.Thus g∈Sel(w); i.e., z∈T(w). Equivalently, graph(T) is closed, and T is closed on Δ. Based on the above discussion, it can be concluded that T is u.s.c. on Δ (see [23]).
In the following, we are to find an open setW, which satisfies the conditions of Theorem 12. Let μ∈(0,1), w∈μT(w). Then there exist w∈∂W and g∈Sel(w) such that (48)wt=μ1-χtC1ΓC1+1hw+μχ∫0tt-ηC1-1ΓC1wηdη+μ∫0tt-ην1-1Γν1gηdη.From (h1) and (HG3), one can obtain (49)wt≤1+χΓC1+1hw+χΓC1∫0tt-ηC1-1wηdη+1Γν1∫0tt-ην1-1gηdη≤1+χΓC1+1c1w+c2+χΓC1+1w+ΣwΓν1∫0tt-ην1-1lηdη≤1+χΓC1+1c1w+c2+χΓC1+1w+ΣwlpΓν1C.Consequently, (50)Γν1ΓC1+1-ΓC1+1+χc1-χwΓν1ΓC1+1+χc2+ΓC1+1ΣwlpC≤1.It follows from (32) that there exists L which satisfies w≠L. Denote (51)W=w∈CB¯,E:w<L.By the definition of W, for every μ∈(0,1), there exists no w∈∂W satisfying w∈μT(w). Since T:W¯→K(C(B¯,E)) is compact and u.s.c., according to Theorem 12, T has at least one fixed point in W¯.Next, we assume the nonlocal itemh is Lipschitz continuous; that is,(h2) for everyx1,x2∈C(B¯,E), there is k satisfying hx1-hx2≤kx1-x2.Now, we give the existence criterion for the generalized Bagley-Torvik type fractional differential inclusions (1) when the nonlocal item satisfies (h2).Theorem 14.
Suppose conditions (h2) and(HG1)-(HG4) are satisfied. If there exists L>0, such that (52)Γν1ΓC1+1-ΓC1+1+χk-χLΓν1ΓC1+1+χh0+ΓC1+1ΣLlpC>1,and (53)1+χΓC1+1k+4χΓC1+1+4CbpΓν1-1<0,then the generalized Bagley-Torvik type differential inclusion (1) has a solution in C(B¯,E).Proof.
According to Theorem13, the multioperator T is upper semicontinuous. Suppose Δ⊆Br is bounded, and belongs to co¯({0}∪T(Δ)). We need to prove γ(Δ)=0.
From (h2) and Lemma11, one can infer that (54)γTΔt≤γShwgwt:w∈Δ,gw∈Selw≤1+χΓC1+1kγΔ+2χΓC1+1γΔ+4Γν1∫0tt-ην1-1bsγΔηdη≤1+χΓC1+1k+4χΓC1+1+4CbpΓν1γΔ.Thus, one has (55)γΔ≤γco¯0∪TΔ=γTΔ≤1+χΓC1+1k+4χΓC1+1+4CbpΓν1γΔ.Together with (53), one can conclude that γ(Δ)=0.
From a proof similar to Theorem13, we can prove that the multioperator T has at least one fixed point, which is a solution of the generalized Bagley-Torvik type fractional differential inclusions (1).Remark 15.
AssumeΩ(x)=k1xp+k2, ∀x>0 for some k1,k2>0 and 0<p<1; then conditions (32) and (52) are automatically satisfied.Subsequently, we will investigate the case whereG is Lipschitz-type about the Hausdorff metric. Before proceeding, let us introduce the following definition.Suppose(E,d) is metric spaces, corresponding to (E,·). Let A1,A2∈Kbf(E), a1∈A1. Denote (56)Da1,A2=inf⁡da1,a2:a2∈A2,ρA1,A2=sup⁡Da1,A2:a1∈A1.A function H:Kbf(E)×Kbf(E)→R+ is called the Hausdorff metric on E, if (57)HA1,A2=max⁡ρA1,A2,ρA2,A1.Now, we make the following assumption:( H G 5 ) there exists ω∈Lp(B¯,R+), satisfying (58)HGs,z1s,Gs,z2z≤ωsz1-z2,∀s∈B¯,z1,z2∈CB¯,E,and for almost every s∈B¯, G(s,z(s))≤ω(s)(1+z), z∈C(B¯,E).Noting that if(HG5) is satisfied, (HG3) and (HG4) are automatically satisfied, in this case, the following theorems automatically hold.Theorem 16.
Leth and G satisfy conditions (h1), (HG1), (HG2), and (HG5). If there exists L>0, such that (59)Γν1ΓC1+1-ΓC1+1+χc1-χ-(ΓC1+1ωpCLΓν1ΓC1+1+χc2+ΓC1+1ωpC>1,and (60)4χΓC1+1+4CωpΓν1-1<0,then the fractional differential inclusion (1) has at least one solution.Theorem 17.
Assumeh and G satisfy conditions (h2), (HG1), (HG2), and (HG5). If there exists L>0, such that (61)Γν1ΓC1+1-Γν+1+χk-χ-(ΓC1+1ωpCLΓν1ΓC1+1+χg0+ΓC1+1ωpC>1,and (62)1+χΓC1+1k+4χΓC1+1+4CωpΓν1<1,then the fractional differential inclusion (1) has at least one solution.


## 4. Example

Consider the following nonlocal problem of fractional differential inclusion.Example 18.
Discuss the following differential inclusion:(63)Dczs-π16cD1/2zs∈Gs,zs,s∈Bz0=∑j=1mβjzsj,where sj(j=1,2,…,m) are certain constants, with 0<s1<s2<⋯<sm<1, ν1=1, ν2=1/2, C1=1/2, ∑j=1mβj=1/90, χ=π/16. Let p=2 and G(s,z)=[0,(1/3)s4z+1/6].
Clearly, for alls∈B¯, G(s,z)≤1/3z+1/6=l(s)Σ(z), and then l2=1, Σ(z)=(1/3)z+1/6.
Further, we supposeh(z)=∑j=1mβjz(sj). Obviously, h(0)=0. Moreover, (64)HGs,z1,Gs,z2≤13s4z1-z2,∀z1,z2∈CB¯,R,s∈B¯,hz1-hz2≤∑j=1mβjz1sj-z2sj≤∑j=1mβjmaxsj∈B¯⁡z1sj-z2sj,z1,z2∈CB¯,R.Take b(s)=(1/3)s4, k=∑j=1mβj=1/90. Then, one can easily obtain that b2=1/9.
It is easy to check that (h2) and(HG1)-(HG4) hold. Further, we can find L=3, such that (65)Γν1ΓC1+1-ΓC1+1+χk-χLΓν1ΓC1+1+χh0+ΓC1+1ΣLlpC=3×π/2-π/160-π/16π/21/3×3+1/6>1,and (66)1+χΓC1+1k+4χΓC1+1+4CbpΓν1=180+12+49<1.Therefore, according to Theorem 14, there exists at least one solution in C(B¯,R) for the problem (63).


## 5. Conclusion

This paper has studied the generalized Bagley-Torvik type fractional order differential inclusions with nonlocal conditions. By employing the noncompactness measure of Hausdorff and the nonlinear alternative for Kakutani maps fixed point theorem, the existence results have been derived when the nonlocal item is compact and Lipschitz continuous and multifunction is compact and Lipschitz. An example has been used to illustrate applications of the main result. Future research directions include the extension of the present results to other relevant cases, for example, controllability and topological structure of the solution set [12, 13].



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*Source: 2761321-2018-09-02.xml*