Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces

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DOI: 10.24193/subbmath.2017.4.02

Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces

John R. Graef, Nassim Guerraiche and Samira Hamani

Abstract. The authors establish sufficient conditions for the existence of solutions to boundary value problems for fractional differential inclusions involving the Hadamard type fractional derivative of order α ∈ (1,2] in Banach spaces.

Their approach uses M¨onch’s fixed point theorem and the Kuratowski measure of noncompacteness.

Mathematics Subject Classification (2010):26A33, 34A08, 34A60, 34B15.

Keywords:Fractional differential inclusion, Hadamard-type fractional derivative, fractional integral, M¨onch’s fixed point theorem, Kuratowski measure of noncompacteness.

1. Introduction

In this paper we are concerned with the existence of solutions to boundary value problems (BVP for short) for fractional order differential inclusions. In particular, we consider the boundary value problem

HD^ry(t)∈F(t, y(t)), for a.e. t∈J = [1, T], 1< r≤2, (1.1)

y(1) = 0, y(T) =yT, (1.2)

where ^HD^r is the Hadamard fractional derivative, (E,| · |) is a Banach space, P(E) is the family of all nonempty subsets ofE, F : [1, T]×E → P(E) is a multivalued map, andy_T ∈R.

Differential equations of fractional order are valuable in modeling phenomena in various fields of science and engineering. They can be found in viscoelasticity, electro- chemistry, control, porous media, electromagnetism, etc. The monographs of Hilfer [18], Kilbaset al.[19], Podlubny [23], and Momaniet al.[21] are very good sources on the background mathematics and various applications of fractional derivatives. The literature on Hadamard-type fractional differential equations has not undergone as

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much development as it has for the Caputo and Riemann-Liouville fractional derivatives; see, for example, the papers of Ahmed and Ntouyas [2], Benhamida, Graef, and Hamani [10], and Thiramanus, Ntouyas, and Tariboon [24].

The fractional derivative that Hadamard [16] introduced in 1892 differs from other fractional derivatives in the sense that the kernel of the integral in the definition of the Hadamard derivative contains a logarithmic function with an arbitrary exponent. A detailed description of the Hadamard fractional derivative and integral can be found in [11, 12, 13].

In this paper, we present existence results for the problem (1.1)-(1.2) in the case where the right hand side is convex valued. This result relies on the set-valued analog of M¨onch’s fixed point theorem combined with the technique of measure of noncompactness. Recently, this has proved to be a valuable tool in studying fractional differential equations and inclusions in Banach spaces; for additional details, see the papers of Laosta et al. [20], Agarwal et al. [1], and Benchohra et al. [7, 8, 9]. Our results here extend to the multivalued case some previous results in the literature and constitutes what we hope is an interesting contribution to this emerging field. We include an example to illustrate our main results.

2. Preliminaries

This section contains definitions, concepts, lemmas, and preliminary facts that will be used in the remainder of this paper. Let C(J, E) be the Banach space of all continuous functions fromJ into Ewith the norm

kyk_∞= sup{|y(t)|:t∈J},

and let L¹(J, E) be the Banach space of Lebesgue integrable functions y : J → E with the norm

kykL¹ = Z T

1

|y(t)|dt.

The spaceAC¹(J, E) is the space of functionsy:J →E that are absolutely continuous and have an absolutely continuous first derivative.

For any Banach spaceX, we set

P_cl(X) ={Y ∈ P(X) :Y is closed}, P_b(X) ={Y ∈ P(X) :Y is bounded}, P_cp(X) ={Y ∈ P(X) :Y is compact}, and

P_cp,c(X) ={Y ∈ P(X) :Y is compact and convex}.

A multivalued map G:X → P(X) isconvex (closed)valued ifG(X) is convex (closed) for allx∈X. We say thatGisbounded on bounded setsifG(B) =∪_x∈BG(x) is bounded in X for allB∈P_b(X) (i.e., sup_x∈B{sup{|y|:y∈G(x)}}is bounded).

The mappingG isupper semi-continuous (u.s.c) onX if for each x₀ ∈X, the setG(x₀) is a nonempty closed subset ofX, and for each open setN ofX containing G(x₀), there exists an open neighborhoodN₀of x₀ such thatG(N₀)⊂N. A mapG is said to becompletely continuousifG(B) is relatively compact for everyB∈Pb(X).

If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c if and only if Ghas a closed graph (i.e.,xn →x_∗, yn →y_∗,

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yn ∈ G(xn) imply y_∗ ∈ G(x_∗)). The mapping G : X → P(X) has a fixed point if there exists x∈ X such that x∈ G(x). The set of fixed points of the multivalued operator Gwill be denoted byF ix G. A multivalued mapG:J →Pcl(X) is said to be measurable if for everyy∈X, the function

t→d(y, G(t)) =inf{|y−z|:z∈G(t)}

is measurable.

Definition 2.1. A multivalued mapF :J×E→ P(E) is said to be Carath´eodory if:

(1) t→F(t, u) is measurable for eachu∈E;

(2) u→F(t, u) is upper semicontinuous for a.e.t∈J.

For eachy ∈AC¹(J, E), define the set of selections ofF by SF,y={v∈L¹(J, E) :v(t)∈F(t, y(t)) a.e.t∈J}.

Let (X, d) be a metric space induced from the normed space (X,| · |). The function H_d :P(X)× P(X)→R+∪ {∞}given by

Hd(A, B) = max{sup

a∈A

d(a, B),sup

b∈B

d(A, b)}

is known as the Hausdorff-Pompeiu metric.

For more details on multivalued maps, see the books of Aubin and Cellina [4], Aubin and Frankowska [5], Castaing and Valadier [14], and Deimling [15].

For convenience, we first recall the definitions of the Kuratowski measure of noncompacteness and summarize the main properties of this measure.

Definition 2.2. ([3, 6]) LetEbe a Banach space and let ΩEbe the bounded subsets of E.The Kuratowski measure of noncompactness is the map β : ΩE →[0,∞) defined by

β(B) = inf{ >0 :B⊂

m

[

j=1

Bj anddiam(Bj)≤}.

Properties:The Kuratowski measure of noncompactness satisfies the following properties (for more details see [3, 6]):

(P₁) β(B) = 0 if and only if B is compact (B is relatively compact).

(P2) β(B) =β(B).

(P3) A⊂B impliesβ(A)≤β(B).

(P4) β(A+B)≤β(A) +β(B).

(P₅) β(cB) =|c|β(B), c∈R. (P6) β(convB) =β(B).

Here B and conv B denote the closure and the convex hull of the bounded set B, respectively.

For a given setV of functionsu:J →E, we set V(t) ={u(t) :u∈V}, t∈J,

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and

V(J) ={u(t) :u∈V(t), t∈J}.

Theorem 2.3. ([17], [22, Theorem 1.3]) Let E be a Banach space and let C be a countable subset of L¹(J, E) such that there exists h∈ L¹(J,R+) with |u(t)| ≤ h(t) for a.e.t∈J and everyu∈C. Then the functionϕ(t) =β(C(t))belongs toL¹(J,R+) and satisfies

β (Z T

0

u(s)ds:u∈C )!

≤2 Z T

0

β(C(s))ds.

Lemma 2.4. ([20, Lemma 2.6])LetJ be a compact real interval,F be a Carath´eodory multivalued map, and let θ be a linear continuous map from L¹(J, E) → C(J, E).

Then the operator

θ◦SF,y:L¹(J, E)→Pcp,c(C(J, E)), y→(θ◦SF,y)(y) =θ(SF,y) is a closed graph operator inL¹(J, E)×C(J, E).

In what follows, log(·) = log_e(·), andn = [r] + 1 where [r] denotes the integer part ofr.

Definition 2.5. ([19]) The Hadamard fractional integral of orderr for a functionh : [1,+∞)→Ris defined by

I^rh(t) = 1 Γ(r)

Z t

1

log t

s r−1

h(s)

s ds, r >0, provided the integral exists.

Definition 2.6. ([19]) For a functionhon the interval [1,+∞), the Hadamard fractional derivative ofhof orderris defined by

(^HD^rh)(t) = 1 Γ(n−r)

td

dt ⁿZ t

1

logt

s

^n−r−1h(s)

s ds, n−1< r < n, n= [r] + 1.

Let us now recall M¨onch’s fixed point theorem.

Theorem 2.7. ([22, Theorem 3.2]) Let K be a closed and convex subset of a Banach spaceE,U be a relatively open subset ofK, andN :U → P(K). Assume that graphN is closed,N maps compact sets into relatively compact sets, and for somex0∈U, the following two conditions are satisfied:

(i) M ⊂ U, M ⊂ conv(x0∪N(M)), M = C, with C a countable subset of M, impliesM is compact;

(ii) x6∈(1−λ)x₀+λN(x)for all x∈U \U, λ∈(0,1).

Then there existsx∈U with x∈N(x).

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3. Main results

Let us start by defining what we mean by a solution of the problem (1.1)-(1.2).

Definition 3.1. A function y ∈ AC¹(J, E) is said to be a solution of (1.1)-(1.2) if there exist a function v ∈ L¹(J, E) with v(t) ∈ F(t, y(t)) for a.e. t ∈ J, such that

HD^αy(t) =v(t) onJ, and the conditionsy(1) = 0 andy(T) =yT are satisfied.

Lemma 3.2. Let h:J →E be a continuous function. A function y is a solution of the fractional integral equation

y(t) = 1 Γ(r)

Z t

1

logt

s r−1

h(s)ds

s +(logt)^r−1 (logT)^r−1

"

yT − 1 Γ(r)

Z T

1

logT

s r−1

h(s)ds s

#

(3.1) if and only if y is a solution of the fractional BVP

HD^ry(t) =h(t), for a.e.t∈J = [1, T], 1< r≤2, (3.2)

y(1) = 0, y(T) =yT. (3.3)

Proof. Applying the Hadamard fractional integral of order r to both sides of (3.2), we obtain

y(t) =c1(logt)^r−1+c2(logt)^r−2+^HI^rh(t). (3.4) From (3.3), we havec2= 0 and

c1= 1

(logT)^r−1[yT −^HI^rh(T)].

Hence, we obtain (3.1). Conversely, it is clear that if y satisfies equation (3.1), then

(3.2)-(3.3) hold.

Theorem 3.3. Let R > 0,B ={x∈E :kxk ≤ R}, U ={x∈ C(J, E) :kxk ≤ R}, and assume that:

(H1) F :J×E→ P_cp,p(E) is a Carath´eodory multi-valued map;

(H2) For each R >0, there exists a functionp∈L¹(J, E) such that kF(t, u)kP = sup{|v|:v(t)∈F(t, y)} ≤p(t) for each (t, y)∈J×E with|y| ≥R, and

lim inf

R→∞

RT 0 p(t)dt

R =δ <∞;

(H3) There exists a Carath´eodory function ψ:J ×[1,2R]→R+ such that β(F(t, M))≤ψ(t, β(M))a.e.t∈J and each M ⊂B;

(H4) The functionϕ= 0 is the unique solution in C(J,[1,2R]) of the inequality ϕ(t)≤2

( 1 Γ(r)

Z t

1

log t

s r−1

ψ(s, ϕ(s))ds s + (logt)^r−1

(logT)^r−1

"

y_T + 1 Γ(r)

Z T

1

logT

s r−1

ψ(s, ϕ(s))ds s

#)

fort∈J. (3.5)

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Then the BVP (1.1)-(1.2) has at least one solution inC(J, B), provided that

δ < Γ(r+ 1)

(logT)^r. (3.6)

Proof. We wish to transform the problem (1.1)-(1.2) into a fixed point problem, so consider the multivalued operator

N(y) = (

h∈C(J,R) :h(t) = 1 Γ(r)

Z t

1

log t

s r−1

v(s)ds s + (logt)^r−1

(logT)^r−1

"

yT − 1 Γ(r)

Z T

1

logT

s r−1

v(s)ds s

#

, v∈SF,y

) . Clearly, from Lemma 3.2, the fixed points of N are solutions to (1.1)-(1.2). We shall show thatN satisfies the assumptions of M¨onch’s fixed point theorem. The proof will be given in several steps. First note thatU =C(J, B).

Step 1:N(y) is convex for eachy∈C(J, B).

Takeh₁, h₂∈N(y); then there existv₁,v₂∈S_F,ysuch that for eacht∈J, we have h_i(t) = 1

Γ(r) Z t

1

logt

s r−1

v_i(s)ds s + (logt)^r−1

(logT)^r−1

"

yT − 1 Γ(r)

Z T

1

logT

s r−1

vi(s)ds s

#

fori= 1, 2. Let 0≤d≤1; then for eacht∈J, (dh₁+ (1−d)h₂)(t) = 1

Γ(r) Z t

1

logt

s r−1

[dv₁+ (1−d)v₂]ds s + (logt)^r−1

(logT)^r−1

"

yT− 1 Γ(r)

Z T

1

logT

s r−1

[dv1+ (1−d)v2]ds s

# . SinceS_F,y is convex (becauseF has convex values), we have

dh1+ (1−d)h2∈N(y).

Step 2:N(M)is relatively compact for each compactM ⊂U.

Let M ⊂ U be a compact set and let {h_n} be any sequence of elements of N(M). We will show that{h_n} has a convergent subsequence by using the Arzel`a- Ascoli criterion of compactness in C(J, B). Since hn ∈ N(M), there exist yn ∈ M andvn ∈SF,y such that

hn(t) = 1 Γ(r)

Z t

1

logt

s r−1

vn(s)ds s + (logt)^r−1

(logT)^r−1

"

y_T − 1 Γ(r)

Z T

1

logT

s r−1

v_n(s)ds s

#

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for n ≥ 1. Using Theorem 2.3 and the properties of the Kuratowski measure of noncompactness, we have

β({hn(t)})≤2 ( 1

Γ(r) Z t

1

β (

logt s

^r−1vn(s) s :n≥1

)!

ds

+ (logt)^r−1 (logT)^r−1

"

yT + 1 Γ(r)

Z T

1

β (

logT s

r−1

vn(s) s :n≥1

)!

ds

#) . (3.7) On the other hand, since M(s) is compact in E, the set{v_n(s) :n≥1}is compact.

Consequently,β({vn(s) :n≥1}) = 0 for a.e. s∈J. Furthermore, β

( logt

s r−1

vn(s) s

)!

=

log t s

r−1

1

sβ({vn(s) :n≥1}) = 0 and

β (

logT s

r−1

v_n(s) s

)!

=

logT s

r−1

1

sβ({v_n(s) :n≥1}) = 0

for a.e.t, s∈J. Hence, from this and (3.7),{hn(t) :n≥1}is relatively compact in B for eacht∈J. In addition, for eacht1, t2∈J witht1< t2, we have

|hn(t2)−hn(t1)|=

1 Γ(α)

Z t₁

1

"

logt₂ s

α−1

−

logt₁ s

α−1# v_n(s)

s ds

+ 1

Γ(α) Z t2

t₁

logt₂

s α−1

v_n(s) s ds

≤ p(t) Γ(α)

Z t1

1

"

logt2

s α−1

−

logt1

s α−1#

ds s + p(t)

Γ(α) Z t₂

t₁

logt₂

s α−1

ds s.

Ast1→t2, the right hand side of the above inequality tends to zero. This shows that {hn : n≥1} is equicontinuous. Consequently,{hn :n≥1} is relatively compact in C(J, B).

Step 3:N has a closed graph.

Lety_n →y_∗,h_n∈N(y_n), andh_n →h_∗. We need to show thath_∗∈N(y_∗). Now h_n∈N(y_n) means that there existsv_n∈S_F,y such that, for eacht∈J,

hn(t) = 1 Γ(r)

Z t

1

log t

s r−1

(logT)^r−1

"

y_T − 1 Γ(r)

Z T

1

logT

s r−1

v_n(s)ds s

# .

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Consider the continuous linear operatorθ:L¹(J, E)→C(J, E) defined by

θ(v)(t)→hn(t) = 1 Γ(r)

Z t

1

logt

s r−1

(logT)^r−1

"

y_T− 1 Γ(r)

Z T

1

logT

s r−1

v_n(s)ds s

# .

Clearly,khn(t)−h_∗(t)k →0 as n→ ∞. From Lemma 2.4 it follows that θ◦SF is a closed graph operator. Moreover,hn(t)∈θ(SF,y_n). Sinceyn→y, Lemma 2.4 implies

h(t) = 1 Γ(r)

Z t

1

logt

s r−1

(logT)^r−1

"

yT − 1 Γ(r)

Z T

1

logT

s r−1

v(s)ds s

# .

Step 4:M is relatively compact inC(J, B).

SupposeM ⊂U, M ⊂conv({0} ∪N(M)), and M =C for some countable set C ⊂ M. Using an argument similar to the one used in Step 2 shows that N(M) is equicontinuous. Then, sinceM ⊂conv({0} ∪N(M)), we see thatM is equicontinuous as well. To apply the Arzel`a-Ascoli theorem, it remains to show thatM(t) is relatively compact inE for eacht∈J. SinceC⊂M ⊂conv({0} ∪N(M)) andC is countable, we can find a countable set H ={hn : n ≥1} ⊂N(M) with C ⊂ conv({0} ∪H).

Then, there existyn∈M andvn∈SF,y_n such that

h_n(t) = 1 Γ(r)

Z t

1

log t

s r−1

v_n(s)ds s + (logt)^r−1

(logT)^r−1

"

yT − 1 Γ(r)

Z T

1

logT

s r−1

vn(s)ds s

# .

Since M ⊂C ⊂conv({0} ∪H)), from the properties of the Kuratowski measure of noncompactness, we have

β(M(t))≤β(C(t))≤β(H(t)) =β({hn(t) :n≥1}).

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Using (3.7) and the fact thatvn(s)∈M(s), we obtain β(M(t))≤2

( 1 Γ(r)

Z t

1

β (

logt s

r−1

vn(s) s :n≥1

)!

ds

+ (logt)^r−1 (logT)^r−1

"

yT + 1 Γ(r)

Z T

1

β (

logT s

r−1

vn(s) s :n≥1

)!

ds

#)

≤2 ( 1

Γ(r) Z t

1

log t

s r−1

β(M(s))ds s + (logt)^r−1

(logT)^r−1

"

y_T + 1 Γ(r)

Z T

1

logT

s r−1

β(M(s))ds s

#)

≤2 ( 1

Γ(r) Z t

1

log t

s r−1

ψ(s, β(M(s)))ds s + (logt)^r−1

(logT)^r−1

"

yT + 1 Γ(r)

Z T

1

logT

s r−1

ψ(s, β(M(s)))ds s

#) .

We also have that the function ϕ given by ϕ(t) = β(M(t)) belongs to C(J,[1,2R]). Consequently, by (H4), ϕ = 0; that is, β(M(t)) = 0 for all t ∈ J. Now, by the Arzel`a-Ascoli theorem,M is relatively compact inC(J, B).

Step 5:Leth∈N(y) withy∈U. We claim thatN(U)⊂U. If this were not the case, then in view of (H2), there exists functions v∈S_F,y andp∈L¹(J, E) such that

h(t) = 1 Γ(r)

Z t

1

logt

s r−1

(logT)^r−1

"

yT − 1 Γ(r)

Z T

1

logT

s r−1

v(s)ds s

# ,

and

R <kN(y)k_P ≤ 1 Γ(r)

Z t

1

logt

s r−1

|v(s)|ds s + (logt)^r−1

(logT)^r−1

"

|yT|+ 1 Γ(r)

Z T

1

logT

s r−1

|v(s)|ds s

#

≤ (logT)^r Γ(r+ 1)

Z t

1

p(s)ds+ (logT)^r Γ(r+ 1)

Z T

1

p(s)ds

≤2(logT)^r Γ(r+ 1)

Z t

1

p(s)ds.

Dividing both sides byRand taking the lim inf asR→ ∞, we have 2

(logT)^r Γ(r+ 1)

δ≥1

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which contradicts (3.6). Hence,N(U)⊂U.

As a consequence of Steps 1-5 and M¨onch’s Theorem (Theorem 2.7 above), N has a fixed pointy∈C(J, B) that in turn is a solution of problem (1.1)-(1.2).

4. An example

We conclude this paper with an example to illustrate our main result, namely, Theorem 3.3 above.

Consider the fractional differential inclusion

HD^αy(t)∈F(t, y(t)), for a.e.t∈J= [1, e], 0< α≤1, (4.1)

y(1) = 0, y(e) = 1. (4.2)

Here, F: [1, e]×R→ P(R) is a multivalued map satisfying F(t, y) ={v∈R:f₁(t, y)≤v≤f₂(t, y)},

where f1, f2 : [1, e]×R→R, f1(t,·) is lower semi-continuous (i.e., the set{y ∈R: f1(t, y)> µ} is open for eachµ∈R), andf2(t,·) is upper semi-continuous (i.e., the set the set{y∈R:f2(t, y)< µ} is open for eachµ∈R). We assume that there is a functionp∈L¹(J,R) such that

kF(t, u)k_P = sup{|v|, v(t)∈F(t, y)}

= max(|f1(t, y)|,|f2(t, y)|} ≤p(t), t∈[1, e], y∈R. It is clear thatF is compact and convex valued, and is upper semi-continuous.

ChooseC(s) to be the space of linear functions and chooseϕ(t) =β(C(t)) such that β(u(s)) = u(s)

2 with

u(s) =as, a >0, 2

a ≤s≤4R a . For (t, y)∈J×Rwith|y| ≥R, we have

lim inf

R→∞

Re 0 p(t)dt

R =δ <∞.

Finally, we assume that there exists a Carath´eodory functionψ:J[1,2R]→R+such that

β(F(t, M))≤ψ(t, β(M))a.e. t∈J and eachM ⊂B ={x∈R:|x| ≤R}, andϕ= 0 is the unique solution inC(J,[1,2R]) of the inequality

ϕ(t)≤2 ( 1

Γ(r) Z t

1

logt

s r−1

ψ(s, ϕ(s))ds s + (logt)^r−1

1 + 1

Γ(r) Z e

1

loge s

^r−1

ψ(s, ϕ(s))ds s

. fort∈J.

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Since all the conditions of Theorem 3.3 are satisfied, problem (4.1)-(4.2) has at least one solutiony on [1, e].

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John R. Graef

Department of Mathematics

University of Tennessee at Chattanooga Chattanooga, TN 37403-2504, USA e-mail:[email protected] Nassim Guerraiche

Laboratoire des Mathématiques Appliqués et Pures Université de Mostaganem

B.P. 227, 27000, Mostaganem, Algerie e-mail:hamani [email protected] Samira Hamani

Laboratoire des Mathématiques Appliqués et Pures Université de Mostaganem

B.P. 227, 27000, Mostaganem, Algerie e-mail:[email protected]