Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations

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http://www.sciencepublishinggroup.com/j/ajam doi: 10.11648/j.ajam.20190706.13

ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online)

Boundary Value Problems of Nonlinear Variable Coefficient

Fractional Differential Equations

Badawi Hamza Elbadawi Ibrahim

1,3

_{, Qixiang Dong}

2

_{, Zhengdi Zhang}

1,∗

1_{Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang, China} 2_{School of Mathematical Sciences, Yangzhou University, Yangzhou, China}

3_{Department of Mathematics, Faculty of Education, University of Khartoum, Sudan}

Email address:

[email protected] (B. H. E. Ibrahim), [email protected] (Zhengdi Zhang)

∗_{Corresponding author}

To cite this article:

Badawi Hamza Elbadawi Ibrahim, Qixiang Dong, Zhengdi Zhang. Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations.American Journal of Applied Mathematics. Vol. 7, No. 6, 2019, pp. 157-163. doi: 10.11648/j.ajam.20190706.13.

Received:October 24, 2019;Accepted:November 19, 2019;Published:Published: December 30, 2019

Abstract:

It is recognized that the theory of boundary value problems for fractional order-differential equations is one of the rapidly developing branches of the general theory of differential equations. As far as we know, most of the papers studied the fractional Riemann-Liouville derivative with respect to boundary values that are zero. However, for the purpose of this study, we concern ourselves with Captou type derivative of the orderα ∈ (2,3), with respect to boundary values that are nonzero. We establish sufficient conditions for the existence of solutions for boundary value problem of nonlinear variable coefficient of fractional order. On the other hand, the boundary value problem is formulated as follows:

c_Dα_u₍_t_{) +}_p₍_t₎_f₍_{t, u}₍_t_{)) +}_q₍_t_{) = 0}_{, u}_{(0) =}_{a , u}0_{(0) =}_{b, u}_{(1) =}_d

Wherea, b, d∈Rare constants. In this paper, we investigate the existence and uniqueness of solutions for a class of boundary value problem of the nonlinear variable coefficient of fractional differential equations. The existence of solutions involving Captuo fractional derivatives is discussed under the assumption that the bounded conditions are constants. By means of the Banach contraction mapping principle and Larry- Schauder alternative, the existence of solutions are obtained. Finally, some examples are discussed to illustrate the results, which are generalized to nonlinear fractional derivatives with variable coefficients.

Keywords:

Fractional Derivatives, Fixed Point Theorem, Boundary Value Problem

1. Introduction

In recent years, the theory of fractional differential equations has become an important area of study, see [1, 2, 3, 4]. Boundary value problems of fractional differential equations have applications in various fields of science such as physics, mechanics, chemistry, engineering, etc.[7, 6, 5, 8].

Also, it has received great attention and a variety of results concerning the existence of solutions, based on different kinds of analytic techniques, which can be found in [11, 10, 9, 12]. In [13] the authors considered the existence of multiple positive solutions for the following fractional differential equations with a negatively perturbed term

(

−Dα_u₍_t_{) =}_p₍_t₎_f₍_{t, u}₍_t₎₎₋_q₍_t₎_, ₀_{< t <}₁_,

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Where Dα _{is the standard Riemann-Liouville derivative,} 2 < α ≤ 3 is a real number, q : (0,1) → [0,∞] is Lebesgue integrable and does not vanish identically on any subinterval of(0,1). They established the existence results by Krasnoselskii’s fixed point theorem in a cone. Cui [14] studied the following boundary value problem

(

Dp_x₍_t_{) +}_p₍_t₎_f₍_{t, x}₍_t_{)) +}_q₍_t_{) = 0}_, ₀_{< t <}₁_,

x(0) =x0(0) = 0, x(1) = 0,

whereDp _{is the standard Riemann-Liouville derivative,}₂ _<

p ≤ 3 is a real number, q : (0,1) → R is continuous and Lebesgue integrable,p : (0,1) → [0,∞) is continuous and does not vanish identically on any subinterval of(0,1). Under the assumption thatf(t, x)is a Lipschitz continuous function, by usingu0-positive operator, he deduced that the Lipschitz

constant is related to the first eigenvalues corresponding to the relevant operators.

In this paper we study the following boundary value problems for fractional differential equation (BVP in short)

(_c

Dα_u₍_t_{) +}_p₍_t₎_f₍_{t, u}₍_t_{)) +}_q₍_t_{) = 0}_, ₀_{< t <}₁_,

u(0) =a, u0(0) =b, u(1) =d, (1)

wherec_Dα_{is Caputo fractional derivatives with}₂ _{< α} _≤₃_,

a, b, d ∈ Rare constants,q : (0,1) → Ris continuous and Lebesgue integrable andp: (0,1)→[0,∞)is continuous and does not vanish identically on any subinterval of(0,1).

2. Preliminaries

In this section, we state some definitions and results that we are going to use throughout this paper, we investigate the problem (1) with non-homogenous boundary value conditions. By using Banach contraction mapping principle and Larry-Schauder alternative, the existence of solutions of the BVP (1) is obtained.

Definition 2.1. Letp >0be a fixed number. The Riemann-Liouville fractional integral of order p > 0 of a function h∈C([a, b])is defined by

I_aph(t) = 1 Γ(p)

Z t a

(t−s)p−1h(s)ds, t∈[a, b]

provided the right side is point-wisely defined, where Γ(·)

denotes the well-known gamma function, i.e., Γ(z) = R∞

0 e

−t_tz−1_d_t_.

Definition 2.2. Letp≥0andn= [p] + 1. Ifh∈ACn[a, b]

then the Caputo fractional derivative of order pofh at the pointtis defined by

c_Dp ah(t) =

1 Γ(n−p)

Z t a

(t−s)n−p−1h(n)(s)ds, t∈[a, b]

c_Dp

ais also called the Caputo fractional differential operator.

For simplicity, whena= 0, we denotecDα

0 andI0αbycDα

andIα_{, respectively.}

Lemma 2.3. Letp, q >0andn= [p]+1. Then the following relations hold.

c_Dp_tq−1₌ Γ(q) Γ(q−p)t

q−p−1_{, q > n}

and

c_Dp_tk_{= 0}_{, k}_{= 0}_,₁_,₂_{, ..., n}₋₁_.

Lemma 2.4. Letp > 0, h(t) ∈ C(0,1). The homogenous fractional differential equation

c_Dα_h₍_t_{) = 0}

has a solution

h(t) =c0+c1t+c2t2+...+cn−1tn−1,

whereci ∈R, i= 0,1,2, ...., n−1are some constants.

Lemma 2.5. Let α, β ≥ 0 and h ∈ L1[a, b]. Then Iα_Iβ_h₍_t_{) =} _Iα+β_h₍_t_{) =} _Iβ_Iα_h₍_t₎_and c_Dα_Iα_h₍_t_{) =} _h₍_t₎

for allt∈[a, b].

Lemma 2.6.(Leray-Schauder nonlinear alternative). LetF be a Banach space andΩa bounded open subset ofF,0∈Ω

and

T : Ω → F be a completely continuous operator. Then, either

(i)there existsu∈∂Ω, λ >1such thatT(u) =λu, or

(ii)there exists a fixed point u∗∈Ω.

Lemma 2.7. Leta, b, c ∈ Rand y ∈ C[0,1]. The unique solution of the boundary value problem

(_c

Dα_u₍_t_{) +}_y₍_t_{) = 0}_, ₀_{< t <}₁_,

u(0) =a, u0(0) =b, u(1) =d, (2)

is given by

u(t) =h(t) + Z 1

0

G(t, s)y(s)ds, (3)

whereh(t) = (d−b−a)t2₊_bt₊_{a, and}_G₍_{t, s}₎_{is the Green’s}

function given by

G(t, s) = 1 Γ(α)

(

t2₍₁₋_s₎α−1₋₍_t₋_s₎α−1_, ₀_≤_s_≤_t_≤₁_,

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Proof.Applying Lemma 2.4 and 2.5, the Eq (2) is equivalent to the integral equation

u(t) =− 1

Γ(α) Z t

0

(t−s)α−1y(s)ds−c0−c1t−c2t2 (5)

The boundary conditionu(0) =a,givesc0=−a. Differentiating (5), we get

u0(t) =− 1

Γ(α−1) Z t

0

(t−s)α−2y(s)ds−c1−2c2t.

Sinceu0(0) =b, we deduce thatc1=−b. Further, conditionu(1) =dimplies that

c2=a+b−d− 1 Γ(α)

Z 1

0

(1−s)α−1y(s)ds.

Substitutingc0, c1, c2into Eq (5), we obtain

u(t) = (d−b−a)t2+bt+a+ 1 Γ(α)

Z 1

0

t2(1−s)α−1y(s)ds− 1

Γ(α) Z t

0

(t−s)α−1y(s)ds, (6) or

u(t) = (d−b−a)t2+bt+a+t2Iαy(1)−Iαy(t)

Thus we get

u(t) =h(t) + 1 Γ(α)

Z 1

0

G(t, s)y(s)ds

The proof is completed.

3. Existence Results

In this section, Now we study the existence of solutions to BVP (1). Let us denote byC([a, b], R)the Banach space of all continuous functionsu: [0,1]→Rendowed with supremum normkuk= maxt∈[0,1]|u(t)|.

Definition 3.1. A functionu: [0,1]→Ris said to be a solution to (1), ifusatisfies

u(t) =h(t) + Z 1

0

G(t, s)[p(s)f(s, u(s)) +q(s)]ds (7)

fort∈[0,1].

Define an operatorT :C([0,1], R)→C([0,1], R)by

T u(t) =h(t) + Z 1

0

G(t, s)[p(s)f(s, u(s)) +q(s)]ds

foru∈ C([0,1], R)andt ∈ [0,1]. Then we transform the existence of solutions to the fixed point problem. We first list the following hypothesis.

(H1)f : [0,1]×R→Ris continuous.

(H2) There exist nonnegative functiong∈L1_([0_,_1]_{, R}

+)such that

|f(t, u)−f(t, v)| ≤g(t)|u−v|

for allu, v∈Randt∈[0,1].

(H3) There exist nonnegative functionφ ∈ L1_([0_,_1]_{, R}

+), and a continuous nondecreasing functionψ : [0,∞)→ (0,∞)

such that

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for all(t, u)∈[0,1]×R.

Lemma 3.2. Letα >0andf ∈L1([a, b], R+), Then for allt∈[a, b], we have

I_aα+1f(t)≤ kI_aαfkL1.

Proof.Letf ∈L1([a, b], R+). Then

kI_aαfkL1 = Z 1

0

Iαf(r)dr≥ 1

Γ(α) Z t

a Z r

a

(r−s)α−1f(s)dsdr

= 1 Γ(α)

Z t a

Z t s

(r−s)α−1f(s)dr

ds

= 1

αΓ(α) Z t

a

(t−s)αf(s)ds=Iaα+1f(t).

Theorem 3.3.Suppose that the condition (H1) and (H2) are satisfied. If

N <1 (8)

then the BVP (1) has a unique solution inC([0,1], R),whereN = 2kIα−1_pg_k_.

Proof.Define an operatorT :C([0,1], R)→C([0,1], R)by

T u(t) =h(t) + Z 1

0

G(t, s)[p(s)f(s, u(s)) +q(s)]ds

foru∈C([0,1], R)andt∈[0,1]. Thenu∈C([0,1], R)is a solution to the BVP (1) if and only ifuis a fixed point ofT. Foru, v∈C([0,1], R), applying (6), we obtain

T u(t)−T v(t)

≤

Z 1

0

G(t, s)p(s)

f(s, u(s))−f(s, v(s)) ds

= [Iαp(1)(f(1, u(1))−f(1, v(1))]−[Iαp(t)(f(t, u(t))−f(t, v(t))].

The condition(H2)implies that

T u(t)−T v(t)

≤ max 0≤t≤1

u(t)−v(t)

t2Iα|p(1)|g(1) +Iα|p(t)|g(t)

.

It follows from Lemma 3.2 that

kT u−T vk ≤ ku−vk

kIα−1pgkL1+kI α−1_pg_k

L1

= 2kIα−1pgkku−vk

Fort∈[0,1]. Hence

kT u−T vk ≤Nku−vk.

The assumption (8) shows that T is a contraction. By Banach contraction principle, T has a unique fixed point in

C([0,1], R), which is the solution to the BVP (1). The proof is completed.

Next, we prove an existence result by using Larry-Schouder’s nonlinear alternative. For simplicity, let k = max0≤t≤1|h(t)| = |(d − b − a) + b + a| = |d|,

M1 = 2kIα−1pφkL1, M2 = 2kI α−1_q_k

L1 +k, and M = max{M1, M2}, whereφis the function appearing in condition

(H3).

Theorem 3.4. Suppose that (H1) and (H3) are satisfied. If

lim sup r→+∞

Mψ(r) r <1,

then the BVP (1) has at least one solution inC([0,1], R).

Proof. First let us prove thatT is completely continuous. It is clear thatT is continuous sincef andGare continuous. Sincelim supr→+∞M

ψ(r)

r <1, there exists a numberr >0

such thatM(ψ(r) + 1) < r. LetBr = {u ∈ C([0,1], R) :

kuk ≤r}. ThenBris a bounded subset inC([0,1], R). For

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T u(t)

= Z 1 0

G(t, s)[p(s)f(s, u(s) +q(s)]ds+h(t) ≤ Z 1 0

|G(t, s)|[|p(s)|φ(s)ψ(kuk) +|q(s)|] +|h(t)|

≤ ψ(r) Z 1

0

|G(t, s)|ds|p(s)|φ(s) + Z 1

0

|G(t, s)|ds|q(s)|+|h(t)|

≤ ψ(r)

Iα|p(1)|φ(1) +Iα|p(s)|φ(s)

+Iα|q(1)|+Iα|q(s)|+|h(t)| ≤ ψ(r)

kIα−1pφkL1+kIα−1pφk_L1

+kIα−1qkL1+kIα−1qk_L1+k

≤ ψ(r)2kIα−1pφkL1+ 2kIα−1qk_L1+k

≤ (ψ(r)M1+M2)≤M(ψ(r) + 1)

HenceT(Br)is uniformly bounded. For allt1, t2∈[0,1], t1< t2andu∈Br, we have,

T u(t1)−T u(t2)

≤

Z 1

0

|G(t1, s)−G(t2, s)||p(s)f(s, u(s))|ds+|h(t1)−h(t2)|

≤

Z 1

0

|G(t1, s)−G(t2, s)||p(s)|φ(s)ψ(kuk)|ds+|h(t1)−h(t2)|

≤ kpkψ(r)

Γ(α) Z t1

0

|G(t1, s)−G(t2, s)|φ(s)ds+ Z t2

t1

|G(t1, s)−G(t2, s)|φ(s)ds

+ Z 1

t2

|G(t1, s)−G(t2, s)|φ(s)ds

+|h(t1)−h(t2)|

≤ kpkψ(r)

Γ(α) Z t1

0

|t21(1−s) α−1

−(t1−s)α−1−t22(1−s) α−1

+ (t2−s)α−1|ds

+ Z t2

t1

|t2₁(1−s)α−1−t2₂(1−s)α−1+ (t2−s)α−1|ds

+ Z 1

t2

|t2₁(1−s)α−1−t2₂(1−s)α−1|ds

+|h(t1)−h(t2)|

≤ kpkψ(r)

Γ(α) Z t1

0

|t2₁−t2₂|(1−s)α−1+ (t2−s)α−1−(t1−s)α−1ds

+ Z t2

t1

|t2₁−t2₂|(1−s)α−1+ (t2−s)α−1ds

+ Z 1

t2

|t2₁−t2₂|(1−s)α−1

ds+|h(t1)−h(t2)|

≤ kpkψ(r)

Γ(α+ 1)

|t2₁−t2₂|(1− |1−t1|α) +|tα2 −t α

1| − |t2−t1|α + |t2₁−t₂2|(|1−t1|α− |1−t2|α) +|t2−t1|α

+ |t2₁−t2₂||1−t2|α

+|h(t1)−h(t2)|.

It follows that

T u(t1)−T u(t2)

≤

kpkψ(r) Γ(α+ 1)

|t21−t 2 2|+|t

α 2 −t

α 1|

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It is easy to see thatT u(t1)−T u(t2)

tend to0ast1−t2→ 0, and the convergence is independent ofu∈Br. This show

thatT(Br)is equicontinuous. By Arzela- Ascolli Theorem we

deduce thatT is completely continuous.

Now letΩ ={u∈B :kuk < r}. ThenΩis an open and bounded subset inB and0 ∈ Ω. If there is au ∈ ∂Ωsuch thatu=λT ufor someλ∈(0,1)and eacht∈[0,1], then we have

|u(t)|=λ|T u(t)| ≤ |T u(t)| ≤M(ψ(r) + 1)< r.

This is contradictory to the fact that u ∈ ∂Ω. Hence Lemma 2.6 (Leray-Schauder nonlinear alternative) allows us to conclude thatT has a fixed point u∗ ∈ Ω. Therefore the BVP (1) has at least a solutionu∗ ∈ B. This completes the proof.

4. Example

Example 4.1. Consider the following fractional boundary value problem

 

 c_D11

4 u(t) = Γ(11 4)

_t2 5u+t+3 (1−t)34

+_(1+t)1

, 0< t <1,

u(0) =a, u0(0) =b, u(1) =d,

In this case we have

f(t, x) =t 2

5u+t+ 3, 2< α= 11

4 <3

p(t) = Γ( 11

4) (1−t)34

, q(t) = Γ( 11

4) (1 +t).

|f(t, x)−f(t, y)| ≤ t

2

5|x−y|

therefore

|f(t, x)−f(t, y)| ≤g(t)|x−y|,∀x, y∈R

and

g(t) = t 2

5

kIα−1pgkL1 =Iαp(t)g(t) = 1 Γ(α)

Z 1

0

(1−s)α−1p(s)g(s)ds= 1 5

Z 1

0

(1−s)s2ds=Γ(2)Γ(3)

5Γ(5) = 0.01667<1.

Example 4.2.Consider the following problem

( c_D7

3u(t) = √1

1+t

(1+t3)e−tu3

216 +

t2

(1−t), 0< t <1,

u(0) =a, u0(0) =b, u(1) =d,

has at least one solution . Applying Theorem(3.4), we haveα= 7₃and

|f(t, x)| = √ 1

1 +t

(1 +t3₎_e−t_x3 216 +

t2 (1−t)

≤ √ 1

1 +t(1 +t

3₎ _|_x_|

6 3

+ t 2

(1−t)

≤ p(t)φ(t)ψ(|x|) +q(t).

wherep(t) =_√1

1+t, φ(t) = (1 +t

3₎_{, ψ}₍_|_x_|_{) =}

|x|

6 3

, q(t) = t2 (1−t).

Let us evaluateM(ψ(r) + 1), some computations lead to

kIα−1pφkL1 = Iαp(t)φ(t) = 1 Γ(α)

Z 1

0

(1−s)α−1p(s)φ(s)ds= 1 Γ(7

3) Z 1

0

(1−s)43 1 (1 +s)12

(1 +s3)ds

≤ 2

Γ(7₃) Z 1

0

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And

kIα−1qkL1 = 1 Γ(7₃)

Z 1

0

(1−s)43 s 2

(1−s)ds=

Γ(4₃)Γ(3)

Γ(7₃)Γ(13₃)= 0.2223.

In fact

h(t) = (d−b−a)t2+bt+a

we takea=b=d= 0, there for

k=|h(t)|= 0, M1= 2kIα−1pφkL1 = 1.4398

M2= 2kIα−1qkL1+k= 0.4446, thenM = 1.8844

we see that(H4)is equivalent to

1.8844

(r 6)

3 + 1

−r <0f orr= 2.

5. Conclusion

In this work we studied the existence and uniqueness of solutions for a class of boundary value problem of nonlinear variable coefficient of fractional differential equations. Employing Banach contraction theorem and Larry-Schauder alternative, the existence of solutions were obtained. Some examples were discussed to illustrate our results which generalized to nonlinear Fractional derivatives with variable coefficients.

Funding

This work was supported by Key Program of the National Natural Science Foundation of China (Grant No. 11872189) and the National Natural Science Foundation of China (Grant No. 11472116).

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