Successive iteration and positive solutions for nonlocal higher-order fractional differential equations with p-Laplacian

Successive iteration and positive solutions for nonlocal higher-order fractional differential equations with p-Laplacian

QIUYAN ZHONG Jining Medical College

Department of Information Engineering Hehua Road No. 16, 272067, Jining

CHINA

[email protected]

XINGQIU ZHANG Jining Medical College

Department of Information Engineering Hehua Road No. 16, 272067, Jining

CHINA

[email protected]

Abstract: In this paper, we investigate the existence of positive solutions for higher-order fractional differential equations with p-Laplacian operator and nonlocal boundary conditions. By means of the properties of the corre- sponding Green function together with monotone iterative technique, we obtain not only the existence of positive solutions for the problems, but also establish iterative schemes for approximating the solutions. The nonlinearity permits singularities at t = 0 and/or t = 1.

Key–Words: Fractional differential equations; Integral boundary value problem; p-Laplacian; Positive solution;

Successive iteration

1 Introduction

The purpose of this paper is to consider the existence of positive solutions for the following singular frac- tional differential equations involving p-Laplacian op- erator (PFDE, for short) and integral conditions











D₀₊^β (ϕ_p(D₀₊^α u(t))) + a(t)f (t, u(t)) = 0, t ∈ J, u(0) = u⁰(0) = · · · = u⁽ⁿ⁻²⁾(0) = 0,

D₀₊^α u(0) = 0, u⁽ⁱ⁾(1) = λ Z η

0

h(t)u(t)dt,

(1)

where D₀₊^α , D^β₀₊ are standard Riemann-Liouville derivative, ϕp(s) = |s|^p−2s, p > 1, a ∈ C((0, 1), R⁺), J = (0, 1), f ∈ C((0, 1) × R⁺, R⁺), f (t, u) may be singular at t = 0 and/or t = 1, R⁺ = [0, +∞), h ∈ L¹[0, 1] is nonnegative and h(t) may be singular at t = 0 and t = 1, i ∈ [1, n − 2] is a fixed integer, n − 1 < α ≤ n, n ≥ 3, 0 < β ≤ 1 0 < η ≤ 1, 0 ≤ λ^R₀^ηh(t)t^α−1dt < ∆, here

∆ = (α − 1)(α − 2) · · · (α − i).

Due to both by the intensive development of the theory of fractional calculus itself and by the wide ap- plications such as in control, porous media, aerody- namics, electrodynamics of complex medium, poly- mer rheology, electromagnetic, and so on, fractional differential equations have attracted more and more researchers’s much attention in recent years. We refer the readers to [1-3] for an extensive collection of such results.

In [4-12], by means of the fixed point index the- ory, fixed point theory together with the relevant re-

sults on u0bounded operator and lattice structure, the authors investigated the existence and multiplicity of positive and nontrivial solutions for fractional differ- ential equation

D₀₊^α u(t) + a(t)f (t, u(t)) = 0, 0 < t < 1 (A) subject to different boundary conditions. A natural question is “How can we find the solutions when they are known to exist?”. There are few results on the computation of positive solutions for fractional differ- ential equations at present, see [8, 9, 12-15]. In [16], using the fixed point index theorem in cones, under some weak conditions concerning the first eigenvalue corresponding to the relevant linear operator, the au- thor obtained the existence and multiplicity of positive solutions for some singular higher-order fractional d- ifferential equations. Motivated by above papers, the aim of this paper is to give the iterative procedure of positive solutions for BVP (1).

This paper has the following three new features.

First, compared with [4-11], the boundary conditions contain the i-th order of the unknown function and a parameter λ. Particularly, a Lebesgue integrable func- tion h is involved in the boundary condition. Second, compared with [16], p-Laplacian operator is involved in differential operator. This means that the problems discussed in this paper have more general form. Fi- nally, we obtain not only the existence of positive so- lutions for the problems, but also establish iterative schemes for approximating the solutions. The itera- tive sequences begin with simple functions which is

convenient in applications. Obviously, it is possible to replace the Riemann integrals in the boundary condi- tions by Riemann-Stieltjes integrals with minor mod- ifications.

The rest of this paper is organized as follows. In Section 2, we give some preliminaries and lemmas.

The main result is formulated in section 3 and an ex- ample is given in section 4 to illustrate how to use the main result.

2 Preliminaries and several lemmas

Let E = C[0, 1], kuk = max_0≤t≤1|u(t)|, then (E, k · k) is a Banach space. For the reader’s convenience, we present some necessary definitions and lemmas from fractional calculus theory which can be found in the recent literature, see [1-3].

Definition 1 The Riemann-Liouville fractional inte- gral of orderα > 0 of a function y : (0, ∞) → R is given by

I₀₊^α y(t) = 1 Γ(α)

Z t 0

(t − s)^α−1y(s)ds provided the right-hand side is pointwise defined on (0, ∞).

Definition 2 The Riemann-Liouville fractional derivative of order α > 0 of a continuous function y : (0, ∞) → R is given by

D^α₀₊y(t) = 1 Γ(n − α)

d dt

nZ t 0

y(s) (t − s)^α−n+1ds wheren = [α] + 1, [α] denotes the integer part of the numberα, provided that the right-hand side is point- wise defined on(0, ∞).

Now, we consider the following fractional differ- ential equation











D^α₀₊u(t) + y(t) = 0, 0 < t < 1, u(0) = u⁰(0) = · · · = u⁽ⁿ⁻²⁾(0) = 0, u⁽ⁱ⁾(1) = λ

Z η 0

h(t)u(t)dt.

(2)

Lemma 3 [16] Assume that λ^R₀^ηh(t)t^α−1dt 6= ∆.

Then for anyy ∈ L¹[0, 1], the unique solution of the boundary value problems (2) can be expressed in the form

u(t) = Z 1

0

G(t, s)y(s)ds, t ∈ [0, 1], where

G(t, s) = G₁(t, s) + G₂(t, s), (3)

G₁(t, s) =



















t^α−1(1 − s)^α−1−i− (t − s)^α−1

Γ(α) ,

0 ≤ s ≤ t ≤ 1, t^α−1(1 − s)^α−1−i

Γ(α) , 0 ≤ t ≤ s ≤ 1,

G2(t, s) = λt^α−1

∆ − λ^R₀^ηh(t)t^α−1dt Z η

0

h(t)G1(t, s)dt.

Here,G(t, s) is called the Green function of BVP (2).

Obviously,G(t, s) is continuous on [0, 1] × [0, 1].

Lemma 4 [16] If 0 ≤ λ^R₀^ηh(t)t^α−1dt < ∆, then the functionG(t, s) defined by (3) satisfies

(a1) G(t, s) ≥ m₁t^α−1s(1 − s)^α−1−i, ∀ t, s ∈ [0, 1];

(a2) G(t, s) ≤ M1t^α−2s(1 − s)^α−1−i, ∀ t, s ∈ [0, 1];

(a3) G(t, s) ≤ M₁t^α−1(1 − s)^α−1−i, ∀ t, s ∈ [0, 1];

(a4) G(t, s) > 0, ∀ t, s ∈ (0, 1);

where m₁ = _Γ(α)^{1 }1 + ^λ

∆−λRη

0 h(t)t^α−1dt

Rη

0 h(t)t^α−1dt^, M1 = _Γ(α)^{n }1 + ^λ

∆−λRη

0 h(t)t^α−1dt

Rη

0 h(t)t^α−2dt^.

Let q > 1 satisfy ¹_p + ¹_q = 1. Then, ϕ⁻¹_p (s) = ϕ_q(s). To study the PFDE (1), we first consider the associated linear PFDE











D^β₀₊(ϕ_p(D^α₀₊u(t))) + y(t) = 0, 0 < t < 1, u(0) = u⁰(0) = · · · = u⁽ⁿ⁻²⁾(0) = 0, D^α₀₊u(0) = 0, u⁽ⁱ⁾(1) = λ

Z η 0

h(t)u(t)dt, (4)

for y ∈ L¹[0, 1] and h ≥ 0.

Lemma 5 The unique solution for the associated lin- ear PFDE (4) can be written by

u(t) = ^ 1 Γ(β)

q−1Z 1 0

G(t, s) ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1y(τ )dτ^ds.

(5)

Proof. Let w = D^α₀₊u, v = ϕp(w). Then, the initial value problem

( D^β₀₊v(t) + y(t) = 0, t ∈ (0, 1),

v(0) = 0 (6)

has the solution v(t) = c1t^β−1− I^βy(t), t ∈ [0, 1].

Noticing that v(0) = 0, 0 < β ≤ 1, we have that c1 = 0. As a consequence,

v(t) = −I^βy(t), t ∈ [0, 1]. (7)

Considering that D₀₊^α u = w, w = ϕ⁻¹_p (v), we have from (7) that











D^α₀₊u(t) = ϕ⁻¹_p (−I^β(y(t))), 0 < t < 1, u(0) = u⁰(0) = · · · = u⁽ⁿ⁻²⁾(0) = 0, u⁽ⁱ⁾(1) = λ

Z η 0

h(t)u(t)dt.

(8)

By Lemma 3, the solution of (8) can be expressed by (5).

We list below assumptions used in this paper for convenience.

(H₁) h ∈ L¹[0, 1] is nonnegative;

(H2) f (t, 0) 6≡ 0 on [0, 1], f : (0, 1)×R⁺→ R⁺ is continuous and nondecreasing on x, and there exists constant r > 0 such that, for any t ∈ (0, 1), u ∈ R⁺,

f (t, cu) ≥ c^rf (t, u), ∀ 0 < c ≤ 1; (9) (H₃) a : (0, 1) → R⁺ is continuous, a(t) 6≡ 0 with 0 < ^R₀¹(1 − s)^α−1−iϕ⁻¹_p ^R₀^s(s − τ )^β−1f (τ, 1)dτ^a(s)ds < +∞.

Remark 6 If c ≥ 1, then it is not difficulty to see that (9) is equivalent to

f (t, cu) ≤ c^rf (t, u), ∀ c ≥ 1. (10) Define a subset P in E as follows

P = {u ∈ C([0, 1], R⁺) : there exist two positive numbers lu < 1 < Lusuch that l_ut^α−1≤ u(t) ≤ L_ut^α−1, t ∈ [0, 1]}.

(11)

Clearly, P is nonempty since t^α−1 ∈ P.

Now define an operator A as follows:

(Au)(t) =^ 1 Γ(β)

q−1Z 1 0

G(t, s)ϕ⁻¹_p ^ Z s

0

(s − τ )^β−1f (τ, u(τ ))dτ^ds, t ∈ [0, 1].

(12)

Lemma 7 Suppose that (H₁) − (H₃) hold. Then A : P → P is completely continuous and nondecreasing.

Proof. For any u ∈ P , there exist two positive num- bers 0 < lu < 1 < Lusuch that

l_ut^α−1 ≤ u(t) ≤ L_ut^α−1, t ∈ [0, 1]. (13)

Thus, we have from Lemma 4, (9), (10), (H2), (H3) that

(Au)(t)

=^ 1 Γ(β)

q−1Z 1 0

G(t, s) ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1f (τ, u(τ ))dτ^ds

≤^ 1 Γ(β)

q−1

M1t^α−1 Z 1

0

(1 − s)^α−1−i ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1f (τ, Luτ^α−1)dτ^ds

≤^ 1 Γ(β)

q−1

M1t^α−1 Z 1

0

(1 − s)^α−1−i ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1f (τ, L_u)dτ^ds

≤^ 1 Γ(β)

q−1

M₁t^α−1 Z 1

0

(1 − s)^α−1−i ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1f (τ, 1)L^r_udτ^ds

=^ 1 Γ(β)

q−1

M1L^r(q−1)_u Z 1

0

(1 − s)^α−1−i ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1f (τ, 1)dτ^ds · t^α−1

< +∞,

(14)

(Au)(t)

=^ 1 Γ(β)

q−1Z 1 0

G(t, s) ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1f (τ, u(τ ))dτ^ds

≥^ 1 Γ(β)

q−1

m₁t^α−1 Z 1

0

s(1 − s)^α−1−i ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1f (τ, luτ^α−1)dτ^ds

≥^ 1 Γ(β)

q−1

m1t^α−1 Z 1

0

s(1 − s)^α−1−i ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1l^r_uτ^r(α−1)f (τ, 1)dτ^ds

=^ 1 Γ(β)

q−1

m₁l^r(q−1)_u Z 1

0

s(1 − s)^α−1−i ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1τ^r(α−1)f (τ, 1)dτ^ds · t^α−1 (15)

which means that A is well defined, uniformly bound- ed and A(P ) ⊂ P . By standard argument, according to the Lebesgue dominated convergence theorem and the Arzela-Ascoli theorem, it is not difficult to see that A is completely continuous. Noticing the monotonic- ity of f on x, we know that A is nondecreasing.

3 Main result

For notational convenience, we denote Λ =^ 1

βΓ(β)

q−1

M₁ Z 1

0

(1 − s)^α−1−is^β(q−1)ds. (16)

Theorem 8 Suppose that conditions (H1) − (H3) hold. In addition, if there existsa > 0, such that

f (t, u) ≤ ϕ_p^a Λ

, (t, u) ∈ [0, 1] × [0, a]. (17) Then the BVP (1) has two positive solutions u∗ and u^∗; and there exist two positive numbersl_i < L_i (i = 1, 2) such that

l₁t^α−1≤ u∗(t) ≤ L₁t^α−1,

l2t^α−1≤ u^∗(t) ≤ L2t^α−1, t ∈ [0, 1]. (18) Moreover, for initial values u0 = 0, v0 = at^α−1, monotone sequences {u_n}^∞_n=1 and {v_n}^∞_n=1 satisfy limn→∞un = u∗, limn→∞vn = u^∗, whereun+1 = Aun, vn+1 = Avn, n = 0, 1, 2, · · · .

Proof. Let Pa = {u ∈ P : kuk ≤ a}. For u ∈ Pa, we have 0 ≤ u(s) ≤ kuk ≤ a, s ∈ [0, 1]. Thus, for t ∈ [0, 1], it follows from Lemma 4, (16) and (17) that

(Au)(t)

=^ 1 Γ(β)

q−1Z 1 0

G(t, s) ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1f (τ, u(τ ))dτ^ds

≤^ 1 Γ(β)

q−1

M1t^α−1 Z 1

0

(1 − s)^α−1−i ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1dτ · ϕ_p^a Λ

ds

=^ 1 Γ(β)

q−1

M₁ Z 1

0

(1 − s)^α−1−i

·ϕ⁻¹_p ^1

βs^βds · a Λ

=^ 1 βΓ(β)

q−1

M₁ Z 1

0

(1 − s)^α−1−i s^β(q−1)ds · a

= a, Λ

(19)

which implies that kAuk ≤ a, i.e., A(Pa) ⊂ Pa. Let u0(t) = 0, t ∈ [0, 1]. Similar to (19), by (17) we get that u1 = Au0∈ P_a. Denote

u_n+1= Au_n= Aⁿ⁺¹u₀, n = 1, 2, · · · . It follows from A(Pa) ⊂ Pa that un ∈ P_a (n = 1, 2, · · ·). By Lemma 7, we know that {u_n} is a se- quentially compact set.

Since u1 = Au0= A0 ∈ Pa, we have

u1(t) = (Au0)(t) = (A0)(t) ≥ 0 = u0(t), t ∈ [0, 1].

By induction, we get

u_n+1(t) ≥ u_n(t), n = 1, 2, · · · .

Consequently, there exists u∗ ∈ P_a such that un → u∗. Let n → ∞, by the continuity of A and un+1 = Au_n, we know that Au∗ = u∗. This is to say that u∗ ≥ 0 is a fixed point of A, i.e., u∗ is a nonnegative solution for BVP (1). By (H2), it is not difficult to see that 0 is not the solution for BVP (1). Thus, u∗ is a positive solution for BVP (1).

Let v0(t) = at^α−1, t ∈ [0, 1], then v0 ∈ P_a. It follows from A(Pa) ⊂ P_athat v1 ∈ P_a. Denote

vn+1= Avn= Aⁿ⁺¹v0, n = 1, 2, · · · . Similarly, we get that

vn∈ P_a, n = 0, 1, 2, · · · . On the other hand, we have

v1(t) = (Av0)(t) =^ 1 Γ(β)

q−1Z 1 0

G(t, s) ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1f (τ, v₀(τ ))dτ^ds

≤ ^ 1

Γ(β)

q−1

M₁t^α−1 Z 1

0

(1 − s)^α−1−i ϕ⁻¹_p ^

Z s 0

(s − τ )^β−1dτ · ϕ_p^a Λ

ds

≤ ^ 1

βΓ(β)

q−1

M₁t^α−1 Z 1

0

(1 − s)^α−1−i

·s^β(q−1)ds · a

= at^α−1= v₀(t).Λ

By Lemma 7, we know that v2 = Av1 ≤ Av₀ = v1. By induction, we get that

v_n+1 ≤ v_n, n = 0, 1, 2, · · · .

Consequently, there exists u^∗ ∈ P_a such that vn → u^∗. Let n → ∞, by the continuity of A and v_n+1 = Avn, we know that Au^∗ = u^∗. This is to say that u^∗ ≥ 0 is a fixed point of A, i.e., u^∗ is a nonneg- ative solution for BVP (1). Moreover, considering that the zero function is not a solution of the prob- lem, we have that u^∗ is a positive solution for BVP (1). Since u∗, u^∗ ∈ P_a, there exist positive numbers li < Li(i = 1, 2) such that (18) holds.

Remark 9 The iterative sequences in Theorem 8 be- gin with simple functions which is significant for com- putational purpose.

4 An example

Consider the following singular fractional differential equations













 D

1 4

0+(ϕ3(D

5 2

0+u(t))) + ^√_1−t¹ u³² = 0, 0 < t < 1, u(0) = u⁰(0) = 0, D

5 2

0+u(0) = 0, u⁰(1) = 2

3 Z ³₄

0

√1

tu(t)dt.

(20)

Let α = ⁵₂, β = ¹₄, p = 3, q = ³₂, λ = ²₃, η = ³₄, i = 1, n = 3, a(t) = ^√1

1−t, h(t) = ^√1

t, f (t, u) = u³², then ∆ = ³₂, 0 < λ^R₀^ηh(t)t^α−1dt = ₁₆³ < ∆. Clear- ly, (H1), (H2) hold for h(t) = ^√1_t, r = ³₂. On the other hand, we have

0 <

Z 1 0

(1 − s)^α−1−iϕ⁻¹_p ^ Z s

0

(s − τ )^β−1 f (τ, 1)dτ^a(s)ds

≤ Z 1

0

ϕ⁻¹_p ^1

βs^βds < ϕ⁻¹_p ^1 β

< +∞

which implies that (H3) also holds. By simple com- putation, we get

M1 = n Γ(α)

1 + λ

∆ − λ^R₀^ηh(t)t^α−1dt Z η

0

h(t)t^α−2dt^

≈ 3.1166,

Λ =^ 4 Γ(¹₄)

¹

2×3.1166×^ Z 1

0

(1−s)¹²s¹⁸ds^≈ 1.8752.

Take a = 10³, then f (t, u) = u³² ≤ (10³)³² = 3.1623 × 10⁴ < 2.8438 × 10⁵ = ϕ₃^_1.8752^{103 }. Thus, (17) holds. It follows from Theorem 8 that BVP (20) has the positive minimal and maximal solution- s u∗ and u^∗; and there exist some positive numbers l_i < L_i(i = 1, 2) such that (18) holds.

Acknowledgements: The project is supported fi- nancially by the Natural Science Foundation of Shandong Province of China (ZR2015AL002), a Project of Shandong Province Higher Education- al Science and Technology Program (J15LI16), the Foundations for Jining Medical College Natural Sci- ence (JYQ14KJ06, JY2015KJ019, JY2015BS07), and the National Natural Science Foundation of Chi- na (11571197, 11571296, 11371221, 11071141). The corresponding author of this paper is: Dr. Xingqiu Zhang.

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