# Monotone iterative technique and existence results for fractional functional differential equations.

1. INTRODUCTION

The purpose of this paper is to investigate the following boundary value problem of fractional functional differential equation with p-Laplacian operator

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 1 < [alpha] [less than or equal to] 2, 0 < [beta] [less than or equal to] 1, [sup.c][D.sup.[alpha]] and [sup.c][D.sup.[beta]] are the Caputo fractional derivatives, 0 < [tau], [theta] < 1, a, d [greater than or equal to] 0, b, c > 0 are real constants satisfying b > and [2-[alpha]/[alpha]-1] a and [[phi].sub.p] (x) is a p-Laplacian operator defined by [[phi].sub.p] (x) = [[absolute value of x].sup.p-2] x, p > 1, [[phi].sub.q] = [[phi].sub.p.sup.- 1], 1/p + 1/q = 1.

We will suppose that the following assumptions are satisfied:

(A1) f [member of] C([0,1] x [0, + [infinity]) x [0, + [infinity]), [R.sup.+]), f(t, u, v) > 0 for all (t, u, v) [member of] [0, 1] x [0, + [infinity]) x [0, + [infinity]).

(A2) [eta] [member of] C([-[tau], 0], [0, [infinity])), [xi] [member of] C([1, 1 + [theta]], [0, [infinity])) and [eta](0) = [xi](1) = 0.

Recently, the study of nonlinear fractional boundary value problems has gained much attention because of their applications in various research areas of applied sciences and engineering. In particular, many authors have investigated the existence results of positive solutions of nonlinear boundary value problems for fractional differential equations. (See [1, 2, 8, 9, 10, 11, 12] and the references therein.) But there are relatively few works available for the existence of positive solutions for fractional functional differential equations. For instance, in , Li et al. considered the following boundary value problem of fractional functional Sturm-Liouville differential equation

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 1 < [alpha] [less than or equal to] 2 and [D.sup.[alpha]] is the Caputo fractional derivative. By means of the Guo Krasnoselskii fixed point theorem, they obtained the existence of positive solutions for the fractional functional BVP (1.2).

In , by means of fixed point theorems on cones, Zhao et al. investigated the following fractional functional boundary value problem

(1.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In , by using the Guo Krasnoselskii fixed point theorem on cones, Li et al. established the positive solutions for the following fractional functional differential equation

(1.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 1 < [alpha] [less than or equal to] 2, 2 < [beta] [less than or equal to] 3, [D.sup.[alpha]] and [D.sup.[beta]] are the Caputo fractional derivatives.

We notice that all the results in the papers mentioned above are obtained by means of fixed point theorems on cones. Motivated by these papers, but taking completely different technique from [3, 4, 5], we will consider the functional fractional boundary value problem (1.1). Here, we will use the monotone iterative technique to establish the existence results of positive solutions for the fractional BVP (1.1). We not only get the existence results of positive solutions, but also construct two iterative schemes for approximating the solutions. Furthermore, the technique does not require the existence of upper and lower solutions. To the author's knowledge, few works were done in the literature concerning the existence of positive solutions for boundary value problems of fractional functional differential equations with p-Laplacian operator by means of the monotone iterative method. Therefore, the aim of this paper is to fill this gap.

The plan of this paper is as follows. In section 2, we give some definitions and lemmas that are used throughout the paper. In section 3, we establish our main results by using the monotone iterative technique. Finally, in section 4, an example is worked out to demonstrate the applicability of our main result.

2. PRELIMINARIES

In this section, we present some definitions and lemmas which are useful for the proof of our main result.

Definition 2.1 ([6, 7]). The Riemann Liouville fractional integral of order [alpha] [member of] [R.sup.+] for a continuous function h : (0, [infinity]) [left arrow] R is defined by

(2.1) [I.sup.[alpha]] h(t) = 1/[GAMMA]([alpha]) [[integral].sup.t.sub.0] [(t - s).sup.[alpha]-1] h(s)ds.

where [GAMMA](.) is the Euler Gamma function, provided that the integral exists.

Definition 2.2 ([6, 7]). If h [member of] [C.sup.n] [0, 1], then the Caputo fractional derivative of order a is defined by

(2.2) [sup.c][D.sup.[alpha]] h(t) = 1/[GAMMA](n - [alpha]) [[integral].sup.t.sub.0] - [(t - s).sup.n-[alpha]-1] [h.sup.(n)] (s)ds = [I.sup.n- [alpha]][h.sup.(n)](t), n - 1 < [alpha] < n,

where n = [[alpha]] + 1 and [[alpha]] denotes the integer part of the real number [alpha].

Remark 2.3. If [alpha] = n [member of] [N.sub.0], then the Caputo derivative coincides with a conventional n-th order derivative of the function h(t).

Lemma 2.4 ([6, 7]). Let n = [[alpha]] + 1 for [beta] [not member of] N and n = [beta] for [alpha] [member of] N. If y (t) [member of] [C.sup.n] [0, 1], then

([I.sup.[alpha]][sup.c][D.sup.[alpha]]y)(t) = y(t) - [n- 1.summation over (i=0)] [y.sup.i](0)/i! [t.sup.i].

Lemma 2.5 ([6, 7]). Let [alpha] > 0 and n = [[alpha]] + 1 for [alpha] [not member of] N and n = [alpha] for [alpha] [member of] N. If h(t) [member of] C[0, 1], then the homogeneous fractional differential equation

[sup.c][D.sup.[alpha]] h(t) = 0

has a solution

h(t) = [c.sub.1] + [c.sub.2] t + [c.sub.3] [t.sup.2] + ... + [c.sub.n] [t.sup.n-1],

where [c.sub.i] [member of] R, (i = 1, 2,..., n).

Lemma 2.6 (). If 1 < [alpha] [less than or equal to] 2 and f [member of] C([0,1] x [0, + [infinity]) x [0, + [infinity]), [R.sup.+]), then the boundary value problem for fractional functional differential equation

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is equivalent to the integral equation

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(2.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [rho] = bc + ac + ad.

Lemma 2.7. If 1 < [alpha] [less than or equal to] 2, 0 < [beta] [less than or equal to] 1 and f [member of] C([0, 1] x [0, + [infinity]) x [0, + [infinity]), [R.sup.+]), then the boundary value problem for fractional functional differential equation

(2.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is equivalent to the integral equation

(2.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where u(-[tau], t), u([theta], t) and G(t, s) are defined by (2.5)-(2.7) respectively.

Proof. For any f [member of] C([0, 1] x [0, + [infinity]) x [0, + [infinity]), [R.sup.+]), by Lemma 2.4, we have

[[phi].sub.p]([sup.c][D.sup.[alpha]] u(t)) = -[I.sup.[beta]] f(t, u(t - [tau]), u(t + [theta])) + [c.sub.0], [c.sub.0] [member of] R.

Using the boundary condition [sup.c][D.sup.[alpha]] u(0) = 0, we get [c.sub.0] = 0. Hence, we obtain

(2.10) [sup.c][D.sup.[alpha]] u(t) + [[phi].sub.q] ([I.sup.[beta]] f (t, u(t - [tau]), u(t + [theta]))) = 0.

By means of Lemma 2.6, a solution of (2.10) with the boundary conditions of (2.8) can be expressed as

(2.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in which u(-[tau], t), u([theta], t) and G(t,s) are given by (2.5)-(2.7) respectively.

Now, we will present the properties of the Green's function:

Lemma 2.8 (). The function G(t, s) given by (2.7) verifies the following properties:

(i) G(t, s) is continuous on [0, 1] x [0, 1).

(ii) For b > 2-[alpha]/[alpha]-1 a, we get G(t, s) > 0 for t, s [member of] (0, 1).

(iii) G(t, s) [less than or equal to] G(s, s) for t, s [member of] (0, 1).

Throughout this paper, let [x.sub.0](t) be a solution of the BVP (1.1) with f [equivalent to] 0, then it satisfies

(2.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[x.sub.0] (-[tau], t) = [e.sup.(a/b)t]/b [[integral].sup.0.sub.t] [e.sup.-(a/b)s] [eta(s)ds, t [member of] [-[tau], 0]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assume that u(t) is a solution of the BVP (1.1) and x(t) = u(t) - [x.sub.0](t). Since x(t) [equivalent to] u(t) for 0 [less than or equal to] t [less than or equal to] 1, x(t) verifies

(2.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

(x + [x.sub.0])(s - [tau]) = x(s - [tau]) + [x.sub.0] (s - [tau]),

(x + [x.sub.0])(s + [theta]) = x(s + [theta]) + [x.sub.0](s + [theta]),

x (-[tau], t) = [e.sup.(a/b)t] x(0), t [member of] [-[tau], 0]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let B = C[-[tau], 1 + [theta]] be endowed with the norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then it is clear that B is a Banach space. Define a cone K [subset] B as follows:

K = {x [member of] B : x(t) [greater than or equal to] 0 for any t [member of] [-[tau]t, 1 + [theta]]} .

Consider the operator T : K [right arrow] K

(2.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easy to see that u is a positive solution of the BVP (1.1) if and only if x = u - [x.sup.0] is a nontrivial fixed point of T, where [x.sup.0] is given by (2.12).

Lemma 2.9. Assume that (A1) and (A2) hold. Then T : K [right arrow] K is completely continuous.

Proof. From the definition of T, it is obvious that Tx(t) [greater than or equal to] 0 for t [member of] [-[tau], 1 + [theta]], i.e., T x [member of] K, [for all]x [member of] K. Also, using the Arzela Ascoli theorem and the standard arguments, one can easily show that T : K [right arrow] K is completely continuous operator.

Remark 2.10. Note that for any t [member of] [-[tau], 0] and t [member of] [1, 1 + [theta]], Tx(t) [less than or equal to] Tx(0) and Tx(t) [less than or equal to] Tx(1) hold respectively. So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

To guarantee the existence of positive solutions, we will assume the following condition:

* (H1) There exists [delta] > 0 such that 0 [less than or equal to] [u.sub.1] [less than or equal to] [u.sub.2] [less than or equal to] [delta] + [[parallel][x.sub.0] [parallel].sub.[-[tau],0]], 0 [less than or equal to] [v.sub.1] [less than or equal to] [v.sub.2] [less than or equal to] [delta] + [[parallel][x.sub.0][parallel].sup.[1,1+[theta]]] and t [member of] [0, 1] imply f (t, [u.sub.1], [v.sub.1]) [less than or equal to] f (t, [u.sub.2], [v.sub.2]).

Lemma 2.11. Suppose that (A1), (A2) and (H1) hold. Then for any [x.sub.1], [x.sub.2] [member of] [[bar.K].sub.[delta]] with [x.sub.1](t) [less than or equal to] [x.sub.2](t), t [member of] [-[tau], 1 + [theta]] implies (T[x.sub.1])(t) [less than or equal to] (T[x.sub.2])(t).

Proof. Let [x.sub.1], [x.sub.2] [member of] [[bar.K].sub.[delta]]. Then, for any v [member of] [0, 1], we have

(2.15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows from (2.15) and (H1) that

f (v, ([x.sub.1] + [x.sub.0])(v-[tau]), ([x.sub.1] + [x.sub.0])(v + [theta])) [less than or equal to] f (v, ([x.sub.2] + [x.sub.0])(v-[tau]), ([x.sub.2] + [x.sub.0])(v + [theta])), v [member of] [0, 1], thus we have

[I.sup.[beta]] (v, ([x.sub.1] + [x.sub.0])(v - [tau]), ([x.sub.1] + [x.sub.0])(v + [theta])) [less than or equal to] [I.sup.[beta]] f (v, ([x.sub.2] + [x.sub.0])(v - [tau]), ([x.sub.2] + [x.sub.0])(v + [theta]))

Since [[phi].sub.q] is increasing on R, we derive that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so, we obtain

(2.16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence for any t [member of] [-[tau], 1 + [theta]], by (2.14) and (2.16) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, (T[x.sub.1] )(t) [less than or equal to] (T[x.sub.2])(t) is satisfied for t [member of] [-[tau], 1 + [theta]]. The proof is completed.

3. MAIN RESULT

In this section, we obtain the existence of positive solutions and its monotone iterative scheme for the fractional BVP (1-1).

For convenience, let us denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 3.1. Assume that (A1), (A2) and (H1) hold. Suppose also that there exists [delta] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the BVP (1.1) has two positive solutions [w.sup.*](t) + [x.sub.0](t) and [v.sup.*](t) + [x.sub.0](t) satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [v.sub.0](t) = 0, -[tau] [less than or equal to] t [less than or equal to] 1 + [theta].

Proof. Let x [member of] [[bar.K].sub.[delta]]. Then for any t [member of] [0, 1] we have

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (3.2), it follows that

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

thus we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, we get T[[bar.K].sub.[delta]] [subset] [[bar.K].sub.[delta]]. Let

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then [parallel][w.sub.0][parallel] [less than or equal to] [delta] and [w.sub.0](t) [member of] [[bar.K].sub.[delta]]. Let [w.sub.1] = T[w.sub.0], then [w.sub.1] [member of] [[bar.K].sub.[delta]]. We denote

(3.5) [w.sub.n+1] = T[w.sub.n] = [T.sup.n+1] [w.sub.0] (n = 0, 1, 2,...).

Since T[[bar.K].sub.[delta]] [subset] [[bar.K].sub.[delta]], we get [w.sub.n] [member of] [[bar.K].sub.[delta]] (n = 0, 1, 2,...). By Lemma 2.9, T is compact, we assert that [{[w.sub.n]}.sup.[infinity].sub.n=1] has a convergent subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and there exists [w.sup.*] [member of] [[bar.K].sub.[delta]] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From the definition of T, (3.4) and (3.5), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, [w.sub.1](t) [less than or equal to] [w.sub.0](t). By means of Lemma 2.11, we obtain T[w.sub.1](t) [less than or equal to] T[w.sub.0](t), i.e., [w.sub.2](t) [less than or equal to] [w.sub.1](t), t [member of] [-[tau], 1 + [theta]]. Thus, we have

[w.sub.n+1](t) [less than or equal to] [w.sub.n](t), t [member of] [-[tau], 1 + [theta]], (n = 0, 1, 2,...).

Therefore, [w.sub.n] [right arrow] [w.sup.*]. Let n [right arrow] [infinity] in (3.5). Then we get T[w.sup.*] = [w.sup.*] since T is continuous. Evidently, [w.sup.*] is a fixed point of the operator T, that is [y.sub.1](t) = [w.sup.*](t) + [x.sub.0](t) is a positive solution of the BVP (1.1).

Let [v.sub.0](t) = 0, t [member of] [-[tau], 1 + [theta]], then [v.sub.0](t) [member of] [[bar.K].sub.[theta]]. Let [v.sub.1] = T[v.sub.0], then [v.sub.1] [member of] [[bar.K].sub.[theta]], we denote

[v.sub.n+1] = T[v.sub.n] = [T.sup.n+1][v.sub.0] (n = 0,1, 2,...).

Similar to [{[w.sub.n]}.sup.[infinity].sub.n=1], we claim that [{[v.sub.n]}.sup.[infinity].sub.n=1] has a convergent subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and there exists [v.sup.*] [member of] [[bar.K].sub.[theta]] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which means [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [v.sub.1] = T[v.sub.0] = T0 [member of] [[bar.K].sub.[theta]], we have

[v.sub.1](t) = T[v.sub.0](t) = (T0)(t) [greater than or equal to] 0,

that is

[v.sub.2](t) = (T[v.sub.1])(t) [greater than or equal to] (T0)(t) = [v.sub.1](t), t [member of] [-[tau], 1 + [theta]].

By induction, it is obvious that

[v.sub.n+1](t) [greater than or equal to] [v.sub.n](t), t [member of] [-[tau], 1 + [tau]] (n = 0, 1, 2, ...),

so, we have [v.sub.n] [right arrow] [v.sup.*] in norm [parallel]*[parallel] and T[v.sup.*] = [v.sup.*]. Therefore, T has fixed points [w.sup.*] and [v.sup.*], which means that [y.sub.1](t) = [w.sup.*](t) + [x.sub.0](t) and [y.sub.2](t) = [v.sup.*] (t) + [x.sub.0](t) are positive solutions of the fractional BVP (1.1). The proof is completed.

Remark 3.2. It is obvious that [w.sup.*] + [x.sub.0] and [v.sup.*] + [x.sub.0] are the maximal and minimal solutions of the BVP (1.1). If they coincide then (1.1) has a unique positive solution in [K.sub.[theta]].

Corollary 3.3. Assume that (A1), (A2) and (H1) hold. Suppose also that there exist 0 < [[theta].sub.1] < [[theta].sub.2] < ... < [[theta].sub.n] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then the BVP (1.1) has 2n positive solutions [[w.sup.*].sub.k](t) + [x.sub.0](t) and [[v.sup.*].sub.k](t) + [x.sub.0](t) satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4. AN EXAMPLE

Consider the following fractional functional boundary-value problem:

(4.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

f{t, u, v) = t + 1 + + 1/40 (u + v), (t, u, v) [member of] [- 1/6, 8/7]) x [0, [infinity]),

and a = d = 0, 6 = c = 1, p = 2, q = 2, [alpha] = 3/2, [beta] = 1/2, [tau] = 1/6, [theta] = 1/7. Notice that [eta](t) = - sin([pi]t), and [xi](t) = [e.sup.1-t] - 1 are nonnegative functions satisfying [eta](0) = [xi](1) = 0. By easy calculation, we evaluate [x.sub.0](t) = 2/[pi] [sin.sup.2]([pi]/2t), for t [member of] [- 1/6, 0] and [x.sub.0](t) = [e.sup.1-t] - 1 for t [member of] [1, 8/7], so [[parallel][x.sub.0][parallel].sub.[- 1/6, 0]] = 2/[pi] [sin.sup.2]([pi]/12]), [[parallel][x.sub.0][parallel].sub.[1,8/7]] = 0. Choosing [delta] = 20, we get A = 1/2. Moreover, it is obvious that f(t, u, v) satisfies

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, by means of Theorem 3.1, the BVP (4.1) has two positive solutions [w.sup.*] + [x.sub.0] and [v.sup.*] + [x.sub.0]. For n = 0, 1, 2,..., the two iterative schemes are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

in which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

FULYA YORUK DEREN, NUKET AYKUT HAMAL, AND TUGBA SENLIK CERDIK

Department of Mathematics, Ege University, 35100, Bornova, Izmir

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