Approximating solutions of nonlinear hybrid Caputo fractional integro-differential equations via Dhage iteration principle Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 5, No. 1, 2019, pp. 3-12

DOI: 10.15826/umj.2019.1.001

APPROXIMATING SOLUTIONS OF NONLINEAR HYBRID CAPUTO FRACTIONAL INTEGRO DIFFERENTIAL EQUATIONS VIA DHAGE ITERATION PRINCIPLE

Abdelouaheb Ardjouni

Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, Souk Ahras, 41000, Algeria abd_ardjouni@yahoo.fr

Ahcene Djoudi

Department of Mathematics, University of Annaba, P.O. Box 12, Annaba, 23000, Algeria adjoudi@yahoo.com

Abstract: In this article, we prove the existence and approximation of solutions of the initial value problems of nonlinear hybrid Caputo fractional integro-differential equations. The main tool employed here is the Dhage iteration principle in a partially ordered normed linear space. An example is also given to illustrate the main results.

Keywords: Approximating solutions, Initial value problems, Dhage iteration principle, Hybrid fixed point theorem.

1. Introduction

Fractional differential equations with and without delay arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1-4, 9, 11-13, 15-20] and the references therein.

Hybrid differential equations involve the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. This class of equations arises from a variety of different areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on [1, 2, 5, 7, 9, 10, 20].

Dhage and Lakshmikantham [10] discussed the existence of solutions for the following first-order hybrid differential equation

(Knity ='<♦•*<«»

[ x (¿0) = X0 € R,

where ¿0,T € R with T > 0, g : [¿0,¿0 + T]xR ^ R\ {0} and f : [¿0,¿0 + T] xR ^ R are continuous

functions. By using the fixed point theorem in Banach algebra, the authors obtained the existence results.

Let J = [0, a] be a closed and bounded interval of the real line R for some a € R with a > 0. The hybrid fractional differential equation

ni^iik)=/<t-T(t>> a-e't€J-x (0) = 0,

has been investigated in [20], where Da is the Riemann-Liouville fractional derivative of order 0 < a < 1, g : J x R ^ R\ {0} and f : J x R ^ R are continuous functions. By employing the fixed point theorem in Banach algebra, the authors obtained the existence of a solution.

Dhage et al. [9] studied the existence and approximation of the solutions of the following nonlinear fractional integro-differential equation

cr (zm^pWgM) =}(tMt), ik(.M.))i.), tej,

g (t,x (t)) J V ' Jo

x (0) = xo € R+,

where CDa is the Caputo fractional derivative of order 0 < a < 1, is the Riemann-Liouville fractional integral of order 0 < P < 1, g : J x R ^ R\ {0}, h, k : J x R ^ R and f : J x R x R ^ R are continuous functions. By using the Dhage iteration principle, the authors obtained the existence and approximation of solutions.

Inspired and motivated by the works mentioned above and some recent studies on hybrid fractional differential equations, we consider the existence and approximation of solutions for the following initial value problem (in short IVP) of the nonlinear hybrid Caputo fractional integro-differential equation

CD«(----)=f(t,x(t)), t £ J,

W) + F77n / {t-sf-lcAs,x{s))dJ (1-1)

r (p) Jo x (0) = p (0) e,

where 0 < a < 1, 0 <P < 1, e € R, g,f : J x R ^ R are given functions and p : J ^ R is a given function.

By a solution of the IVP (1.1) we mean a function x € C(J, R) that satisfies the corresponding integral equation of (1.1), where C(J,R) is the space of continuous real-valued functions defined on J.

The purpose of this paper is to use the Dhage iteration principle to show the existence and approximation of solutions of (1.1) under weaker partially continuity and partially compactness type conditions.

The article is organized as follows. In Section 2 we give some preliminaries and key fixed point theorem that will be used in later sections. In Section 3 we prove some sufficient conditions of the existence and approximation of solutions of (1.1) by using the Dhage iteration principle. For details on the Dhage iteration principle we refer the reader to [6]. Finally, an example is given to illustrate our main results.

2. Preliminaries

We introduce some necessary definitions, lemmas and theorems which will be used in this paper. For more details, see [15, 19].

Definition 1 [15, 19]. The fractional integral of order a > 0 of a function x : J — R is given

1 ft

Iax(t) = —— / (t - s)a~1x(s)ds, r(a) y o

provided the right-hand side is pointwise defined on J.

Definition 2 [15, 19]. The Caputo fractional derivative of order a > 0 of a function x : J — R is given by

C Dax(t) = Da

where

k=0 '

n = [a] + 1 for a / N0, n = a for a € N0, (2.1)

and Da is the Riemann-Liouville fractional derivative of order a defined by

1 dn f1

Dax(t) = Dnr~ax{t) = —-r-f- / {t - s)n-a~lx{s)ds.

w w T(n - a) dtn Jo

The Caputo fractional derivative CDax exists for x belonging to ACn (J, R) the space of functions which have continuous derivatives up to order (n — 1) on J such that x(n-1) € AC1 (J, R). AC1 (J, R) also denoted AC (J, R) is the space of absolutely continuous functions. In this case, the Caputo fractional derivative is defined by

1 ft

cDax(t) = Tn~ax{n)(t) = —-- / (t - s)n-a~lx{n)(s)ds.

r(n — a) Jo

Remark that when a = n, we have CDax(t) = x(n)(t).

Lemma 1. [15, Lemma 2.22] Let a > 0 and let n be given by (2.1). If x € ACn (J, R), then

n- 1

(Ia cDax) (t) = x(t) - Ys^-JT1^

k=0 '

(k)(0)

In particular, when 0 < a < 1, (/a CDax) (t) = x(t) — x(0).

Let E denote a partially ordered real normed linear space with an order relation ■< and the norm || ■ ||. It is known that E is called regular if {xn} is a nondecreasing (resp. nonincreasing) sequence in E such that xn ^ x* as n to, then xn ■ x* (resp. xn ^ x*) for all n € N. The conditions guaranteeing the regularity of E may be found in Heikkila and Lakshmikantham [14] and the references therein.

Definition 3. A mapping A : E — E is called isotone or monotone nondecreasing if it preserves the order relation that is, if x ■ y implies Ax ■ Ay for all x, y € E. Similarly, A is called monotone nonincreasing if x ■ y implies Ax ^ Ay for all x, y € E. Finally, A is called monotonic or simply monotone if it is either monotone nondecreasing or monotone nonincreasing on E.

Definition 4. An operator A on a normed linear space E into itself is called compact if A(E) is a relatively compact subset of E. A is called totally bounded if for any bounded subset S of E, A(S) is a relatively compact subset of E. If A is continuous and totally bounded, then it is called completely continuous on E.

Definition 5 [6]. A mapping A : E ^ E is called partially continuous at a point a € E if for e > 0 there exists a 5 > 0 such that ||Ax — Aa|| < e whenever x is comparable to a and ||x — a|| < 5. A called partially continuous on E if it is partially continuous at every point of it. It is clear that if A is partially continuous on E, then it is continuous on every chain C contained in E.

Definition 6 [5, 6]. An operator A on a partially normed linear space E into itself is called partially bounded if A(C) is bounded for every chain C in E. A is called uniformly partially bounded, if all chains A(C) in E are bounded by a unique constant. A is called partially compact if A(C) is a relatively compact subset of E for all totally ordered sets or chains C in E. A is called partially totally bounded if for any totally ordered and bounded subset C of E, A(C) is a relatively compact subset of E. If A is partially continuous and partially totally bounded, then it is called partially completely continuous on E.

Definition 7 [5]. The order relation — and the metric d on a non-empty set E are said to be compatible if {xn} is a monotone, that is, monotone nondecreasing or monotone nondecreasing sequence in E and if a subsequence {xnk} of {xn} converges to x* implies that the whole sequence {xn} converges to x*. Similarly, given a partially ordered normed linear space (E, —, || ■ ||), the order relation — and the norm || ■ || are said to be compatible if — and the metric d defined through the norm || ■ || are compatible.

Clearly, the set R of real numbers with usual order relation < and the norm defined by the absolute value function has this property.

Theorem 1 [6]. Let (E, —, || ■ ||) be a regular partially ordered complete normed linear space such that the order relation — and the norm || ■ || are compatible in every compact chain of E. Let A : E ^ E be a partially continuous, nondecreasing and partially compact operator. If there exists an element x0 € E such that x0 — Ax0 or x0 ^ Ax0, then the operator equation Ax = x has a solution x* in E and the sequence {Anx0} of successive iterations converges monotonically to x*.

Remark 1. [9] The compatibility of the order relation — and the norm || ■ || in every compact chain of E is held if every partially compact subset of E possesses the compatibility property with respect to — and || ■ ||.

Remark 2. [5] Note that every compact mapping in a partially normed linear space is partially compact and every partially compact mapping is partially totally bounded, however the reverse implications do not hold. Again, every completely continuous mapping is partially completely continuous and every partially completely continuous mapping is continuous and partially totally bounded, but the converse may not be true. Then, the hypothesis concerning the partially continuous and partially compact operator in Theorem 1 may be replaced by the continuous and compact operator.

3. Main results

The equivalent integral formulation of the IVP (1.1) is considered in the function space C(J, R) of continuous real-valued functions defined on J. We define a norm || ■ || and the order relation < in C(J, R) by

||x|| = sup |x(t)|, (3.1)

teJ

x < y ^ x(t) < y(t), (3.2)

for all t € J. Clearly, C(J, R) is a Banach space with respect to above supremum norm, regular and also partially ordered with respect to the above partially order relation <. It is known that the partially ordered Banach space C(J, R) has some nice properties with respect to the above order relation in it.

Lemma 2 [8]. Let (C(J, R), <, || ■ ||) be a partially ordered Banach space with the norm || ■ || and the order relation < defined by (3.1) and (3.2) respectively. Then || ■ || and < are compatible in every partially compact subset of C(J, R).

We need the following definition in what follows.

Definition 8. A function u € C(J, R) is said to be a lower solution of the IVP (1.1) if it satisfies the corresponding integral inequality of

(

C no ( Ht)

Da[ ---^- }<f(t,u(t)), t £ J,

1

ÜÖ) Jo u (0) < p (0) 0.

Similarly, an upper solution v € C(J, R) for the IVP (1.1), by reversing the above inequalities.

We consider the following set of assumptions: (B1) g, f : J x R ^ R are continuous functions and p : J ^ R is a continuous function such that

1

1 r

and

1 ^

V (t) + Y^j (t - sf~lg{s, x{s))ds > 0, for all t € J and x € C(J, R). (B2) There exist constants Ka ,Kf > 0 such that

|g(t,x)| < Kg and |f (t,x)| < Kf for all t € J and x € R.

(B3) There exists constant Kp > 0 such that

|p (t2) - P (ti)| < Kp |t2 - ti| for all ti,t2 € J.

(B4) g(t,x) and f (t,x) are monotone nondecreasing functions in x for all t € J. (B5) The IVP (1.1) has a lower solution u € C(J, R).

Lemma 3. Let h € C (J, R) and q € C (J, (0, to)). If x/q € AC (J, R), then the IVP

Cnaf

teJ,

x (0) = q (0) 0, is equivalent to the integral equation

1

x(t) = q (t) (d + yj^ fQ(t- s)a~1h(s)ds^ , ie J.

Theorem 2. Assume that hypotheses (B1)-(B5) hold. Then the IVP (1.1) has a solution x* defined on J and the sequence {xn} of successive approximations defined by

Xn+i(t)= (p (t) jf (t-sf-'gis, xn(s))<te) Jo (t-sf-1/(s, xn(s)) , (3.3)

for all t € J, where x0 = u converges monotonically to x*.

Proof. Set E = C(J, R). Then by Lemma 2, every compact chain in E is compatible with respect to the norm || ■ || and order relation <. Define the operator A on E by

(Ax) (t) = (p(t) + ^JQ(t- s)l3~1g(s, x(s))<tej x(e + sr-V (s, x(s)) ds^j , t € J.

From the continuity of the integral, it follows that A defines the map A : E ^ E. Now, by Lemma 3, the IVP (1.1) is equivalent to the operator equation

(Ax) (t) = x(t), t € J.

We shall show that the operator A satisfies all the conditions of Theorem 1. This is achieved in the series of following steps.

Step I: A is a nondecreasing operator on E. Let x,y € E be such that x < y. Then by hypothesis (B4), we obtain

1 f ' ■ — a(s, x(s))ds\ (в -—7—- f (t.-s)0-1! (s,x(s)

(Ax) (t) = (p (t) + ^ f(t - sf-lg(s, x(s))ds^j (в + ^ f(t - sT~lf (s, x(s)) ds^j < (p (t) + ^ f(t - sf~lg(s, y(s))ds^j + ^ fit - sY~lf (s, y(s)) ds^j = (Ay) (t),

Г (ß) ./0 / V Г ytij J0

for all t € J. This shows that A is nondecreasing operator on E into E.

Step II: A is a partially continuous operator on E. Let {xn} be a sequence in a chain C in E such that xn У x when n —у то. Then, by dominated convergence theorem, we have

lim (Axn) (t) = lim

и^те и^те

(p (t) + Y^ßj ft- s)ß~1g(s, xn(s))ds^j + f^) f{t ~ s)a~lf (s'Xn{s)) ds)

= (p(t) f(t - sf~l [¿im^s,^))] (ö+f^y f[t~s)a~l fe 7 (s"Tra(s))] dS)

= (p^+y^ ¡\t-s)ß-1g(s,x(s))ds^j jft-s)"-1 f(s,x(s))ds^j= (Ax) (t),

T(ß)Jo v 7 v 77 ) V ГI

for all t € J. This shows that {Axn} converges to Ax pointwise on J.

x

Next, we show that {Axn} is an equicontinuous sequence of functions in E. Let ti,t2 € J be arbitrary with t1 < t2. Then

I (Axn) (i2) - (Axn) (¿i) | < (jp (ii)| + ^ J 1 (ii - sf~l \g(s, xra(s))| ds x J*2 (¿2 - r1 / (s. M*)) - J*1 (il - / (s. ^„(s)) )

+ (\p(h)-p(t1)\+ 1

rt2 rtl

/ (t2 - s)?-1g(s,x„(s))ds - (ti - s)?-1g(s,x„(s))ds Jo 7o

rtf)

+ .C ~ S)a~l 17 (S' ^(S))I ds)

~ (|p(tl)l + Ttf + 1)) {fja) C ((tl " Sr_1 " ~ ST~1) I'(*'*»(*))! ds + ^ f\t2-sT-l\f{s,Xn{s))\ds^

+ (kp\t2 - ill + ^ fo 1 ((ii - s)'3-1 - (i2 - ¿O^"1) \g (s,xn(s))\ds

a? \ 2Kf

uniformly for all n € N. This shows that the convergence Axn ^ Ax is uniformly and hence A is partially continuous on E.

Step III: A is a partially compact operator on E. Let C be an arbitrary chain in E. We show that A(C) is a uniformly bounded and equicontinuous set in E. First we show that A(C) is a uniformly bounded. Let x € C be arbitrary. Then

|(Ar) (i)| < (|p(!)| + (t - s)""1 l9(s.*«)l ds)

X (l^ + f^)/*4-8)'""'!/*8.^))!^ < (AV + b(0)| + ^ J\t-sf-Us) (W + JiL.J\t-sr-^s)

for all t € J. Taking supremum over t, we obtain ||Ax|| < r for all x € C. Hence A(C) is a uniformly bounded subset of E. Next, we will show that A(C) is an equicontinuous set in E. Let t1,t2 € J with t1 < t2. Then

/ k \ 2K

|Ux) (i2) - (At) (¡i)| < (jp (i,)| + j t(alr) (¡2 - ¡0"

uniformly for all x € C. Hence A(C) is a relatively compact subset of E and consequently A is a partially compact operator on E into itself.

Step IV: u satisfies the operator inequality u < Au. By hypothesis (B5), the IVP (1.1) has a lower solution u on J. Then we have

CDa (-j-^- ] < f(t,u(t)), t € J, (3.4)

satisfying

P{t) + T{f3) jo (t-sf'^MsMs'

u (0) < p (0) e.

Applying Ia to both sides of (3.4) and by using Lemma 1, we obtain FTP) f(t-s)^g{sMs))ds) (W^

u (t) <(p(t) + fLjJ (t-s)/3~1g(s, U(s))(tej (d + -J^j I V- u(s)) dsy te J.

The definition of the operator A implies that u(t) < (Au) (t) for all t € J. Consequently, u is a lower solution to the operator equation x = Ax.

Thus A satisfies all the conditions of Theorem 1 with x0 = u and we apply it to conclude that the operator equation Ax = x has a solution. Consequently the integral equation and the IVP (1.1) has a solution x* defined on J. Furthermore, the sequence {xn} of successive approximations defined by (3.3) converges monotonically to x*. This completes the proof. □

Remark 3. The conclusion of Theorem 2 also remains true if we replace the hypothesis (B5) with the following one

(B5') The IVP (1.1) has an upper solution v € C(J, R).

Example 1. Given a closed and bounded interval J = [0,1] in R, consider the IVP,

CD1'2 (----j = tanh x(t), t € J,

\7r + sin t + ———— / (t — s)~'2^3 arct.an.T (s) ds' r (1/3) J0

x (0) = n,

where a = 1/2, ft = 1/3, e = 1, g (t,x) = arctan x, f (t,x) = tanh x and p (t) = n + sin t. Clearly, the functions g and f are continuous on J x R, p is continuous on J and

7r + sin t + ,0\ [ (t ~ s)~axctan x (s) ds > 0, r (1/3) io

and

1 fT

1 + ^7777^/ (t-s)-1/2 tanhx(s)d,s > 0, 1 (1/2) Jo

r(1/2)

for all t € J and x € C(J, R). The functions g and f satisfy the hypothesis (B2) with Kg = n/2 and Kf = 1. The function p satisfies the hypothesis (B3) with Kp = 1. Moreover, the functions g and f are nondecreasing in x for each t € J and so the hypothesis (B4) is satisfied. Finally the IVP (3.5) has a lower solution

, , ( ntl/3 \ ( tl/2

u(t) = TT + sm t-——— 1-

2r (4/3) M r (3/2) j '

defined on J. Thus all hypotheses of Theorem 2 are satisfied. Hence we apply Theorem 2 and conclude that the IVP (3.5) has a solution x* defined on J and the sequence {xn} defined by

xn+i(t) = + sini + r ^ J (t- s)"2/3 arctanxn (s) ds^j x(l + p / ^ ~ S')_1/2 tanhxn(s)ds^j > for all t € J, where x0 = u, converges monotonically to x*.

Remark 4■ In view of Remark 3, the existence of the solutions x* of the IVP (3.5) may be obtained under the upper solution

,, i nt1/3 \( t1/2

v(t) = 7T + smi + ——— 1 +

2r (4/3) M r (3/2) I ' defined on J and the sequence {xn} defined by

xn+i(t) = + sini + ^ ^^ J (t - s)"2/3 arctanxn (s) ds^j

T(l/2) / ^ ~ s)~1/2tanhxn{s)ds

x 1 +

for all t € J, where x0 = v, converges monotonically to x*

Acknowledgements

The authors would like to thank the anonymous referee for his/her valuable comments and good advice.

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