IS S N 1 3 0 3 –5 9 9 1
POSITIVE SOLUTIONS FOR A FRACTIONAL BOUNDARY VALUE PROBLEM
A. GUEZANE-LAKOUD, S. KOUACHI AND F. ELLAGGOUNE
Abstract. We discuss the existence of positive solutions for a fractional bound-ary value problem by the help of some …xed point theorems and under suitable conditions on the nonlinear term. Two examples are also included to illustrate that the corresponding assumptions are satis…ed.
1. Introduction
The purpose of the present work is to investigate su¢ cient conditions for the existence of three positive solutions for the following fractional boundary value problem (P):
cDq
0+u(t) = a(t)f (u(t)) ; 0 < t < 1;
u (0) = u0(0) = 0; u00(0) = u (1) ; where f : R ! R is a given function, 2 < q < 3; cDq
0+ denotes the Caputo’s
fractional derivative, a 2 C([0; 1]; R): We show that under certain growth conditions on the nonlinear term f; the fractional boundary value problem (P ) has at least one or at least three positive solutions.
Fractional di¤erential equations have recently proved to be valuable tools in the modelling of many phenomena in various …elds of science and engineering, physics and economics. We can …nd numerous applications in viscoelasticity, elec-trochemistry, electrical networks, control theory, biosciences, electromagnetic, sig-nal processes, mechanics and di¤usion processes see [20, 21, 22, 23]. Signi…cant developments in fractional di¤erential equations can be …nd in the monographs of Kilbas et al. [20], Miller and Ross [22], Lakshmikantham et al. [21], Podlubny [23]. Ordinary di¤erential equations and fractional di¤erential equations have been
Received by the editors Nov.21, 2013; accepted: Sept.10, 2014. 2000 Mathematics Subject Classi…cation. 05C38, 15A15, 15A18.
Key words and phrases. Fractional boundary value problem, positivity of solution, …xed point theorem.
*The main results of this paper were presented in part at the conference Algerian-Turkish International Days on Mathematics 2013 (ATIM’ 2013) to be held September 12–14, 2013 in ·
Istanbul at the Fatih University.
c 2 0 1 4 A n ka ra U n ive rsity
studied by many authors by means of …xed point theory see [1, 2, 3, 4, 13, 14, 15, 16, 17, 18, 19, 24, 25].
Note that numerous works [6, 7, 8, 9, 10, 11, 12, 26] were dedicated to the research questions of local and non local boundary value problems for partial di¤erential equations with boundary operators of high (integer and fractional) order. In [6], the initial boundary value problem for partial di¤erential equations of higher order with the caputo fractional derivative was studied in the case when the order of the fractional derivative belongs to the interval (0,1).
In [16], El-Shahed consider the following nonlinear fractional boundary value problem
Dq0+u(t) + a(t)f (t; u(t)) = 0; 0 < t < 1;
u (0) = u0(0) = u0(1) = 0;
here 2 < q 3; and D0q+ denotes the Riemann-Liouville fractional derivative.
Using Krasnoselskii’s …xed point theorem on cone, he proved the existence and nonexistence of positive solutions for the above fractional boundary value problem. In [14], Bai and Lu investigated the existence and multiplicity of positive solu-tions for nonlinear fractional di¤erential equation boundary value problem of type:
Dq0+u(t) + f (t; u(t)) = 0; 0 < t < 1;
u (0) = u(1) = 0;
where 1 < q 2; and D0q+ denotes the Riemann-Liouville fractional derivative.
Applying …xed point theorem on cone, they proved some existence and multiplicity results of positive solutions.
The organization of this paper is as follows. In Section 2, we introduce some de…nitions notations that will be used later. In the third Section, we discuss the existence of at least one positive solution of problem (P) by using Guo-Krasnosel’skii …xed point theorem in cone, then, under some su¢ cient conditions on the nonlinear source term, we apply Avery-Peterson theorem to prove the existence of at least three positive solutions. At the end of this section, we give two examples illustrating the previous results.
2. Preliminaries
In this section, we present some de…nitions and lemmas from fractional calculus theory, which will be needed later.
De…nition 2.1. If g 2 C([a; b]) and > 0, then the Riemann-Liouville fractional integral is de…ned by Ia+g(t) = 1 ( ) Z t a g(s) (t s)1 ds:
De…nition 2.2. Let 0; n = [ ] + 1: If f 2 Cn[a; b] then the Caputo fractional
derivative of order of f de…ned by cD
a+g(t) = (n1 )
Rt a
gn(s)
(t s) n+1ds exists
almost everywhere on [a; b] ([ ] is the entire part of ):
Lemma 2.3. For > 0; g 2 C([0; 1] ; R); the homogenous fractional di¤erential equation cD
a+g(t) = 0 has a solution g(t) = c1+ c2t + c3t2+ + cntn 1; where
ci 2 R, i = 0; : : : ; n; and n = [ ] + 1:
De…ne E = C [0; 1] equipped with the norm kuk = maxt2[0;1]ju (t)j :
Lemma 2.4. Let p; q 0; f 2 L1[a; b]: Then, I0p+I q 0+f (t) = I p+q 0+ f (t) = I q 0+I p 0+f (t) andcDq a+I q
0+f (t) = f (t); for all t 2 [a; b]:
Now we present the necessary de…nition from the theory of cone in Banach spaces.
De…nition 2.5. A nonempty subset P of a Banach space E is called a cone if P is convex, closed and satis…es the following conditions:
(i) x 2 P for all x 2 P and 2 R+;
(ii) x; x 2 P implies x = 0:
De…nition 2.6. A mapping is called completely continuous if it is continuous and maps bounded sets into relatively compact sets.
We start by solving an auxiliary problem which allows us to get the expression of the solution.
Lemma 2.7. Assuming that 6= 2 and y 2 C([0; 1]; R): Then, the problem (P0) cDq
0+u(t) = y(t); 0 < t < 1;
u (0) = u0(0) = 0; u00(0) = u (1) ; has a unique solution given by:
u(t) = 1 (q) Z 1 0 G (t; s) y(s)ds; where G(t; s) = ( (t s)q 1+ 2 t2(1 s) q 1 ; 0 s t; 2 t 2(1 s)q 1 ; 0 t s 1:
Proof. Using Lemmas 2.3 and 2.4, we get
u(t) = I0q+y(t) + a + bt + ct2: (2.1)
The boundary condition u (0) = 0 implies that a = 0: Di¤erentiating both sides of (2.1) and using the initial condition u0(0) = 0; it yields b = 0: The condition
u00(0) = u(1); u00(0) = 2c = u(1); 2c = [Iq 0+y(1) + c]; 2c c = I q 0+y(1); and c = 2 I q
0+y(1): Substituting a; b and c by their values in (2.1), we obtain
u (t) = I0q+y(t) + 2 t 2Iq 0+y(1) = 1 (q) Z t 0 (t s)q 1y(s)ds + 2 1 (q)t 2 Z 1 0 (t s)q 1y(s)ds = 1 (q) Z 1 0 G(t; s)y(s)ds:
3. Existence of positive solutions In this section we assume that 0 < < 2 and :
(H1) a 2 C ([0; 1]; RR +) and for all such that 0 < < 1 then 1
(1 s)q 1a(s)ds 6= 0: (H2) f 2 C (R+; R+) :
De…ne the integral operator T : E ! E by T (u)(t) = 1
(q) Z 1
0
G (t; s) a(s)f (u(s)) ds; (3.1)
that can be written as :
T (u)(t) = I0q+a (t) f (u(t) +
2 t
2Iq
0+a(1)f (u(1) : (3.2)
De…nition 3.1. A function u is called positive solution of problem (P ) if u(t) 0; 8t 2 [0; 1] and it satis…es the boundary condition in (P ):
Let us introduce the following notation A0 = lim
u!0
f (u)
u ; A1 = limu!1
f (u) u : The case A0 = 0 and A1 = 1 is called superlinear case and the case A0 = 1 and
A1= 0 is called sublinear case.
Lemma 3.2. If 0 < < 2, then the function G has the following properties: (1) G(t; s) 0; for all t; s 2 [0; 1]:
(2) For all t 2 [ ; 1] and s 2 [0; 1]; > 0; 0 < < 1; we have
0 2 (s) G(t; s) 2 (s); (3.3)
and where (s) = (1 s)2 q 1. Proof. Let t 2 [0; 1] ; then we have
G(t; s) (1 s)q 1 2
and if t 2 [ ; 1] then G(t; s) (1 s)q 1 t 2 2 (1 s) q 1 2 2 = 2 (s): (3.4)
Lemma 3.3. The solution of fractional boundary value problem (P ) satis…es min u(t)t2[ ;1]
2
2 kuk : (3.5)
Proof. The proof is easy, then we omit it.
Theorem 3.4. Assuming that (H1) (H2) holds, then the fractional boundary
value problem (P ) has at least one positive solution in the both cases superlinear as well as sublinear.
To prove Theorem 3.4, we apply the well-known Guo-Krasnosel’skii …xed point theorem on cone.
Theorem 3.5. [13] Let E be a Banach space, and let K E; be a cone. Assume
1 and 2 are open subsets of E with 0 2 1; 1 2 and let
A : K \ 2n 1 ! K;
be a completely continuous operator such that
(i) jjAujj jjujj ; u 2 K \ @ 1; and jjAujj jjujj ; u 2 K \ @ 2; or
(ii) jjAujj jjujj ; u 2 K \ @ 1; and jjAujj jjujj ; u 2 K \ @ 2:Then A has a
…xed point in K \ 2n 1 :
Proof. Denote E+= fu 2 E; u(t) 0; 8t 2 [0; 1]g and de…ne the cone K by
K = u 2 E+; min
t2[ ;1]u(t) 2
2 kuk ; (3.6)
It is easy to check that K is a nonempty closed and convex subset of E, hence it is a cone. One can check that T K K: It is obvious that T is continuous since G; a and f are continuous. Let us prove that T : K ! E is completely continuous mapping on K:
Claim 1. T (Br) is uniformly bounded, where Br= fu 2 K; kuk rg:
Since the functions a and f are continuous, then there exists a constant c such that maxt2[0;1]ja(t)f(u(t)j = c for any u 2 Br: By virtue of Lemma 3.2 we obtain
jT u(t)j 2c
(2 ) (q): (3.7)
Hence T is uniformly bounded.
jT0u (t)j = 1 (q) R1 0 (q 1) (t s)q 2a (s) f (u (s)) ds + (q)1 R012t2 (1 s)q 1a (s) f (u (s)) ds c (q 1) Z 1 0 (1 s)q 2ds + 4c q (2 ) Z 1 0 (1 s)q 1ds (3.8) = c q 1 + 4 (2 ) = c1 q: Therefore jT u (t2) T u (t1)j = Z t2 t1 T0u (t) dt c1(t2 t1) (q) : (3.9)
Consequently T is equicontinuous. From Arzela-Ascoli theorem we deduce that T is completely continuous operator.
Let us consider the superlinear case. First, A0 = 0, for any " > 0; there exists
R1 > 0; such that if 0 < u R1 then f (u) "u. Let 1 = fu 2 E; kuk < R1g ;
Letting u 2 K \ @ 1, then we have
T u(t) = 1 (q) Z 1 0 G (t; s) a(s)f (u(s)ds; 2" kuk (q) Z 1 0 (s)a(s)ds; (3.10)
Then if we choose " = (q) =2R01 (s)a(s)ds; we get jjT ujj jjujj ; for any u 2 K \ @ 1:
Second, since A1 = 1; then for any M > 0; there exists R2 > 0; such
that f (u) M u for u R2. Let R = max 2R1;2R22 ; and denote by 2 =
fu 2 E : jjujj < Rg : If u 2 K \ @ 2 then min t2[ ;1]u(t) 2 2 kuk = 2 2 R R2: (3.11)
Using the left-hand side of Lemmas 3.2 and 3.3, we obtain T u(t) 2M (q) Z 1 (s)a(s)u (s) ds; (3.12) thus T u(t) 2 4M kuk 2 (q) Z 1 (s)a(s)ds; (3.13)
we can choose M = 2 (q) = 2 4R1 (s)a(s)ds, then we get jjT ujj jjujj ; 8u 2
K \ @ 2:The …rst statement of Theorem 3.5 implies that T has a …xed point in
K \ 2 1 such that R2 jjujj R: Applying similar techniques as above, we
Let us introduce the following functionals. De…ning on K; the nonnegative, continuous, and concave functional by (u) = mint2[ ;1]ju(t)j ; then (u)
kuk : De…ning the nonnegative, continuous, and convex functional ' and on K by ' (u) = (u) = kuk and the nonnegative continuous functional on K by
(u) = kuk ; then (ku) k kuk for 0 k 1:
Theorem 3.6. Assume that (H1) (H2) hold, and that there exists positive
con-stants a; b; c; d; ; and such that a < b; > 2
(2 ) (q) R1 0 (1 s) q 1 a (s) ds; < (2 ) (q)2 R1(1 s)q 1a (s) ds, and
(i) f (u) d for u 2 [0; d] : (ii) f (u) b for u 2 [b; c] : (iii) f (u) a for u 2 [0; a] :
Then the problem (P ) has at least three positive solutions u1; u2; u3 2 K ('; d)
such that
' (ui) d for i = 1; 2; 3; b < (u1) ; a < (u2) with (u2) < b and (u3) < a:
To prove the existence of three positive solutions, we apply (Avery and Peterson …xed point Theorem).
Theorem 3.7. [5] Let K be a cone in a real Banach space E: Let ' and be nonnegative, continuous, and convex functional on K; let be a continuous, non-negative and concave functional on K; and let be a continuous and nonnegative functional on K satisfying (ku) k kuk for 0 k 1: De…ne the sets, K ('; d), K ('; ; b; d), K ('; ; ; b; c; d) and R ('; ; a; d) by
K ('; d) = fu 2 K; ' (u) < dg ;
K ('; ; b; d) = fu 2 K; b (u) ; ' (u) dg ;
K ('; ; ; b; c; d) = fu 2 K; b (u) ; (u) c; ' (u) dg ; R ('; ; a; d) = fu 2 K; a (u) ; ' (u) dg :
For M and d positive numbers we have (u) (u) and kuk M ' (u) for any u 2 K ('; d): Assume T : K ('; d) ! K ('; d) is completely continuous and there exists positive numbers a; b and c with a < b such that
(S1) fu 2 K('; ; ; b; c; d); (u) bg 6= ; and (T u) b for u 2 K('; ; ; b; c; d);
(S2) (T u) > b for u 2 K ('; ; b; d) with (T u) > c;
(S3) 0 =2 R ('; ; a; d) and (T u) a for u 2 R ('; ; a; d) with (u) = a: Then T has at least three positive …xed points u1; u2; u32 K ('; d) such that
' (ui) d for i = 1; 2; 3; b < (u1) ; a < (u2) with (u2) < b and (u3) < a:
Proof. Proceeding analogously as in the proof of Theorem 3.4, we prove that the mapping T is completely continuous on K ('; d):
Claim 1. T K ('; d) K ('; d): Letting u 2 K ('; d); then kuk d: Thus with the help of assumption (i) it yields
' (T u) = kT uk = max t2[0;1] 1 (q) Z 1 0 G (t; s) a(s)f (u(s))ds 2 (2 ) (q) Z 1 0 (1 s)q 1a (s) f (u(s)) ds d 2 (2 ) (q) Z 1 0 (1 s)q 1a (s) ds < d; and hence T u 2 K ('; d):
Claim 2. (S1) holds, that is nu 2 K '; ; ; b;1b ; d ; (u) > bo 6= ; and (T u) > b for u 2 K '; ; ; b;1b ; d : Let y (t) = b 1 with
1
2 < < 1;
then
(y) = ' (y) = kyk = b 1 < b
1 : Moreover we have (y) = min t2[ ;1]y (t) = b 1 > b > (1 ) kyk : Thus y 2 K '; ; ; b;1b ; d ; so n u 2 K '; ; ; b;1b ; d ; (u) > b o 6= ;: Letting u 2 K '; ; ; b;1b ; d ; then b u (t) b 1 ; thus by virtue of (3.3)
and assumption (ii), we obtain
(T u) = min t2[ ;1]jT u(t)j 2 (2 ) (q) Z 1 (1 s)q 1a (s) f (u(s)) ds 2 (2 ) (q) b Z 1 (1 s)q 1a (s) ds > b: So condition (S1) is satis…ed.
Claim 3. (S2) holds. Letting u 2 K ('; ; b; d) such that (T u) = kT uk > c; then
(T u) = min
t2[ ;1]jT u(t)j b;
this implies that (S2) holds.
Claim 4. (S3) holds. Letting u 2 R ('; ; a; d), then 0 < a kuk d; and so 0 =2 R ('; ; a; d) with (u) = kuk = a; using Lemma 3.2 and assumption (iii) it yields
(T u) = max t2[0;1] 1 (q) Z 1 0 G (t; s) a(s)f (u(s)) ds 2 (2 ) (q) Z 1 0 (1 s)q 1a (s) f (u(s)) ds a 2 (2 ) (q) Z 1 0 (1 s)q 1a (s) ds < a; Then (S3) is satis…ed.
Now we give two examples to illustrate Theorem 3.4 and 3.6
Example 3.8. Let us consider the following fractional boundary value problem
cD83
0+u(t) = a(t)f (u(t)) ; 0 < t < 1;
where q = 8 3; = 1 2; f (u) = exp ( u) ; a (t) = t; = 4 5; by calculus we obtain R0;8 0 a (s) ds = R0;8
0 sds = 0; 32 6= 0: The assumptions (H1) (H2) holds and that
A0= 1; A1= 0; applying Theorem 3.4, we deduce that there exists at least one
positive solution.
Example 3.9. Let us consider the following fractional boundary value problem
cD94
0+u(t) = a(t)f (u(t)); 0 < t < 1;
where q = 9 4; = 1; a (t) = p 1 + t; = 9 10; f (u) = 8 < : u3 2 ; 0 u 3; 7u2 2 18 ; 3 u 4; 38 ; u 4:
It is easy to see that (H1) and (H2) are satis…ed. Let us check the assumptions of
Theorem 3.6 > 2 (0; 1) 5 4 9 4 Z 1 0 p 1 + sds = 2:151 7; < (0; 9) 2 (0; 1)54 9 4 Z 1 0;9 p 1 + sds = 0; 87149:
If we choose = 2; 30; = 0; 5; a = 2; b = 3; c = 0; 1; d 127; 65; then the assumptions of Theorem 3.6 are satis…ed, consequently, there exists at least three positive solutions u1; u2; u32 K ('; d) such that
kuik d = 128; 3 < mint2[9
10;1] u1(t) ; 2 < ku2k ; with mint2[109;1] u2(t) < 3 and
Conclusion
In this paper, we have proved the existence of at least one positive solution of problem (P) by using Guo-Krasnosel’skii …xed point theorem in cone, then under some su¢ cient conditions on the nonlinear source term, we have applied Avery-Peterson theorem to prove the existence of at least three positive solutions. One can prove the existence of multiple positive solutions by using other …xed theorems.
Acknowledgement
We would like to thank the anonymous referees for making a number of valuable comments.
References
[1] R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Positive Solutions Of Di¤ erential Di¤ erence and Integral Equations, Kluwer Academic Publisher, Boston, 1999.
[2] B. Ahmed, J.J. Nieto, Anti-periodic fractional boundary value problems, Comput. Math. Appl. 62 (3) (2011), 1150-1156.
[3] B. Ahmed, J.J. Nieto, J. Pimentel, Some boundary value problems of fractional di¤ erential equations and inclusions, Comput. Math. Appl. 62 (3) (2011), 1238-1250.
[4] A. Ashyralyev, Y.A. Sharifov, Existence and uniqueness of solutions for the system of non-linear fractional di¤ erential equations with nonlocal and integral boundary conditions, Abstr. Appl. Anal. 2012, Article ID 594802, 14pp.
[5] R.I. Avery, A.C. Peterson, Three positive …xed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 42 (35) (2001), 313-322.
[6] D. Amanov, Solvability of boundary value problems for equation of higher order with frac-tional derivatives, in boundary value problem for Di¤erential Equations, The collection of Proceeding no. 17, 204–209 chernovsti, Russia, 2008.
[7] D. Amanov, A. Ashyralyev, Initial boundary value problem for fractional partial di¤ erential equations of higher order, Abstr. Appl. Anal. 2012, doi. 10.1155/2012/973102.
[8] A. Ashyralyev, A note on fractional derivatives and fractional powers of operators, J. Math. Anal. Appl. 357 (1) (2009), 232-236.
[9] A. Ashyralyev, A well-posedness of fractional parabolic equations, Bound. Value Probl. 2013, doi. 10.1186/1687-2770-2013-31.
[10] A. Ashyralyev, F. Dal, Finite di¤ erence and iteration methods for fractional hyperbolic partial di¤ erential equations with the Neumann condition Discrete Dyn. Nat. Soc. 2012, doi. 10. 1155/2012/434976.
[11] A. Ashyralyev, B. Hicdurmaz, A note on the fractional Schrodinger di¤ erential equations, Kybernetes 40 (5-6) (2011), 736-750.
[12] A. Ashyralyev, Z. Cakir, FDM for fractional parabolic equations with the Neumann condition, Adv. Di¤er. Equ. 2013, 2013:120.
[13] Z. Bai, On positive solutions of nonlocal fractional boundary value problem, Nonlinear Analy-sis 72 (2) (2010), 916-924.
[14] Z. Bai, H. Lu, Positive solutions for boundary value problem of a nonlinear fractional di¤ er-ential equation, J. Math. Anal. Appl. 311 (2) (2005), 495-505.
[15] B. Bonilla, M. Rivero, L. Rodriguez-Germa, J.J. Trujillo, Fractional di¤ erential equations as alternative models to nonlinear di¤ erential equations, Appl. Math. Comput. 187 (1) (2007), 79-88.
[16] M. El-Shahed, Positive solutions for boundary value problem of nonlinear fractional di¤ er-ential equation, Abstr. Appl. Anal. 2007, Article ID 10368, 8 pages.
[17] L.J. Grimm, Existence and continuous dependence for a class of nonlinear neutral-di¤ erential equations, Proc. Amer. Math. Soc. 29 (1971), 525-536.
[18] A. Guezane-Lakoud, R. Khaldi, Solvability of two-point fractional boundary value problem, J. Nonlinear Sci. Appl. 5 (2012), 64-73.
[19] A. Guezane-Lakoud, R. Khaldi, Positive solution to a higher order fractional boundary value problem with fractional integral condition, Romanian J. Math. Comput. Sci. 2 (2012), 28-40. [20] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Di¤ erential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, The Netherlands, 2006.
[21] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional di¤ erential equations, Nonlinear Anal., Theory Methods Appl. 69 (8) (2008), 2677-2682.
[22] K.S. Miller, B. Ross, An introduction to the fractional calculus and di¤ erential equations, John Wiley, New York, 1993.
[23] I. Podlubny, Fractional Di¤ erential Equations, 198 Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
[24] J.R.L. Webb, G. Infante, D. Franco, Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions, Proc. R. Soc. Edinb., Sect. A, Math. 138 (2) (2008), 427-446.
[25] J.R.L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, Nonlinear Di¤er. Equ. Appl. NoDEA, 15 (1-2) (2008), 45-67.
[26] A. Yakar, M.E. Koksal, Existence results for solutions of nonlinear fractional di¤ erential equations, Abst. Appl. Anal. 2012, Article ID 267108, 12 pages.
Current address : A. Guezane-Lakoud andS. Kouachi; Laboratory of Advanced Materials, Fac-ulty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, ALGERIA, F. Ellaggoune; Laboratory of Applied Mathematics and Modeling, University 8 mai 1945 - Guelma, P.O. Box 401, Guelma 24000, ALGERIA
E-mail address : a_guezane@yahoo.fr, sa.sa.kouachi@yahoo.fr, fellaggoune@gmail.com URL: http://math.science.ankara.edu.tr
