FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE. Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi

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1 Opuscula Math. 37, no. 2 27), Opuscula Mathematica FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi Communicated by Jean Mawhin Abstract. In this paper, we focus on the solvability of a fractional boundary value problem at resonance on an unbounded interval. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin. The obtained results are illustrated by an example. Keywords: boundary value problem at resonance, existence of solution, unbounded interval, coincidence degree of Mawhin, fractional differential equation. Mathematics Subject Classification: 34B4, 34B5.. INTRODUCTION In this paper, we are concerned with the existence of solutions for the following fractional differential equation subject to the nonlocal conditions D q xt) = ft, xt)) et), t, ),.) I 2 q x) =, D q x ) = xη),.2) ηq where D q denotes the Riemann Liouville fractional derivative of order q, < q < 2, η >, f : [, ) R R and e : [, ) R are given functions satisfying some conditions. The problem.).2) happens to be at resonance in the sense that the dimension of the kernel of the linear operator Lx = D q x is not less than one under boundary conditions.2). It should be noted that in recent years, there have been many works related to boundary value problems at resonance for ordinary differential equations. We refer the reader to [7 9,, 3, 4, 6] and the references therein. c Wydawnictwa AGH, Krakow

2 266 Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi However, the articles on the existence of solutions of fractional differential equations on the half-line are still few in number, and most of them deal with problems under nonresonance conditions. For some recent articles investigating resonant and nonresonant fractional problems on the unbounded interval, see [ 5,, 7]. Recently fractional differential equations have been investigated by many researchers and different methods have been used to obtain such fixed point theory, upper and lower solutions method, coincidence degree theory, etc. The rest of this paper is organized as follows. In Section 2, we give some necessary notations, definitions and lemmas. In Section 3, we study the existence of solutions of.).2) by the coincidence degree theory due to Mawhin. Finally, we give an example to illustrate the obtained results. 2. PRELIMINARIES First recall some notation, definitions and theorems which will be used later. Let X and Y be two real Banach spaces and let L : dom L X Y be a linear operator which is a Fredholm map of index zero, define the continuous projections P and Q, respectively, by P : X X, Q : Y Y such that Im P = ker L, ker Q = Im L. Then X = ker L ker P, Y = Im L Im Q, thus L dom L ker P : dom L ker P Im L is invertible, denote its inverse by K P. Let Ω be an open bounded subset of X such that dom L Ω, the map N : X Y is said to be L-compact on Ω if QN Ω ) is bounded and K P I QN) : Ω X is compact. Since Im Q is isomorphic to ker L then, there exists an isomorphism J : Im Q ker L. It is known that the coincidence equation Lx = Nx is equivalent to x = P JQN)x KP I Q)Nx. Theorem 2. [5]). Let L be a Fredholm operator of index zero and N be L-compact on Ω. Assume that the following conditions are satisfied: ) Lx λnx for every x, λ) [dom L\ ker L) Ω], ), 2) Nx / Im L for every x ker L Ω, 3) deg QN ker L, Ω ker L, ), where Q : Y Y is a projection such that Im L = ker Q. Then the equation Lx = Nx has at least one solution in dom L Ω. Theorem 2.2 [6]). Let C = {y C [, )) : lim t y t) exists} equipped with the norm y = sup t [, ) yt). Let F C. Then F is relatively compact if the following conditions hold: ) F is bounded in C, 2) the functions belonging to F are equicontinuous on any compact subinterval of [, ), 3) The functions from F are equiconvergent at.

3 Fractional boundary value problems on the half line 267 Let X be the space { X = x C [, )) : D q x C [, )), } lim t e t xt) and lim e t D q t xt) exist endowed with the norm x = sup t e t xt) sup e t D q t then X is a Banach space. Let Y = L [, ) equipped with the norm y = yt) dt. xt), Define the operator L : dom L X Y by Lx = D q x, where dom L = { x X : D q x Y, I 2 q x) =, D q x ) = xη) } X, then L maps dom L into Y. Let N : X Y be the operator Nxt) = ft, xt)) et), t [, ), then the problem.).2) can be written as Lx = Nx. We recall the definition of the Riemann-Liouville fractional derivative and Riemann-Liouville fractional integral, we can find their properties in [2]. Definition 2.3. The Riemann-Liouville fractional integral of order p > is given by I p gt) = t Γp) provided that the right-hand side exists. gs) ds, t >, t s) p Definition 2.4. The Riemann-Liouville fractional derivative D p g of order p > is given by ) n t D p d gs) gt) = ds, t >, Γ n p) dt t s) p n where n = [p] [p] is the integer part of p), provided that the right-hand side is point-wise defined on, ). Lemma 2.5. The homogenous fractional differential equation D q a gt) = has a solution gt) = c t q c 2 t q 2... c n t q n, where c i R, i =,..., n and n = [q].

4 268 Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi Lemma 2.6. Let p, q, f L [, ). Then the following assertions hold:. I p I q ft) = I pq f t) = I q I p ft) and D q I q ft) = ft) for all t [, ). 2. If p > q >, then the formula D q I p ft) = I p q ft) holds almost everywhere on t [, ), for f L [, ) and it is valid at any point t [, ) if f C[, ). 3. If q and p >, then I q t p ) x) = Γp) Γpq) xpq, D q t p ) x) = Γp) Γp q) xp q, D q t q j) x) =, j =, 2,..., n, where n = [q]. 3. MAIN RESULTS We give the first result on the existence of a solution for the problem.).2). Theorem 3.. Assume that the following conditions are satisfied: H ) There exists nonnegative functions α, β L [, ) such that for all x R t [, ) we have ft, x) et) e t α t) x βt). 3.) H 2 ) There exists a constant M > such that for x dom L if D q xt) > M for all t [, ) fs, xs)) e s)) ds η s) q f s, x s)) es)) ds. 3.2) H 3 ) There exists a constant M > such that for any xt) = c t q ker L with c > M either c fs, xs)) es)) ds η s) q f s, x s)) es)) ds < or c fs, xs)) es)) ds 3.3) η s) q f s, x s)) es)) ds >. 3.4) Then for every e Y, the problem.).2) has at least one solution in X, provided 2A α ) >, where A = M, M = max t e t t q ) [ e q ) ] q =.

5 Fractional boundary value problems on the half line 269 In order to prove Theorem 3. we need some lemmas. Lemma 3.2. The operator L : dom L X Y is a Fredholm operator of index zero. Furthermore, the linear projector operator Q : Y Y is defined by Qyt) = Ae t ys)ds η s) q ys)ds 3.5) and the linear operator K P : Im L dom L ker P can be written as Morover, where K p yt) = I q yt), y Im L. K p y A y, y Im L, 3.6) A = Proof. Using Lemma 2.5, we get η s) q e s ds. xt) = at q bt q 2, t >, 3.7) as a solution of Lxt) =. Applying I 2 q to both sides of the equality 3.7), then using condition I 2 q x) =, we get b =. Thanks to condition D q x ) = η xη), we q conclude ker L = { x dom L : xt) = at q, a R, t [, ) }. Now the problem D q xt) = yt) 3.8) has a solution x that satisfies the conditions I 2 q x ) =, D q x ) = η xη) q if and only if ys)ds η s) q ys)ds =. 3.9) In fact from 3.8) and together with the boundary condition I 2 q x) = we get xt) = I q yt) at q. According to D q x ) = x η), we obtain On the other hand, if 3.9) holds, setting ys)ds = Iq yη). xt) = I q yt) at q,

6 27 Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi where a is an arbitrary constant, then xt) is a solution of 3.8). Hence Im L = y Y : ys)ds Now taking 3.5) into account, it yields Q 2 y = Q Qy) = A 2 e t y s) ds e s ds η s) q ys)ds =. η s) q ys)ds η s) q e s ds = Qy, thus Q is a continuous projector and Im L = ker Q. Rewrite y = y Qy) Qy, then y Qy ker Q = Im L, Qy Im Q and Im Q Im L = {}, then Y = Im L Im Q. Thus dim ker L = = dim Im Q = co dim Im L =, this means that L is a Fredholm operator of index zero. Now define a projector P from X to X by setting P xt) = Dq x) t q, Γ q) and the generalized inverse K P : Im L dom L ker P of L as K p yt) = I q yt) = t t s) q ys)ds. Γ q) Obviously, Im P = ker L and P 2 x = P x. It follows from x = x P x) P x that X = ker P ker L. By simple calculation, we can get ker L ker P = {}. Hence X = ker L ker P. Let us show that the generalized inverse of L is K P. In fact, for y Im L, we have LK p ) yt) = D q K p y t)) = D q I q yt) = y t), and for x dom L ker P, we obtain K p L) xt) = K p ) D q xt) = I q D q xt) = xt) Dq x) t q I2 q x) Γ q) Γ q ) tq 2. In view of x dom L ker P, then I 2 q x ) = and P x =, thus K p L)xt) = xt).

7 Fractional boundary value problems on the half line 27 This shows that K p = L dom L ker P ). From the definition of K p, we have t e t K p y) t) e t t s) q ys) ds that leads to This completes the proof. Lemma 3.3. Let Then Ω is bounded. e t t q ys) ds max t D q K p y) t) = I yt) e t t q ys) ds = y, K p y M ) y = A y. ) y = M y. Ω = {x dom L\ ker L : Lx = λnx for some λ [, ]}. Proof. Suppose that x Ω and Lx = λnx. Thus λ and QNx =, so fs, xs)) e s)) ds η s) q f s, x s)) es)) ds =. Thus, by condition H 2 ), there exists t R such that D q x t ) M. It follows from D q xt) = DD q xt) and the fact that D q x Y, that D q t x) = Dq x t ) D q x s) ds. Then, we have D q x) M Lxs) ds M Nxs) ds = M Nx. 3.) On the other hand, since x dom L\ ker L, then I P ) x dom L ker P and LP x =, thus from Lemma 3.2, we get I P ) x = K p LI P )x A LI P )x 3.) = A Lx A Nx.

8 272 Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi So D q x P x I P ) x = A x) A Nx, 3.2) thanks to inequalities 3.) and 3.), 3.2) becomes x A M 2A Nx. 3.3) On the other hand, by 3.), we have Nx = f s, x s)) es) ds x α β. 3.4) Thus x A M 2A x α 2A β. Since 2A α ) >, we obtain x which proves that Ω is bounded. A M 2A α 2A β 2A α <, Lemma 3.4. The set Ω 2 = {x ker L : Nx Im L } is bounded. Proof. Let x Ω 2, then x ker L implies xt) = ct q, c R and QNx =, therefore f s, cs q ) es) ) ds η s) q f s, cs q ) es) ) ds =. From condition H 2 ) there exists t such that D q x t ) M and so c M On the other hand, ) x = sup t so Ω 2 is bounded. e t xt) sup t xt) c D q c A MA <, sup e t t q t [ ) Lemma 3.5. Suppose that the first part of Condition H 3 ) of Theorem 3. holds. Let Ω 3 = {x ker L : λjx λ) QNx =, λ [, ]}, where J : ker L Im Q is the linear isomorphism given by J ct q ) = ce t for all c R, t. Then Ω 3 is bounded..

9 Fractional boundary value problems on the half line 273 Proof. Let x Ω 3, then x t) = c t q and λjx = λ) QNx that is equivalently written as λc = λ) A f s, c s q ) es) ) ds η s) q f s, c s q ) es) ) ds ). If λ =, then c =. Otherwise, if c > M, in view of 3.3) one has λc 2 = λ)ac f s, c s q ) es) ) ds η s) q f s, c s q ) es) ) ) <, which contradicts the fact that λc 2. So c M, moreover ) x = c sup e t t q t c A M Γ q) A, thus Ω 3 is bounded. Lemma 3.6. Under the second part of Condition H 3 ) of Theorem 3., the set Ω 3 is bounded. Proof. The proof is similar to the one of Lemma 3.5. Lemma 3.7. Suppose that Ω is an open bounded subset of X such that dom L Ω. Then N is L-compact on Ω. Proof. Suppose that Ω X is a bounded set. Without loss of generality, we may assume that Ω = B, r), then for any x Ω, x r. For x Ω, and by condition 3.), we obtain

10 274 Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi QNx Ae t Ae t Ae t r fs, xs)) es) ds η s) q fs, xs)) e s) ds e s αs) xs) βs)ds η s) q e s αs) x s) βs) ) ds αs)ds βs)ds r Γ q) η s) q βs)ds Ae t ) r α β ). η s) q α s) ds Thus QNx A ) r α β ), 3.5) which implies that QN Ω ) is bounded. Next, we show that K P I Q) N Ω ) is compact. For x Ω, by 3.) we have Nx = f s, x s)) es) ds r α β. 3.6) On the other hand, from the definition of K P 3.6) one gets and together with 3.6), 3.5) and K P I Q) Nx A I Q)Nx A [ Nx QNx ] A A )) r α β ). It follows that K P I Q) N Ω ) is uniformly bounded.

11 Fractional boundary value problems on the half line 275 Let us prove that K P I Q) N Ω ) is equicontinuous. For any x Ω, and any t, t 2 [, T ], t < t 2 with T >, we have e t K P I Q) Nx) t ) e t2 K P I Q) Nx) t 2 ) = = t e t t s) q I Q) Nxs)ds t 2 e t2 t 2 s) q I Q) Nxs)ds t 2 t t e t2 t 2 s) q e t t s) q ) I Q) Nxs) ds e t2 t 2 s) q I Q) Nxs) ds e t2 t q 2 e t t q 2 t q t 2 t ) t I Q) Nxs) ds I Q) Nxs) ds, as t t 2. On the other hand, we have e t D q K P I Q) Nx) t ) e t D q K P I Q) Nx) t 2 ) t t 2 = e t I Q) Nxs)ds e t2 I Q) Nxs)ds t e t e t2) I Q) Nxs) ds t 2 t ) t t 2 I Q) Nxs) ds t 2 t e t2 I Q) Nx s) ds t I Q) Nxs) ds, as t t 2. So K P I Q) N Ω ) is equicontinuous on every compact subinterval of [, ). In addition, we claim that K P I Q)N Ω ) is equiconvergent at infinity. In fact, e t K p I Q) Nx) t) t e t t s) q I Q) Nxs) ds

12 276 Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi e t t q e t t q t On the other hand, we have Thus I Q) Nxs) ds e t t q I Q) Nx Nx QNx ) e t t q 2A) [r α β ]. e t D q K p I Q) Nx) t) e t t I Q) Nxs) ds e t Nx QNx ) e t A )) r α β ). lim t e t K p I Q) Nx) t) =, lim t e t D q K p I Q) Nx) t) =, consequently K P I Q) N Ω ) is equiconvergent at. Now we are able to give the proof of Theorem 3.. Proof of Theorem 3.. In what follows, we shall prove that all conditions of Theorem 2. are satisfied. Let Ω to be an open bounded subset of X such that 3 i= Ω i Ω. From Lemma 3.2, we know that L is a Fredholm operator of index zero. By Lemma 3.7, we have N is L-compact on Ω. By virtue of the definition of Ω, it yields i) Lx λnx for all x, λ) [dom L\ ker L) Ω], ), ii) Nx / Im L for all x ker L Ω. Now we prove that condition iii) of Theorem 2. is satisfied. Let H x, λ) = ±λjx λ) QNx. Since Ω 3 Ω, then H x, λ) for every x ker L Ω. By the homotopy property of degree, we get deg QN ker L, Ω ker L, ) = deg H, ), Ω ker L, ) = deg H, ), Ω ker L, ) = deg ± J, Ω ker L, ). So, the third assumption of Theorem 2. is fulfilled and Lx = Nx has at least one solution in dom L Ω, i.e..).2) has at least one solution in X. The proof is completed. Example 3.8. Consider the fractional boundary value problem { D q xt) = ft, xt)) e t), t, ), I 2 q x) =, D q x ) = η xη), q where q = 3 2, ft, x) = 4 e t e t x 8e t ), et) = 3e t. Choosing αt) = 4 e t and βt) = 3e t 2e 2t, then α, β are nonnegative and belong to L [, ). For all x R, t [, ) we have ft, x) et) e t αt) x βt), P )

13 Fractional boundary value problems on the half line 277 so hypothesis H ) of Theorem 3. is satisfied. We claim that condition H 3 ) is satisfied, indeed, since fs, xs)) e s)) ds η fs, xs)) es)) ds, η s) q f s, x s)) es)) ds then for M = 8, η = and any xt) = c t q ker L with c > M, we have fs, xs)) e s)) ds ) ) 4 e s e s c s 3 2 8e s 3e s ds 2e 2s s 2e s 3e s) ds = >. η s) q f s, x s)) es)) ds As a result, if c >, then c fs, xs)) es)) ds η s) q f s, x s)) es)) ds >, or if c <, then c fs, xs)) es)) ds η s) q f s, x s)) es)) ds <.

14 278 Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi Now, we have = fs, xs)) e s)) ds 4 e s e s x s) 8e s) ) 3e s ds η s) q f s, x s)) es)) ds ) 3 Γ s) e s e s xs) 8e s) ) 3e s ds 4 e s e s x s) 8e s) 3e s ds =.968 3, 3e s 2e 2s) ds consequently, H 2 ) is satisfied for any constant M >. Finally, a simple calculus gives where 2A α ) = >, A = [ e q ) ] q = and α = 4 e t ds = 4. We conclude from Theorem 3. that the problem P) has at least one solution in X. Acknowledgments The authors would like to thank the referee for his her) helpful comments and suggestions. REFERENCES [] R.P. Agarwal, M. Benchohra, S. Hamani, S. Pinelas, Boundary value problems for differential equations involving Riemann-Liouville fractional derivative on the half line, Dyn. Contin. Discrete Impuls. Syst. Ser. A. Math. Anal. 8 2) 2, [2] R.P. Agarwal, D. O Regan, Infinity Interval Problems for Difference and Integral Equations, Kluwer Academic Publisher Dordrecht, 2. [3] R.P. Agarwal, D. O Regan, Infinite interval problems modeling phenomena which arise in the theory of plasma and electrical potential theory, Stud. Appl. Math. 23) 3, [4] A. Arara, M. Benchohra, N. Hamidi, J.J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Anal. 72 2),

15 Fractional boundary value problems on the half line 279 [5] Y. Chen, X. Tang, Positive solutions of fractional differential equations at resonance on the half-line, Boundary Value Problems 22) 22:64, 3 pp. [6] C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, 973. [7] Y. Cui, Solvability of second-order boundary-value problems at resonance involving integral conditions, Electron. J. Differ. Equ. 22) 22:45, 9 pp. [8] W. Feng, J.R.L. Webb, Solvability of three-point boundary value problems at resonance, Nonlinear Anal. Theory, Methods and Appl ), [9] D. Franco, G. Infante, M. Zima, Second order nonlocal boundary value problems at resonance, Math. Nachr ) 7, [] A. Guezane-Lakoud, A. Kilickman, Unbounded solution for a fractional boundary value problem, Advances in Difference Equations 24) 24:54. [] C.P. Gupta, S.K. Ntouyas, P.Ch. Tsamatos, On an m-point boundary-value problem for second-order ordinary differential equations, Nonlinear Anal ), [2] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, vol. 24, 26. [3] N. Kosmatov, Multi-point boundary value problems on an unbounded domain at resonance, Nonlinear Anal ) 8, [4] R. Ma, Existence of positive solutions for second-order boundary value problems on infinity intervals, Appl. Math. Lett. 6 23), [5] J. Mawhin, Topological degree methods in nonlinear boundary value problems, NSFCBMS Regional Conference Series in Mathematics, Am. Math. Soc, Providence, 979. [6] D. O Regan, B. Yan, R.P. Agarwal, Solutions in weighted spaces of singular boundary value problems on the half-line, J. Comput. Appl. Math ), [7] X. Su, S. Zhang, Unbounded solutions to a boundary value problem of fractional order on the half-line, Comput. Math. Appl. 6 2), Assia Frioui frioui.assia@yahoo.fr University Guelma Laboratory of Applied Mathematics and Modeling P.O. Box 4, Guelma 24, Algeria Assia Guezane-Lakoud a_guezane@yahoo.fr University Badji Mokhtar-Annaba Faculty of Sciences Laboratory of Advanced Materials P.O. Box 2, 23, Annaba, Algeria

16 28 Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi Rabah Khaldi University Badji Mokhtar-Annaba Faculty of Sciences Laboratory of Advanced Materials P.O. Box 2, 23, Annaba, Algeria Received: March, 26. Revised: May 3, 26. Accepted: June 2, 26.