D Motreanu

F Atici

G Bonanno

G Jumarie

G Molica Bisci

H Gray

HA Jalab

Hamid A Jalab

JB Diaz

L Jiang

LH Bian

P Candito

Rabha W Ibrahim

Boundary Value Problems

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Ibrahim and Jalab Boundary Value Problems  (2015) 2015:23 DOI 10.1186/s13661-015-0287-7RESEARCH Open AccessDiscrete boundary value problem based onthe fractional Gâteaux derivativeRabha W Ibrahim1* and Hamid A Jalab2*Correspondence:rabhaibrahim@yahoo.com1Institute of Mathematical Sciences,University of Malaya, Kuala Lumpur,50603, MalaysiaFull list of author information isavailable at the end of the articleAbstractThis article aims to organize positive solutions to discrete fractional boundary valueproblems for continuous Gâteaux diﬀerentiable functions. A generalized Gâteauxderivative is introduced using a fractional discrete operator for a Jumarie fractionaloperator. The method of ﬁnding solutions is based on critical point theorems of ﬁnitedimensional Banach spaces.1 IntroductionIn , Diaz and Osler [] imposed a discrete fractional diﬀerence operator based on aninﬁnite series. In , Gray and Zhang [] presented a class of fractional diﬀerence oper-ators and introduced the Leibniz formula. In , Atici and Eloe [] introducedmethodsfor composing fractional sums and diﬀerences. Mathematicians, physicists, and engineershave recently employed fractional calculus to solve varieties ofmodels of applied problemsin diﬀerent ﬁelds. In a previous work, we developed various classes of discrete fractionaloperators with one or two parameters for image and signal processing [–].Researchers in the ﬁelds of computer science, neural networks, control systems, foodprocessing, and economics rely on mathematical modeling because it naturally entailsnonlinear diﬀerence equations. Therefore, many authors have widely developed variousprocedures and patterns, such as upper and lower solutions, ﬁxed point theorems, andthe Brouwer degree, to study discrete models [–]. Critical point theory has recentlyattracted the attention of many researchers. The theory guarantees the outcomes of bothordinary and partial diﬀerential problems. Hence, critical point theory is the main tool forﬁnding solutions to fractional discrete nonlinear equations.This study aims to establish positive solutions to discrete fractional boundary valueproblems for continuous Gâteaux diﬀerentiable functions. A generalized Gâteaux deriva-tive is introduced using a fractional discrete operator. Some results are reﬁned in this di-rection. The method for ﬁnding solutions is based on critical point theorems of ﬁnite di-mensional Banach spaces. The applications are illustrated to study the abstract outcomes.2 MainmethodsThis section deals with preliminaries and some concepts.© 2015 Ibrahim and Jalab; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly credited.Ibrahim and Jalab Boundary Value Problems  (2015) 2015:23 Page 2 of 10Deﬁnition . [] For a continuous function g :R→R and a constant  > , the forwardoperator FW () is deﬁned by the equalityFW ()g(x) := g(x + ).The fractional diﬀerence on the right and of the order ℘ ,  < ℘ <  of g(x) is deﬁned by theformula℘g(x) := (FW () – )℘g(x)with its fractional derivative on the rightg(℘)+ (x) = lim→℘[g(x) – g()]℘.Deﬁnition . [] Let (,‖ ·‖) be a real Banach space with the dual space∗. A functionϕ : →R is called Gâteaux diﬀerentiable at x ∈  if χ := ϕ′(x) ∈ ∗ satisﬁeslim→+ϕ(x + y) – ϕ(x)= ϕ′(x)(y), ∀y ∈ .Deﬁnition . [] A function ϕ :  → R is called Gâteaux diﬀerentiable and ver-iﬁes the Palais-Smale condition ((PS)-condition) if any bounded sequence {xn} withlimn→∞ ‖ϕ′(xn)‖∗ =  has a convergent subsequence.Combining Deﬁnitions . and ., we obtain a fractional Gâteaux derivative.Deﬁnition . Let (,‖ · ‖) be a real Banach space with the dual space ∗. A functionϕ : →R has a fractionalGâteaux derivative, of the order  < ℘ <  at x ∈  if ϕ(℘)(x) ∈ ∗exists such that for a constant  > , the forward operator FWy(), y ∈  is deﬁned by theequalityFWy()ϕ(x) := ϕ(x + y),with the fractional diﬀerence on the right℘y ϕ(x) :=(FWy() – )℘ϕ(x),and its fractional derivative on the rightlim→+℘y [ϕ(x) – ϕ()]℘= ϕ(℘)(x)(y), ∀y ∈ .The next results show some properties of the fractional Gâteaux derivative, which basi-cally are generalizations of some results given in []. Therefore, we skip the proofs.Theorem . Suppose that  is a real Banach space and F ,G :  → R are two continu-ously fractional Gâteaux diﬀerentiable functions. Assume thatT := F –GIbrahim and Jalab Boundary Value Problems  (2015) 2015:23 Page 3 of 10and that x ∈  and ρ,ρ ∈R exist, with ρ < F(x) < ρ, thereby obtainingsupy∈F– ]ρ,ρ[G(y) <G(x) + ρ – F(x), ()supy∈F– ]–∞,ρ]G(y) <G(x) + ρ – F(x). ()Presume that T satisﬁes the (PS)-condition. Then y ∈ F– ]ρ,ρ[ such that T(y) < T(y),y ∈ F– ]ρ,ρ[, and T ′(y) = .Theorem . Let  be a real Banach space, and let F ,G : →R be two continuous frac-tional Gâteaux diﬀerentiable functions. Assume thatT = F –Gand that x ∈  and ρ ∈R such that ρ > F(x)supy∈F– ]–∞,ρ[G(y) <G(x) + ρ – F(x). ()We infer that T satisﬁes the (PS)-condition. Then y ∈ F– ]–∞,ρ[ such that T(y) < T(y),y ∈ F– ]–∞,ρ[, and T ′(y) = .Theorem . Let  be a real Banach space and let F ,G : → R be two continuous frac-tional Gâteaux diﬀerentiable functions. Assume thatT = F –G,where T is bounded from below, and x ∈  and ρ ∈R with ρ < F(x) such thatsupy∈F– ]–∞,ρ]G(y) <G(x) + ρ – F(x). ()We deduce that T satisﬁes the (PS)-condition. Then y ∈ F– ]ρ, +∞[ such that T(y) ≤T(y), y ∈ F– ]ρ, +∞[, and T ′(y) = .We apply the above critical point theorems to ﬁnd solutions to the fractional diﬀerenceequation, taking the type–℘y(ψp(℘y μk–))+ σkψp(μk) = h(k,μk), k ∈ [,N], ()μ = μN+ = ,where h : [,N] × R → R is a continuous function, σk ≥ , and ψp(ς ) := |ς |p–ς , < p < +∞.Deﬁne the Banach space asB := {μ : [,N + ]→R : μ = μn+ = }Ibrahim and Jalab Boundary Value Problems  (2015) 2015:23 Page 4 of 10endowed with a discrete norm‖μ‖ :=(N+∑k=∣∣℘y,k–μ∣∣p + σk|μ|p)/psuch that (see [, Lemma .])maxk∈[,N]|μk| ≤ (N + )(p–)/p ‖μ‖, ∀μ ∈ B.LetF(μ) := ‖μ‖pp , G(μ) :=N∑k=H(k,μk), T(μ) := F(μ) –G(μ), ∀μ ∈ B,whereH(k, t) :=∫ th(k,u)du, ∀(k, t) ∈ [,N]×R.Note that T ∈ C(B,R), T() = , and that all the critical points of T are the solutionsof ().3 Main ﬁndingsIn this section, we introduce at least one solution to problem () with the following results.Theorem . Consider three positive constants C, C, and C such thatC <( + σ )/p(N + )(p–)/pC < C, ()where σ :=∑Nk= σk . Ifβ :=∑Nk= maxt H(k, t) –∑Nk=H(k,C)(C)p – ( + σ )(N + )p–Cp≤ p(N + )p–andβ :=∑Nk= maxt H(k, t) –∑Nk=H(k,C)(C)p – ( + σ )(N + )p–Cp≤ p(N + )p– ,where (Ci)p = ( + σ )(N + )p–Cp, i = , , then () has at least one non-trivial solution μ∗such thatC(N + )(p–)/p <∥∥μ∗∥∥ < C(N + )(p–)/p .Proof We aim to apply Theorem . on T , because the critical points of T are solutionsto (). Choose ω ∈ B and deﬁneωk :={C, k ∈ [,N],, k = ,k =N + .Ibrahim and Jalab Boundary Value Problems  (2015) 2015:23 Page 5 of 10Thus, we obtainF(ω) =  + σp Cp.According to condition (), a computation implies thatρ :=(C)pp(N + )p– < F(ω) <(C)pp(N + )p– := ρ.We proceed to consider conditions () and ():supy∈F– ]–∞,ρ](G(y) –G(ω)ρ – F(ω))≤∑Nk= maxt H(k, t) –∑Nk=H(k,C)(C)pp(N+)p– –+σCpp= p(N + )p–∑Nk= maxt H(k, t) –∑Nk=H(k,C)(C)p – ( + σ )(N + )p–Cp:= p(N + )p–β ≤ .Similarly, we havesupy∈F–]ρ,ρ[(G(y) –G(ω)ρ – F(ω))≤∑Nk= maxt H(k, t) –∑Nk=H(k,C)(C)pp(N+)p– –+σCpp= p(N + )p–∑Nk= maxt H(k, t) –∑Nk=H(k,C)(C)p – ( + σ )(N + )p–Cp:= p(N + )p–β ≤ .Thus, in virtue of Theorem ., we conclude that T has at least one critical point μ∗ suchthatρ < F(μ∗)< ρ,and we obtainC(N + )(p–)/p <∥∥μ∗∥∥ < C(N + )(p–)/p . By letting C = , C = C, we obtain the following result.Corollary . Consider two positive constants C and C with( + σ )/p(N + )(p–)/p < . ()Ifβ :=∑Nk=H(k,C) –∑Nk= maxt H(k, t)( + σ )(N + )p– ≤Cpp(N + )p–Ibrahim and Jalab Boundary Value Problems  (2015) 2015:23 Page 6 of 10andβ :=∑Nk= maxt H(k, t) –∑Nk=H(k,C)p – ( + σ )(N + )p– ≤Cpp(N + )p– ,where p = ( + σ )(N + )p– = , then () has at least one non-trivial solution μ∗ such that∥∥μ∗∥∥ < C(N + )(p–)/p .Theorem . Consider two positive constants C and  such that( + σ )/p(N + )(p–)/pC < . ()Ifβ :=∑Nk= maxt H(k, t) –∑Nk=H(k,C)()p – ( + σ )(N + )p–Cp ≤p(N + )p– ,where ()p = ( + σ )(N + )p–Cp, then () has at least one non-trivial solution μ∗ suchthat∥∥μ∗∥∥ ≤ (N + )(p–)/p .Proof We aim to employ Theorem . on T , where all critical points of T are solutionsto (). Choose ω ∈ B as in Theorem .. We haveF(ω) =  + σp Cp.According to condition (), a calculation yieldsF(ω) < ()pp(N + )p– := ρ.We proceed to consider condition ():supy∈F– ]–∞,ρ[(G(y) –G(ω)ρ – F(ω))≤∑Nk= maxt H(k, t) –∑Nk=H(k,C)()pp(N+)p– –+σCpp= p(N + )p–∑Nk= maxt H(k, t) –∑Nk=H(k,C)()p – ( + σ )(N + )p–Cp:= p(N + )p–β ≤ .Thus, in view of Theorem ., we observe that T has at least one critical point μ∗ suchthatF(μ∗)< ρ,Ibrahim and Jalab Boundary Value Problems  (2015) 2015:23 Page 7 of 10and we obtain∥∥μ∗∥∥ ≤ (N + )(p–)/p .This completes the proof. Theorem . Consider two positive constants C and γ such thatγ < ( + σ )/p(N + )(p–)/pC . ()Ifη :=∑Nk= maxt H(k, t) –∑Nk=H(k,C)(γ )p – ( + σ )(N + )p–Cp ≤p(N + )p– ,where (γ )p = ( + σ )(N + )p–Cp, then () has at least one non-trivial solution.Proof Our aim is to employ Theorem . on T , because the critical points of T are solu-tions to (). We put ω ∈ B as in Theorem . such thatF(ω) =  + σp Cp.According to condition (), a manipulation leads toρ := (γ )pp(N + )p– < F(ω).Thus, we consider condition ():supy∈F– ]–∞,ρ](G(y) –G(ω)ρ – F(ω))≤∑Nk= maxt H(k, t) –∑Nk=H(k,C)(γ )pp(N+)p– –+σCpp= p(N + )p–∑Nk= maxt H(k, t) –∑Nk=H(k,C)(γ )p – ( + σ )(N + )p–Cp:= p(N + )p–η ≤ .Therefore, in virtue of Theorem ., we conclude that T has at least one critical point.The boundedness of the solution is due to the (SP)-condition. The above completes theproof. 4 Multiplicity ﬁndingsWe impose multiplicity solutions in this section.Theorem . Let the hypothesis of Theorem . hold. Assume that h(k, ) = , k ∈ [,N].Consider two constants κ > p and κ >  such that for |χ | ≥ κ, one has < κH(k,χ )≤ χh(k,χ ). ()If T ′ is a Lipschitz continuous functional in B, then () admits at least two solutions.Ibrahim and Jalab Boundary Value Problems  (2015) 2015:23 Page 8 of 10Proof In view of Theorem ., () has at least one solution, say μ. We use the mountainpass theorem to identify the second solution. The functional T = F – G is obviously inclass C(B,R) with T() = . Assume that μ is a strict local minimum for T in B. Thus aconstant  >  satisﬁesinf‖μ–μ‖=T(μ) > T(μ), ∀μ ∈ B.After integrating (), we observe that there are two positive constants b and b such thatH(k, t)≥ b|t|κ – b, ∀k ∈ [,N], t ∈R.We then proceed to identify the second solution of (). Let ν ∈ B\{}. A calculation thenyieldsT(tν) = (F –G)(tν)= p‖tν‖p –N∑k=H(k, tνk)≤ tpp ‖ν‖p – tκbN∑k=|νk|κ + bN .Given that κ > p, we havetpp ‖ν‖p – tκbN∑k=|νk|κ + bN → –∞, as t → +∞.This formula implies that the functional T has a critical point μ := infν∈B maxt∈RT(ν(t)),which is the second solution for (). This condition completes the proof. In the same manner as in Theorem ., we have the following results.Theorem . Let the hypothesis of Theorem . be satisﬁed. Assume that h(k, ) = , k ∈[,N]. Consider the assertion of (). If T ′ is a Lipschitz continuous functional in B, then() admits at least two solutions.Theorem . Let the hypothesis of Theorem . hold. Assume that h(k, ) = , k ∈ [,N].Consider the inequality (). If T ′ is a Lipschitz continuous functional in B, then () admitsat least two solutions.5 Positive ﬁndingsWe focus on the positive results of problem (). For this purpose, we assumeh(k, t) :={λk(t), t ≥ ,, t < ,()where λ : [,N] → R is a non-negative, non-zero function and  : [, +∞) → R is a con-tinuous function with () = . We have the following result.Ibrahim and Jalab Boundary Value Problems  (2015) 2015:23 Page 9 of 10Theorem . Let h(k, t) be deﬁned in () and () let be satisﬁed. Then the problem–℘y(ψp(℘y μk–))+ σkψp(μk) = λk(μk), k ∈ [,N], ()μ = μN+ = has at least one positive solution μk .Proof By Theorem ., we conclude that () has at least one solution μ∗. If σk =  andμj := mink∈[,N] μk , then we obtain–℘y(ψp(℘y μk–)) ≥ , k ∈ [,N].However, ψp is strictly monotone, and μj – μj– =  and μj+ – μj = . Consequently, wehave a positive solution. Let σk = , we attainψp(℘y μj) ≥ ∣∣℘y μj∣∣p–(μj+ –μj)℘ –∣∣℘y μj–∣∣p–(μj –μj–)℘≥ andσjψp(μj)≥ |μj|p–(μj+ –μj)℘ – |μj–|p–(μj –μj–)℘ ,≥ so μj ≥  and μ ≥ . This completes the proof. In the same manner as Theorem ., we have the following.Theorem . Let h(k, t) be deﬁned in () and let () hold. Then problem () has at leastone positive solution μk .6 ExampleAssume the problem–℘y (μk–) + σkμk =λ + |μk|μk , k ∈ [,N], ()μ = μN+ = .Let h(ζ ) := λ + |ζ |ζ , ζ ∈ R. It is clear that h() = , when λ =  and limζ→+ h(ζ )ζ = +∞.Moreover, we let κ =  > p =  and κ = ⇒ |ζ | ≥ . Thus, we conclude that < H(ζ )≤ ζh(ζ ), ∀|ζ | ≥ ,λ > .Obviously, T ′(μk) = ( ‖μ‖pp –∑Nk=H(k,μk))′ is a Lipschitz continuous functional. Hence inview of Theorem ., problem () has at least two solutions.Ibrahim and Jalab Boundary Value Problems  (2015) 2015:23 Page 10 of 10Now let σk =  and λ = . We then have the following problem:–℘y (μk–) = |μk|μk , k ∈ [,N], ()μ = μN+ = such that (t) := t|t|, t ∈ [,∞). Hence, in view of Theorem ., where inequality () issatisﬁed for σ = , problem () has at least one positive solution.Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsBoth authors jointly worked on deriving the results and approved the ﬁnal manuscript.Author details1Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur, 50603, Malaysia. 2Faculty of Computer Scienceand Information Technology, University of Malaya, Kuala Lumpur, 50603, Malaysia.AcknowledgementsThe authors would like to thank the referees for giving useful suggestions for improving the work. This research issupported by Project No.: RG312-14AFR from the University of Malaya.Received: 13 September 2014 Accepted: 13 January 2015References1. Diaz, JB, Osler, TJ: Diﬀerences of fractional order. Math. Comput. 28, 185-202 (1974)2. Gray, H, Zhang, N: On a new deﬁnition of the fractional diﬀerence. Math. Comput. 50, 513-529 (1988)3. Atici, F, Eloe, P: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137, 981-989 (2009)4. Jalab, HA, Ibrahim, RW: Texture enhancement based on the Savitzky-Golay fractional diﬀerential operator. Math.Probl. Eng. 2013, Article ID 149289 (2013)5. Jalab, HA, Ibrahim, RW: Texture enhancement for medical images based on fractional diﬀerential masks. Discrete Dyn.Nat. Soc. 2013, Article ID 618536 (2013)6. Jalab, HA, Ibrahim, RW: On generalized fractional diﬀerentiator signals. Discrete Dyn. Nat. Soc. 2013, Article ID 795954(2013)7. Jiang, L, Zhou, Z: Three solutions to Dirichlet boundary value problems for p-Laplacian diﬀerence equations. Adv.Diﬀer. Equ. 2008, Article ID 345916 (2008)8. Candito, P, Molica Bisci, G: Existence of two solutions for a second-order discrete boundary value problem. Adv.Nonlinear Stud. 11, 443-453 (2011)9. Molica Bisci, G, Repovs, D: On sequences of solutions for discrete anisotropic equations. Expo. Math. (2014).doi:10.1016/j.exmath.2013.12.00110. Molica Bisci, G, Repovs, D: Nonlinear algebraic systems with discontinuous terms. J. Math. Anal. Appl. 398, 846-856(2013)11. Bian, LH, Sun, HR, Zhang, QG: Solutions for discrete p-Laplacian periodic boundary value problems via critical pointtheory. J. Diﬀer. Equ. Appl. 18, 345-355 (2012)12. Jumarie, G: Fractional Diﬀerential Calculus for Non-Diﬀerentiable Functions: Mechanics, Geometry, Stochastics,Information Theory. Lambert Academic Publishing, Saarbrücken (2013)13. Motreanu, D, Râdulescu, V: Variational and Non-Variational Methods in Nonlinear Analysis and Boundary ValueProblems. Nonconvex Optimization and Applications. Kluwer Academic, Dordrecht (2003)14. Bonanno, G: A critical point theorem via the Ekeland variational principle. Nonlinear Anal. 75, 2992-3007 (2012)

Discrete boundary value problem based on the fractional Gâteaux derivative

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Discrete boundary value problem based on the fractional Gâteaux derivative

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