Mild solutions for a problem involving fractional derivatives in the nonlinearity and in the non-local conditions

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1 Tatar Advances in Difference Equations :18 RESEARCH Open Access Mild solutions for a problem involving fractional derivatives in the nonlinearity in the non-local conditions Nasser-eddine Tatar Correspondence: tatarn@kfupm. edu.sa Department of Mathematics Statistics King Fahd University of Petroleum Minerals Dhahran Saudi Arabia Abstract A second-order abstract problem of neutral type with derivatives of non-integer order in the nonlinearity as well as in the nonlocal conditions is investigated. This model covers many of the existing models in the literature. It extends the integer order case to the fractional case in the sense of Caputo. A fixed point theorem is used to prove existence of mild solutions. AMS Subject Classification 26A33 34K4 35L9 35L7 35L15 35L7 Keywords: Cauchy problem Cosine family Fractional derivative Mild solutions Neutral second-order abstract problem 1 Introduction In this paper we investigate the following neutral second-order abstract differential problem d [ u t + gt ut u t ] = Aut+f t u t C D α u t t I =[T] dt u = u + p u C D β ut u = u 1 + q u C D γ u t 1 with a b g 1. Here the prime denotes time differentiation C D = a b g denotes fractional time differentiation in the sense of Caputo. The operator A is the infinitesimal generator of a strongly continuous cosine family Ct t ofbounded linear operators in the Banach space X f g are nonlinear functions from R + X X to X u u 1 are given initial data in X. The functions p :[CI; X] 2 X q :[CI; X] 2 X are given continuous functions see the example at the end of the paper. This problem has been studied in case a b g are or 1 see [1-8]. Well-posedness has been established using different fixed point theorems the theory of strongly continuous cosine families in Banach spaces. We refer the reader to [791] for a good account on the theory of cosine families. Fractional non-local conditions are the natural generalization of the integer order non-local conditions as studied by Hernez [5] others. They include the discrete case where the solution is prescribed at some finite number of times. 211 Tatar; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use distribution reproduction in any medium provided the original work is properly cited.

2 Tatar Advances in Difference Equations :18 Page 2 of 12 Time delay is a natural phenomena which occurs in many problems see [1112]. It is caused for instance by the finite switching speed of amplifiers in electronic networks or finite speed for signal propagation in biological networks. We can trace problems with delays back to Volterra who introduced past states in population dynamics. It has been also introduced by Boltzmann in viscoelasticity in the form of a convolution. When there is a dependence on all past states we usually call such a delay a distributed delay. There are in fact several types of delays. The importance of delays has been pointed out by many researchers we are now witnessing a growing interest in such problems. An important class of delayed differential equations or functional differential equations is the class of neutral differential equations. In this type of problems the delayed argument occurs in the derivative of the state variable. This is the case for instance when a growing population consumes more or less food than matured one or when this term appears in the constitutive relationship between the stress the strain. In fact neutral differential equations arise naturally in biology ecology electronics economics epidemiology control theory mechanics [11-18]. More precisely they appear in the study of oscillatory systems electrical networks containing lossless transmission line high-speed computers distributed non-lumped transmission line lossless transmission line terminated by a tunnel diode lumped parallel capacitor [ ] vibrating masses attached to an elastic bar [1112] automatic control neuro-mechanical systems some variational problems Euler equations [141617]. For the sake of simplicity since the case where time delay exists in the function g has been already studied before at least for some types of delays we shall focus on the distributed delay present in the nonlinearity f. We consider the case g prove existence of mild solutions under different conditions on the different data. In particular this work may be viewed as an extension of the work in [6] to the fractional order case. Indeed the work in [6] is concerned with the first-order derivatives whereas here we treat the fractional order case where some difficulties arise because of the non-local nature of the fractional derivatives. In addition to that to the best of the author s knowledge fractional derivatives are introduced here for such problems for the first time. The next section of this paper contains some notation preliminary results needed in our proofs. Section 3 treats the existence of a mild solution in the space of continuously differentiable functions. An example is provided to illustrate our finding. 2 Preliminaries In this section we present some notation assumptions preliminary results needed in our proofs later. Definition 1. The integral Ia+ α hx = 1 x htdt Ɣα a x t 1 α x > a is called the Riemann-Liouville fractional integral of h of order a > when the right side exists.

3 Tatar Advances in Difference Equations :18 Page 3 of 12 Here Γ is the usual Gamma function Ɣz := e s s z 1 ds z >. Definition 2. The fractional derivative of h of order a > in the sense of Caputo is given by C D α a hx = 1 x Ɣn α a In particular C D β a hx = 1 x Ɣ1 β a h n tdt α n+1 x t x > a n =[α]+1. h tdt β x t x > a <β<1. See [19-22] for more on fractional derivatives fractional integrals. We will assume that H1 A is the infinitesimal generator of a strongly continuous cosine family Ct t Î R of bounded linear operators in the Banach space X. The associated sine family St t Î R is defined by Stx := Csxds t R x X. It is known see [781] that there exist constants M 1 ω such that Ct Me ω t t R St St t M e ω s ds t t R. For simplicity we will designate by M Ñ bounds for Ct St oni =[T] respectively. If we define E := x X : Ctx is once continuously differentiable on R} then we have Lemma 1. see [781] Assume that H1 is satisfied. Then t i StX E t Î R ii StE DA t Î R iii d Ctx = AStx x E t R dt iv d2 Ctx = ACtx = CtAx x DA t R. dt2 Lemma 2. see [781] Suppose that H1 holds v : R X a continuously differentiable function qt = q t = St svsds. Then qt Î DA q t = Ct sv sds + Ctv = Aqt+vt. Ct svsds

4 Tatar Advances in Difference Equations :18 Page 4 of 12 Definition 3. A continuously differentiable function u satisfying the integro-differential equation ut =Ct [ u + pu C D β ut ] + St [ u 1 + q u C D γ u t g u + pu C D β ut u 1 + q u C D γ u t ] Ct sg s u s u s ds + St sf s u s C D α u s ds is called a mild solution of problem 1. This definition follows directly from the definition of the cosine family 1 see [67]. 3 Existence of mild solutions In this section we prove existence of a mild solution in the space C 1 I; X. Before we proceed with the assumptions on the different data we recall that E is a Banach space when endowed with the norm x E = x + sup t 1 AStx x Î E see [23]. It is also well-known that ASt :E X is a bounded linear operator. By B r x X we will denote the closed ball in X centered at x of radius r. The assumptions on f g p q are H2 i ft.. : X X X is continuous for a. e. t Î I. t I 2 ii For every x y Î X X the function f.x y :I X is strongly measurable. iii There exist a nonnegative continuous function K f t a continuous nondecreasing positive function Ω f such that f t x y K f t f x + y for t x y Î I X X. iv For each r> the set fi B r X 2 is relatively compact in X. H3 i The function g takes its values in E g : I X X X is continuous. ii There exist a nonnegative continuous function K g t a continuous non-decreasing positive function Ω g two positive constants C 1 C 2 such that gt x y E K g t g x + y gt x y C 1 x + y + C2 for t x y Î I X X. iii The family of functions t gt u v; u v Î B r CI; X} is equicontinuous on I. iv For each r> the set gi B r X 2 is relatively compact in E.

5 Tatar Advances in Difference Equations :18 Page 5 of 12 H4 u +p :[CI; X] 2 E takes its values in E q :[CI; X] 2 X are completely continuous. The positive constants N p N q will denote bounds for u + pu v E qu v respectively. To lighten the statement of our result we denote by l := 1 C 1 max 1 T 1 α Ɣ2 α } A 1 = MN p + Ñ [ u 1 + N q + C 1 u 1 + N p + N q + C2 ] A 2 = N p + M [ u 1 + N q + C 1 u 1 ] + N p + N q + C2 + C2 δ = A 3 = l A 1 T 1 α } 1 + A 2 max 1 A 4 = l 1 M +max 1 T 1 α Ɣ2 α Ɣ2 α } A 5 = l Ñ 1 T + M 1 α } max 1. Ɣ2 α We are now ready to state prove our result. Theorem 1. Assume that H1-H4 hold. If l > max A 4 K g s A 5 K f s } ds < δ ds f s+ g s 3 then problem 1 admits a mild solution u Î C 1 [ T]. Proof. Note that by our assumptions for u v Î C[ T]; the maps u vt :=Ct [ u + pu I 1 β vt ] + St [ u 1 + q u I 1 γ v t g u + pu I 1 β vt u 1 + q u I 1 γ v t ] Ct sg s u s v s ds + St sf s u s I 1 α v s ds t I u vt :=ASt [ u + pu I 1 β vt ] + Ct [ u 1 + q u I 1 γ v t g u + pu I 1 β vt u 1 + q u I 1 γ v t ] g t u t v t ASt sg s u s v s ds + Ct sf s u s I 1 α v s ds t I 5 are well defined map [C[ T]] 2 into C[ T]. These maps are nothing but the right h side of 2 its derivative. We would like to apply the Leray-Schauder alternative [which states that either the set of solutions of 6 below is unbounded or we have a fixed point in D containing zero a convex subset of X provided that the mappings F Ψ are completely continuous]. To this end we first prove that the set of solutions u l v l of u λ v λ = λ u λ v λ u λ v λ <λ<1 6

6 Tatar Advances in Difference Equations :18 Page 6 of 12 is bounded. Then we prove that this map is completely continuous. Therefore there remains the alternative which is the existence of a fixed point. We have from 4 u λ t MN p + Ñ [ u 1 + N q + C 1 u 1 + N p + N q + C2 ] from 5 Then + M + Ñ K g s g uλ s + v λ s ds K f s f u λ s + s 1 α Ɣ2 α sup v λ z ds z s v λ t N p + M [ u 1 + N q + C 1 u 1 + N p + N q + C2 ] + C 1 uλ t + v λ t + C M t I K g s g uλ s + v λ s ds s K f s f u 1 α λ s + Ɣ2 α sup v λ z ds t I. z s u λ t A 1 + M + Ñ K g s g u λ s +max K f s f u λ s +max T 1 α } 1 Ɣ2 α T 1 α } 1 Ɣ2 α % sup v λ z z s ds sup v λ z z s ds t I 7 1 C 1 v λ t A 2 + C 1 u λ t + + M K g s g u λ s +max K f s f u λ s +max 1 T 1 α } 1 Ɣ2 α sup v λ z z s ds T 1 α } sup v λ z ds Ɣ2 α z s t I where A 1 = MN p + Ñ[ u 1 + N q + C 1 u 1 + N p + N q +C 2 ] A 2 = N p + M[ u 1 + N q + C 1 u 1 + N p + N q +C 2 ]+C 2. T 1 α } Taking the sup in the relation 7 max 1 sup in the relation 8 Ɣ2 α adding the resulting expressions we end up with sup z A 1 + M z t +max 1 K g s g λ s ds + Ñ T 1 α Ɣ2 α } A 2 + K f s f λ s ds K g s g λ s ds + M K f s f λ s ds

7 Tatar Advances in Difference Equations :18 Page 7 of 12 where Λz is equal to the expression T 1 C 1 max 1 1 α Ɣ2 α } u λ z +1 C 1 max T 1 α } } λ s = sup u λ z +max 1 v λ z z s Ɣ2 α or simply t λ t A 3 + A 4 K g s g λ s t ds + A 5 With A 3 = l A 1 T 1 α } 1 + A 2 max 1 Ɣ2 α A 4 = l 1 T M 1 α } +max 1 Ɣ2 α } } T 1 1 α Ɣ2 α v λ z K f s f λ s ds t I 9 A 5 = l Ñ 1 T + M 1 α } max 1 Ɣ2 α provided that l := 1 C 1 max T 1 α } 1 >. Ɣ2 α If we designate by l t the right h side of 9 then ϕ λ = A 3 T =: δ Θ l t l t t Î I ϕ λt A 4 K g t g ϕλ t + A 5 K f t f ϕλ t We infer that ϕ λ t δ max A 4 K g t A 5 K f t }[ f ϕλ t + g ϕλ t ] t I. ds t f s+ g s max A 4 K g s A 5 K f s } ds t I. This with 3 shows that Θ l t thereafter the set of solutions of 6 is bounded in [CI; X] 2 : It remains to show that the maps F Ψ are completely continuous. From our hypotheses it is immediate that 1 u vt :=Ct [ u + pu I 1 β vt ] + St [ u 1 + q u I 1 γ vt g u + pu I 1 β vt u 1 + q u I 1 γ v t ] is completely continuous. To apply Ascoli-Arzela theorem we need to check that 1 B 2 r := 1u v :u v B 2 r }

8 Tatar Advances in Difference Equations :18 Page 8 of 12 is equicontinuous on I. Let us observe that 1 u vt + h 1 u vt t Ct + h s Ct s g s u s v s ds t+h + Ct sg s u s v s ds + t St + h s St s f s u s I 1 α v s ds t+h + St sf s u s I 1 α v s ds for t Î I h such that t + h Î I. In virtue of H1 H3 for t Î I ε > given there exists δ > such that Cs + h Csgt s ut s vt s < ε for s Î [ t] u v B 2 r when h < δ. This together with H2 H3 the fact that St is Lipschitzian imply that 1 u vt + h 1 u vt εt + M g 2r +Ñ f r + t+h t rt1 α Ɣ2 α K g sds + N l h f r + t+h t K f sds rt1 α Ɣ2 α t K f sds for some positive constant N l : The equicontinuity is therefore established. On the other h for t Î I assξ Ct -sξ is continuous from [ t] gi X 2 to X [ t] gi X 2 is relatively compact 2u vt := Ct sgs us vsds u v B 2 r X is relatively compact as well in X. AsforF 3 := F - F 1 + F 2 we decompose it as follows k 1 s i+1 3 u vt = Ss Ss i f t s u t s I 1 α v t s ds i=1 s i k 1 s i+1 + Ss i f t s ut s I 1 α vt s ds i=1 s i select the partition s i } k i=1 of [ t] in such a manner that for a given ε > Ss Ss f t s u t s I 1 α vt s <ε

9 Tatar Advances in Difference Equations :18 Page 9 of 12 for u v B 2 r X whens s Î [s i s i+1 ]forsomei = 1... k - 1: This is possible in as much as f t s u t s I 1 α vt s s [ t] u v B 2 r X} is bounded by H2iii the operator S is uniformly Lipschitz on I. This leads to where k 1 3 u vt εb T X+ s i+1 s i cout s i r Ut s i r i=1 := Ss i f t s u t s I 1 α vt s s [ t] u v B 2 r X} cout s i r designates its convex hull. Therefore 3 B 2 r t is relatively compact in X. By Ascoli-Arzela Theorem 3 B 2 r is relatively compact in CI; X consequently F 3 is completely continuous. Similarly we may prove that Ψ is completely continuous. We conclude that F Ψ admits a fixed point in [C[ T]] 2. Remark 1. In the same way we may treat the more general case d [ u t+gt ut u t ] = Aut+f t ut C D α 1 ut... C D α n ut dt u = u + p u C D β 1 ut... C D β m ut u = u 1 + q u C D γ 1 ut... C D γ r ut where a i b j g k 1 i = 1... n j = 1... m k = 1...r. Remark 2. If g does not depend on u t that is for gt ut we may avoid the condition that g must be an E-valued function. We require instead that g be continuously differentiable apply Lemma 2 to to obtain instead of Ct sgs usds Ct sg s usds + Ctg u ASt sgs usds + gt ut in 5. Example As an example we may consider the following problem [ ut t x+gt x ut x u t t x ] = u xx t x t +Ft x ut x C D α ut x t I =[T] x [ π] ut =ut π = t I T u x =u x+ Pus C D β us xds x [ π] T u t x =u 1 x+ Qus C D γ us xds x [ π] 1

10 Tatar Advances in Difference Equations :18 Page 1 of 12 in the space X = L 2 [ π]. This problem can be reformulated in the abstract setting 1. To this end we define the operator Ay = y with domain DA := y H 2 [ π] : y = yπ =}. The operator A has a discrete spectrum with -n 2 n = as eigenvalues z n s = 2/π sinns n = as their corresponding normalized eigenvectors. So we may write Ay = n 2 y z n z n n=1 y DA. Since -A is positive self-adjoint in L 2 [ π] the operator A is the infinitesimal generator of a strongly continuous cosine family Ct t Î R which has the form Cty = cosnty z n z n y X. n=1 The associated sine family is found to be Cty = n=1 sinnt y z n z n y X. n One can also consider more general non-local conditions by allowing the Lebesgue measure ds to be of the form dμs dhs Lebesgue-Stieltjes measures for nondecreasing functions μ h or even more general: μ h of bounded variation that is u x = u x+ u t x = u 1 x+ T T P us C D β usxdμs Q us C D γ usxdηs. These continuous non-local conditions cover of course the discrete cases u x =u x+ u t x =u 1 x+ n m α i ut i x+ β C i D β ut i x i=1 r γ i ut i x+ i=1 i=1 i=1 k λ C i Dγ u t i x which have been extensively studied by several authors in the integer order case. For u v Î C[ T]; X x Î [ π] defining the operators T pu vx := Pus vs xds T qu vx := Qus vs xds gtuvx := Gtxutxvtx f tuvx : = Ftxutxvtx 11

11 Tatar Advances in Difference Equations :18 Page 11 of 12 allows us to write 1 abstractly as d [ u t+gt ut u t ] = Aut+f t ut C D α ut dt u = u + pu C D β ut u = u 1 + qu C D γ ut. Under appropriate conditions on F G P Q which make H2-H4 hold for the corresponding f g p q Theorem 1 ensures the existence of a mild solution to problem 1. Some special cases of this problem may be found in [24-28]. They model some phenomena with hereditary properties. See also [29-33] for some problems with fractional boundary conditions. 4 Competing interests The author declares that they have no competing interests. Acknowledgements The author is very grateful for the financial support provided by King Fahd University of Petroleum Minerals through the project No. IN 17. Received: 21 November 21 Accepted: 29 June 211 Published: 29 June 211 References 1. Benchohra M Ntouyas SK: Existence of mild solutions of second order initial value problems for delay integrodifferential inclusions with nonlocal conditions. Math Bohemica : Benchohra M Ntouyas SK: Existence results for the semiinfinite interval for first second-order integrodifferential equations in Banach spaces with nonlocal conditions. Acta Univ Palacki Olomuc Fac Rer Nat Math 22 41: Benchohra M Ntouyas SK: Existence results for multivalued semilinear functional differential equations. Extr Math : Byszewski L Laksmikantham V: Theorems about the existence uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl Anal : Hernez ME: Existence results for a second order abstract Cauchy problem with nonlocal conditions. Electron J Differ Equ : Hernez EM Henríquez HR McKibben MA: Existence of solutions for second order partial neutral functional differential equations. Integr Equ Oper Theory 28 62: Travis CC Webb GF: Cosine families abstract nonlinear second order differential equations. Acta Math Acad Sci Hung : Travis CC Webb GF: An abstract second order semilinear Volterra integrodifferential equation. SIAM J Math Anal : Fattorini HO: Second Order Linear Differential Equations in Banach Spaces. North-Holl Mathematics Studies North- Holl Amsterdam Travis CC Webb GF: Compactness regularity uniform continuity properties of strongly continuous cosine families. Houston J Math : El sgol tz LE: Introduction to the Theory of Differential Equations with Deviating Arguments. Holden-Day San Francisco; Hale JK: Theory of Functional Differential Equations. Springer-Verlag New York; El sgol tz LE: Qualitative Methods in Mathematical Analysis. Transaction on Mathematical Monographs American Mathematical Society Providence Hughes DK: Variational optimal control problems with delayed arguments. J Optim Theory Appl : Nagumo J Shimura M: Self-oscillation in a transmission line with a tunnel diode. Proc IRE : Rubanik VP: Oscillations of Quasilinear Systems with Retardation. Nauk Moscow; 1969 Russian. 17. Sabbagh LD: Variational problems with lags. J Optim Theory Appl : Shimura M: Analysis of some nonlinear phenomena in a transmission line. IEEE Trans Circ Theory : Miller KS Ross B: An Introduction to the Fractional Calculus Fractional Differential Equations. Wiley New York; Oldham KB Spanier J: The Fractional Calculus. Academic Press New York; Podlubny I: Fractional Differential Equations. In Mathematics in Sciences Engineering. Volume 198. Academic Press San Diego; Samko SG Kilbas AA Marichev OI: Fractional Integrals Derivatives. Theory Applications. Gordon Breach Yverdon Kisyński J: On cosine operator functions some parameter group of operators. Stud Math :93-15.

12 Tatar Advances in Difference Equations :18 Page 12 of Chen W Holm S: Modified Szabo s wave equation models for lossy media obeying frequency power law. J Acoust Soc Am : Fukunaga M Shimizu N: Role of prehistories in the initial value problem of fractional viscoelastic equations. Nonlinear Dyn 24 38: Haddar H Li J-R Matignon D: Efficient solution of a wave equation with fractional-order dissipative terms. J Comput Appl Math : Hedrich KS Filipovski A: Longitudinal creep vibrations of a fractional derivative order rheological rod with variable cross section. Facta Univ Ser Mech Autom Control Robot : Orsingher E Beghin L: Time-fractional telegraph equations telegraph processes with Brownian time. Probab. Theory Relat Fields : Kirane M Tatar N-e: Absence of local solutions to an elliptic system with time-fractional dynamical boundary conditions. Sib Math J : Labidi S Tatar N-e: Blow up for the Euler-Bernoulli beam problem with a fractional boundary condition. Dyn Syst Appl 28 17: Labidi S Tatar N-e: Unboundedness for the Euler-Bernoulli beam equation with fractional boundary dissipation. Appl Math Comp : Labidi S Tatar N-e: Breakdown for a Kirchhoff type beam with a fractional boundary feedback. J Dyn Control Syst : Mbodje B: Wave energy decay under fractional derivative controls. IMA J Math Control Inf : doi:1.1186/ Cite this article as: Tatar: Mild solutions for a problem involving fractional derivatives in the nonlinearity in the non-local conditions. Advances in Difference Equations :18. Submit your manuscript to a journal benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com