Infinitely many solutions for impulsive nonlinear fractional boundary value problems

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1 Heidarkhani et al. Advances in Difference Equations (26) 26:96 DOI.86/s y R E S E A R C H Open Access Infinitely many solutions for impulsive nonlinear fractional boundary value problems Shapour Heidarkhani * AmjadSalari Giuseppe Caristi 2 * Correspondence: s.heidarkhani@razi.ac.ir Department of Mathematics Faculty of Sciences Razi University Kermanshah 679 Iran Full list of author information is available at the end of the article Abstract Based on variational methods critical point theory the existence of infinitely many classical solutions for impulsive nonlinear fractional boundary value problems is ensured. MSC: Primary A8; secondary B7; 58E5; 58E; 26A Keywords: classical solution; infinitely many solutions; fractional differential equation; impulsive condition; variational methods; critical point theory Introduction his paper aims at ensuring the existence of infinitely many classical solutions for the following impulsive nonlinear fractional boundary value problem: td α c ( D α t u(t)) + a(t)u(t)=λf ( t u(t) ) + h ( u(t) ) t t j a.e.t [ ] (D λμ ) ( td α ( c D α t u)) ( (t j )=μi j u(tj ) ) j =...n u() = u()= where α (/2 ] a C([ ]) such that there are two positive constants a a 2 such that < a a(t) a 2 λ >μ f :[] R R is an L -Carathéodory function h : R R is a Lipschitz continuous function with the Lipschitz constant L >i.e. h(x ) h(x 2 ) L x x 2 for every x x 2 R h() = = t < t < < t n < t n+ = ( t D α (c Dα t u))(t j)= td α (c Dα t u)(t+ j ) td α (c Dα t u)(t j ) td α ( t D α (c Dα t u)(t)) (c Dα t u)(t+ j )=lim t t j + (c Dα t u)(t)) I j : R R j =...narecontinuousfunc- td α (c Dα t u)(t j )=lim t t j tions. ( t D α Fractional differential equations (FDEs) are a simplification of ordinary differential equations integration to arbitrary non-integer orders. FDEs have recently established themselves as precious tools in modeling many events in different fields of science engineering. We can also observe plentiful applications in such fields as electrochemistry 26 Heidarkhani et al. his article is distributed under the terms of the Creative Commons Attribution. International License ( which permits unrestricted use distribution reproduction in any medium provided you give appropriate credit to the original author(s) the source provide a link to the Creative Commons license indicate if changes were made.

2 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page 2 of 9 chemistry electromagnetic mechanics biology electricity economics polymer rheology control theory regular variation in thermodynamics signal image processing wave propagation aerodynamics electrodynamics of complex medium blood flow phenomena biophysics viscoelasticity damping etc. (see [ 8]).herehavealsobeenimportant advances in the theory of fractional calculus fractional ordinary partial differential equations recently; for instance see [9 2]. Many researchers have explored the existence of solutions for nonlinear FDEs with various tools such as fixed-point theorems the method of upper lower solutions critical point theory the topological degree theory variational methods for instance see [ 25]. We also cite [26] in which Zhou Peng were concerned with the Navier-Stokes equations with time-fractional derivative of order α ( ). his type of equations can be used to simulate anomalous diffusion in fractal media. hey established the existence uniqueness of local global mild solutions proved the existence regularity of classical solutions. On the other h impulsive differential equations have become important in recent years as mathematical models of phenomena in both the physical the social sciences. For example many biological phenomena involving thresholds bursting rhythm models in medicine biology optimal control models in economics frequency modulated systems do exhibit impulsive effects. For the background applications of the theory of impulsive differential equations to different areas we refer the reader to the classical monograph [27]. For the general aspects of impulsive differential equations we refer the reader to [28 2]. he existence of multiple solutions of impulsive problems has been studied also using the variational methods critical point theory (see []). Both FDEs impulsive differential equations have drawn intense attention from researchers in the last decades due to the numerous applications. he idea that combining these two classes of differential equations may yield an interesting promising object of investigation viz. impulsive FDEs prompted numerous papers. For the recent developments in theory applications of impulsive FDEs we refer the reader to [ 6] the references therein. Impulsive problems for fractional equations have been treated by topological methods in [7 ]. In [ ] based on variational methods critical point theory the authors studied the existence multiplicity of solutions for the problem (D λμ ) in the case h(x) = for all x R. We also cite [2 6] in which fractional systems have been studied. In [5 6]through variational methods critical point theory the existence of multiple solutions for coupled systems of nonlinear fractional differential equations was analyzed. In [] using Ricceri s variational principle the existence of one weak solution for a class of fractional differential systems was argued. In [] employing Ricceri svariational principle theexistence of an infinite number of weak solutions for a class of impulsive fractional differential systems was guaranteed. In [2] using variational methods critical point theory the multiplicity results of solutions for a class of impulsive fractional differential systems was established. Motivated by the researches above this paper employs a smooth version of [7] heorem 2. which is a more precise version of Ricceri s variational principle [8] heorem 2.5. his is undertaken under several hypotheses on the behavior of nonlinear terms at infinity under conditions on f impulsive terms I j j =...n where we demonstrate the existence of definite intervals about λ μ in which the problem (D λμ ) admits a sequence of classical solutions which is unbounded in the space E α that will be introduced in the

3 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page of 9 next section. Moreover some of the consequences of heorem. are outlined. Replacing the conditions at infinity of the nonlinear terms by a similar one at zero the same findings hold besides the sequence of classical solutions strongly converges to zero; see heorem.. wo examplesofapplicationsarepointed out (see Examples..2). For an argument regarding the existence of infinitely many solutions for boundary value problems through Ricceri s variational principle [8] the reader may refer to [9 5]. We also refer the reader to [52 5] in which the existence of solutions to boundary value problems for FDEs has been studied. 2 Preliminaries In this section we will introduce several basic definitions notations lemmas propositions used throughout this paper. Definition 2. ([5]) For a function f defined on [a b] α >theleftright Riemann-Liouville fractional integrals of order α for the function f are defined by ad α t f (t)= Ɣ(α) td α b f (t)= Ɣ(α) t a b t (t s) α f (s)ds t [a b] (s t) α f (s)ds t [a b] while the right-h sides are point-wise defined on [a b] where Ɣ(α) is the gamma function. Definition 2.2 ([5]) Let a b R AC([a b]) be the space of absolutely continuous functions on [a b]. For <α f AC([a b]) left right Riemann-Liouville Caputo fractional derivatives are defined by ad α t f (t) d dt a D α t f (t)= td α b f (t) d dt t D α b d Ɣ( α) dt f (t)= Ɣ( α) t a b c a Dα t f (t) c D α a +f (t):= ad α t f (t)= Ɣ( α) t (t s) α f (s)ds (s t) α f (s)ds t c t Dα b f (t) c D α b f (t):= td α b f (t)= Ɣ( α) a (t s) α f (s)ds b t (s t) α f (s)ds where Ɣ(α) is the gamma function. Note that when α = c a D t f (t) =f (t) c t D b f (t) = f (t) We have the following property of fractional integration. Proposition 2. ([5 8]) We have the following property of fractional integration: b a [ad γ t f (t) ] g(t)dt = b a [td γ b g(t)] f (t)dt γ >

4 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page of 9 provided that f L p ([a b] R N ) g L q ([a b] R N ) p q /p +/q +γ or p q /p +/q =+γ. o create suitable function spaces apply critical point theory to explore the existence of solutions for the problem (D λμ ) we require the following essential notations findings which will be used in establishing our main results. Let < α < p < E αp ( ) be the Banach space which has closure of C ([ ]) with respect to the norm u p E αp = () c a Dα t u(t) p L p () + u p L p ().Itisanestablished fact that E αp ( ) is a reflexive separable Banach space (see [2] Proposition.). In short E α2 =Eα by. the norms in L 2 ( ) C([ ]): u 2 = u(t) 2 dt u L 2 ( ) ( ) u = max u(t) u C [ ]. t [] E α is a Hilbert space with inner product (u v) α = (c Dα t u(t)c Dα t v(t)+u(t)v(t)) dt the norm u 2 α = ( c Dα t u(t) 2 + u(t) 2 )dt.notethatifa C([ ]) there are two positive constants a a 2 sothat<a a(t) a 2 an equivalent norm in E α is u 2 aα = ( c D α t u(t) 2 dt + a(t) u(t) 2) dt. Proposition 2.2 ([2]) Let <α. For u E α we have u α c Ɣ(α +) D α t u. () In addition for 2 < α u α /2 Ɣ(α)(2α ) /2 c D α t u. By () we can take E α with the norm ( /2 u α = c D α t u(t) dt) 2 = c Dα t u u E α in the following. By Proposition 2.2whenα > /2 for every u E α we have where ( /2 u k c D α t u(t) dt) 2 = k u α < k u aα (2) k = α 2 Ɣ(α) 2α.

5 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page 5 of 9 Now we put C := 2( Lk 2 ) C 2 := 2( +Lk 2 ). () We suppose that the Lipschitz constant L > ofthefunctionh satisfies the condition Lk 2 <. Definition 2. Afunction { u u AC ( [ ] ) : tj+ is said to be a classical solution of problem (D λμ )if t j ( c D α t u(t) 2 + } u(t) 2) dt < j =...n td α ( c D α t u(t)) + a(t)u(t)=λf ( t u(t) ) + h ( u(t) ) a.e.t [ ]\{t...t n } the limits t D α (c Dα t u)(t+ j ) td α (c Dα t u)(t j )exist ( td α (c Dα t u))(t j)=μi j (u(t j )) u() = u()=. We have the following definition of a weak solution for the problem (D λμ ). Definition 2. Afunctionu E α is said to be a weak solution of the problem (D λμ ) if for every v E α [( c D α t u(t))( c D α t v(t)) + a(t)u(t)v(t) ] dt + μ j= = λ f ( t u(t) ) v(t)dt + h ( u(t) ) v(t)dt. n ( I j u(tj ) ) v(t j ) Lemma 2. ([] Lemma 2.) he function u E α is a weak solution of (D λμ ) if only if u is a classical solution of (D λμ ). Our basic tool to guarantee the existence of infinitely many classical solutions for the problem (D λμ ) is a smooth version of heorem 2. of [7] which is a more precise version of Ricceri s variational principle [8] which we recall here. his result has relevance for the celebrated three critical points theorem of Pucci Serrin [55 56]. heorem 2. Let X be a reflexive real Banach space let : X R be two Gâteaux differentiable functionals such that is sequentially weakly lower semi-continuous strongly continuous coercive is sequentially weakly upper semi-continuous. For every r > inf X let us put ϕ(r):= inf u ( r) sup v ( r) (v) (u) r (u)

6 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page 6 of 9 γ := lim inf ϕ(r) δ := lim inf ϕ(r). r + r (inf X ) + hen one has: (a) For every r > inf X every λ ] [ the restriction of the functional ϕ(r) I λ = λ to (] r[) admits a global minimum which is a critical point (local minimum) of I λ in X. (b) If γ <+ then for each λ ] [ the following alternative holds: γ either (b ) I λ possesses a global minimum or (b 2 ) there is a sequence {u n } of critical points (local minima) of I λ such that lim (u n)=+. n + (c) If δ <+ then for each λ ] [ the following alternative holds: δ either (c ) there is a global minimum of which is a local minimum of I λ or (c 2 ) there is a sequence of pairwise distinct critical points (local minima) of I λ which weakly converges to a global minimum of. Corresponding to the functions f h I j j =...n we introduce the functions F : [ ] R R H : R RJ j :[] R R j =...n respectively as follows: ξ F(t ξ):= f (t x)dx for all ξ R ξ H(ξ):= h(x)dx for all ξ R J j (x)= x I j (ξ)dξ j =...nforeveryx R. A specific case of our main result is the following theorem. heorem 2.5 Let α (/2 ] t ( ) let f : R R be a non-negative continuous function put F(x)= x f (ξ)dξ for all x R. Assume that F(ξ) F(ξ) lim inf = lim sup =+. ξ + ξ + hen for every continuous function I : R R whose J(x)= x I(ξ)dξ for every x R is a non-positive function satisfying the condition sup x ξ ( J(x)) J := lim <+ ξ

7 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page 7 of 9 for every μ [ μ [ where μ := C k 2 J ( k2 C lim inf ξ + F(ξ) ) the problem td α c ( D α t u(t)) + u(t)=f ( u(t) ) + h ( u(t) ) t t a.e. t [ ] ( td α ( c D α t u)) (t )=μi ( u(t ) ) u() = u() = has an unbounded sequence of classical solutions. Main results First we set A(α):= ( ) 2α 6α 2 9α +6 Ɣ 2 ( α) ( α) 2 (2 α)( 2α). Now we formulate our main result as follows. heorem. Assume that (A) F(t x) for all t [ ] [ ] x R; sup (A2) lim inf x ξ F(tx)dt ξ + hen for each λ ]λ λ 2 [ where λ := λ 2 := (A(α)+ 2 a )C 2 lim sup ξ + C < F(tξ)dt k 2 sup lim inf x ξ F(tx)dt ξ + C (A(α)+ 2 a )k 2 C 2 lim sup ξ + F(tξ)dt. for all continuous functions I j : R R j =...n for which J j (x)= x I j(ξ)dξ j =...n for every x R are non-positive functions satisfying the condition sup n x ξ j= J := lim J j(x) < () ξ for every μ [ μ Jλ [ where μ Jλ := C k 2 J ( λ λ 2 ) the problem (D λμ ) has an unbounded sequence of classical solutions. Proof Fix λ ]λ λ 2 [ assume that I j j =...n are the functions satisfying the condition (). Since λ < λ 2 onehasμ Jλ >.Fixμ [ μ Jλ [putν := λ ν 2 := C λ 2.IfJ C + μ λ λ 2k 2 = clearly ν = λ ν 2 = λ 2 λ ]ν ν 2 [. If J sinceμ < μ J Jλ we obtain λ λ 2 + μk2 J C <so C λ 2 C + μ λ λ 2k 2 J > λnamelyλ < ν 2.Hencesinceλ > λ = ν one has λ ]ν ν 2 [. Now set Q(t x)=f(t x) μ λ n j= J j(x) for all (t x) [ ] R. ake

8 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page 8 of 9 X =E α define on X two functionals by setting for each u X (u):= 2 u 2 aα H ( u(t) ) dt (u)= F ( t u(t) ) dt μ λ n ( J j u(tj ) ). j= It is clear that is a Gâteaux differentiable functional sequentially weakly upper semicontinuous whose Gâteaux derivative at the point u X is the functional (u) X given by (u)v = f ( t u(t) ) v(t)dt μ λ n ( I j u(tj ) ) v(t j ) j= for every v X : X X is a compact operator (see []). Moreover is a Gâteaux differentiable functional of which the Gâteaux derivative at the point u X is the functional (u) X givenby (u)v = [( c D α t u(t))( c D α t v(t)) + a(t)u(t)v(t) ] dt h ( u(t) ) v(t)dt for every v X. Moreover by the sequentially weakly lower semi-continuity of u aα the continuity of H is sequentially weakly lower semi-continuous in X. Now from the facts L ξ h(ξ) L ξ for every ξ R taking (2) () into account for every u X we have C u 2 aα (u) C 2 u 2 aα. (5) Put I λ := λ. Now we are to demonstrate that γ <+ whereγ has been defined in heorem 2.. Let{ξ n } be a real sequence such that ξ n >foralln N ξ n + as n lim n sup x ξ n Q(t x)dt = lim inf sup x ξ Q(t x)dt. ξn 2 ξ + Put r n = C ξn 2 for all n N.Sinceξ k 2 n >r n >foralln N.Nowletu ( r n ) owing to (5) we have C u 2 aα (u)<r n. (6) By (2)(6)wehave u ξ n.hus ( r n ) { u : u ξ n }.

9 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page 9 of 9 Hence since () = () = for every n large enough one has ϕ(r n )= inf u ( r n ) (sup v ( r n ) (v)) (u) r n (u) sup x ξ n Q(t x)dt C n k 2 = sup x ξ n F(t x)dt C n k 2 + Moreover by Assumption (A2) one has sup v ( r n ) (v) r n sup x ξ n (F(t x) μ λ n j= J j(x)) dt C n k 2 μ sup n x ξn j= J j(x) λ C n k 2. lim inf ξ + which implies sup x ξ C k 2 F(t x)dt <+ lim n sup x ξ n F(t x)dt C n k 2 <+. (7) hen regarding ()(7) we have lim n sup x ξ n F(t x)dt C n k 2 + lim n μ sup n x ξn j= J j(x) <+ λ C n k 2 from which follows lim n sup x ξ n (F(t x) μ λ n j= J j(x)) dt C n k 2 <+. herefore Since γ lim inf n + ϕ(r n) lim n sup x ξ n (F(t x) μ λ n j= J j(x)) dt sup x ξ n Q(t x)dt sup x ξ n F(t x)dt + μ ξn 2 ξn 2 λ C n k 2 <+. (8) sup n x ξn j= J j(x) n taking () into account one has lim inf ξ + sup x ξ Q(t x)dt lim inf sup x ξ F(t x)dt + μ ξ + λ J. (9)

10 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page of 9 Moreover J j j =...n are non-positive we have lim sup ξ + Q(t ξ)dt lim sup ξ + F(t ξ)dt. () herefore from (9)() from Assumption (A2) (8) one has λ ]ν ν 2 [ ] (A(α)+ 2 a )C 2 lim sup ξ + ] [. γ F(tξ)dt C k 2 lim inf ξ + sup x ξ F(tx)dt [ For the fixed λtheinequality(8) guarantees that the condition (b) of heorem 2. can be used either I λ has a global minimum or there is a series {u n } of weak solutions of the problem (D λμ ) so that lim n u aα =+. he other step is to investigate that the functional I λ has no global minimum. Since λ < lim sup ξ + F(t ξ)dt (A(α)+ 2 a )C 2 ξ 2 we can consider a real sequence {γ n } a positive constant τ so that γ n + as n λ < τ < F(t γ n )dt (A(α)+ 2 a )C 2 γ 2 n () for each n N large enough. Let {w n } be a sequence in X defined by γ n t ift [ ) w n (t)= γ n ift [ ] (2) γ n ( t) if t ( ]. Obviously one has γ n w ift ( ) n (t)= if t ( ) ift ( ) γ n ( c D α t w(t) ) = (t s) α w (s)ds Ɣ( α) = Ɣ( α) γ n t α γ n ( ) α γ n α ift [ ) α ift [ ] α [( ) α (t ( )) α ] if t ( ]

11 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page of 9 so that w n 2 aα = A(α)γ 2 n + particularly considering (5) it followsthat a(t) wn (t) ( 2 dt A(α)+ 2 a ) γn 2 ( (w n ) A(α)+ 2 a ) C 2 γn 2. () On the other h based on the non-positivity of J j j =...nweobservethat (w n ) F(t γ n )dt. () So from () () ()wehave ( I λ (w n )= (w n ) λ (w n ) A(α)+ 2 a ( <( λτ) A(α)+ 2 a ) C 2 γn 2 ) C 2 γ 2 n λ ( ) F(t γ n )dt for every n N large enough. Hence the functional I λ is unbounded from below it shows that I λ has no global minimum. herefore heorem 2. ensures that there is asequence{u n } X of critical points of I λ so that lim n (u n )=+ whichfrom(5) shows that lim n u n aα =+. SincethecriticalpointsofI λ are the weak solutions of the problem (D λμ ) thanks to Lemma 2. considering that they are classical solutions we have the conclusion. Remark. he condition non-positivity of J j j =...n can be replaced by the following one: (I) I j () = I j (s)s <for all j =...n. In fact by this assumption for every j =...n we have J j (x)= x I j(ξ)dξ forx R. Remark.2 Under the conditions lim inf ξ + sup x ξ F(t x)dt = lim sup ξ + F(t ξ)dt = heorem. ensures that for every λ > for each μ [ J [theproblem(d λμ )admits infinitely many classical solutions. Moreover if J = the result holds for every λ > μ. Now we give an application of heorem.. Example. Let α = 5 9 =n =t = 2 put a n := 2n!(n +2)! 2n!(n +2)!+ b n := (n +)! (n +)!

12 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page 2 of 9 for every n N. Consider the problem 9 td 5 ( c 9 D 5 t u(t) ) + a(t)u(t)=λf ( t u(t) ) + h ( u(t) ) t t a.e.t [ ] ( td ( 9 c 9 D 5 t u ))( ) ( ( )) = μi u 2 2 (5) u() = u() = where a(t)=2+cos t for all t [ ] f (t x)= { 2t(n+)! 2 [(n+)! 2 n! 2 ] (x n!(n+2) ) π 6(n+)! if (t x) [ ] n N [a n b n ] elsewhere h(x)= Ɣ2 ( 9 5 ) ln( + x 2 ) for all x R I 6 (x)=e x (x )(2 x )wherex = min{x} for all x R. Accordingtothedataabovewehavek =Ɣ ( 5 9 ) a =L = Ɣ2 ( 9 5 ) C 8 =.25 C 2 =.75A( 5 9 )=.8259 Ɣ 2 ( 9 ) (n+)! n! f ( x)dx =(n +)! 2 n! 2 for every n N. F(a hen one has lim n ) n + a 2 n F(ξ) lim sup ξ =. herefore lim inf ξ + =lim n + F(b n ) b 2 n sup x ξ F(t x)dt = lim inf t sup x ξ F( x)dt = ξ 2 ξ + ξ 2 =.Solim inf ξ F(ξ) = C (A( 5 9 )+ 2 a lim sup )k 2 C 2 ξ + = F(t ξ)dt ξ 2 Ɣ 2 ( 5 9 ) F( ξ) 5( lim sup 2Ɣ 2 ( )) 9 ξ + ξ 2 2Ɣ 2 ( 5 9 = ) 27( Ɣ 2 ( 9 )). Hence using heorem. since sup x ξ ( J (x)) J := lim =< ξ the problem (5) foreveryλ > 27( Ɣ 2 ( 9 )) 2Ɣ 2 ( 9 5 ) sequence of classical solutions. μ [ + ) has an unbounded Remark. he following condition: (A 2) there exist two sequence {θ n } {η n } with η n >for every n N ( A(α)+ 2 a ) C 2 θn 2 < C k 2 η2 n

13 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page of 9 for all n N lim n + η n =+ such that lim n + sup x η n F(t x)dt F(t θ n )dt C k 2 η 2 n (A(α)+ 2 a )C 2 θ 2 n < lim sup ξ + F(t ξ)dt (A(α)+ 2 a )C 2 is more general than condition (A2) of heorem.. Infactbychoosingθ n =forall n N from (A 2) we obtain (A2). If we assume (A 2) instead of (A2) choose r n = C η 2 n k 2 for all n N by the same arguing as for heorem.weobtain ϕ(r n ) sup v (] r n ]) (v) F(t w n(t)) dt r n 2 w n 2 aα + H(w n(t)) dt t sup x η n F(t x)dt F(t θ n )dt C k 2 η 2 n (A(α)+ 2 a )C 2 θ 2 n where w n (t)isthesameas(2)butγ n replaced by θ n wehavethesameconclusionas in heorem. with the interval ]λ λ 2 [replacedbytheinterval = ] lim sup ξ + F(tξ)dt (A(α)+ 2 a )C 2 k 2 lim n + Here we point out a simple consequence of heorem.. sup x ηn F(tx)dt F(tθ n )dt C ηn (A(α)+ 2 2 a )k 2 C 2 θn 2 Corollary.2 AssumethatAssumption (A) holds. Furthermore suppose that sup (B) lim inf x ξ F(tx)dt ξ + < C ; k 2 F(tξ)dt (B2) lim sup ξ + >(A(α)+ 2 a )C 2 for all continuous functions I j : R R j =...n for which J j (x)= x I j(ξ)dξ j =...n for every x R are non-positive functions satisfy the condition () for every μ [ μ J [ where μ J := C ( k2 lim inf k 2 J C ξ + the problem sup x ξ F(t x)dt ) [. td α c ( D α t u(t)) + a(t)u(t)=f ( t u(t) ) + h ( u(t) ) t t j a.e. t [ ] ( td α ( c D α t u)) ( (t j )=μi j u(tj ) ) j =...n u() = u()= has an unbounded sequence of classical solutions. Remark. heorem 2.5is animmediately consequenceofcorollary.2when μ =. We here give the following consequence of the main result.

14 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page of 9 Corollary. Let f :[] R be a non-negative continuous function let F (x)= x f (ξ)dξ for all x R. Assume that (D) lim inf ξ + F (ξ) (D2) lim sup ξ + F (ξ) <+ ; =+. hen for every b i L ([ ]) for i n with min x [] {b i (x) i n} > with b for any non-negative continuous functions f i : R R with F i (x)= x f i(ξ)dξ for all x R for 2 i n satisfying { max sup ξ R } F i (ξ); 2 i n { min lim inf ξ + } F i (ξ) ξ ;2 i n > 2 for each λ ] [ C k 2 F lim inf (ξ) ξ + for all continuous function I j : R R j n for which J j (x) = x I j(ξ)dξ j n for every x R are non-positive functions satisfy the condition () for every μ [ μ Jλ [ where μ Jλ := C ( λk2 lim inf k 2 J C ξ + the problem ) F (ξ) td α ( c D α t u(t)) + a(t)u(t)=λ n ( ) ( ) b i (t)f i u(t) + h u(t) ( td α ( c D α t u)) ( (t j )=μi j u(tj ) ) j =...n u() = u()= i= t tj a.e. t [ ] has an unbounded sequence of classical solutions. Proof Set F(ξ)= n i= b(t)f i(ξ) for all ξ R.Assumption(D2) along with the condition ensures { min lim inf ξ R } F i (ξ) ξ ;2 i n > 2 lim sup ξ + F(t ξ)dt = lim sup ξ + n i= F i(ξ) b i (t)dt =+. Moreover from the assumption (D) the condition { } max F i (ξ); 2 i n sup ξ R

15 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page 5 of 9 we obtain lim inf ξ + sup ( x ξ F(t x)dt ) F (ξ) b(t)dt lim inf <+. ξ + Hence the conclusion follows from heorem.. Arguingasintheproofofheorem. but using conclusion (c) of heorem 2. instead of (b) one establishes the following result. heorem. AssumethatAssumption (A) holds. Furthermore suppose that sup x ξ F(tx)dt < (E) lim inf ξ + hen for each λ ]λ λ [ where λ := (A(α)+ 2 a )C 2 lim sup ξ + λ := k 2 lim inf ξ + F(tξ)dt C sup x ξ F(tx)dt C (A(α)+ 2 a )k 2 C 2 lim sup ξ + F(tξ)dt. for all continuous functions I j : R R j =...n for which J j (x)= x I j(ξ)dξ j =...n for every x R are non-positive functions satisfy the condition J := lim ξ + sup x ξ G(t x)dt <+ (6) for every μ [ μ Jλ [ where μ Jλ := C ( λk2 sup k 2 J C lim inf x ξ F(tx)dt ξ + ) the problem (D λμ ) has a sequence of pairwise distinct classical solutions which strongly converges to in E α. Proof Fix λ ]λ λ [leti j j =...n are the functions satisfying the condition (6). λ Since λ < λ 2 onehasμ Jλ >.Fixμ ] μ Jλ [setν := λ ν := C.If C + μ λ λ k 2 J J = clearly ν = λ ν = λ λ ]ν ν [. If J sinceμ < μ Jλ onehas so λ λ + μk2 J C < C λ C + μ λ λ k 2 J > λ namely λ < ν.hencerecallingthatλ > λ = ν onehasλ ]ν ν [.

16 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page 6 of 9 Now put Q(t x)=f(t x) μ λ n j= J j(x) for all x R t [ ]. Since sup x ξ Q(t x)dt taking (6) into account one has lim inf ξ + sup x ξ F(t x)dt + μ λ sup x ξ n j= J j(x) sup x ξ Q(t x)dt lim inf sup x ξ F(t x)dt + μ ξ + λ J. (7) Moreover since J j j =...n are non-positive we have lim sup ξ + Q(t ξ)dt lim sup ξ + F(t ξ)dt. (8) herefore from (7)(8) we obtain λ ]ν ν [ ] (A(α)+ 2 a )C 2 lim sup ξ + F(tξ)dt k 2 lim inf ξ + C sup x ξ F(tx)dt [ ]λ λ [. We take X I λ as in the proof of heorem.. Weprovethatδ <+. Forthis purpose let {ξ n } be a sequence of positive numbers such that ξ n + as n + lim n sup x ξ n F(t x)dt <+. ξn 2 Put r n = C ξn 2 for all n N. Let us show that the functional I k 2 λ has no local minimum at zero. For this purpose let {γ n } be a sequence of positive numbers τ >suchthatγ n + as n λ < τ < F(t γ n )dt (A(α)+ 2 a )C 2 γ 2 n (9) for each n N large enough. Let {w n } be a sequence in X defined by (2). So owing to () () (9)weobtain ( I λ (w n )= (w n ) λ (w n ) A(α)+ 2 a ( <( λτ) A(α)+ 2 a ) C 2 γn 2 < ) C 2 γ 2 n λ F(t γ n )dt for every n N large enough. Since I λ () = is not a local minimum of the functional I λ. Hence the part (c) of heorem 2. ensures that there exists a sequence {u n } in X of critical points of I λ such that u n aα asn the proof is complete.

17 Heidarkhani et al. Advances in Difference Equations (26) 26:96 Page 7 of 9 Remark.5 Applying heorem. resultssimilartoremark. Corollaries.2. can be obtained. We end this paper by giving the following example as an application of heorem.. Example.2 Let α =.625 =n =2t = t 2 = 2 a(t) = t+ for all t [ ] f : [ ] (R \{}) R be the function defined by ( ( ( )) ( )) ( ( ( ( )))) x f (x)=2x ln ln ln sin 2 ln ln ln x 2 2 x 2 ( ) ( ( ( ( )))) xln sin ln ln ln x 2 ( ( )) +8xln )(+ln 2 x 2 x 2 x 2 f (t x)= { e t f (x) if(t x) [ ] (R \{}) if(t x) [ ] {} let h(x)= Ɣ2 (.625) arctan x for all x R I 2 (x)= x I 2(x)= 2x for all x R. A direct calculation shows F(t x)= { e t x 2 ln(ln( x 2 )) sin 2 (ln(ln(ln( x 2 )))) + x 2 ln ( x 2 ) if (t x) [ ] (R \{}) if(t x) [ ] {}. Accordingtothedataabovewehavek =2Ɣ (.625) L = Ɣ2 (.625) 2 C =.25>C 2 =.75 lim inf ξ + sup x ξ F(t x)dt F(t ξ)dt = lim sup =+. ξ + Hence using heorem. since J := lim ξ + sup x ξ ( J (x) J 2 (x)) = 2 < the problem td.625 ( c D.625 t u(t) ) + u(t) t + = λf ( t u(t) ) + Ɣ2 (.526) 2 t t 2 a.e. t [ ] ( td.75 ( c D.625 t u ))( ) = μ ( u ( td.75 ( c D.625 t u ))( 2 u() = u() = ( ) = 2μ )) ( u ( )) 2 arctan ( u(t) )

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