EXISTENCE OF SOLUTIONS TO INFINITE ELASTIC BEAM EQUATIONS WITH UNBOUNDED NONLINEARITIES

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1 Electronic Journl of Differentil Eqution, Vol. 17 (17), No. 19, pp ISSN: URL: or EXISTENCE OF SOLUTIONS TO INFINITE ELASTIC BEAM EQUATIONS WITH UNBOUNDED NONLINEARITIES HUGO CARRASCO, FELIZ MINHÓS Abtrct. Thi rticle concern the exitence of unbounded olution to fourthorder boundry-vlue problem on the hlf-line with two-point boundry condition. One-ided Ngumo condition ply pecil role it llow n ymmetric unbounded behvior on the nonlinerity. The rgument re bed on the Schuder fixed point theorem nd lower nd upper olution method. A n ppliction, n exmple i given with non-ordered lower nd upper olution, to prove our reult. 1. Introduction Fourth-order differentil eqution cn model the bending of n eltic bem nd, in thi ene, we refer them bem eqution. They hve received increed interet from everl field of cience nd engineering, either on bounded domin [3, 5, 8,, 1] either on the rel line [1, 7, 1, 13, 17, 18]. We tudy the fully nonliner bem eqution on the hlf line, u (4) (t) = f(t, u(t), u (t), u (t), u (t)), t [, + ), (1.1) where f : [, + ) R 4 R i n L 1 -Crthéodory function, nd the boundry condition re of Sturm-Liouville type, u() = A, u () = B, u () + u () = C, u (+ ) = D, (1.) A, B, C, D R, < nd u (+ ) := lim t + u (t). The non-compctne of the intervl require delicte technique to obtin ufficient condition for the olvbility of boundry vlue problem on the hlf-line. A exmple, we refer the extenion of continuou olution on the correponding finite intervl under digonliztion proce, fixed point theory in ome Bnch pce nd lower nd upper olution method (ee [, 4, 14, ] nd the reference therein). We preent here n pproch bed on n dequte pce of weighted function. Lower nd upper olution method i very dequte technique to del with boundry vlue problem it provide not only the exitence of bounded or unbounded olution but lo their locliztion nd, from tht, ome qulittive dt bout olution, their vrition nd behvior (ee [6, 9, 15, 16, 19]). In thi pper we ue not necerily ordered lower n upper olution, generlizing, in thi wy, 1 Mthemtic Subject Clifiction. 34B1, 34B15, 34B4. Key word nd phre. Hlf line problem; Schuder fixed point theorem; unbounded nd nonordered upper nd lower olution; one-ided Ngumo condition. c 17 Tex Stte Univerity. Submitted Jnury 19, 17. Publihed Augut 1, 17. 1

2 H. CARRASCO, F. MINHÓS EJDE-17/19 the et of dmiible lower nd upper function. A fr we know, it i the firt time where uch function re pplied to boundry vlue problem defined on the hlf line, to obtin unbounded olution. An importnt tool i the Ngumo condition, ueful to get priori etimte on ome derivtive of the olution. The uul growth condition of the Ngumo type ued in the literture i bilterl one. However the me etimtion hold with imilr one-ided umption, which llow unbounded nonlineritie in boundry vlue problem. Therefore, it generlize the two-ided condition, it i proved in [8, 1]. In hort, thi work h the following noveltie relted to the exitent literture in thi field: The nonlinerity f i n L 1 -Crthéodory function, llowing dicontinuitie in time; From the unilterl Ngumo growth condition umed on f, eqution (1.1) cn del with unbounded nonlineritie; The lower nd upper olution do not need to be well ordered, or even ordered, nd, moreover, their boundry condition re more generl, mking eier to obtin lower nd upper olution to the problem. The non-compctne of the ocited opertor i overcome by conidering n dequte pce of weighted function. The pper i orgnized it follow: In Section ome uxiliry reult re defined uch the pce, the weighted norm, the unilterl Ngumo condition nd lower nd upper olution to be ued. Section 3 contin the min reult: n exitence nd locliztion theorem, where it i proved the exitence of olution, nd ome bound on the firt nd econd derivtive well. Finlly, n exmple, which i not covered by the exitent reult, how the pplicbility of the min theorem.. Definition nd preliminry reult In thi work we conider the pce X = x C 3 [, + ) : lim t + x (i) (t) exit in R, i =, 1,, 3} 1 + t3 i with the norm x X := mx x, x, x, x }, where ω (i) = t<+ ω(i) (t), i =, 1,, t3 i It cn be proved tht (X, X ) i Bnch pce (ee [16]). The following definition etblihe the umption umed on the nonlinerity. Definition.1. A function f : [, + ) R 4 R i clled n L 1 -Crthéodory function if it tifie: (i) for ech (x, y, z, w) R 4, t f(t, x, y, z, w) i meurble on [, + ); (ii) for lmot every t [, + ), (x, y, z, w) f(t, x, y, z, w) i continuou in R 4 ; (iii) for ech ρ >, there exit poitive function ϕ ρ L 1 [, + ) uch tht for ll (x, y, z, w) R 4 with (x, y, z, w) X < ρ, then f(t, x, y, z, w) ϕ ρ (t),.e. t [, + ).

3 EJDE-17/19 INFINITE ELASTIC BEAM EQUATIONS 3 Solution of the liner problem ocited with (1.1)-(1.) re defined with Green function, which cn be obtined by tndrd clculu. Lemm.. Let t 3 η L 1 [, + ). Then the liner boundry vlue problem u (4) (t) + η(t) =, t [, + ), (.1) with boundry condition (1.), h unique olution in X. Moreover, thi olution cn be expreed where u(t) = A + Bt + C D t + D 6 t3 + G(t, ) = 3 6 t + t t, t t + t3 6, t < +. G(t, )η()d (.) To pply fixed point theorem it i importnt to hve n priori etimtion for u. In the literture thi bound i obtined from bilterl Ngumo-type growth. In thi pper it i ued more generl one-ided Ngumo condition, which llow unbounded nonlineritie on (1.1) (for more detil ee [11, 1, 3]. Remrk tht, it i proved in [1] function cn verify n unilterl Ngumo condition but not two-ided one. Let γ i, Γ i C[, + ), γ i (t) Γ i (t), i =, 1, nd define the et E = (t, x, x 1, x, x 3 ) [, + ) R 4 : γ i (t) x i Γ i (t), i =, 1, }. Definition.3. An L 1 -Crthéodory function f : E R i id to tify the one-ided Ngumo-type growth condition in E if it tifie either f(t, x, y, z, w) ψ(t)h( w ), (t, x, y, z, w) E, (.3) or f(t, x, y, z, w) ψ(t)h( w ), (t, x, y, z, w) E, (.4) for ome poitive continuou function ψ, h, nd ome ν > 1, uch tht ψ()d < +, t<+ ψ(t)(1 + t) ν < +, Next lemm provide n priori bound. d = +. (.5) h() Lemm.4. Let f : [, + ) R 4 R be n L 1 -Crthéodory function tifying (.3), or (.4), nd (.5) in E. Then for every r > there exit R > (not depending on u) uch tht every u olution of (1.1), (1.) tifying γ (t) u(t) Γ (t), γ 1 (t) u (t) Γ 1 (t), γ (t) u (t) Γ (t), (.6) for t [, + ), tifie u < R. Proof. Let u be olution of (1.1), (1.) uch tht (.6) hold. Conider r > uch tht r > mx C Γ (), C γ } (), D. (.7) By thi inequlity we cnnot hve u (t) > r for ll t [, + ), becue u () = C u () mx C Γ (), C γ } () < r (.8) nd u (+ ) = D < r.

4 4 H. CARRASCO, F. MINHÓS EJDE-17/19 If u (t) r for ll t [, + ), tking R > r/ the proof i complete u = t<+ u (t) r < R. If there exit t (, + ) uch tht u (t) > r, then by (.5) we cn tke R > r uch tht R h() d > M mx Γ (t) ν M t ν 1, M γ (t) ν } 1 inf t<+ 1 + t ν 1 r t<+ Γ (t) (1+t) ν inf t<+ γ (t) (1+t) ν. with M := t<+ ψ(t)(1+t) ν nd M 1 := t<+ Aume tht growth condition (.3) hold. By (.7), poe tht there re, t + (, + ) uch tht u ( ) = r, u (t) > r for ll t (, t + ]. Then u (t +) u () h() d = = u () h(u ()) u(4) ()d f(, u(), u (), u (), u ()) h(u u ()d ()) ψ() u () d M + ( u () ) νu () = M + (1 + ) ν (1 + ) ( u (t + ) = M (1 + t + ) ν u ( ) (1 + ) ν + ( M M 1 + < R r h() d. t<+ Γ (t) 1 + t u () (1 + ) ν d d 1+ν + νu () ) (1 + ) 1+ν d ν ) (1 + ) ν d So u (t + ) < R nd, t + re rbitrry in (, + ), we hve u (t) < R for ll t [, + ). By the me technique in (.8), nd conidering t nd uch tht u ( ) = r, u (t) < r for ll t [t, ], it cn be proved tht u (t) > R for ll t [, + ) nd, therefore u < R/ < R for ll t [, + ). If f tifie (.4), following imilr rgument, the me concluion i chieved. Next reult will ply key role to pply fixed-point theorem. Lemm.5 ([1, Theorem 6..]). A et M X i reltively compct if the following three condition hold: (1) ll function from M re uniformly bounded; () ll function from M re equicontinuou on ny compct intervl of [, + ); (3) ll function from M re equiconvergent t infinity, tht i, for ny given ɛ >, there exit t ɛ > uch tht u(i) (t) lim 1 + t3 i t + u (i) (t) < ɛ, for ll t > tɛ 1 + t 3 i, x M nd i =, 1,, 3.

5 EJDE-17/19 INFINITE ELASTIC BEAM EQUATIONS 5 The function conidered lower nd upper olution for the initil problem re defined it follow. Definition.6. Given < nd A, B, C, D R, function α C 4 [, + ) X i id to be lower olution of problem (1.1),(1.) if α (4) (t) f(t, α(t), α (t), α (t), α (t)), t [, + ), nd α () B, α () + α () C, α (+ ) < D, (.9) where α(t) := α(t) α() + A. A function β i n upper olution if it tifie the revered inequlitie with β(t) := β(t) β() + A. We point out tht α nd β need not to be well ordered or even ordered. 3. Min reult In thi ection we prove the exitence of t let one olution for problem (1.1), (1.). Theorem 3.1. Let f : [, + ) R 4 R be n L 1 -Crthéodory function, nd α, β lower nd upper olution of (1.1), (1.), repectively, uch tht nd α (t) β (t), t [, + ). (3.1) If f tifie the one-ided Ngumo condition (.3), or (.4), in the et E = (t, x, y, z, w) [, + ) R 4 : α(t) x β(t), α (t) y β (t), } α (t) z β (t), f(t, α(t), α (t), z, w) f(t, x, y, z, w) f(t, β(t), β (t), z, w), (3.) for (t, z, w) fixed nd α(t) x β(t), α (t) y β (t), then problem (1.1), (1.) h t let olution u C 4 (, + ) X nd there exit R > uch tht α(t) u(t) β(t), α (t) u (t) β (t), α (t) u (t) β (t), R < u (t) < R, t [, + ). Proof. Integrting (3.1) nd (.9), we hve α (t) β (t) nd α(t) β(t), for t [, + ). Therefore we cn conider the modified nd perturbed eqution u (4) (t) = f(t, δ (t, u), δ 1 (t, u ), δ (t, u ), δ 3 (t, u )) t u (t) δ (t, u ) 1 + u (t) δ (t, u ), t [, + ), (3.3) where the function δ j : [, + ) R R, j =, 1,, 3 re given by β(t), x > β(t) δ (t, x) = x, α(t) x β(t) α(t), x < α(t), β (i) (t), y i > β (i) (t) δ i (t, y i ) = y i, α (i) (t) y i β (i) (t) i = 1,, α (i) (t), y i < α (i) (t),

6 6 H. CARRASCO, F. MINHÓS EJDE-17/19 R, w > R δ 3 (t, w) = w, R w R R, w < R. For clerne, we do the proof in everl tep. Step 1: Every olution of (3.3), (1.) tifie α (t) u (t) β (t) for ll t [, + ). Let u be olution of the modified problem (3.3), (1.) nd poe, by contrdiction, tht there exit t (, + ) uch tht α (t) > u (t). Therefore inf t<+ (u (t) α (t)) <. By (.9) thi infimum cn not be ttined t +. If inf t<+ (u (t) α (t)) := u ( + ) α ( + ) <, then the following contrdiction i chieved u ( + ) α ( + ) = C u () = 1 (u () α ()) <. If there i (, + ) then, we cn define + α () C min t<+ (u (t) α (t)) := u ( ) α ( ) <, with u ( ) = α ( ) nd u (4) ( ) α (4) ( ). Therefore by (3.) nd Definition.6 we get the contrdiction u (4) ( ) α (4) ( ) = f(, δ (, u( )), δ 1 (, u ( )), δ (, u ( )), δ 3 (, u ( ))) + 1 u ( ) δ (, u ( )) 1 + t 1 + u ( ) δ (, u ( )) α(4) ( ) = f(, δ (, u( )), δ 1 (, u ( )), α ( ), α ( )) + 1 u ( ) α ( ) 1 + t 1 + u ( ) α ( ) α(4) ( ) u ( ) α ( ) 1 + u ( ) α ( ) < t So u (t) α (t), t [, + ). Anlogouly it cn be hown tht u (t) β (t), t [, + ). A α () B β () nd u () = B, integrting on [, + ), α(t) α() = α (t) α () = α ()d β ()d = β (t) β (), u ()d = u (t) B α (t) α (t) α () + B u (t) β (t) β () + B β (t), α ()d u ()d = u(t) A α(t) u(t) β(t). β ()d = β(t) β(),

7 EJDE-17/19 INFINITE ELASTIC BEAM EQUATIONS 7 Step : Problem (3.3), (1.) h t let one olution. Let u define the opertor T : X X by with T u(t) = A + Bt + C D t + D 6 t3 G(t, )F (u())d, F (u()) := f(, δ (, u), δ 1 (, u ), δ (, u ), δ 3 (, u )) u () δ (, u ) 1 + u () δ (, u ). By Lemm., the fixed point of T re olution of problem (3.3), (1.). So it i ufficient to prove tht T h fixed point. (i) T : X X i well defined. Let u X. A f i n L 1 -Crthéodory function, o, for ρ > mx u X, α X, β X }, we obtin F (u()) d φ ρ () u () δ (, u ) 1 + u () δ (, u ) d φ ρ () + 1 d < +, 1 + tht i, F i lo n L 1 -Crthéodory function. By the Lebegue Dominted Convergence Theorem, nd nlogouly for (T u)(t) lim t t 3 = D 6 (T u) (t) lim t + = D t, lim t + lim t + (T u) (t) 1 + t G(t, ) 1 + t 3 F (u())d F (u())d < +, nd lim t + (T u) (t). Therefore T u X. (ii) T i continuou. For ny convergent equence u n u in X, there exit ρ > uch tht n u n X < ρ, we hve T u n T u X = mx T u n T u, (T u n ) (T u), } (T u n ) (T u), (T u n ) (T u) where qudn +, M = mx t<+ t< t M F (u n ()) F (u()) d, G(t, ), 1 + t3 t<+ } G(t, ) t, 1. 1 G(t, ) 1 + t, t

8 8 H. CARRASCO, F. MINHÓS EJDE-17/19 (iii) T i compct. Let B X be ny bounded ubet, therefore there i R > uch tht u X < R for ll u B. For ech u B, one h T u = t<+ + T u(t) A + B + C D + D 1 + t3 t<+ G(t, ) 1 + t 3 F (u()) d A + B + C D + D + M(φ R () ) < +. By the me rgument it cn be proved tht (T u) (i) < +, for i = 1,, 3, nd, therefore, T u X = mx T u, (T u), (T u), (T u) } < +, tht i, T B i uniformly bounded. T B i equicontinuou, becue, for L > nd t 1, t [, L], we hve T u(t 1) 1 + t 3 1 T u(t ) 1 + t 3 A + Bt 1 + C D t 1 + D 6 t t 3 A + Bt + C D t + D 6 t t 3 + G(t 1, ) 1 + t 3 G(t, ) F (u()) d t 3 A + Bt 1 + C D t 1 + D 6 t t G(t 1, ) 1 + t 3 1 t 1 t. By the me technique one how tht (T u)(i) (t 1 ) 1 + t 3 i 1 A + Bt + C D t + D 6 t3 1 + t 3 G(t, ) ( 1 + t 3 φ R () + 1 ) d, 1 + (T u)(i) (t ), 1 + t 3 i uniformly, for i = 1,, 3, t 1 t. Moreover T B i equiconvergent t infinity, becue T u(t) 1 + t 3 lim T u(t) t t 3 C D A + Bt + t + D 6 t3 1 + t 3 D 6 + C D A + Bt + t + D 6 t3 1 + t 3 D 6 t +, nd nlogouly for + G(t, ) 1 + t G(t, ) 1 + t F (u()) d (T u)(i) (t) (T u) (i) (t) 1 + t 3 i lim, t +. t t 3 i ( φ ρ1 + 1 ) d, 1 + So, by Lemm.5, the et T B i reltively compct. A T i completely continuou then by Schuder Fixed Point Theorem, T h t let one fixed point u X.

9 EJDE-17/19 INFINITE ELASTIC BEAM EQUATIONS 9 4. Exmple Conider the fourth-order differentil eqution (1 + t )u (4) (t) = u(t) u (t) 6 e u (t) e t (6t + u (t)), t >, (4.1) with the boundry condition u() = A, u () =, u () + u () =, u (+ ) = D, (4.) where A, 1 3 < nd < D < 6. We remrk tht the bove problem i prticulr ce of (1.1), (1.) with B = C = nd f(t, x, y, z, w) = x w 6 ew e t (6t + z) 1 + t. (4.3) Moreover the function α(t) nd β(t) = t 3 +t 1 re, repectively, non ordered lower nd upper olution for (4.1), (4.)), with α(t) = A nd β(t) = t 3 + t + A, f tifie the Ngumo condition (.3) with ψ(t) = 1, 1 < ν <, h( w ) 1, 1 + t on E = (t, x, y, z, w) [, + ) R 4 : A x t 3 + t + A, y 3t + t, } z 6t +, nd tifie the umption of Theorem 3.1. Therefore, there i t let non trivil olution u of (4.1), (4.), nd R >, uch tht A u(t) t 3 + t + A, u (t) 3t + t, u (t) 6t +, u R, t [, + ). We remrk tht, thi olution i unbounded nd, from the loction prt, we notice tht u i nondecreing nd convex. It i importnt to tre tht the nonlinerity (4.3) doe not tify the uul two-ided Ngumo-type condition. In fct, if there exit ψ, h C(R +, R+ ) tifying with f(t, x, y, z, w) ψ (t)h ( w ), (t, x, y, z, w) E, h () d = +, then, in prticulr, f(t, x, y, z, w) ψ (t)h ( w ), nd, for t [, + ), x = 1, y 3t + t, z = 6t +, nd w R, w 6 ew f(t, 1, y, 6t +, w) = 1 + t ψ (t)h ( w ), For ψ (t) = 1/(1 + t ) we hve w 6 e w h ( w ) nd the following contrdiction hold + > + ( 6)e d d = +. h () Acknowledgment. Thi work w ported by Ntionl Found through FCT- Fundção pr Ciênci e Tecnologi prt of project : SFRH/BSAB/11446/16. The uthor re grteful to the nonymou referee for hi/her comment nd uggetion.

10 1 H. CARRASCO, F. MINHÓS EJDE-17/19 Reference [1] R. P. Agrwl, D. O Regn; Infinite Intervl Problem for Differentil, Difference nd Integrl Eqution, Kluwer Acdemic Publiher, Glgow 1. [] R. P. Agrwl, D. O Regn; Non-liner boundry vlue problem on the emi-infinite intervl: n upper nd lower olution pproch, Mthemtik 49, no. 1-, () [3] D. Anderon, F. Minhó; A dicrete fourth-order Lidtone problem with prmeter, Applied Mthemtic nd Computtion, 14 (9) [4] C. Bi nd C. Li; Unbounded upper nd lower olution method for third-order boundry-vlue problem on the hlf-line, Electronic Journl of Differentil Eqution, 119 (9), 1 1. [5] A. Cbd, G. Figueiredo; A generliztion of n extenible bem eqution with criticl growth in R N, Nonliner Anl.-Rel World Appl., (14), [6] A. Cbd, F. Minhó; Fully nonliner fourth order eqution with functionl boundry condition, J. Mth. Anl. Appl., Vol. 34/1 (8), [7] S.W. Choi, T.S. Jng; Exitence nd uniquene of nonliner deflection of n infinite bem reting on non-uniform nonliner eltic foundtion, Boundry Vlue Problem 1, 1:5, 4 pge. [8] J. Filho, F. Minhó; Exitence nd loction reult for hinged bem with unbounded nonlineritie, Nonliner Anl., 71 (9), [9] J. Gref, L. Kong, F. Minhó; Higher order boundry vlue problem with φ-lplcin nd functionl boundry condition, Computer nd Mthemtic with Appliction, 61 (11), [1] M. R. Groinho, F. Minhó, A. I. Snto; A note on cl of problem for higher order fully nonliner eqution under one ided Ngumo type condition, Nonliner Anl., 7 (9), [11] M. R. Groinho, F. Minhó, A. I. Snto; A third-order boundry vlue problem with oneided Ngumo condition, Nonliner Anl., 63 (5), [1] T. S. Jng, H. S. Bek, J. K. Pik; A new method for the non-liner deflection nlyi of n infinite bem reting on non-liner eltic foundtion, Interntionl Journl of Non-Liner Mechnic, 46 (11), [13] T. S. Jng; A new emi-nlyticl pproch to lrge deflection of Bernoulli Euler-v.Krmn bem on liner eltic foundtion: Nonliner nlyi of infinite bem, Interntionl Journl of Mechnicl Science 66 (13) 3. [14] H. Lin, P. Wng, W. Ge; Unbounded upper nd lower olution method for Sturm-Liouville boundry vlue problem on infinite intervl, Nonliner Anl., 7 (9), [15] H. Lin, J. Zho; Exitence of Unbounded Solution for Third-Order Boundry Vlue Problem on Infinite Intervl, Dicrete Dynmic in Nture nd Society, 1, Article ID , 14 pge. [16] H. Lin, J. Zho, R. P. Agrwl; Upper nd lower olution method for nth-order BVP on n infinite intervl, Boundry Vlue Problem 14, 14:1, 17 pge. [17] X. M, J. W. Butterworth, G. C. Clifton; Sttic nlyi of n infinite bem reting on tenionle Pternk foundtion, Europen Journl of Mechnic A/Solid 8 (9), [18] P. Mhehwrin, S. Khtri; Nonliner nlyi of infinite bem on grnulr bed-tone column-reinforced erth bed under moving lod, Soil nd Foundtion, 5 (1), [19] F. Minhó; Loction reult: n under ued tool in higher order boundry vlue problem, Interntionl Conference on Boundry Vlue Problem: Mthemticl Model in Engineering, Biology nd Medicine, Americn Intitute of Phyic Conference Proceeding, 114 (9), [] F. Minhó, T. Gyulov, A. I. Snto; Lower nd upper olution for fully nonliner bem eqution, Nonliner Anl., 71 (9), [1] Y. P. Song; A nonliner boundry vlue problem for fourth-order eltic bem eqution, Boundry Vlue Probl., 191, 14. [] B. Yn, D. O Regn, R. P. Agrwl; Unbounded olution for igulr boundry vlue problem on the emi-infinite intervl: Upper nd lower olution nd multiplicity, J. Comput. Appl. Mth., 197 (6)

11 EJDE-17/19 INFINITE ELASTIC BEAM EQUATIONS 11 [3] J. Zhng, W. Liu, J. Ni, T. Chen; The exitence nd multiplicity of olution of three-point p-lplcin boundry vlue problem with one-ided Ngumo condition, Journl of Applied Mthemtic nd Computing, 4, Iue 1, (7) 9. Hugo Crrco Deprtmento de Mtemátic, Ecol de Ciênci e Tecnologi, Univeridde de Évor, Ru Romão Rmlho, 59, Évor, Portugl E-mil ddre: hugcrrco@gmil.com Feliz Minhó Deprtmento de Mtemátic, Ecol de Ciênci e Tecnologi, Centro de Invetigção em Mtemátic e Aplicçõe (CIMA-UE), Intituto de Invetigção e Formção Avnçd, Univeridde de Évor, Ru Romão Rmlho, 59, Évor, Portugl E-mil ddre: fminho@uevor.pt