INVERSE NODAL PROBLEM FOR A p-laplacian STURM-LIOUVILLE EQUATION WITH POLYNOMIALLY BOUNDARY CONDITION

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1 Electroic Joural of Differetial Equatios, Vol. 8 8), No. 4, pp. 9. ISSN: URL: INVERSE NODAL PROBLEM FOR A p-laplacian STURM-LIOUVILLE EQUATION WITH POLYNOMIALLY BOUNDARY CONDITION HIKMET KOYUNBAKAN, TUBA GULSEN, EMRAH YILMAZ Commuicated by Ira Herbst Abstract. I this article, we exted solutio of iverse odal problem f oe-dimesioal p-laplacia equatio to the case whe the boudary coditio is polyomially eigeparameter. To fid the spectral data as eigevalues ad odal parameters, a Prüfer substitutio is used. The, we give a recostructio fmula of the potetial fuctio by usig odal legths. This method is similar to used i [4], ad our results are me geeral.. Itroductio Cosider p-laplacia Sturm-Liouville eigevalue problem y p ) ) = p ) λ qm x) ) y p ), x,.) with the boudary coditios y) =, y ) =, y, λ) fλ)y, λ) =, where p >,.) fλ) = a λ a λ) a m λ) m, a i R, a m, m Z,.3) λ is a spectral parameter ad y p ) = y p ) y. Throughout this study, we suppose that q m x) is a real-valued C[, ]-fuctio defied o the iterval x f each m Z ad yx, λ) deotes the solutio of the problem.)-.). Whe p =, Equatio.) becomes the well-kow Sturm-Liouville equatio. The idea of iverse eigevalue problems with a eigeparameter together with the boudary coditios is of great iterest to may problems of mathematical physics ad mechaics. These type problems have may physical applicatios. F istace, Sturm- Liouville equatio icludig spectral parameter with the boudary coditios arises i heat ad oe-dimesioal wave equatio by seperatio of variables. There are may literatures o these type of problems see [, 3, 6, 7, 8, 9, 8, 9,, 5]). Mathematics Subject Classificatio. 34A55, 34L5, 34L. Key wds ad phrases. Iverse odal problem; Prüfer substitutio; Sturm-Liouville equatio. c 8 Texas State Uiversity. Submitted February, 7. Published Jauary, 8.

2 H. KOYUNBAKAN, T. GULSEN, E. YILMAZ EJDE-8/4 Iverse spectral problem ivolves recoverig differetial equatio from its spectral parameters like eigevalues, mig costats ad odal poits zeros of eigefuctios). These type of problems have bee divided ito two parts; iverse eigevalue problem ad iverse odal problem. They play a imptat role ad also have may applicatios i applied mathematics. Iverse odal problem was firstly studied by McLaughli i 988. She showed that the kowledge of a dese subset of odal poits is sufficiet to determie the potetial fuctio of Sturm-Liouville problem up to a costat [6]. Also, some umerical results about this problem were give i []. Nowadays, may auths have give some iterestig results about iverse odal problems f differet type of operats see [4,, 3, 5, 7,, 6]). I this study, we devote our efft with the iverse odal problem f p-laplacia Sturm-Liouville equatio with boudary coditio polyomially depedet o spectral parameter. Essetially, we give asymptotics of eigeparameters ad recostructio fmula f potetial fuctio. Note that iverse eigevalue problems f differet p-laplacia operats have bee studied by several auths see [, 5,, 4,, ]). The zero set X = {x } j= of the eigefuctio y,mx) crespodig to λ,m is called the set of odal poits. Ad, l = x x is referred as the odal legth of y,m. The eigefuctio y,m x) has exactly odal poits i, ), say = x ),m < x),m < < x),m < x),m =. Let us ow recall some imptat results. Firstly, we eed to itroduce the geeralized sie fuctio S p which is the solutio of the iitial value problem ) S p ) p = p )S p ) p,.4) S p ) =, S p) =. S p ad S p are periodic fuctios which satisfy the idetity S p x) p S px) p =, f ay x R. These fuctios are p-aalogues of classical sie ad cosie fuctios. It is well kow that = is the first zero of S p i positive axis [5]. Lemma. [5]). a) F S p, b) dt = π t p ) p p si π p ), S p) = S p S p p S p. S p S p ) p ) = S p p p )S p p = p S p p = p) p S p p. Usig S p x) ad S px), the geeralized taget fuctio T p x) ca be defied as follows [5] T p x) = S px) S px), f x k ). The remaiig part of this study is gaized as follows; I sectio, we give some asymptotic fmulas f eigevalues ad odal parameters f p-laplacia

3 EJDE-8/4 INVERSE NODAL PROBLEM FOR p-laplacian 3 Sturm-Liouville eigevalue problem.)-.) with boudary coditio polyomially depedet o spectral parameter by usig modified Prüfer substitutio. I sectio 3, we give a recostructio f the potetial fuctio of the problem.)-.).. Asymptotic behavi of some eigeparameters I this sectio, we preset some results o.)-.). Oe of them is the Prüfer s trasfmatio which is oe of the most powerful method f solvig iverse problem. Recall that the Prüfer s trasfmatio f a ozero solutio y of.) takes the fm yx) = Rx)S p λ /p θx, λ)),.) y x) = λ /p Rx)S pλ /p θx, λ)), y x) yx) = S pλ /p θx, λ)) λ/p S p λ /p θx, λ)),.) where Rx) is amplitude ad θx) is the Prüfer variable [3]. Stadard maipulatios [] yield θ x, λ) = q mx) λ Sp p λ /p θx, λ) )..3) Lemma. []). Defie θx, λ ) as i.) ad φ x) = S p pλ /p θx, λ )) p. The, f ay g L, ), φ x)gx)dx =. Theem.. The eigevalues λ,m of the p-laplacia Sturm-Liouville eigevalue problem give i problem.)-.) have the fm λ /p, = a ) p as. p) p λ /p, = a ) p a ) p p) p O p ), f m =, λ /p,m = a ) p a m ) mp p) p q x)dx O ), f m =,.4) p q x)dx q m x)dx O p ), f m 3,.5).6) Proof. Let θ, λ) = f.)-.). Itegratig both sides of.3) with respect to x from to, we obtai By Lemma., θ, λ) = λ q m x)s p pλ /p θx, λ))dx. q m x){spλ p /p θx, λ)) }dx = o), as. p

4 4 H. KOYUNBAKAN, T. GULSEN, E. YILMAZ EJDE-8/4 Hece, we obtai θ, λ) = pλ q m x)dx O λ )..7) Let λ,m be a eigevalue of the problem.)-.). F m =, by.), we have λ /p, R)S pλ /p, θ, λ,)) a λ, R)S p λ /p, θ, λ,) ) =, λ p, = S pλ /p, θ, λ,)) a S pλ /p, θ, λ,)) = T /p p λ, θ, λ,) ). As is sufficietly large, it follows that λ /p, θ, λ,) = Tp λ p ), a By cosiderig.7) ad.8) together, we obtai λ /p, = a ) p p) p = λ p, a oλ p, q x)dx O p ). )..8) F m =, by.), usig the same process as i m =, we ca easily obtai λ /p, R)S pλ /p, θ, λ,)) a λ, a λ, ) )R)S p λ /p, θ, λ,)) =, λ p, a λ, a λ, ) = S pλ Therefe, /p, θ, λ,)) S pλ /p, θ, λ,)) = T pλ /p λ /p, = a ) p a ) p p) p Fially, by.), we have λ /p,mr)s pλ /p,mθ, λ,m )) a λ,m... a m λ,m ) m )R)S p λ /p,mθ, λ,m )) =,, θ, λ,))..9) q x)dx O p ). λ p,m a λ,m a m λ,m ) = S pλ /p,mθ, λ,m )) m S pλ /p,mθ, λ,m )) = T p λ /p θ, λ,m ) ), f m 3, by cosiderig.7) ad.) together, we deduce that λ /p,m = p) p a ) p a m ) mp q m x)dx O ). p.)

5 EJDE-8/4 INVERSE NODAL PROBLEM FOR p-laplacian 5 Theem.3. The odal poits f problem.)-.) satisfy the followig asymptotic estimates: x j, = j j a p j j, q t) p p p p q t)dt ) p Sp pdt O j ), p.) f m =, x = j as. x j, = j j a p p a p j p j, p p p q t) ) p Sp pdt O j ), f m =, p j a p p a m mp mp j p p p q m t) ) p Sp pdt O j ), f m 3, p Proof. Itegratig.3) from to x ad lettig θx, λ) = x = j λ /p,m F m =, from.4), we deduce that λ /p, = a ) p q t)dt q m t)dt j, we have λ /p,m.).3) q m t) S λ pdt. p.4),m p) p q x)dx O p ),.5) ad therefe, we obtai fmula.) by usig.4) ad.5). F m =, from fmula.5), the asymptotic estimate of eigevalues /λ /p, is cosidered as λ /p, = a ) p a ) p p) p ad, we coclude fmula.) by usig.4) ad.6). F m 3, from the fmula.6), it ca easily be show that = λ /p,m a ) p a m ) mp p) p O ), p ad, we obtai fmula.3) by usig.4) ad.7). q x)dx O p ),.6) q m x)dx.7) Theem.4. Asymptotic estimate of the odal legths f the problem.)-.) satisfies lj, = a p ) p p j, x j, p p p q t)dt q t)s p pdt O p ), f m =,.8)

6 6 H. KOYUNBAKAN, T. GULSEN, E. YILMAZ EJDE-8/4 lj, = a p p a p p ) p j, x j, l = a p p a m mp ) p x p p p q t)dt q t)s p pdt O p ), f m =, mp p p p q m t)s p pdt O p ), f m 3. q m t)dt.9).) Proof. F a large N, itegratig.3) o [x, x ] ad usig the defiitio of odal legths, we have λ /p,m l = = x x pλ,m x q m t)s p pdt q m t) Sp p.) ) dt, λ,m x p λ /p,m q m t)s pλ pdt p O ).,m x λ,m F m =, m = ad m 3, we ca easily obtai.8),.9) ad.) by usig the fmulas.5),.6),.7) ad.), respectively. 3. Recostructio of the potetial fuctio I this sectio, we give a explicit fmula f the potetial fuctio by usig the odal legths. The method used i the proof of the theem is similar to classical problems; p-laplacia Sturm-Liouville eigevalue problem ad p-laplacia eergydepedet Sturm-Liouville eigevalue problem see [, 4,, ]). Theem 3.. Let q m x) be a real-valued C[, ]-fuctio o the iterval x. The /p λ q m x) = lim pλ,ml ),m, 3.) f j = j x) = max{j : x < x} ad m Z. Proof. We eed to cosider Theem.3 f the proof. From.), we have pλ /p,m l = pλ,m λ/p,m x q m t)dt pλ/p,m x q m t)s p p p )dt. The, we ca use similar procedure as those i [4] f j = j x) = max{j : x < x} to show λ /p,m q m t)dt q m x), x ad pλ /p,m q m t) Sp p dt, x p)

7 EJDE-8/4 INVERSE NODAL PROBLEM FOR p-laplacian 7 poitwise almost everywhere. Hece, we obtai /p λ q m x) = lim pλ,ml ),m. Theem 3.. Let {l ) : j =,,..., } = be a set of the odal legths of problem.)-.), where q m is a real-valued C[, ]-fuctio. Let us defie F, x) = p) p l ) j, ) p ) p/ q t)dt, f m =. 3.) a F, x) = p) p l ) j, ) F,m x) = p) p l ) ) f m 3. p) p/ a a ) p/ p) p/ a a m ) mp p q t)dt, f m =. 3.3) q m t)dt, 3.4) The {F,m x)} coverges to q m poitwise almost everywhere i L, ), f all cases. Proof. We prove this theem oly f m =. Other cases ca be show similarly. F m =, by the asymptotic fmulas of eigevalues.4) ad odal legths.8), we obtai /p λ, pλ l ) j, ), = pλ, l j, ) p a π )p/ l ) j, l) j, Cosiderig l ) j, = o), as, we have p) p l ) j, ) p a ) p/ q x) poitwise almost everywhere i L, ). q t)dt, q t)dto). Coclusio. I this study, we give some asymptotic estimates f eigevalues, odal parameters ad potetial fuctio of the p-laplacia Sturm-Liouville eigevalue problem.)-.). We show that the obtaied results are the geeralizatios of the classical problem. Ackowledgemets. The auths is deeply idebted to the reviewer, who made remarks which cotributed to the improvemets i the text ad i the trasparecy of the results. Refereces [] Bidig, P.; Drábek, P; Sturm Liouville they f the p-laplacia, Stud. Sci. Math. Hug., 44), ). [] Browe, P. J.; Sleema, B. D.; Iverse odal problems f Sturm Liouville equatios with eigeparameter depedet boudary coditios, Iverse Probl., 4), ). [3] Buteri, S. A.; O iverse spectral problem f o-selfadjoit Sturm Liouville operat o a fiite iterval, J. Math. Aal. Appl., 335), ). [4] Buteri, S. A.; Shieh, C. T.; Icomplete iverse spectral ad odal problems f differetial pecils, Results Math., 6-), ). [5] Che, H. Y.: O geeralized trigoometric fuctios, Master of Sciece, Natioal Su Yatse Uiversity, Kaohsiug, Taiwa, 9).

8 8 H. KOYUNBAKAN, T. GULSEN, E. YILMAZ EJDE-8/4 [6] Elbert, A.; O the half-liear secod der differetial equatios, Acta Math. Hugar., 493-4), ). [7] Eskitascioglu, E. I.; Acil, M.; A iverse Sturm-Liouville problem with a geeralized symmetric potetial, Electro. J. Differ. Equ., 74), 7 7). [8] Freilig, G.; Yurko, V. A.; Iverse problems f Sturm Liouville equatios with boudary coditios polyomially depedet o the spectral parameter, Iverse Probl., 65), 553 ). [9] Guliyev, N. J.; Iverse eigevalue problems f Sturm-Liouville equatios with spectral parameter liearly cotaied i oe of the boudary coditio, Iverse Probl., 4), ). [] Hald, O. H.; McLaughli, J.R.; Solutio of iverse odal problems, Iverse Probl., 53), ). [] Koyubaka, H.; Iverse odal problem f p-laplacia eergy-depedet Sturm-Liouville equatio, Boud. Value Probl., 3:7 3) Erratum: Iverse odal problem f p- Laplacia eergy-depedet Sturm-Liouville equatio, Boud. Value Probl., 4: 4). [] Koyubaka, H.; Paakhov, E. S.; A uiqueess theem f iverse odal problem, Iverse Probl. Sci. Eg., 56), ). [3] Koyubaka, H.; Yilmaz, E.; Recostructio of the potetial fuctio ad its derivatives f the diffusio operat, Z. Naturfsch. A, 633-4), 7-3 8). [4] Law, C. K., Lia, W. C., Wag, W. C.; Iverse odal problem ad the Ambarzumya problem f the p-laplacia, Proc. Roy. Soc. Ediburgh Sect. A Math., 396), ). [5] Law, C. K.; Yag, C. F.; Recostructig the potetial fuctio ad its derivatives usig odal data, Iverse Probl., 43), ). [6] McLaughli, J. R.; Iverse spectral they usig odal poits as data-a uiqueess result, J. Differ. Equatios, 73), ). [7] Ozka, A. S.; Keski, B.; Iverse odal problems f Sturm-Liouville equatio with eigeparameter depedet boudary ad jump coditios, Iverse Probl. Sci. Eg., 38), ). [8] Sadovichii, V. A.; Sultaaev, Y. T.; Akhtyamov, A. M.; Solvability theems f a iverse oself-adjoit Sturm Liouville problem with oseparated boudary coditios, Differ. Equ., 56), ). [9] Walter, J; Regular eigevalue problems with eigevalue parameter i the boudary coditios, Math. Z., 334), ). [] Wag, W. C.; Direct ad iverse problems f oe dimesioal p-laplacia operats, Natioal Su Yat-se Uiversity, PhD Thesis, ). [] Wag, W. C.; Cheg, Y. H.; Lia, W. C.; Iverse odal problems f the p-laplacia with eigeparameter depedet boudary coditios, Math. Comput. Model., 54-), ). [] Wag, Y. P.; Shieh, C. T.; Iverse problems f Sturm Liouville equatios with boudary coditios liearly depedet o the spectral parameter from partial ifmatio, Results Math., 65-), 5-9 4). [3] Yatır, A.; Oscillatio they f secod der differetial equatios ad dyamic equatios o time scales, Master of Sciece, Izmir istitute of Techology, Izmir, 4). [4] Yag, C. F.; Yag, X.; Ambarzumya s theem with eigeparameter i the boudary coditios, Acta Math. Sci., 34), ). [5] Yag, C. F.; Yag, X. P.; Iverse odal problem f the Sturm Liouville equatio with polyomially depedet o the eigeparameter, Iverse Probl. Sci. Eg., 97), ). [6] Yurko, V. A.; Iverse odal problems f Sturm-Liouville operats o star-type graphs, J. Iverse Ill-Posed Probl., 67), ). Hikmet Koyubaka Firat Uiversity, Departmet of Mathematics, 39, Elazıg, Turkey address: hkoyubaka@gmail.com

9 EJDE-8/4 INVERSE NODAL PROBLEM FOR p-laplacian 9 Tuba Gulse Firat Uiversity, Departmet of Mathematics, 39, Elazıg, Turkey address: tubagulse87@hotmail.com Emrah Yilmaz Firat Uiversity, Departmet of Mathematics, 39, Elazıg, Turkey address: emrah3983@gmail.com