Existence of Countably Many Positive Solutions for Nonlinear Boundary Value Problems on Time Scales

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1 Appl. Math. Inf. Sci. 8, No. 5, Applied Mathematic & Information Science An International Journal Exitence of Countably Many Poitive Solution for Nonlinear Boundary Value Problem on Time Scale Abdulkadir Dogan Department of Applied Mathematic, Faculty of Computer Science, Abdullah Gul Univerity, Kayeri, 3839, Turkey Received: 4 Aug. 3, Revied: Nov. 3, Accepted: Nov. 3 Publihed online: Sep. 4 Abtract: In thi paper, we conider the exitence of countably many poitive olution for nonlinear ingular boundary value problem on time cale. By uing the fixed-point index theory and a new fixed-point theorem in cone, the ufficient condition for the exitence of countably many poitive olution are etablihed. Keyword: Time cale, Boundary value problem, Singularity, Poitive olution, Fixed point theorem Introduction The theory of time cale, which ha recently received a lot of attention, wa introduced and developed by Aulbach and Hilger in 988. It ha been created in order to unify continuou and dicrete analyi, and it allow a imultaneou treatment of differential and difference equation, extending thoe theorie to o-called dynamic equation. Further, the tudy of time cale ha led to everal important application, e.g., in the tudy of inect population model, heat tranfer, neural network, phytoremediation of metal, wound healing, and epidemic model. In thi paper, we are intereted in the exitence of countably many poitive olution for ingular multipoint boundary value problem on time cale, φu t at fut, t,t T, ubject to following boundary condition u, ut b i uξ i, where φ : R R i an increaing homeomorphim and poitive homomorphim and φ. ξ i,t T with < ξ < ξ < < ξ < T, and b i atifie b i,t T, < b i <,at :,T T, and ha countably many ingularitie in,t T. The uual notation and terminology for time cale a can be found in,3, will be ued here. A projection φ : R R i called an increaing homeomorphim and poitive homomorphim if the following condition are atified: i if x y, then φx φy, x,y R; ii φ i a continuou bijection and it invere mapping i alo continuou; iii φxyφxφy, x,y R. If the above condition hold, then it implie that φ i homogeneou and generate a p-laplacian operator, i.e., φu u p u, for ome p>. In recent year, there i much attention focued on on the exitence of poitive olution of boundary value problem on time cale, ome author have found many reult; for detail, ee,5,6,7,8,9,,3,9,,,, 3, 4, 5 and the reference therein. But for the exitence of countable many poitive olution for boundary value problem on time cale, few work were done a far a we know 4,6. We would like to mention the reult of Ma et al. 7, Liang and Zhang 5 and Ji et al.. Ma et al. 7 tudied the exitence of monotone poitive olution for the BVP φ p u qt ft,u, t,, u n α i u ξ i, u n β i uξ i, where ξ i, and α i,β i < atify n α i, n β i <.The main tool i the monotone iterative technique. Correponding author abdulkadir.dogan@agu.edu.tr

2 78 A. Dogan: Exitence of Countably Many Poitive Solution for... Liang and Zhang 5 conidered the exitence of countably many poitive olution for ingular BVP ϕ p u at fut, t,, u α i uξ i, ϕu β i ϕu ξ i, where ϕ : R R i a increaing homeomorphim and poitive homomorphim and ϕ, ξ i, with < ξ < ξ < < ξ <, α i,β i,, < α i <, < β i <. They howed that there exit countably many poitive olution by uing the fixed-point index theory and a new fixed-point theorem in cone. Ji et al. found the exitence of countably many poitive olution for a ingular multipoint BVP φ p u tat fut, t,, u α i uξ i, u α i uη i, where φ p p, p>,φ p φ q and p q. They provided ufficient condition for the exitence of countably many poitive olution by uing fixed-point index theory and the Leggett-William fixed point theorem. However, to the bet of our knowledge, no work ha been done for BVP and. The aim of thi paper i to fill the gap in the relevant literature. Throughout the paper, we will uppoe that the following condition are atified: H f :,, i continuou; H There exit a equence {t i } uch that < t i < t i < T, lim i t i t < T, and t,t T. lim t ti,,,..., and < a <. Moreover, at doe not vanih identically on any ubinterval of,t T. The ret of paper i arranged a follow. In Section, we tate ome definition, notation, lemma and prove everal preliminary reult. Section 3 i devoted to the preentation and proof of our main reult. In lat ection 4, we preent an example of a family of function at that atify condition H. Preliminarie In thi ection, we provide ome background material from theory of cone in Banach pace. Definition.. Let E be a real Banach pace. A nonempty, cloed, convex et P E i a cone if it atifie the following two condition: i x P, λ imply λx P; ii x P, x P imply x. Every cone P E induce an ordering in E given by x y if and only if yx P. Definition.. A map α i aid to be a nonnegative continuou concave functional on a cone P of a real Banach pace E if α : P, i continuou, and αtxty tαxtαy for all x,y P and t,. Definition.3. Given a nonnegative continuou functional γ on a cone P of E, for each d > we define the et Pγ,d{x P:γx<d}. The following fixed point theorem will play an important role in the proof of our main reult. Theorem.. 4. Let E be a Banach pace and P E be a cone in E. For r >, define Ω r {u P : u < r}. Aume that T : P Ω r P i a completely continuou operator uch that Tu u for u Ω r ; a If Tu u for u Ω r, then it,ω r,p; b If Tu u for u Ω r, then it,ω r,p. Theorem.. 8. Let P be a cone in a Banach pace E. Let α,β and γ be three increaing, nonnegative and continuou functional on P, atifying for ome c > and M > uch that γu βu αu, u Mγu for all u Pγ, c. Suppoe there exit a completely continuou operator T : Pγ,c P and < a < b < c uch that S γtu<c, for all u Pγ,c; S βtu>b, for all u Pβ,b; S3 Pα,a /, and αtu<a, for all u Pα,a. Then T ha at leat three fixed point u,u, u 3 Pγ,c uch that αu <a<αu, βu <b<βu 3, γu 3 <c. Lemma.. If b i, then for h C ld,t T and h, φu t ht, t,t, 3 u, ut b i uξ i 4 ha the unique olution ut φ hr r A B, 5

3 Appl. Math. Inf. Sci. 8, No. 5, / 79 where A hr r, ξi B b b i φ i φ hr r A. hr r A Proof. Let u be a in 5, taking the delta derivative of 5, we have u tφ hr r A, moreover, we get φu t hr r A, t t taking the nabla derivative of thi expreion yield φu t ht. Routine calculation verify that u atifie the boundary value condition in 4, o that u given in 5 i a olution of 3 and 4. It i eay to ee that BVP φu, u, ut b iuξ i ha only the trivial olution. Thu u in 5 i the unique olution of 3 and 4. The proof i complete. Lemma.. The olution of BVP 3 and 4 atifie ut, for t,t T. Proof. Let Since ϕ φ hr r hr r. hr r hr r, it follow that ϕ. According to Lemma., we get u B b ξi i ϕ ϕ b i b i ϕ ξ i ϕ b i ϕ ϕ b T i ξ i ϕ b i and ut ϕ b ξi i ϕ ϕ b i b i ϕ b i ξi ϕ b i b T i ϕ ξ i ϕ b i b Tξi i ϕ b. i If t,t, we have ut ϕ ϕ b i b i ξi b i b i ξi b i b i ϕ ϕ ϕ ϕ ϕ ϕ ξi b i ϕ b i ϕ b i ϕ ϕ ξ i ϕ b i b T i ξ i ϕ b. i So ut, t,t. The proof i complete. Lemma.3. If u P, then ut θ u, t θ,tθ, T where u up t,t T ut. Proof. Let inf { ξ,t : up utuξ t,t T }. Cae i.,θ. It follow from the concavity of ut that each point on chord between,u and T,uT

4 8 A. Dogan: Exitence of Countably Many Poitive Solution for... i below the graph of ut. Thu, Hence, ut u utu t, t θ,t θ. T ut min t θ,tθ u utu t T u utu T θ T T θ ut θ T T u θ T u, which implie that ut θ T u. Cae ii. θ,t θ. If t θ,, imilarly, we have Thu, ut u uu t, t θ,. ut min t θ, u uu t u uu θ θ u θ u θ T u. If t,t θ, imilarly, Thu, ut u utu t, t,t θ. T ut min t,tθ u utu t T u utu T θ T θ T u T θ ut θ T T u. Therefore, we find ut θ u, t θ,t θ. T Cae iii. T θ,t. Similarly, we have ut u utu t, t θ,t θ. Thu, ut min t θ,tθ u uu t u uu θ θ u θ u θ T u, which yield Thi complete the proof. ut θ u, t θ,t θ. T 3 Exitence of poitive olution Let the Banach pace E C ld,t T,R with norm u up,tt ut and define the cone P E by P {u E : uti a nondecreaing concave and nonnegative function on,t T }. Let θ k < r k < T θ k, we define the nonnegative, increaing, continuou functional γ k,β k, and α k by γ k u max t θ k,r k T utur k, β k u α k u min utur k, t r k,tθ k T max utut θ k. t θ k,tθ k T It i eay to ee that for each u P, γ k u β k u α k u. Moreover, by Lemma.3, for each u P, γ k uur k θ k T u. Define the operator T : P E by Tut φ ar fur r ar fur r b ξi i φ b i ar fur r b i φ ar fur r ar fur r b i ar fur r b i It i eay to ee that Tut, Tu, TuT b ituξ i and.

5 Appl. Math. Inf. Sci. 8, No. 5, / 8 φtu t at fut. Thi how T P P. One may how that T : P P i completely continuou. Lemma 3.. Aume condition H hold. Then there exit a contant θ max{t T : < t < T } atifie θ < a <. θ Furthermore, the function At t t b i φ t t ar r ar r t φ ar r ar r b i t φ ar r ar r b i i a poitive continuou function on t,t t and ha a minimum on t,t t, therefore there exit L > uch that At, t t,t t. Proof. At firt, it i eaily een that At i continuou on t,t t. Next, let A t t A t b i t φ t t ar r ar r, t φ ar r ar r b i t φ ar r b i ar r. Then, from condition H, we have the function A t i trictly monotone decreaing on t,t t and A T t, the function A t i trictly monotone increaing on t,t t and A t. Becaue A t and A t are not equal to zero at the ame time. Therefore the function AtA ta t i poitive on t,t t, which implie Lmin t t,tt At>. The proof i complete. For convenience, we denote by λ /L, { λ / φ ar r ξi b b i φ i } φ ar r. ar r Theorem 3.. Aume that H and H hold. Let {θ k } k be uch that θ k t k,t k,k,,... Let {r k } k and {R k} k be uch that R k < θ k T r k < r k < mr k < R k, mr k MR k, k,,... Furthermore for each natural number k, we aume that f atifie: H3 fu φmr k, for all u θ k T r k,r k, H4 fu φmr k, for all u,r k, where m λ,, M,λ. Then BVP and ha infinitely many olution {u k } k uch that r k u k R k, k,,... Proof. Becaue < t t k < θ k < t k < T, k,,..., then for any k N, u P, by Lemma.3, we get ut θ k T u, t θ k,t θ k. 6 We conider the equence {Ω,k } k open ubet of E defined by and {Ω,k } k of Ω,k {u P: u <r k }, k,,..., Ω,k {u P: u <R k }, k,,... For a fixed k and u Ω,k, by 6 we have r k u up ut t T up ut θ k θ k t Tθ k T u θ k T r k, for all t θ k,t θ k. By condition H3, we have fu φmr k, for all t θ k,t θ k. Sincet,Tt θ k,tθ k, if H hold, we conider three poibilitie: i If ξ t,t t, then for u Ω,k, by H3 and

6 8 A. Dogan: Exitence of Countably Many Poitive Solution for... Lemma 3., we have Tu φ ar fur r ar fur r b ξi i φ b i ar fur r b i φ ar fur r b i ar fur r t φ ξ b i ar fur r t ar fur r t ar fur r b i t φ ar fur r b i ar fur r b i t φ ar fur r b i ar fur r mr k b i t ξ φ t ar r b i ξ t t ar r φ ξ ar r b i ξ ar r b i ξ t φ ξ ar r ξ ar r b i mr k Aξ mr k L>r k u. ii If ξ T t,t, then for u Ω,k, by H3 and Lemma 3., we have Tu b i t t φ Tt ar fur r b i t ar fur r t t b i φ Tt ar fur r b i t ar fur r b i mr k AT t >mr k L>r k u. iii If ξ,t, then for u Ω,k, by H3 and Lemma 3., we have Tu φ ar fur r ar fur r t t ar fur r t φ t ar fur r t t mr k φ ar r t t ar r mr k At mr k L>r k u. Thu, in all cae, an application of Theorem. implie that it,ω,k,p. 7 On the other hand, let u Ω,k, we have ut u R k, by H4 we have fut φmr k, for all t,t.

7 Appl. Math. Inf. Sci. 8, No. 5, / 83 Therefore Tu φ ar fur r ar fur r b ξi i φ b i ar fur r b i φ ar fur r ar fur r b i ar fur r b i φ ar fur r b ξi i φ ar fur r b i φ ar fur r b i MR k φ ar r b ξi i φ ar r b i φ ar r b i R k u. Hence Theorem. implie that it,ω,k,p. 8 Becaue r k < R k for k N, 7 and 8, it follow from the additivity of the fixed point index that it,ω,k \Ω,k,P, for k N. Thu, T ha a fixed point in Ω,k \Ω,k uch that r k u k R k. Since k N wa arbitrary, the proof i completed. For notational convenience, we denote ρ k and η k ρ k φ ar r b ξi i φ ar r b i η k φ ar r rk φ θ k b i θk ar r, θk ar r. Theorem 3.. Suppoe that H and H hold, and let {θ k } k be uch that θ k t k,t k,k,,... Let {a k } k, {b k} k and {c k} k be uch that and c k < a k < θ k T b k < b k < c k, ρ k b k < η k c k, for k,,... Furthermore for each natural number k we aume that f atifie: H5 fu<φ c k ρ k, for all ut T θ k c k ; H6 fu>φ b k η k, for all b k ut T θ k b k ; H7 fu<φ a k ρ k, for all ut T θ k a k. Then the BVP and ha three infinite familie of olution {u k } k, {u k} k and {u 3k } k atifying α k u k <a k < α k u k, β k u k <b k < β k u 3k, γu 3k <c k, for k N. Proof. By the definition of the completely continuou operator T, it i eay to check that T : Pγ k,c k P, for k N. We prove that all the condition of Theorem. are atified. In order to make ue of property i of Theorem., we choe u Pγ k,c k. Then γ k u max t θk,r k T ut ur k c k, thi implie ut c k for t,r k T. If we recall that u T θ γ k k u T θ c k k. Therefore we get ut T γ k c k, t,t T. Then aumption H5 implie fu<φ c k ρ k, t,t T. by

8 84 A. Dogan: Exitence of Countably Many Poitive Solution for... So γ k Tu max t γ k,r k T TutTur k rk φ ar fur r ar fur r b ξi i φ b i ar fur r b i φ ar fur r ar fur r b i ar fur r b i T φ ar fur r b ξi i φ ar fur r b i φ ar fur r b i < c k φ ar r ρ k b ξi i φ ar r b i φ ar r b i c k. Conequently, condition i i atified. Secondly, we how that ii of Theorem. i fulled. For thi we chooe u Pβ k,b k. Then β k u min t rk,tθ k T ut ur k b k, thi mean ut b k, for t r k,t θ k T. Therefore we have u b k, for t r k,t θ k T. Noticing that u T θ k γ k u T θ k β k u T θ k b k, we get By H6 we get So b k ut T θ k b k for t r k,t θ k T. fu>φ b k η k, for t r k,t θ k T. β k Tu min TutTur k t r k,tθ k T > rk φ ar fur r ar fur r b ξi i φ b i ar fur r b i φ ar fur r ar fur r b i ar fur r rk φ θ k b i θk ar fur r θk ar fur r b rk θk k φ ar r η k θ k b k. θk ar r Thu, condition ii i atified. Latly we verify that iii of Theorem. i alo atified. We note that ut a k 4, t T i a member of Pα k,a k and α k u a k 4 < a k. Therefore Pα k,a k. Now let u Pα k,a k. Then α k u max t θk,tθ k T ut ut θ k a k. Thi implie that ut a k for t θ k,t θ k T.

9 Appl. Math. Inf. Sci. 8, No. 5, / 85 Together with u T θ k γ k u T θ k α k u T θ k a k. Then we have ut T θ k a k, t,t T. By H7 we have Therefore α k Tu fu<φ a k ρ k, t,t T. max TutTuT θ k t θ k,tθ k T θk φ ar fur r ar fur r b ξi i φ b i ar fur r b i φ ar fur r ar fur r b i ar fur r b i T φ ar fur r b ξi i φ ar fur r b i φ ar fur r < a k φ ar r ρ k b i b ξi i φ ar r b i φ ar r b i a k. Hence condition iii of Theorem. i atified. Becaue all hypothee of Theorem. are atified, claim follow. If we add the condition of at ft,,, t,t T, to Theorem 3. we can get three infinite familie of poitive olution {u k } k, {u k} k and {u 3k } k atifying 4 Example <α k u k <a k < α k u k, β k u k <b k < β k u 3k, γu 3k <c k, for k N. There exit the function at atifying condition H. Let T and δ π /39/4, t 5 n 6, t n t i 4, n,,... i We conider the function at :,, i given by at n a nt, t,, where nnt n t n, t < t nt n, δt a n t n t /, t n t n t < t n, δtt n /, t n < t t nt n, nnt n t, t nt n n < t. At firt, it i eaily een that t /4</, t n t n,n,,... and note that n 4 π4 n 4 9, t lim t n 5 n 6 i i π π4 9 > 5.

10 86 A. Dogan: Exitence of Countably Many Poitive Solution for... Next, ince π, we have n a n t t n Hence, n δ at t 6 nn n t n t n t n / t n t / tn t n / t t n tt n / δ n t n t n / δ n π δ π δ n a n t t t n t n / n π 3. n Thi implie that condition H hold. Acknowledgement n 6 a n t t <. The author would like to thank the editor and the anoymou referee for their helpful comment and uggetion. The project i upported by Abdullah Gul Univerity Foundation of TurkeyProject No. 5. Reference D.R. Anderon, Exitence of olution for nonlinear multipoint problem on time cale, Dynamic Sytem and Application, 5, M. Bohner, A. Peteron, Dynamic Equation on Time Scale: An Introduction with Application, Birkhauer, Boton, Cambridge, MA,. 3 M. Bohner, A. Peteron, Advance in Dynamic Equation on Time Scale, Birkhauer, Boton, Cambridge, MA, 3. 4 K. Deimling, Nonlinear Functional Analyi, Springer- Verlag, New York, A. Dogan, J.R. Graef, L. Kong, Higher order emipoitone multi-point boundary value problem on time cale, Computer and Mathematic with Application, 6, A. Dogan, J.R. Graef, L. Kong, Higher order ingular multipoint boundary value problem on time cale, Proceeding of the Edinburgh Mathematical Society, 54, A. Dogan, Exitence of three poitive olution for an m-point boundary-value problem on time cale, Electronic Journal of Differential Equation, 3, no.49, A. Dogan, Exitence of multiple poitive olution for p- Laplacian multipoint boundary value problem on time cale, Advance in Difference Equation, 3, no.38, W. Han, Z. Jin, S. Kang, Exitence of poitive olution of nonlinear m-point BVP for an increaing homeomorphim and poitive homomorphim on time cale, Journal of Computational and Applied Mathematic, 33, Z. He, Double poitive olution of three-point boundary value problem for p-laplacian dynamic equation on time cale, Journal of Computational and Applied Mathematic, 8, S. Hilger, Analyi on meaure chain-a unified approach to continuou and dicrete calculu, Reult in Mathematic, 8, D. Ji, Z. Bai, W. Ge, The exitence of countably many poitive olution for ingular multipoint boundary value problem, Nonlinear Analyi, 7, E.R. Kaufmann, Poitive olution of a three-point boundary value problem on a time cale, Electronic Journal of Differential Equation, 8, S. Liang, J. Zhang, Z. Wang, Exitence of countably many poitive olution for n th-order m-point boundary value problem on time cale, Electronic Journal of Differential Equation, 3, S. Liang, J. Zhang, The exitence of countably many poitive olution for nonlinear ingular m-point boundary value problem, Journal of Computational and Applied Mathematic, 4, S. Liang, J. Zhang, The exitence of countably many poitive olution for nonlinear ingular m-point boundary value problem on time cale, Journal of Computational and Applid Mathematic, 3, D. Ma, Z. Du, W. Ge, Exitence and iteration of monotone poitive olution for multipoint boundary value problem with p-laplacian operator, Computer and Mathematic with Application, 5, J.L. Ren, W.G. Ge, B.X. Ren, Exitence of poitive olution for quai-linear boundary value problem, Acta Mathematicae Applicatae Sinica,, in Chinee 9 Y. Sang, H. Su, F. Xu, Poitive olution of nonlinear m-point BVP for an increaing homeomorphim and homomorphim with ign changing nonlinearity on time cale, Computer and Mathematic with Application, 58, Y. Sang, H. Su, Several exitence theorem of nonlinear m-point boundary value problem for p-laplacian dynamic equation on time cale, Journal of Mathematical Analyi and Application, 34, Y.H. Su, Arbitrary poitive olution to a multi-point p- Laplacian boundary value problem involving the derivative on time cale, Mathematical and Computer Modelling, 53,

11 Appl. Math. Inf. Sci. 8, No. 5, / 87 H.R. Sun, Triple poitive olution for p-laplacian m-point boundary value problem on time cale, Computer and Mathematic with Application, 58, H.R. Sun, W.T. Li, Multiple poitive olution for p- Laplacian m-point boundary value problem on time cale, Applied Mathematic and Computation, 8, Y. Yang, F. Meng, Poitive olution of the ingular emipoitone boundary value problem on time cale, Mathematical and Computer Modelling, 5, Y. Zhu, J. Zhu, The multiple poitive olution for p- Laplacian multipoint BVP with ign changing nonlinearity on time cale, Journal of Mathematical Analyi and Application, 344, Abdulkadir Dogan received hi Ph.D. in applied mathematic from Univerity of Wale, U.K. He wa a potdoctoral fellow at Leiceter Univerity, U.K. He purued potdoctoral reearch at Univerity of Tenneee at Chattanooga, USA. He joined Abdullah Gul Univerity, Faculty of Computer Science in applied mathematic in December. He became head of the Applied Mathematic Department in. Dr Dogan reearch interet include finite element method, numerical olution of partial differential equation, boundary value problem for ordinary differential equation, and dynamic equation on time cale. He ha publihed a lot of article in SCI Journal and hi article have been highly cited. He gave many invited talk in variou pretigiou cientific meeting and academic intitution.