Existence Results for Semipositone Boundary Value Problems at Resonance

Transcription

1 Advances in Dynamical Systems and Applications ISSN , Volume 13, Number 1, pp ) Existence Results for Semipositone Boundary Value Problems at Resonance Fulya Yoru Deren, Nuet Ayut Hamal and Tugba Senli Cerdi Ege University Department of Mathematics 351, Bornova, Izmir, Turey Abstract This wor is concerned with the existence of three positive solutions for semipositone boundary value problems of three-point boundary conditions. The analysis is based upon a fixed point theorem on a cone. AMS Subject Classifications: 34B1, 34B18. Keywords: Positive solution, semipositone, fixed point theorem. 1 Introduction Nonlinear boundary value problems BVPs) for ordinary differential equations ODEs) with nonlocal boundary conditions BCs) have been well studied over the past decades. The study of nonlocal BCs for ODEs goes bac, as far as we now, to Picone [13] and have been widely investigated during the years by many authors [, 1, 15]. The research of nonlinear boundary value problems with multi-point boundary conditions at nonresonance case has a significant part in both theory and applications [4, 1, 16]. Moreover, it is nown that the research of existence of solutions for nonlinear boundary value problems is not easy for the resonant case. Lately, the multipoint boundary value problems at resonance for ordinary differential equations have been investigated in detail and many successful results have been attained, for instance, see [1, 6, 7, 11]. Nevertheless, to our nowledge, the corresponding results for secondorder with integral boundary conditions, are seldom seen [3,8,9] and very little research Received March 14, 17; Accepted August 1, 17 Communicated by Mehmet Ünal

2 46 Fulya Yoru Deren, Nuet Ayut Hamal, Tugba Senli Cerdi has been done on semipositone problems for nonlinear integral boundary value problems at resonance [5]. In [9], Liu and Ouyang established solutions for the nonlinear boundary value problems with integral boundary conditions z t) + ft, zt)) =, < t < 1, z) =, z1) = α η zs)ds, where η, 1), 1 αη = 1 and f C[, 1] R, R). The authors obtained a sufficient condition for the existence results of solutions. In [5], Henderson and Kosmatov consider the Neumann boundary value problem at resonance { z t) = ft, zt)), < t < 1, z ) = z 1) =, where β : [, 1] R +, α and ft, z) + α z + βz), for t [, 1] and z. Here positive solutions are obtained by means of the Guo Krasnosel sii fixed point theorem. Motivated by the papers mentioned above, we study the existence of positive solutions for the following boundary value problem with a sign-changing nonlinearity z t) = ft, zt)), t, 1), n z 1.1) ) =, z1) = α zs)ds, where α > 1, n, 1), αn = 1, the continuous function f : [, 1] [, ) R satisfies the inequality ft, z) β z M in [, 1] [, ), for some constant β and a positive constant M >. When a nontrivial solution exists for the homogeneous boundary value problem, the corresponding boundary value problem is called at resonance. Consider z t) + β zt) = F t, zt)), t, 1), n z 1.) ) =, z1) = α zs)ds, where F t, z) = ft, z) + β z. It is easy to see that the BVP 1.1) is equivalent to the BVP 1.) and the BVP 1.) is at nonresonance. Throughout the paper we will assume that the following conditions hold:

3 Semipositone Boundary Value Problems at Resonance 47 H1) β α, π ) ; H) f : [, 1] [, ) R is a continuous function; H3) there exists M >, satisfying F t, z) + M, t, z) [, 1] [, ). Preliminaries Lemma.1. Assume αn = 1. If h C[, 1], R), then the BVP z t) + β zt) = ht), t, 1), z ) =, z1) = α n zs)ds.1) has a unique solution zt) = Gt, s)hs)ds. Here 1 sin βt s), s t 1, β Gt, s) =, t s 1, + β where = α sin βn cos β. β α cos βn s) + β sin β1 s) α, s n, β sin β1 s), n s, Lemma.. Suppose that condition H1) is satisfied. Then i) For any t, s [, 1], Gt, s). ii) There exist d, 1] and φ C[, 1], [, )) satisfying dφs) Gt, s) φs), t, s [, 1]. Proof. By Lemma.1, we show the proof of i) and ii).

4 48 Fulya Yoru Deren, Nuet Ayut Hamal, Tugba Senli Cerdi i) Using cos βn > cos β for β α, π ), > α sin βn cos βn β and tan βn > βn for βn > with αn = 1, we obtain >. Moreover, it is enough to show the case when s [, n] and s t. Let ϕs) = α cos βn s) + β sin β1 s) α, s [, n]. Since ϕ t), by standard calculus, the function ϕ is concave on, n). Using the inequality cos β sin β for β α, π ), we can derive that ϕ) = α cos βn + β sin β α = β sin β α sin βn β sin β α sin β > α sin β sin β )..) Because ϕn) holds, together with.) and the concavity of the function ϕ, ϕs) for s [, n]. Hence, Gt, s), t, s [, 1]. ii) Let Φs) = 1 s, Kt, s) = λφs) Gt, s). Upper bounds. We will prove that for s t and s t, Kt, s), t, s [, 1], when λ > is sufficiently large. Case 1. If s t and s [, n], then If s [n, 1], then βn s) ) Kt, s) = λ1 s) β sin β1 s) α sin β λ1 n) sin β β. Kt, s) = λ1 s) β sin β1 s) β sin β1 s) λ1 s). β

5 Semipositone Boundary Value Problems at Resonance 49 1 Tae λ λ 1 :=, then Kt, s) holds for s t. 1 n) Case. If s t and s [, n], then Kt, s) λ1 s) sin β βn s) ) β sin β1 s) α sin β β λ1 s) sin β β sin β1 s) β β λ1 n) sin β β, so, for λ λ := sin β, we have Kt, s) for s t. If s [n, 1], then 1 n)β Kt, s) λ1 s) 1 cos βn sin β1 s) sin β1 s) β β cos βn λ1 s) 1 s) 1 s) cos βn ) = 1 s) λ 1, cos βn so, for λ λ 3 := 1 +, we have Kt, s) for s t. Let λ max{λ 1, λ, λ 3 }, then Kt, s) for t, s) [, 1] [, 1] and we obtain λ Φs) Gt, s). If we choose φs) := λ Φs), then Gt, s) φs), t, s [, 1]. Lower bounds. We will prove that for s t and s t, Kt, s), t, s [, 1], when λ > is sufficiently small. Case 1. Let ψs) = α cos βn s) + β sin β1 s) α. Note that ψ is concave and ψs) min{ψ), ψn)} for s [, n]. If s t and s [, n], then Kt, s) = λ1 s) λ1 s) cos β β λ cos β β α cos βn s) + β sin β1 s) α) β min{α cos βn + β sin β α, β sin β1 n)} min{α cos βn + β sin β α, β sin β1 n)}, so, for λ λ 4 := cos β min{α cos βn+β sin β α, β sin β1 n)}, we have Kt, s) β for s t. If s [n, 1], then we obtain Kt, s) = λ1 s) λ1 s) cos β β β sin β1 s) β sin β1 s).

6 5 Fulya Yoru Deren, Nuet Ayut Hamal, Tugba Senli Cerdi Since sin x x, < x π, hx) = 1, x = is continuous on [, π ], we can get := min hx) >, x [, π ] Kt, s) λ1 s) cos β β β1 s) = 1 s) λ cos β ), so, for λ λ 5 := cos β, we deduce that Kt, s) for s t. Case. For s t and s [, n], Kt, s) = λ1 s) λ1 s) cos β β λ cos β β α cos βn s) + β sin β1 s) α) β min{α cos βn + β sin β α, β sin β1 n)} min{α cos βn + β sin β α, β sin β1 n)}, so, for λ λ 4 := cos β min{α cos βn+β sin β α, β sin β1 n)}, we have Kt, s) β for s t. For s [n, 1], Kt, s) λ1 s) β λ1 s) cos β = 1 s) λ cos β sin β1 s) β β1 s) ), so, for λ λ 5 := cos β, we have Kt, s) for s t. Choose < λ min{λ 4, λ 5 }. Then Kt, s) for t, s) [, 1] [, 1] and λ Φs) Gt, s). So, dφs) Gt, s) φs), t, s [, 1], where d = λ. Therefore, the proof is completed. λ

7 Semipositone Boundary Value Problems at Resonance 51 Theorem.3 See [14]). Assume that B is a Banach space, K B is a cone in B and set K r = {z K : z < r}, Kϕ, a, b) = {z K : a ϕz), z b}. Suppose that the operator T : K K is completely continuous and ϕ is a nonnegative, continuous, concave functional on K with ϕz) z for any z K r. If there exist < p < q < d c < r such that i) T z c for z K c ; ii) {z Kϕ, q, d) : ϕz) > q} and ϕt z) > q for all z Kϕ, q, d); iii) T z < p for all z p; iv) ϕt z) > q for z Kϕ, q, c) with T z > d; v) T z z for z K r. Then T has at least four fixed points z 1, z, z 3 and z 4 in K r such that z 1 < p, ϕz ) > q, p < z 3 with ϕz 3 ) < q, c < z 4 r. 3 Main Result In this section, by using Theorem.3, we obtain the existence of positive solutions for the BVP 1.1). Here, we study on C[, 1] with the norm z = sup zt), which is a Banach space. { t [,1] } Choose the cone by P = z C[, 1] : zt), min zt) d z t 1. Apparently, P is a cone in C[, 1]. Let ϕ : P [, ) be a nonnegative continuous concave functional defined by It is obvious that ϕz) z, z P. We need the following lemma. ϕz) = min zt), z P. t [,1] Lemma 3.1. Let v be a unique positive solution for the following boundary value problem z t) + β zt) = 1, t, 1), n z 3.1) ) =, z1) = zs)ds.

8 5 Fulya Yoru Deren, Nuet Ayut Hamal, Tugba Senli Cerdi Then in which L = K d, K = φs)ds. vt) dl, t [, 1], Proof. Using the Green s function, we get The proof is complete. vt) = For the readers convenience, let us set Our main result is as follows: Gt, s)ds φs)ds dl. { 1 } = min ML, 1. Theorem 3.. Suppose that H1) H3) are satisfied. Moreover there exist nonnegative constants a, b, c, N with 1 < a < a + d < b < d 1 c < c, λd < N < c b satisfying i) ft, z) + β z + M < a K for t [, 1] and z [, a]; ii) ft, z) + β z + M b K N for t [, 1] and z [ b d, b d ]; iii) ft, z) + β z + M c K for t [, 1] and z [, c]; iv) lim inf z min ft, z) t [,1] z =. Then the BVP 1.1) has at least three positive solutions if λ, ]. Proof. Assume that xt) = λmvt), in which v is a unique solution of the BVP 3.1). Lemma 3.1 implies that xt) = λmvt) λmld d for t [, 1]. Now, we prove that the following boundary value problem z t) + β zt) = F t, zt) xt)), t, 1), n z 3.) ) =, z1) = zs)ds

9 Semipositone Boundary Value Problems at Resonance 53 has a positive solution, where F t, u) = Denote the operator A : E E by Azt) = λ { F t, u) + M, u F t, ) + M, u. Gt, s) F s, zs) xs))ds. We shall prove that A has a fixed point in our cone P. Notice that Azt) is continuous on [, 1], for any z P, and since Gt, s), φs) holds. Then, by Lemma. we have Az λ φs) F s, zs) xs))ds. 3.3) Thus for any z P, we can deduce from 3.3) and Lemma. that min Azt) = min t 1 t 1 d d Az. { λgt, s) F } s, zs) xs))ds λφs) F s, zs) xs))ds So, AP ) P. Then from the definition of A, one can see easily that the operator A is completely continuous, and each fixed point of A in P is a solution of BVP 3.). Firstly, we shall show that A : P c P c. Let z P c, then z c. If max {zt) xt), } = z t) then we get F t, z t)) = ft, z t)) + β z t) + M. We can derive from iii) that F t, z t)) c K for t [, 1]. So, it follows from iii) that Azt) = λ λ c K c. λgt, s) F s, zs) xs))ds φs) F s, zs) xs))ds φs)ds Then, Az c for z P c. So, condition i) of Theorem.3 is satisfied. Similarly, using the above process, we can derive that A : P a P a.

10 54 Fulya Yoru Deren, Nuet Ayut Hamal, Tugba Senli Cerdi To verify ii) of Theorem.3, let zt) = b d, then z P ϕ, b, b ), ϕz) = b d d > b. { Thus, z P ϕ, b, b ) } : ϕz) > b. Furthermore, z P ϕ, b, b ) implies that d d b zt) b d for t [, 1] and b d zt) xt) zt) b for t [, 1]. ii) d implies that F t, zt) xt)) + M b N for t [, 1]. Then K i.e., ϕaz) = min λ t [,1] dλ =dλ Gt, s) F s, zs) xs))ds φs) F s, zs) xs))ds φs)f s, zs) xs)) + M)ds dλ b K N φs)ds = dλnb > b, ϕaz) > b, z P ϕ, b, b ). d On the other hand, if z P ϕ, b, c) with Az b, then we get d ϕaz) = min t [,1] Azt) d Az b d > b. Finally, we will show that condition v) of Theorem.3 holds. Choose a number γ > satisfying γ + β )λd K 1. The hypotheses iv) implies that there is ρ > such that 1 λml 1 ρ and fs, us)) + M γus), us) dρ. 3.4) { } b Set R = max d, ρ and P R = {z K : z < R}. Let z P R. Then Because xt) = λmvt) λmdl λml zt) R zt) xt) ρ λml zt) zt) ρ d z λml zt), t [, 1]. ρ = dr dρ, t [, 1]

11 Semipositone Boundary Value Problems at Resonance 55 together with 3.4), we have Thus, we can obtain F s, zs) xs)) = fs, zs) xs)) + β zs) xs)) + M Azt) dλ γ + β )zs) xs)) γ + β ) dr. φs) F s, zs) xs))ds γ + β )λd 1 φs)ds R R. Then, Az z for z P R. Since all conditions of Theorem.3 are satisfied, F has at least four solutions z 1, z, z 3 and z 4 with z 1 < a, ϕz ) > b, z 3 > a, ϕz ) < b, c < z 4 R. Besides, z t) d z dϕz ) > db > d > xt), t [, 1], and z 3 t) d z 3 da > d > xt), t [, 1], z 4 t) d z 4 dc > d > xt), t [, 1]. Hence z = z x, z3 = z 3 x, z4 = z 4 x are positive solutions of the BVP 1.1). Now we prove that z, z3, and z4 are in fact the positive solutions of our problem 1.1). To see this we have for t [, 1], z represents a fixed point of the operator F. Then z t) = F z t) = = = = = λgt, s) F s, z s) xs))ds λgt, s)[f s, z s) xs)) + M]ds λgt, s)f s, z s) xs))ds + λm λgt, s)f s, z s) xs))ds + λmvt) λgt, s)f s, z s) xs))ds + xt). Gt, s)ds

12 56 Fulya Yoru Deren, Nuet Ayut Hamal, Tugba Senli Cerdi This implies that and z t) xt) = z = λgt, s)f s, z s) xs))ds λgt, s)f s, z )ds. Consequently, z is a positive solution of BVP 1.1). Similarly, we can show that z 3 and z 4 are positive solutions of the BVP 1.1). The proof is completed. Acnowledgements The first author was supported by the Ege University Scientific Research Project No:15- FEN-69). Also, the authors would lie to than the anonymous referees for providing valuable suggestions. References [1] C. Bai, J. Fang, Existence of positive solutions for three point boundary value problems at resonance, J. Math. Anal. Appl. 91 4), [] R. Conti, Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital. 1967), [3] Y. Cui, Solvability of second-order boundary-value problems at resonance involving integral conditions, Electron. J. Differential Equations, 1, No.45, 9 pp. [4] H. Feng, D. Bai, Existence of positive solutions for semipositone multi-point boundary value problems, Math. Comput. Modelling, 54 11), [5] J. Henderson, N. Kosmatov, Positive Solutions of the Semipositone Neumann Boundary Value Problem, Math. Model. Anal. :5, [6] G. Infante, M. Zima, Positive solutions of multi-point boundary value problems at resonance, Nonlinear Anal. 69 8), [7] W. Jiang, B. Wang, Positive solutions for second-order multi-point boundaryvalue problems at resonance in Banach spaces, Electron. J. Differential Equations, vol.11 11), no., pp [8] X. Lin, Existence of solutions to a nonlocal boundary value problem with nonlinear growth, Bound. Value Probl. doi:1.1155/11/

13 Semipositone Boundary Value Problems at Resonance 57 [9] H. Liu, Z. Ouyang, Existence of solutions for second-order three-point integral boundary value problems at resonance, Bound. Value Probl. 13, 13:197. [1] R. Y. Ma, Q. Z. Ma, Positive Solutions for Semipositone m-point Boundary- value Problems, Acta Math. Sin. Vol., No., pp [11] J. J. Nieto, Existence of a solution for a three-point boundary value problem for second-order differential equation at resonance, Bound. Value Probl. 13, 13:13. [1] S. K. Ntouyas, Nonlocal initial and boundary value problems: a survey, in: Handboo of differential equations: ordinary differential equations, Vol. II, Elsevier B. V., Amsterdam, 5, [13] M. Picone, Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine in Italian), Ann. Scuola Norm. Sup. Pisa Cl. Sci ), [14] C. Song, P. Weng, Multiple positive solutions for p-laplacian functional dynamic equations on time scales, Nonlinear Anal. 68 8), [15] W. M. Whyburn, Differential equations with general boundary conditions, Bull. Amer. Math. Soc ), [16] C. Zhai, C. Yang, Multiple Positive Solutions for Semi-positone m-point Boundary Value Problems, Acta Math. Appl. Sinica, Vol.7, No.3 11),