Research Article Positive Solutions of Sturm-Liouville Boundary Value Problems in Presence of Upper and Lower Solutions

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1 International Differential Equations Volume 11, Article ID 38394, 11 ages doi:1.1155/11/38394 Research Article Positive Solutions of Sturm-Liouville Boundary Value Problems in Presence of Uer and Lower Solutions Li Zhang, 1 Xiankai Huang, 1 and Weigao Ge 1 Deartment of Foundation Courses, Beijing Union University, Beijing 111, China Deartment of Mathematics, Beijing Institute of Technology, Beijing 181, China Corresondence should be addressed to Li Zhang, amy zhangli@sina.com Received 17 May 11; Revised 3 Setember 11; Acceted 4 Setember 11 Academic Editor: A. M. El-Sayed Coyright q 11 Li Zhang et al. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. We consider a kind of Sturm-Liouville boundary value roblems. Using variational techniques combined with the methods of uer-lower solutions, the existence of at least one ositive solution is established. Moreover, the uer solution and the lower solution are resented. 1. Introduction The Sturm-Liouville boundary value roblems for short, BVPs have received a lot of attention. Many works have been carried out to discuss the existence of at least one solution or multile solutions. The methods used therein mainly deend on the Leray-Schauder continuation theorem and the Mawhin continuation theorem. Since it is very difficult to give the corresonding Euler functional for Sturm-Liouville BVPs and verify the existence of critical oints for the Euler functional, few eole consider the existence of solutions for Sturm-Liouville BVPs by critical oint theory and many works considered the existence of solutions for Dirichlet BVPs. For examle, by a three-critical-oint theorem due to Ricceri 1, Bonanno considered Dirichlet roblems. Moreover, Afrouzi and Heidarkhani 3 also considered the existence of three solutions for a kind of Dirichlet BVP. By using an aroriate variational framework, the authors 4 considered the existence of ositive solutions for the Dirichlet BVP.

2 International Differential Equations In this aer, using variational methods combined with the methods of uer-lower solutions, we consider the ositive solutions of the following BVP: φ x t atφ x ft, x, t, 1, α 1 x α x, 1.1 β 1 x1 β x 1, where >1, φ x x x, α 1,α,β 1,β, α 1 α >, β 1 β >. The aer is organized as follows. In the forthcoming section, we give the Euler functional of BVP1.1 and some basic lemmas. In Section 3, firstly, we give an uer solution of BVP1.1, then, by the mountain ass lemma, the lower solution of BVP1.1 is obtained. At last, we show the existence of at least one ositive solution of BVP1.1 based on the uer solution and the lower solution we obtain.. Preliminary The Sobolev sace W 1,, 1 is defined by W 1,, 1 { x :, 1 R x is absolutely continuous and x L, 1; R }.1 and is endowed with the norm x xt dt x t 1/ dt.. Then, W 1,, 1 is a searable and reflexive Banach sace 5. Lemma.1 see 6. There exists a ositive constant c such that x x y y, x y c x y,, x y c x y, 1 <<.3 for any x, y R N.Herex, y x y T. For x C, 1, suose that x max t,1 xt, x m min t,1 xt. Lemma. see 7. If x W 1,, 1, then, x x. Lemma.3 see 8. For x X, letx ± max{±x, }; then, the following roerties hold: i x X x,x X; ii x x x ; iii x X x X ;

3 International Differential Equations 3 iv if x n n N uniformly converges to x in C, 1, then, xn n N uniformly converges to x ; v φ xx x, φ xx x. In the following, we state the C condition [9]. C Every sequence x n n N H such that the following conditions hold: i ϕx n n N is bounded; ii 1 x n H ϕ x n H, n has a subsequence which converges strongly in H. With a similar roof of Lemma.5 8, one has the following lemma. Lemma.4. If xt W 1,, 1 is a critical oint of the Euler functional ϕx 1 at x dt 1 x dt Ft, xdt α α α 1x 1 α β β 1 x1 β 1 β,.4 then, xt is a solution of BVP 1.1. Here, Ft, x x ft, sds. Remark.5. While α, the Euler functional ϕx does not include 1/α 1 /α 1 x, while β, ϕx does not include 1/β 1 /β 1 x1. Hence, in order to be convenient, we assume that α 1,α,β 1,β >. With little modification to the roof of Theorem 1.4 in 7, we obtain the following. Remark.6. ϕ is continuously differentiable on W 1,, 1, and, by comutation, one has ϕ x,y atφ xydt φ x y α1 x dt ft, xydt φ y α φ β1 x1 β y1, x,y W 1,, 1..5 Definition.7. u W 1,, 1 is an uer solution of BVP 1.1 if it satisfies φ u t atφ u ft, u, t, 1, α 1 u α u, β 1 u1 β u 1..6 If u is not a solution of BVP1.1, then, u is a strict uer solution. Definition.8. v W 1,, 1 is a lower solution of BVP1.1 if it satisfies φ v t atφ v ft, v, t, 1, α 1 v α v, β 1 v1 β v 1. If v is not a solution of BVP1.1, then, v is a strict lower solution..7 Definition.9. x W 1,, 1 is said to be a ositive solution of BVP1.1 if xt, xt /, t, 1.

4 4 International Differential Equations 3. Existence of Positive Solutions Choose x W 1,, 1 and x t >, t, 1 satisfying φ x 1, then, x t c t φ q s c 1 ds where 1/ 1/q 1,c 1,c are constants. If we choose c 1 1, c α φ q c 1 /α 1, x t satisfies α 1 x α x, β 1x 1 β x 1. Moreover, x t φ q t c 1 is continuous. Lemma 3.1. Assume A 1 ft, x C, 1,, lim x ft, x/φ x <at, t, 1, is satisfied; then, x a 1/ 1 x is a strict uer solution of BVP 1.1. Herea > 1 is some ositive constant. Proof. From A 1, there exists a constant N> such that ft, x φ x <at, x > N. 3.1 Hence, ft, x <atφ x a, t, 1, 3. holds for x and some large ositive constant a > 1. Then, f t, a 1/ 1 x <atφ a 1/ 1 x a 3.3, atφ a 1/ 1 x φ a 1/ 1 x t, 1, that is, φ x atφ x ft, x >, t, 1. Obviously, α 1 x α x, β 1 x1 β x 1. Therefore, from Definition.7, one has that x a 1/ 1 x is a strict uer solution of BVP 1.1. In the following, we assume the following conditions. A There exist δ>andgx :,, satisfying gmx m 1 gx for m<1, g /, ft, x >gxfor x,δ, t, 1. A 3 There exists μ>such that μgx gxx, x, Gx x gsds. Consider the auxiliary BVP φ x t atφ x gx, t, 1, α 1 x α x, 3.4 β 1 x1 β x 1.

5 International Differential Equations 5 Obviously, the corresonding Euler functional of BVP3.4 is ϕ x 1 at x dt 1 β β 1 x1 β 1 β. x dt Gx t gx dt α α α 1x 1 α 3.5 Obviously, ϕ is continuously differentiable on W 1,, 1, and, by comutation, one has ϕ x,y atφ xydt φ β1 x1 β y1, φ x y dt x,y W 1,, 1. gx α1 x ydtφ y α 3.6 Lemma 3.. If xt W 1,, 1 is a solution of BVP 3.4, then, xt. Proof. Let xt W 1,, 1 be a solution of the BVP 3.4. If there exists a subset E, 1, meas E /, xt fort E, then from the BVP 3.4, one has g fort E which contradicts with the assumtions. Moreover, x is an absolutely continuous function on, 1, and so the fundamental theorem of calculus ensures the existence of a set E 1, 1 such that meas, 1 \ E 1 andx is differentiable on E 1, x L 1, 1, φ x t atφ xt gx x dt x 1φ x 1 x φ x φ x t x dt E 1 x 1φ x 1 x φ x x 1φ β 1x1 β min{ a m, 1} x. E 1 x φ α1 x α x 1 dt at x dt min{ a m, 1} x atφ xtx dt 3.7 Therefore, for a.e. t, 1, x. Since xt is absolutely continuous on, 1, then, xt for t, 1. Lemma 3.3. Assume that A, A 3 hold; then, BVP3.4 has a solution x 1, that is, BVP3.4 has a ositive solution x 1. Proof. Assume that x n n N W 1,, 1 satisfies i and ii of the C condition; then, ϕ x n c1, ϕ x n 1 xn ε n. 3.8 Here, c 1 is some ositive constant and ε n, n.

6 6 International Differential Equations First, we show that x n n 1 W 1,, 1 is bounded. Indeed, from 3.8, we have ϕ x n,u εn, u W 1,, Choose u x n; then, ϕ x n, x n at x n dt 1 φ β1 β x n1 xn α1 dt φ x α n gx nx ndt. 3.1 Hence, x n n N is bounded. Moreover, 1 ϕ x n,xn atφ x n xndt φ β1 x n 1 β at x n dt x n1 β1 φ x β n. φ x n x n dt xn 1 dt gxnx ndt α1 x n φ x α n gxnx ndt α1 φ x α n 3.11 For large n, μ 1 c1 c 1 μc 1 μϕ x n ϕ x n,x n μ at x n dt μ 1 x n dt μ Gxntdt μ μα α α 1x n 1 α μβ β 1 x n 1 1 β 1 β at xn dt gxnx ntdt α1 φ x α n β1 φ x β n1 μ at x n dt μ μ 1 1 μ 1 x n dt μ at xn dt 1 min{ a m, 1} x n. at x n dt xn dt gx ntdt xn dt xn dt 3.1

7 International Differential Equations 7 Hence, x n n N is bounded; then, x n n N is uniformly bounded in W 1,, 1. By the comactness of the embedding W 1,, 1C, 1, the sequence x n n N has a subsequence, again denoted by x n n N for convenience, such that x n x x n x weakly in W 1,, 1, strongly in C, Moreover, ϕ x n ϕ x m,x n x m φ x n φ x m x n x m dt at φ x n φ x m x n x m dt gx n gx m x n x m dt α1 x n α1 x m φ φ x n x m α β1 x n 1 φ β α φ β1 x m 1 β x n 1 x m Since x n t xt in C, 1, then, φ α 1 x n /α φ α 1 x m /α x n x m, φ β 1 x n 1/β φ β 1 x m 1/β x n 1 x m 1, gx n gx mx n x m, x nt x m tdt, n, m. Moreover, at φ x n φ x m x n x m dt a x n x m φ x n φ x m dt, as n, m From ϕ x n ϕ x m,x n x m ϕ x n ϕ x m x n x m and x n x m is bounded in W 1,, 1, ϕ x n, ϕ x m, m, n, and one has ϕ x n ϕ x m,x n x m. Hence, φ x n φ x m x n x m dt, n,m If, from Lemma.1, there exists a ositive constant c such that 1 φ x n φ x m x n x m dt c x n x m dt. 3.17

8 8 International Differential Equations If <, by Lemma.1, thehölder inequality, and the boundedness of x n n N in W 1,, 1, one has x n x m dt x n x m x x n x m / n x m / dt x n x m x n x m dt / / x n x m dt c / / φ x n φ x m x n x m dt 1 / x n x m dt / 3.18 c / / φ x n φ x m x n x m dt 1 / x n x m /. From 3.17 and 3.18, we have x n x m dt asn, m. Then, x n x m, that is, x n n N is a Cauchy sequence in W 1,, 1. By the comleteness of W 1,, 1, we have x n x in W 1,, 1. From the discussion above, ϕx satisfies the C condition. For t>, x>, one has d G t 1 x t μ dt t μ 1 μg t 1 x t 1 xg t 1 x, 3.19 that is, Gt 1 xt μ is nonincreasing in t. Assume that M max x,1 Gx, x Gx G x μ M x μ, < x x Hence, ϕ x 1 at x dt 1 x dt 1 min{ a m, 1} x M x μ. Gx dt 3.1 Obviously, there exists ρ> such that, for x ρ, 1/ min{ a m, 1}ρ Mρ μ α>.

9 International Differential Equations 9 On the other hand, Gx G1x μ for x>1; then, by Lemma. ϕ x 1 max{ a, 1} x G1 x μ dt 1 φ α1 α φ β1 β x. gx dt 3. Let e be some large ositive constant. Since μ>, ϕ e <. Moreover, ϕ. From the mountain ass lemma 1, ϕ ossesses a critical value c α, that is, there exists x 1 such that ϕ x 1, ϕ x 1 c α>. Then, from Lemma.4, one has that BVP3.4 has a ositive solution x 1 and x 1 /, t, 1. Lemma 3.4. Assume that A, A 3 hold; then, BVP1.1 has a strict lower solution x βx 1 where β is some ositive constant and x 1 is the ositive solution of BVP3.4 one obtains that in Lemma 3.3. Proof. Assume β, 1 is small enough such that βx 1,δ and xt βx 1 t, xt βx 1 t /, t, 1. Then, φ x β 1 φ x 1 β 1 atφ x 1 β 1 gx 1 atφ x g x 3.3 < atφ x f t, x. Moreover, α 1 x α x, β 1 x1 β x 1. Hence, x is a strictly lower solution of BVP1.1 and x x, x / x, t, 1. Theorem 3.5. Assume that A 1 A 3 hold; then, BVP1.1 has a ositive solution x and x x x. Proof. Let I x, x {x W 1,, 1 x x x}. Make a truncation function of ft, x as ft, x, x > x, ft, x ft, x, x x x, f t, x, x < x, 3.4 and assume that Ft, x x ft, sds. Consider the following BVP: φ x t atφ x ft, x, t, 1, α 1 x α x, β 1 x1 β x The corresonding Euler functional of BVP3.5 is ϕx 1 at x dt 1 x dt Ft, xdt α α α 1x 1 α β β 1 x1 β 1 β. 3.6

10 1 International Differential Equations It is obvious that ϕx is weakly lower semicontinuous. Since x and x are continuous on, 1, Ft, x is continuous, ϕx is coercive. Hence, ϕx can attain its infimum in W 1,, 1. Without loss of generality, we may assume that ϕx attains its infimum in x.inthe following, we show that x is a solution of BVP1.1. Assume that x x has a negative minimum, and let t su{t, 1 x xt min s,1 x xs}. If t, then, x x α 1 α x x <, 3.7 which reaches a contradiction. Similarly, t / 1. If t, 1, there exist an oen interval I and t 1 I with t 1 <t, x t <xt, t I, x t 1 <xt 1. Hence, >φ x t 1 φ x t 1 t [ t 1 t1 t [ φ x s φ x s ] ds asφ x s fs, x s asφ xs f s, xs ] ds 3.8 t t 1 as [ φ xs φ x s ] ds >. From the discussion above, one has x t xt. Similarly, x t xt. Then, x I. Since xt and xt are the strictly lower and uer solutions of BVP1.1, resectively, x txt, x txt. Therefore, we obtain a ositive solution of BVP 1.1. Acknolewdgments This work was suorted by the National Natural Science Foundation of China no and Natural Science Foundation of Beijing Union University zk111x. References 1 B. Ricceri, On a three critical oints theorem, Archiv der Mathematik, vol. 75, no. 3,. 6,. G. Bonanno, A minimax inequality and its alications to ordinary differential equations, Journal of Mathematical Analysis and Alications, vol. 7, no. 1,. 1 9,. 3 G. A. Afrouzi and S. Heidarkhani, Three solutions for a quasilinear boundary value roblem, Nonlinear Analysis, vol. 69, no. 1, , 8. 4 R. P. Agarwal, K. Perera, and D. O Regan, Multile ositive solutions of singular roblems by variational methods, Proceedings of the American Mathematical Society, vol. 134, no. 3, , 6. 5 R. A. Adams, Sobolev Sace[M], Academic Press, New York, NY, USA, J. Simon, Regularite de la solution d une equation non lineaire dans R n, in Lecture Notes in Mathematics, vol. 665,. 5 7, Sringer, New York, NY, USA, J. Mawhin and M. Willen, Critical Point Theorem and Hamiltonian Systems, vol. 74, Sringer, New York, NY, USA, 1989.

11 International Differential Equations 11 8 Y. Tian and W. Ge, Alications of variational methods to boundary-value roblem for imulsive differential equations, Proceedings of the Edinburgh Mathematical Society, vol. 51, no., , 8. 9 G. Cerami, An existence criterion for the critical oints on unbounded manifolds, Istituto Lombardo. Accademia di Scienze e Lettere. Rendiconti A., vol. 11, no., , A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical oint theory and alications, Functional Analysis, vol. 14, , 1973.

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