Three symmetric positive solutions of fourth-order nonlocal boundary value problems

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1 Electronic Journal of Qualitative Theory of Differential Equations 2, No. 96, -; Three symmetric positive solutions of fourth-order nonlocal boundary value problems Fuyi Xu School of Science, Shandong University of Technology, Zibo, 25549, Shandong, China School of Mathematics and Systems Science, Beihang University, Beijing 9, China Abstract. In this paper, we study the existence of three positive solutions for fourth-order singular nonlocal boundary value problems. We show that there exist triple symmetric positive solutions by using Leggett-Williams fixed-point theorem. The conclusions in this paper essentially extend and improve some known results. MSC: 34B6. Keywords: nonlocal boundary condition, symmetric positive solutions, singular, Green s function, cone. Introduction Boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics, the existence of positive solutions for such problems has become an important area of investigation in recent years. To identify a few, we refer the reader to [-3,6,,,3,7,8] and references therein. At the same time, a class of boundary value problems with nonlocal boundary conditions appeared in heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. Such problems include two-point, threepoint, multi-point boundary value problems as special cases and have attracted the attention of Gallardo [], Karakostas and Tsamatos [2], Lomtatidze and Malaguti [3] (and see the references therein. For more information about the general theory of integral equations and their relation to boundary value problems, see for example, [4,5]. Motivated by the works mentioned above, in this paper, we study the existence of three symmetric positive solutions for the following fourth-order singular nonlocal boundary value problem(nbvp: u (t(t g(tf(t, u, < t <, u( u( u ( u ( a(su(sds, b(su (sds, (. where a, b L [, ], g : (, [, is continuous, symmetric on (, and may be singular at t and t, f : [, ] [, [, is continuous and f(, x addresses: zbxufuyi@63.com(f.xu 2 This work was supported financially by the National Natural Science Foundation of China (734. EJQTDE, 2 No. 96, p.

2 is symmetric on [, ] for all x [, +. We show that there exist triple symmetric positive solutions by using Leggett-Williams fixed-point theorem. 2 Preliminaries and Lemmas In this section, we present some definitions and lemmas that are important to prove our main results. Definition 2.. Let E be a real Banach space over R. A nonempty closed set P E is said to be a cone provided that (i u P, a implies au P; and (ii u, u P implies u. Definition 2.2. Given a cone P in a real Banach space E, a functional ψ : P P is said to be increasing on P provided ψ(x ψ(y, for all x, y P with x y. Definition 2.3. Given a nonnegative continuous functional γ on P in a real Banach space E, we define for each d > the following set P(γ, d {x P γ(x < d}. Definition 2.4. The function w is said to be symmetric on [, ], if w(t w( t, t [, ]. Definition 2.5. A function u is called a symmetric positive solution of the NBVP(. if u is symmetric and positive on [, ], and satisfies the differential equation and the boundary value conditions in NBVP(.. Definition 2.6. Given a cone P in a real Banach space E, a functional α : P [, is said to be nonnegative continuous concave on P provided α(tx+( ty tα(x + ( tα(y, for all x, y P with t [, ]. Let a, b, r > be constants with P and α as defined above, we note P r {y P y < r}, P {α, a, b} {y P α(y a, y b}. The main tool of this paper is the following well-known Leggett-Williams fixedpoint theorem. Theorem 2..[5-6] Assume E be a real Banach space, P E be a cone. Let T : P c P c be completely continuous and α be a nonnegative continuous concave functional on P such that α(y y, for y P c. Suppose that there exist < a < b < d c such that (i {y P(α, b, d α(y > b} and α(ty > b, for all y P(α, b, d; (ii Ty < a, for all y a; (iii α(ty > b for all y P(α, b, c with Ty > d. Then T has at least three fixed points y, y 2, y 3 satisfying y < a, b < α(y 2, and y 3 > a, α(y 3 < b. EJQTDE, 2 No. 96, p. 2

3 Lemma 2.. [4] Suppose that d : m(sds, m L [, ], y C[, ], then BVP u (t + y(t, < t <, (2. has a unique solution where H(t, s G(t, s+ d u( u( u(t m(su(sds, (2.2 H(t, sy(sds, (2.3 G(s, xm(xdx, G(t, s Proof. Integrating both sides of (2. on [, t], we have u (t t Again integrating (2.4 from to t, we get In particular, u( u(t t By the boundary value conditions (2.2 we get { t( s, t s, s( t, s t. y(sds + B. (2.4 (t sy(sds + Bt + A. (2.5 ( sy(sds + B + A, u( A. B By G(s, x G(x, s and (2.5, we can obtain A u( m(x m(xu(xdx ( x (x sy(sds + x ( x m(x s( xy(sds + ( m(x G(s, xy(sds ( G(s, xm(xdx ( sy(sds. (2.6 ( m(x x dx + Ad y(sds + Ad. x (x sy(sds + Bx + A dx ( sy(sds dx + A x( sy(sds dx + Ad m(xdx EJQTDE, 2 No. 96, p. 3

4 So, we have A d By (2.5, (2.6 and (2.7, we obtain u(t t t t ( (t sy(sds + Bt + A (t sy(sds + t s( ty(sds + t G(t, sy(sds + d H(t, sy(sds. G(s, xm(xdx y(sds. (2.7 ( sy(sds + ( d t( sy(sds + ( d ( G(s, xm(xdx y(sds This completes the proof of Lemma 2.. It is easy to verify the following properties of H(t, s and G(t, s. Lemma 2.2. If m(t >, and d : m(sds (,, then ( H(t, s, t, s [, ], H(t, s >, t, s (, ; G(s, xm(xdx y(sds G(s, xm(xdx y(sds (2 G( t, s G(t, s, G(t, t G(t, s G(s, s, t, s [, ]; η (3 γh(s, s H(t, s H(s, s, where γ (,, η G(x, xm(xdx. d + η So we may denote Green s functions of the following boundary value problems u (t, < t <, and u( u( a(su(sds u (t, < t <, u( u( b(su(sds, by H (t, s and H 2 (t, s, respectively. By Lemma 2., we know that H (t, s and H 2 (t, s can be written by and H (t, s G(t, s + H 2 (t, s G(t, s + a(sds b(sds G(s, xa(xdx, G(s, xb(xdx. EJQTDE, 2 No. 96, p. 4

5 Obviously, H (t, s and H 2 (t, s have the same properties with H(t, s in Lemma 2.2. Remark 2.. For notational convenience, we introduce the following constants α a(sds, β b(sds, γ η α + η, γ 2 η 2 β + η 2 (,, η G(x, xa(xdx, η 2 G(x, xb(xdx. Lemma 2.3. Assume that α, β, h C[, ], then NBVP u (t h(t, < t <, u( u( a(su(sds, u ( u ( b(su (sds (2.8 has a unique solution u(t H (t, τh 2 (τ, sh(sdsdτ. (2.9 Lemma 2.4. Assume that α, β, h C[, ] is symmetric, then the solution u(t of NBVP (2.8 is symmetric on [, ]. Proof. For notational convenience, we set E (τ a(sds G(τ, xa(xdx, E 2 (s b(sds G(s, xb(xdx. EJQTDE, 2 No. 96, p. 5

6 For t, s [, ], by (2.9 and Lemma 2.2 we have u( t u(t. H ( t, τh 2 (τ, sh(sdsdτ [G( t, τ + E (τ][g(τ, s + E 2 (s]h(sdsdτ G( t, τg(τ, sh(sdsdτ + E (τ[g(τ, s + E 2 (s]h(sdsdτ G( t, τe 2 (sh(sdsdτ G( t, τg( τ, sh( sd( sd( τ G( t, τe 2 (sh(sdsd( τ + G(t, τg(τ, sh(sdsdτ + E (τ[g(τ, s + E 2 (s]h(sdsdτ H (t, τh 2 (τ, sh(sdsdτ G(t, τe 2 (sh(sdsdτ E (τ[g(τ, s + E 2 (s]h(sdsdτ Therefore, the solution u(t of NBVP (2.8 is symmetric on [, ]. Lemma 2.5. Assume that a(t, b(t, and α, β (,, h C + [, ], then the solution u(t of NBVP (2.8 is positive on [, ]. Proof. Set v(t u (t. By v (t h(t, t [, ], we know that v(t is a concave function on [, ]. Thus, by (2.3 we have ( v( v( G(s, xb(xdx h(sds. b(sds On the other hand, due to u (t v(t, t [, ], we deduce that u(t is a concave function on [, ]. It follows that by (2.3 ( u( u( G(s, xa(xdx h(sds, a(sds which implies that the solution u(t. Lemma 2.6. Assume that a(t, b(t, and α, β (,, h C + [, ], then the solution u(t of NBVP (2.8 satisfies min u(t γ u, (2. t [,] EJQTDE, 2 No. 96, p. 6

7 where γ γ γ 2, is the supremum norm on C + [, ]. Proof. By Lemma 2.2 and (2.3, we obtain So, u(t H (t, τh 2 (τ, sh(sdsdτ u On the other hand, by Lemma 2.2 and (2.3 we have u(t γ γ 2 γ H (τ, τh 2 (s, sh(sdsdτ. H (τ, τh 2 (s, sh(sdsdτ. (2. H (t, τh 2 (τ, sh(sdsdτ H (τ, τh 2 (s, sh(sdsdτ. H (τ, τh 2 (s, sh(sdsdτ. Combined (2. with (2.2, we deduce inequality (2.. Now we define an integral operator T : C[, ] C[, ] by (Tu(t H (t, τh 2 (τ, sg(sf(s, u(sdsdτ. (2.2 Define a set P by { } P u C + [, ] : u(t is a symmetric and concave function on [, ], min x(t γ u, t [,] is the supremum norm on C + [, ]. It is easy to see that P is a cone in C[, ]. Clearly, u is a solution of the NBVP (. if and only if u is a fixed point of the operator T. In the rest of the paper, we make the following assumptions: (B a, b L [, ], a(t, b(t, α, β (, ; (B 2 g : (, [, + is continuous, symmetric, and < H 2(s, sg(sds < + ; (B 3 f : [, ] [, + [, + is continuous, and f(, x is symmetric on [, ] for all x [, +. Remark 2.2. (B 2 implies that g(t may be singular at t and t. Remark 2.3. If (B holds, then for all t, s [, ], we have H ( t, s H (t, s, H 2 ( t, s H 2 (t, s. Lemma 2.7. Assume that conditions (B, (B 2 and (B 3 hold. Then T : P P is a completely continuous operator. Proof From Lemma 2.4, Lemma 2.5 and Lemma 2.6, we know that T(P P. Now we prove that operator T is completely continuous. For n 2 define g n by g n (t inf{g(t, g( n }, < t n, g(t, n t n, inf{g(t, g( n }, n t <. EJQTDE, 2 No. 96, p. 7

8 Then, g n : [, ] [, + is continuous and g n (t g(t, t (,. And T n : P P by (T n u(t H (t, τh 2 (τ, sg n (sf(s, u(sdsdτ. Obviously, T n is compact on P for any n 2 by an application of the Ascoli- Arzela Theorem. Let B R {u P : u R}. We claim that T n converges uniformly to T as n on B R. In fact, let M R max{f(s, x : (s, x [, ] [, R]}, M max{h (τ, t : τ [, ]}, then M R, M <. Since < H 2(s, sg(s <, by the absolute continuity of integral, we have lim n e( n H 2 (s, sg(sds, where e( [, ] [, ]. So, for any t [, ], fixed R > and u B n n n R, we have (T n u(t (Tu(t H (t, τh 2 (τ, s(g n (s g(sf(s, u(sdsdτ MM R H 2 (s, s g n (s g(s ds MM R e( n H 2 (s, sg(sds (n, where we have used assumptions (B -(B 3 and the fact that H 2 (t, s H 2 (s, s for t, s [, ]. Hence the completely continuous operator T n converges uniformly to T as n on any bounded subset of P, and therefore T is completely continuous. 3 The Main Results We first define the nonnegative, continuous concave functional ϕ : P [, by Obviously, for every u P we have We shall use the following notation: Λ ϕ(u min t [,] u(t. ϕ(u u. H (τ, τh 2 (s, sg(sdsdτ. Our main result is the following theorem. Theorem 3.. Suppose conditions (B, (B 2 and (B 3 hold, and there exist positive constants a, b and c with < a < b < γc such that EJQTDE, 2 No. 96, p. 8

9 (A f(t, u < Λc, for t [, ], u c; (A 2 f(t, u Λb γ, for t [, ], b u b γ ; (A 3 f(t, u Λa, for t [, ], u a. Then the NBVP(. has at least three symmetric positive solutions u, u 2 and u 3 such that u < a, b < ϕ(u 2, and u 3 > a with ϕ(u 3 < b. Proof. we show that all the conditions of Theorem 2. are satisfied. We first assert that there exists a positive number c such that T(P c P c. By (A we have Tu max t [,] (Tu(t max t [,] Λc Λc c. H (t, τh 2 (τ, sg(sf(s, u(sdsdτ H (t, τh 2 (τ, sg(sdsdτ H (τ, τh 2 (s, sg(sdsdτ Therefore, we have T(P c P c. Especially, if u P a, then assumption (A 3 yields T : P a P a. We now show that condition (i of Theorem 2. is satisfied. Clearly, {u b P(ϕ, b, ϕ(u > b}. Moreover, if u P(ϕ, b, b, then ϕ(u b, so γ γ b u b. By the definition of ϕ and (A γ 2, we obtain ϕ(tu min t [,] (Tu(t min t [,] b. H (t, τh 2 (τ, sg(sf(s, u(sdsdτ γh (τ, τh 2 (s, sg(sf(s, u(sdsdτ γh (τ, τh 2 (s, sg(sγ Λbdsdτ Therefore, condition (i of Theorem 2. is satisfied. Finally, we address condition (iii of Theorem 2.. For this we choose u P(ϕ, b, c with Tu > b. Then from Lemma 2.6, we deduce γ ϕ(tu min (Tu(t γ Tu > b. t [,] Hence, condition (iii of Theorem 2. holds. By Theorem 2., we obtain the NBVP(. has at least three symmetric positive solutions u, u 2 and u 3 such that u < a, b < ϕ(u 2, and u 3 > a with ϕ(u 3 < b. EJQTDE, 2 No. 96, p. 9

10 4 Examples In the section, we present a simple example to explain our results. Example 4.. Consider the following fourth-order singular nonlocal boundary value problems (NBVP u (t 6t( t ( 2 2t + 2 min{u2, 2u}, < t <, u( u( u ( u ( su(sds, su (sds,, (4. where g(t 6t( t, a(t b(t t, and f(t, u 2 2t +2 min{u2, 2u}, then g(t, a(t, b(t and f(t, u satisfy the assumptions (B -(B 3. A direct computation shows α β 24 25, η η , γ 6 25, Λ 2 5. We choose a, b 8, c 8. Obviously, a < b < γc. Moreover, 2 5 (i for (t, x [, ] [, c], we have f(t, u f(, c < Λc ; (ii for (t, x [, ] [b, γ b], we have f(t, u f( 2, b > 2 γ bλ ; (iii for (t, x [, ] [, a], we have f(t, u f(, a < Λa. By Theorem 3., we know the NBVP (4. has at least three positive solutions. Acknowledgments The authors are grateful to the referees for their valuable suggestions and comments. References [] J.M. Gallardo, Second order differential operators with integral boundary conditions and generation of semigroups, Rocky Mountain J. Math. 3 ( [2] G.L. Karakostas and P.Ch. Tsamatos, Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary value problems, Electron. J. Diff. Equ. 3 (22-7. [3] A. Lomtatidze and L. Malaguti, On a nonlocal boundary-value problems for second order nonlinear singular differential equations, Georgian Math. J. 7 ( [4] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, UK, (99. EJQTDE, 2 No. 96, p.

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