Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 2009, 267 277 EXISTENCE OF POSITIVE SOLUTIONS TO SYSTEMS OF NONLINEAR INTEGRAL OR DIFFERENTIAL EQUATIONS Xiyou Cheng Zhitao Zhang Abstract. In this paper, we are concerned with existence of positive solutions for systems of nonlinear Hammerstein integral equations, in which one nonlinear term is superlinear the other is sublinear. The discussion is based on the product formula of fixed point index on product cone fixed point index theory in cones. As applications, we consider existence of positive solutions for systems of second-order ordinary differential equations with different boundary conditions. 1. Introduction In this paper, we consider existence of positive solutions for the following system of nonlinear Hammerstein integral equations u(x) = k 1 (x, y)f 1 (y, u(y), v(y)) dy for x, (S) v(x) = k 2 (x, y)f 2 (y, u(y), v(y)) dy for x, where R n is a bounded domain, k i C(, R + ), f i C( R + R +, R + ) (i = 1, 2) R + = [0, ). 2000 Mathematics Subject Classification. 45G15, 34B18, 47H10. Key words phrases. Positive solution, Hammerstein integral equations, fixed point index, product cone. Supported in part by the NSFC (10671195, 10471056,10831005) of China the NSF (06kJD110092) of Jiangsu Province of China. c 2009 Juliusz Schauder Center for Nonlinear Studies 267
268 X. Cheng Zh. Zhang Definition 1.1. Let C + () = {u C() u(x) 0, for all x }, we say that (u, v) is one positive solution, if (u, v) [C + () \ {θ}] [C + () \ {θ}] satisfies system (S). The study of nonlinear Hammerstein integral equations was initiated by Hammerstein, see [4]. Subsequently a number of papers have dealt with existence of nontrivial solutions of nonlinear Hammerstein integral equations, see the monographs [3], [5] the references therein. To the best of our knowledge, the papers dealing with existence of nontrivial solutions, especially positive solutions for system (S) are few, see [7] [10]. They mainly studied existence of nontrivial or nonnegative solutions for systems of nonlinear Hammerstein integral equations by use of topological methods or fixed point index theory in cones. Recently, in [1] Cheng Zhong considered existence of positive solutions for a super-sublinear system of second-order ordinary differential equations by applying the product formula of fixed point index on product cone fixed point index theory in cones. Motivated by these works, we shall deal with existence of positive solutions for system (S), in which one nonlinear term is superlinear the other is sublinear. As applications, we consider existence of positive solutions to systems of second-order ordinary differential equations with superlinear sublinear nonlinearities under different boundary conditions. Throughout this paper, we suppose that kernel functions k i (x, y) (i = 1, 2) satisfy the following conditions: (i) k i (x, y) = k i (y, x), for all x, y ; (ii) there exist p i C(), 0 p i (x) 1 such that k i (x, y) p i (x)k i (z, y), for all x, y, z ; (iii) max x k i(x, y)p i (y) dy is positive. Lemma 1.2. Let B i : C() C() be defined by B i u(x) = k i (x, y)u(y) dy, i = 1, 2. Then the spectral radius of B i, r(b i ) is positive. here Proof. From the definition of B i the conditions about k i, we have B i p i (x) = k i (x, y)p i (y) dy p i (x) k i (z, y)p i (y) dy, x, B i p i (x) p i (x) B i p i, x, B i p i = max k i (x, y)p i (y) dy > 0. x
Solutions to Systems of Nonlinear Integral or Differential Equations 269 By induction, we get Bi n p i (x) p i (x) B i p i n, x, thus Bi n B ip i n. From the formula of spectral radius, we know that r(b i ) = lim n Bn i 1/n B i p i > 0. (H 1 ) (H 2 ) Definition 1.3. If f 1, f 2 in system (S) satisfy the following assumptions: f 1 (x, u, v) lim sup max u 0 + x u lim inf min v 0 + x f 2 (x, u, v) v < 1 r(b 1 ) > 1 r(b 2 ) uniformly w.r.t. v R + ; uniformly w.r.t. u R +, then we say that f 1 is superlinear with respect to u at the origin that f 2 is sublinear with respect to v at the origin. (H 3 ) (H 4 ) Definition 1.4. If f 1, f 2 in system (S) satisfy the following assumptions: lim inf min f 1 (x, u, v) u x u lim sup max v x f 2 (x, u, v) v > 1 r(b 1 ) < 1 r(b 2 ) uniformly w.r.t. v R + ; uniformly w.r.t. u R +, then we say that f 1 is superlinear with respect to u at infinity that f 2 is sublinear with respect to v at infinity. Our main result is the following. Theorem 1.5. Assume that f 1 is superlinear w.r.t. u at the origin infinity, f 2 is sublinear w.r.t. v at the origin infinity satisfies the following condition: (G) lim sup u max x f 2 (x, u, v) = g(v) uniformly w.r.t. v [0, M] (for all M > 0), here g is a locally bounded function. Then system (S) has at least one positive solution. Remark 1.6. If f 1 is superlinear w.r.t. u at the origin infinity f 2 is also superlinear w.r.t. v at the origin infinity, then system (S) has at least one positive solution, which can be seen from Steps 1 2 in the proof of our theorem. Remark 1.7. If f 1 is sublinear w.r.t. u at the origin infinity f 2 is also sublinear w.r.t. v at the origin infinity, furthermore, there exist locally bounded functions g 1 g 2 such that lim sup v max f 1 (x, u, v) = g 1 (u) x
270 X. Cheng Zh. Zhang uniformly w.r.t. u [0, M] lim sup u + max f 2 (x, u, v) = g 2 (v) x uniformly w.r.t. v [0, M] (for all M > 0), then system (S) has at least one positive solution, which can be seen from Steps 3 4 in the proof of our theorem. The paper is organized as follows: in Section 2, we make some preliminaries; in Section 3, we prove our main result; in Section 4, as applications, we prove existence of positive solutions to systems of second order differential equations with different boundary conditions. 2. Preliminaries In this section, we shall construct a cone which is the Cartesian product of two cones change the problem (S) into a fixed point problem in the constructed cone. At the same time, we will give some useful preliminary results for the proof of our theorem. It is well known that C() is a Banach space with the maximum norm u = max x u(x), C + () is a total cone of C(). Choose bounded def domains i (i = 1, 2) such that δ i = min x i p i (x) > 0, which is feasible by the hypotheses of p i. Now construct sub-cones subsets as following: K i = {u C + () u(x) δ i u, for all x i }, K ri = {u K i u < r i }, K ri = {u K i u = r i }, for all r i > 0. Noticing that B i (i = 1, 2) is completely continuous positive, it follows from Lemma 1.2 Krein Rutman theorem (see [6]) that r(b i ) is one of eigenvalues for B i there exist positive eigenfunctions corresponding to r(b i ). Lemma 2.1. Let ψ i (x) be the positive eigenfunctions of B i corresponding to r(b i ) with ψ i(x) dx = 1, then the following conclusions are valid: (a) ψ i(x)u(x) dx u, for all u K i ; (b) ψ i (x) p i (x) ψ i, for all x ; (c) there exist constants c i > 0 such that ψ i(x)u(x) dx c i u, for all u K i. Proof. (a) Obviously, ψ i (x)u(x) dx ψ i (x) dx u = u. (b) Noticing that k i (x, y) p i (x)k i (z, y), for all x, y, z, we have k i (x, y)ψ i (y) dy p i (x)k i (z, y)ψ i (y) dy, for all x, z
Solutions to Systems of Nonlinear Integral or Differential Equations 271 r(b i )ψ i (x) r(b i )p i (x)ψ i (z), for all x, z, which implies that ψ i (x) p i (x) ψ i, for all x. (c) It follows from (b) the definition of K i. by For λ [0, 1], u, v C + (), we define the mappings: T λ,1 (u, v)(x) = T λ, 2 (u, v)(x) = T λ,1 (, ), T λ,2 (, ): C + () C + () C + (), T λ (, ): C + () C + () C + () C + () k 1 (x, y)[λf 1 (y, u(y), v(y)) + (1 λ)f 1 (y, u(y), 0)] dy, k 2 (x, y)[λf 2 (y, u(y), v(y)) + (1 λ)f 2 (y, 0, v(y))] dy T λ (u, v)(x) = (T λ,1 (u, v)(x), T λ, 2 (u, v)(x)). It is obvious that the existence of positive solutions of system (S) is equivalent to the existence of nontrivial fixed points of T 1 in K 1 K 2. To compute the fixed point index of T 1, we need the following results. Lemma 2.2. T λ : K 1 K 2 K 1 K 2 is completely continuous. Proof. For (u, v) K 1 K 2, we show that T λ (u, v) K 1 K 2, i.e. T λ,1 (u, v) K 1 T λ,2 (u, v) K 2. By the above definitions the conditions about k i (x, y), we have T λ,1 (u, v)(x) = k 1 (x, y)[λf 1 (y, u(y), v(y)) + (1 λ)f 1 (y, u(y), 0)] dy p 1 (x) k 1 (z, y)[λf 1 (y, u(y), v(y)) + (1 λ)f 1 (y, u(y), 0)] dy = p 1 (x)t λ,1 (u, v)(z), for all x, z, which implies that Similarly, T λ,1 (u, v)(x) δ 1 T λ,1 (u, v), x 1. T λ, 2 (u, v)(x) δ 2 T λ, 2 (u, v), x 2. Hence, T λ (K 1 K 2 ) K 1 K 2. By the Arzelà Ascoli theorem, we know that T λ : K 1 K 2 K 1 K 2 is completely continuous. Remark 2.3. Denoting T (λ, u, v)(x) = T λ (u, v)(x), T ([0, 1] K r1 K r2 ) is a compact set by the Arzelà Ascoli theorem.
272 X. Cheng Zh. Zhang Next, we recall some concepts about the fixed point index (see [2], [11]), which will be used in the proof of our theorem. Let X be a Banach space let P X be a closed convex cone in X. Assume that W is a bounded open subset of X with boundary W, let A: P W P be a completely continuous operator. If Au u for u P W, then the fixed point index i(a, P W, P ) is defined. One important fact is that if i(a, P W, P ) 0, then A has a fixed point in P W. The following results are useful in our proof. Lemma 2.4 ([2], [11]). Let E be a Banach space let P E be a closed convex cone in E. For r > 0, denote P r = {u P u < r}, P r = {u P u = r}. Let A: P P be completely continuous. valid: Then the following conclusions are (a) if µau u for every u P r µ (0, 1], then i(a, P r, P ) = 1; (b) if mapping A satisfies the following two conditions: (b1) inf u Pr Au > 0; (b2) µau u for every u P r µ 1, then i(a, P r, P ) = 0. Lemma 2.5 ([1]). Let E be a Banach space let P i E (i = 1, 2) be a closed convex cone in E. For r i > 0 (i = 1, 2), denote P ri = {u P i u < r i }, P ri = {u P i u = r i }. Suppose A i : P i P i is completely continuous. If u i A i u i, for all u i P ri, then i(a, P r1 P r2, P 1 P 2 ) = i(a 1, P r1, P 1 ) i(a 2, P r2, P 2 ), where A(u, v) def = (A 1 u, A 2 v), for all (u, v) P 1 P 2. 3. Proof of Theorem 1.5 We will choose a bounded open set D = (K R1 \K r1 ) (K R2 \K r2 ) in product cone K 1 K 2 verify that a family of operators {T λ } λ I satisfy the sufficient conditions for the homotopy invariance of fixed point index on D. Next, we separate the proof into four steps. Step 1. From the superlinear assumption of f 1 at the origin, there are ε (0, 1/r(B 1 )) r 1 > 0 such that (3.1) λf 1 (x, u, v) + (1 λ)f 1 (x, u, 0) (1/r(B 1 ) ε)u, for all x, u [0, r 1 ] v R +. We claim that (3.2) µt λ,1 (u, v) u, for all µ (0, 1] (u, v) K r1 K 2.
Solutions to Systems of Nonlinear Integral or Differential Equations 273 In fact, if it is not true, then there exist µ 0 (0, 1] (u 0, v 0 ) K r1 K 2, such that µ 0 T λ,1 (u 0, v 0 ) = u 0. In combination with (3.1), it follows that u 0 (x) T λ,1 (u 0, v 0 )(x) k 1 (x, y)(1/r(b 1 ) ε)u 0 (y) dy. Multiplying the both sides of this inequality by ψ 1 (x) integrating on, we get that u 0 (x)ψ 1 (x) dx k 1 (x, y)(1/r(b 1 ) ε)u 0 (y)ψ 1 (x) dy dx, u 0 (x)ψ 1 (x) dx (1/r(B 1 ) ε) k 1 (x, y)ψ 1 (x) dx u 0 (y) dy, that is u 0 (x)ψ 1 (x) dx (1 r(b 1 )ε) u 0 (y)ψ 1 (y) dy. Noticing that u 0(x)ψ 1 (x) dx > 0, hence 1 1 r(b 1 )ε, which is a contradiction! Step 2. By use of the superlinear hypothesis of f 1 at infinity, there exist ε > 0 m > 0 such that (3.3) λf 1 (x, u, v) + (1 λ)f 1 (x, u, 0) (1/r(B 1 ) + ε)u, for all x, u m v R +, thus (3.4) λf 1 (x, u, v) + (1 λ)f 1 (x, u, 0) (1/r(B 1 ) + ε)u C 0, for all x u, v R +, here C 0 = (1/r(B 1 ) + ε)m. We can prove that there exists a R 1 > r 1 such that (3.5) µt λ,1 (u, v) u inf u K R1 T λ,1 (u, v) > 0, for all µ 1, (u, v) K R1 K 2. First, if there are (u 0, v 0 ) K 1 K 2 µ 0 1 such that u 0 = µ 0 T λ,1 (u 0, v 0 ), together with (3.4), we get that u 0 (x) T λ,1 (u 0, v 0 )(x) k 1 (x, y)(1/r(b 1 ) + ε)u 0 (y) dy C. It follows that u 0 (x)ψ 1 (x) dx k 1 (x, y)(1/r(b 1 ) + ε)u 0 (y) dyψ 1 (x)dx C, u 0 (x)ψ 1 (x) dx (1 + r(b 1 )ε) u 0 (y)ψ 1 (y) dy C,
274 X. Cheng Zh. Zhang which yields u 0 (x)ψ 1 (x) dx C r(b 1 )ε. Furthermore, in view of Lemma 2.1(c), we know that u 0 C def = R. c 1 r(b 1 )ε Therefore, as R > R, u µt λ,1 (u, v) for all (u, v) K R K 2 µ 1. In addition, if R > m/δ 1, then by use of (3.3) we know that for all (u, v) K R K 2, T λ,1 (u, v) T λ,1 (u, v)(x)ψ 1 (x) dx k 1 (y, x)(1/r(b 1 ) + ε)u(y) dy ψ 1 (x) dx 1 (1 + r(b 1 )ε) u(y)ψ 1 (y) dy (1 + r(b 1 )ε)mes( 1 )δ1 ψ 2 1 R, 1 which implies that inf u KR T λ,1 (u, v) > 0. Hence, we choose R 1 > max{r 1, R, m/δ 1 }. Step 3. In view of the sublinear assumption of f 2 at the origin, there are ε > 0 r 2 > 0 such that (3.6) λf 2 (x, u, v) + (1 λ)f 2 (x, u, 0) (1/r(B 1 ) + ε)v, for all x, v [0, r 2 ] u R +. By (3.6) the proof similar to Steps 1 2, we can deduce that (3.7) µt λ,2 (u, v) v inf v K r2 T λ,2 (u, v) > 0, for all µ 1, (u, v) K 1 K r2. Step 4. By virtue of the sublinear hypothesis condition (G) of f 2 at infinity, there exist ε (0, 1/r(B 2 )), n > 0 C > 0 such that (3.8) λf 2 (x, u, v) + (1 λ)f 2 (x, 0, v) (1/r(B 2 ) ε)v, for all x, v n u R +, (3.9) λf 2 (x, u, v) + (1 λ)f 2 (x, 0, v) (1/r(B 2 ) ε)v + C, for all x u, v R +. From (3.8), (3.9) the similar argument used in Step 2, it can be proved that if v 0 = µ 0 T λ,2 (u 0, v 0 ) for (u 0, v 0 ) K 1 K 2 µ 0 (0, 1], then v 0 R def = C c 2 r(b 2 )ε.
Solutions to Systems of Nonlinear Integral or Differential Equations 275 Hence, we choose R 2 > max{r 2, R }, then (3.10) µt λ, 2 (u, v) v, for all µ (0, 1] (u, v) K 1 K R2. Now we choose an open set D = (K R1 \ K r1 ) (K R2 \ K r2 ). Based on the expressions (3.2), (3.5), (3.7) (3.10), it is easy to verify that {T λ } λ I satisfy the sufficient conditions for the homotopy invariance of fixed point index on D; on the other h, in combination with the classical fixed point index results (see Lemma 2.4), we have i(t 0,1, K r1, K 1 ) = i(t 0,2, K R2, K 2 ) = 1, i(t 0,1, K R1, K 1 ) = i(t 0,2, K r2, K 2 ) = 0. Applying the homotopy invariance of fixed point index the product formula for the fixed point index (see Lemma 2.5), we obtain that i(t 1, D, K 1 K 2 ) = i(t 0, D, K 1 K 2 ) = = 2 i(t 0,j, K Rj \ K rj, K j ) j=1 2 [i(t 0, j, K Rj, K j ) i(t 0, j, K rj, K j )] = 1. j=1 Therefore, system (S) has at least one positive solution. 4. Applications As applications, we consider existence of positive solutions for the following system of second-order ordinary differential equations u (x) = f 1 (x, u(x), v(x)) for x (0, 1), v (x) = f 2 (x, u(x), v(x)) for x (0, 1), (4.1) u(0) = u(1) = 0, v(0) = v (1) = 0, here f 1, f 2 C([0, 1] R + R +, R + ). Theorem 4.1. Assume that f 2 satisfies condition (G) f 1, f 2 satisfy: (H 1) lim sup u 0 + max x [0,1] f 1 (x, u, v) u < π 2 < lim inf u min x [0,1] f 1 (x, u, v) u uniformly w.r.t. v R + ; (H 2) lim inf v 0 + min x [0,1] f 2 (x, u, v) v > π2 4 > lim sup v max x [0,1] f 2 (x, u, v) v uniformly w.r.t. u R +.
276 X. Cheng Zh. Zhang Then system (4.1) has at least one positive solution. Proof. We know that system (4.1) is equivalent to the following system of nonlinear Hammerstein integral equations 1 u(x) = k 1 (x, y)f 1 (y, u(y), v(y)) dy for x [0, 1], where k 1 (x, y) = v(x) = 0 1 0 k 2 (x, y)f 2 (y, u(y), v(y)) dy for x [0, 1], { x(1 y) if x y, y(1 x) if y x, k 2 (x, y) = { x if x y, y if y x. It is easy to verify that kernel functions k i satisfy conditions (i) (iii). According to Theorem 1.5, we need only show that r(b 1 ) = π 2 r(b 2 ) = 4π 2. On that purpose, we need only to verify that the minimal eigenvalue of B1 1 B2 1 is π 2 π 2 /4, respectively. It follows from the following linear eigenvalue problems: { { u (x) = λ n u(x), v (x) = µ n v(x), u(0) = u(1) = 0, v(0) = v (1) = 0. In fact, λ n = n 2 π 2 µ n = (n 1/2) 2 π 2, n N. Remark 4.2. For instance, f 1 (x, u, v) = max{ sin u, u 2 }(1 + tan 1 v) f 2 (x, u, v) = π 2 (1 + cot 1 u) sin v, then f 1 f 2 satisfy conditions (H 1), (H 2) (G). References [1] X. Cheng C. Zhong, Existence of positive solutions for a second-order ordinary differential system, J. Math. Anal. Appl. 312 (2005), 14 23. [2] D. Guo V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. [3] D. Guo J. Sun, Nonlinear Integral Equations, Shong Press of Science Technology, Jinan, 1987. (in Chinese) [4] A. Hammerstein, Nichtlineare intergralgleichungen nebst anwendungen, Acta Math. 54 (1929), 117 176. [5] M. A. Krasnosel skiĭ, Topological methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford, 1964. [6] M. G. Krein M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Transl. Amer. Math. Soc. 10 (1962), 199 325. [7] J. Sun X. Liu, Computation for topological degree its applications, J. Math. Anal. Appl. 202 (1996), 785 796. [8] J. Wang, Existence of nontrivial solutions to nonlinear systems of Hammerstein integral equations applications, Indian J. Pure Appl. Math. 31 (2001), 1303 1311.
Solutions to Systems of Nonlinear Integral or Differential Equations 277 [9] Z. Yang D. O Regan, Positive solvability of systems of nonlinear Hammerstein integral equations, J. Math. Anal. Appl. 311 (2005), 600 614. [10] Z. Zhang, Existence of nontrivial solutions for superlinear systems of integral equations applications, Acta Math. Sinica 15 (1999), 153 162. [11] C. Zhong, X. Fan W. Chen, An Introduction to Nonlinear Functional Analysis, Lanzhou University Press, Lanzhou, 1998. (in Chinese) Manuscript received September 9, 2008 Xiyou Cheng Academy of Mathematics Systems Science Institute of Mathematics Chinese Academy of Sciences Beijing, 100190, P.R. China Department of Applied Mathematics Nanjing Audit University, Nanjing, 210029, P.R. China E-mail address: chengxy03@163.com Zhitao Zhang Academy of Mathematics Systems Science Institute of Mathematics Chinese Academy of Sciences Beijing, 100190, P.R. China E-mail address: zzt@math.ac.cn TMNA : Volume 34 2009 N o 2