On Systems of Boundary Value Problems for Differential Inclusions

Transcription

1 On Systems of Boundary Value Problems for Differential Inclusions Lynn Erbe 1, Christopher C. Tisdell 2 and Patricia J. Y. Wong 3 1 Department of Mathematics The University of Nebraska - Lincoln Lincoln, NE , USA lerbe@math.unl.edu 2 School of Mathematics The University of New South Wales Sydney 2052, Australia cct@maths.unsw.edu.au 3 School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, Singapore , Singapore ejywong@ntu.edu.sg Abstract. Herein we consider the existence of solutions to second-order, two-point boundary value problems (BVPs) for systems of ordinary differential inclusions. Some new Bernstein - Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow the treatment of systems of BVPs without growth restriction. AMS Subject Classification: 34B10, 34B15 Keywords: Boundary value problem; systems of differential inclusions; existence of solutions; a priori bounds; Bernstein - Nagumo condition Running Head: BVPs for Differential Inclusions This research was supported by the Australian Research Council s Discovery Projects (DP ) and Linkage International (LX ) 1

2 Corresponding Author: C C Tisdell School of Mathematics The University of New South Wales Sydney 2052, Australia cct@maths.unsw.edu.au 2

3 1 Introduction Herein we consider the existence of solutions to the following two-point boundary value problem (BVP) for systems of ordinary differential inclusions, x F (t, x, x ), for a.e. t [0, 1], (1.1) αx(0) βx (0) = C, γx(1) + δx (1) = D, (1.2) where: F : [0, 1] R n R n K(R n ) is a multifunction (i.e., a multivalued or set-valued function) and K(R n ) is the family of all nonempty convex and compact subsets of R n (n > 1); C, D are given constants in R n ; α, β, γ, δ are all non-zero given constants in R and a.e. stands for almost every. Note that (1.2) is the well known Sturm-Liouville condition [1]. We also consider (1.1) subject to either of the particular boundary conditions - both special cases of (1.2) - given by x (0) = 0, γx(1) + δx (1) = D, (1.3) αx(0) βx (0) = C, x (1) = 0. (1.4) In the quest for existence of solutions to differential inclusions, the application of appropriate fixed-point methods such as: Leray-Schauder alternative; Granas topological transversality; Mawhin continuation theorem; (or any other degree-based theory) generally rely on the obtention of a priori bounds on possible solutions to a family of differential inclusions related to (1.1), (1.2). (See, for example: [2], [3], [4], [5].) A question with classical roots associated with the preceding discussion is: If x is an appropriate solution to (1.1) then how can an a priori bound on x be obtained in terms of an a priori bound on x? Sufficient conditions on F that guarantee the desired a priori bound on x are referred to as Bernstein - Nagumo conditions. For BVPs involving systems of ordinary differential equations, Bernstein - Nagumo conditions have become a central feature of existence theory (see [6] and references therein) however, less is known about Bernstein - Nagumo conditions for systems of differential inclusions. Due to the important applications of differential inclusions, 3

4 for example, to the areas of control ([7], [8], [9]) and to the treatment of differential equations with discontinuities in the right - hand side ([7], [9]), it seems interesting, significant and worthwhile to formulate Bernstein - Nagumo conditions for the wider class of BVPs involving systems of differential inclusions. Erbe and Krawcewicz [10] studied systems of BVPs involving differential inclusions (n > 1) and gave a Bernstein - Nagumo condition that involved a growth condition on F in x (as well as additional assumptions on F ). Herein, we formulate new, quite general and easily verifiable alternative Bernstein - Nagumo conditions for (1.1) which do not involve any growth restriction on F in x. The new a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some novel results ensuring the existence of solutions to (1.1), (1.2). Several corollaries to the new results are presented that form new existence results for BVPs involving systems of Carathéodory functions. For more on differential inclusions or BVPs, we refer the reader to: [2], [5], [7], [10], [11], [12], [13], [14], [16], [17], [19], [20], [21], [22]. To present the new results we need to define the necessary notations as follows. Definition 1.1 Suppose that E and G are Banach spaces and X E and Y G are subsets. By K(Y ) we denote the family of all nonempty convex and compact subsets of Y. A multivalued map Γ : X K(Y ) is called upper semi-continuous (u.s.c.) if {x X : Γ(x) U} is an open subset of X for any open U in Y. Definition 1.2 In what follows we will consider the following Banach function spaces: C([0, 1]; R n ) = {u : [0, 1] R n : u is continuous on [0, 1]} with the norm u = sup t [0,1] u(t) where denotes the usual Euclidean norm in R n and, will denote the usual inner product on R n ; L 2 ([0, 1]; R n ) = {u : [0, 1] R n : u(t) is L 2 -integrable} with the norm ( 1 1/2 u 2 = u(t) dt) 2 ; 0 4

5 H k ([0, 1]; R n ) = {u : [0, 1] R n : u has weak derivatives u (i) L 2 ([0, 1]; R n ) for 0 i k} with the norm u 2;k = max{ u (i) 2 : 0 i k}; The spaces H k ([0, 1]; R n ) are the usual Sobolev spaces of vector functions, denoted also by W k,2 ([0, 1]; R n ) (for more details see [16]). We now introduce the notion of a Carathéodory map, or multifunction, which will be central for the results to follow. Definition 1.3 A multifunction F : [0, 1] R m K(R n ) is said to be a Carathéodory multifunction (cf., [10], p.198) in case it satisfies the following conditions: (i) the map t F (t, u) is Lebesgue measurable for each u R m ; (ii) the map u F (t, u) is u.s.c. for each t [0, 1]; (iii) for any r 0 there is a function ψ r L 2 [0, 1] such that for all t [0, 1], u R m with u r and y F (t, u) we have y ψ r (t). The following general existence theorem will be very useful for minimizing the length of existence proofs in the remainder of the paper. Theorem 1.4 ([10, Theorem 3.1]) Suppose that F : [0, 1] R n R n K(R n ) is a Carathéodory multifunction and that a, b, c, d are given constants in R with a 2 +b 2 > 0 and c 2 + d 2 > 0 and that A, B are given constants in R n. If there exists a positive constant R (independent of λ) such that: max{ x, x } < R, (1.5) for all solutions x to x λf (t, x, x ), for a.e. t [0, 1], (1.6) ax(0) bx (0) = λa, cx(1) + dx (1) = λb, (1.7) 5

6 for λ [0, 1]; and if the only solution to (1.6), (1.7) for λ = 0 is the zero solution; then, for λ = 1, (1.6), (1.7) has at least one solution in H 2 ([0, 1]; R n ). 2 A Priori Bounds In this section some new a priori bound results to (1.1) are presented, including some Bernstein - Nagumo conditions. The new conditions do not involve any growth restrictions on F therefore are applicable to a wider class of certain problems than those dealt with in [10]. First, we present some a priori bound results for solutions x. To minimize the statement of theorems, we use the following assumption: Assumption 2.1 Let M > 0 be a constant. Suppose that if x 0 M and x 0, x 0 = 0 then we can find an ε > 0 such that for a.e. t [0, 1] we have inf{ x, w + x 2 : w F (t, x, x ), x x 0 + x x 0 < ε} > 0. (2.1) Lemma 2.2 Let M > 0 be a constant such that Assumption 2.1 holds and let α/β > 0, γ/δ < 0. Then all solutions to (1.1), (1.2) satisfy x(t) max{ C / α, M, D / γ }, for a.e. t [0, 1]. Proof Let x be a solution to (1.1), (1.2). Consider w F (t, x, x ) and consider r(t) = x(t) 2 for t [0, 1]. Let t 0 be such that r(t 0 ) = max t [0,1] r(t). If r(t 0 ) = 0 then obviously x(t) = 0 < M for all t [0, 1] and any M > 0 so assume r(t 0 ) > 0 from now on. If t 0 = 0 then 0 r (0) = 2 x(0), x (0), = 2 x(0), αx(0) C, β = 2 α ( β x(0) 2 1 x(0), C α x(0) 2 ). 6

7 So and thus x(0) C / α. 1 x(0), C x(0) C α x(0) 2 α x(0) 2 If t 0 = 1 then r (1) 0 and a similar argument to the case t 0 = 0 yields x(1) D / γ. Now suppose t 0 (0, 1) and r(t 0 ) M 2. We have 0 = r (t 0 ) = 2 x(t 0 ), x (t 0 ) and there exists a ε > 0 such that x x(t 0 ) + x x (t 0 ) < ε. Now since (r(t), r (t)) r(t 0 ), r (t 0 )) as t t 0, there exists an α > 0, and an η > 0, such that for almost every t such that t t 0 < η, we have inf{ x, w + x 2 : w F (t, x, x ), x x 0 + x x 0 < ε} > α > 0, and so 1/2r (t) = x(t)x (t) + x (t) 2 > 0 for almost every t t 0 < η, which contradicts the maximum principle (cf., [14]). Thus the stated bound on x holds as claimed. Lemma 2.3 Let M > 0 be a constant such that Assumption 2.1 holds and let γ/δ < 0. Then all solutions to (1.1), (1.3) satisfy x(t) max{ D / γ, M}, for a.e. t [0, 1]. Proof The proof is almost identical to that of Lemma 2.2. Consider r(t) and t 0 as in the proof of Lemma 2.2. All that is needed to be shown is that if t 0 = 0 then r(0) < M 2. If t 0 = 0 then because of (1.3), we have 0 r (0) = 2 x(0), x (0) = 0, so x(0), x (0) = 0. Therefore, by assumption we have 0 < 2 inf{ x(0), w(0) + x (0) 2 : w(0) F (0, x(0), x (0))} 2[ x(0), x (0) + x (0) 2 ] = r (0), so r (t) is strictly increasing for t near 0. Therefore 0 = r (0) < r (t) for t near 0. This means that r(t) is increasing for t near 0, that is, r(0) < r(t) and hence r(0) max t [0,1] r(t). 7

8 The case t 0 (0, 1] is identical to the proof of Lemma 2.2. Hence the result follows. Lemma 2.4 Let M > 0 be a constant such that Assumption 2.1 holds and let α/β > 0. Then all solutions to (1.1), (1.4) satisfy x(t) max{ C / α, M}, for a.e. t [0, 1]. Proof The proof combines the argument from the proof of Lemma 2.4 applied at the end point t 0 = 1 in a similar fashion. The result follows by using this argument and that of the proof of Lemma 2.2. Remark 2.5 For single-valued functions F (t, x, x ) = {f(t, x, x )}, condition (2.1) reduces to the Hartman condition [23], [24, Chap. XII] x, f(t, x, x ) + x 2 > 0, for a.e. t [0, 1], x > M, x, x = 0. (2.2) We now present some new vector Bernstein - Nagumo conditions. Lemma 2.6 Let M and N be positive constants. If inf{ x, w : w F (t, x, x ), x M, x = N} > 0, (2.3) then all solutions to (1.1) that satisfy: x(t) M, for a.e. t [0, 1]; x (0) < N, x (1) < N; (2.4) also satisfy x (t) < N, for a.e. t [0, 1]. Proof Let x be a solution to (1.1) satisfying x(t) M for a.e. t [0, 1]. We use proof by contradiction and assume that there exists a t 0 [0, 1] such that x (t 0 ) N. Obviously from (2.4) we see that t 0 (0, 1). Next, define the function r(t) = x (t) 2 N 2, and assume that r attains its non-negative maximum value on [0, 1] at t 0 (0, 1). By (2.4) and the continuity of r, there must exist a t 1 [t 0, 1) such that r(t 1 ) = 0 (so 8

9 x (t 1 ) = N) and 0 r (t 1 ) = 2 x (t 1 ), x (t 1 ) 2 inf{ x (t 1 ), w(t 1 ) : w(t 1 ) F (t 1, x(t 1 ), x (t 1 )), x(t 1 ) M, x (t 1 ) = N} > 0, by (2.3), and a contradiction is reached. Therefore x (t) < N, for a.e. t [0, 1]. Lemma 2.7 Let M and N be positive constants. If sup{ x, w : w F (t, x, x ), x M, x = N} < 0, (2.5) then all solutions to (1.1) that satisfy: x(t) M, for a.e. t [0, 1]; x (0) < N, x (1) < N; (2.6) also satisfy x (t) < N, for a.e. t [0, 1]. Proof The proof is virtually identical to that of Lemma 2.6 except that there exists a t 2 (0, t 0 ] such that r(t 2 ) = 0 and 0 r (t 2 ) with a contradiction arising from (2.5). Remark 2.8 If F is a single-valued Carathéodory vector function, that is, when F (t, x, x ) = {f(t, x, x )} and f : [0, 1] R 2n R n (n > 1), then conditions (2.3) and (2.5) respectively reduce to x, f(t, x, x ) > 0, for a.e. t [0, 1], x M, x = N, (2.7) x, f(t, x, x ) < 0, for a.e. t [0, 1], x M, x = N. (2.8) For continuous, vector valued functions f, these inequalities were utilized in [6]. Notice that Lemmas 2.6, 2.7 and Remark 2.8 all assume that x (0) < N, x (1) < N. (2.9) The following lemmas give conditions involving the magnitude of the constants in (1.7), as well as M, such that (2.9) will hold. 9

10 Lemma 2.9 Let M and N be positive constants with β and γ both non-zero. If (2.3) or (2.5) hold and ( C + α M N > max, β ) D + δ M, (2.10) γ then all solutions to (1.1), (1.2) satisfying x(t) M, for a.e. t [0, 1], also satisfy x (t) < N, for a.e. t [0, 1]. Proof In view of the conditions of Lemmas 2.3 or 2.4 or Remark 2.8, it suffices to show that (2.4) holds. Rearrange (1.2) and take norms to obtain x (0) = C αx(0) β C + α M β < N. A similar rearrangement of the boundary conditions at t = 1 leads to x (1) < N. Therefore the result follows from Lemmas 2.3 or 2.4 or Remark 2.8. In a similar manner, one may establish the following results. We omit the details. Lemma 2.10 Let M and N be positive constants with γ non-zero. If (2.3) or (2.5) hold and N > D + δ M, (2.11) γ then all solutions to (1.1), (1.3) satisfying x(t) M, for a.e. t [0, 1], also satisfy x (t) < N, for a.e. t [0, 1]. Lemma 2.11 Let M and N be positive constants with β non-zero. If (2.3) or (2.5) hold and N > C + α M, (2.12) β then all solutions to (1.1), (1.4) satisfying x(t) M, for a.e. t [0, 1], also satisfy x (t) < N, for a.e. t [0, 1]. 10

11 3 Existence of Solutions In this section some new existence theorems to (1.1), (1.2) are presented. The a priori bound lemmas from Section 2 are utilised, in conjunction with Theorem 1.4, to produce the new results. The conditions in the new existence theorems do not involve any growth restrictions on F and therefore are applicable to a wider class of certain problems than those dealt with in [10]. Theorem 3.1 Let F : [0, 1] R n R n K(R n ) be a Carathéodory multifunction and let M and N be positive constants with α(γ + δ) + βγ 0. If (2.1), (2.3) and (2.10) hold then (1.1), (1.2) has at least one solution in H 2 ([0, 1]; R n ). Proof Consider the family of BVPs x λf (t, x, x ), for a.e. t [0, 1], (3.1) αx(0) βx (0) = λc, γx(1) + δx (1) = λd, (3.2) where λ [0, 1] and note that this is in the form (1.6), (1.7). Also we see that, for λ = 1, system (3.1), (3.2) is equivalent to the BVP (1.1), (1.2). All that is required is to show that the conditions of Theorem 1.4 hold. Since α(γ + δ) + βγ 0 it is easy to see by direct computation that the only solution to (3.1), (3.2) is the zero solution. We now show that the system (3.1), (3.2) satisfies the conditions of Lemmas 2.2, 2.6, 2.9. Consider w F (t, x, x ) and let w 1 = λw where w 1 λf (t, x, x ). See that, for λ = 0 the only solution to (3.1), (3.2) is the zero solution (and thus an a priori bound holds), so assume from now on that λ (0, 1]. Since (2.1) holds we then have, for λ (0, 1] 0 < λ inf{ x, w + x 2 : w F (t, x, x ), x x 0 + x x 0 < ε} = inf{ x, λw + λ x 2 : w F (t, x, x ), x x 0 + x x 0 < ε} inf{ x, w 1 + x 2 : w 1 λf (t, x, x ), x x 0 + x x 0 < ε}. Also notice that max{ λc / α, M, λd / γ } max{ C / α, M, D / γ }. 11

12 So we conclude, by Lemma 2.2 that all solutions to (3.1), (3.2) satisfy x(t) max{ C / α, M, D / γ } := M 1. If λ = 0 then the solution to (3.1) is of the form x = 0 so we have an a priori bound on x. So, assume λ (0, 1]. Since (2.3) holds we then have 0 < λ inf{ x, w : w F (t, x, x ), x M, x = N} = inf{ x, λw : w F (t, x, x ), x M, x = N} = inf{ x, w 1 : w 1 λf (t, x, x ), x M, x = N}. Also, from (2.10), ( C + α M N > max, β ( λc + α M max β ) D + δ M, γ λd + δ M, γ ). So by Lemmas 2.6 and 2.9 we have an a priori bound on x with x (t) < N. Hence, if we define R > 0 by R := max{m 1, N} + 1 then all of the conditions of Theorem 1.4 are satisfied and the family (3.1), (3.2) has a solution for λ = 1. Then so must the BVP (1.1), (1.2). Theorem 3.2 Let F : [0, 1] R n R n K(R n ) be a Carathéodory multifunction and let M and N be positive constants. If (2.1), (2.5) and (2.10) hold then (1.1), (1.2) has at least one solution in H 2 ([0, 1]; R n ). Proof In view of Lemma 2.7, the proof is virtually identical to that of Theorem 3.1. Theorem 3.3 Let F : [0, 1] R n R n K(R n ) be a Carathéodory multifunction and let M and N be positive constants. If (2.1), (2.3) and (2.11) hold then (1.1), (1.3) has at least one solution in H 2 ([0, 1]; R n ). 12

13 Proof Consider the family of BVPs x λf (t, x, x ), for a.e. t [0, 1], (3.3) x (0) = 0, γx(1) + δx (1) = λd, (3.4) where λ [0, 1]. The a priori bounds on solutions x and x to (3.3), (3.4) follow in virtually an identical fashion to the arguments used in the proof of Theorem 3.1 and so are omitted for brevity. We obtain an R > 0 defined by R := max{max{m, D / γ }, N} + 1 so that all of the conditions of Theorem 1.4 are satisfied. The existence of a solution to (1.1), (1.3) now follows. Theorem 3.4 Let F : [0, 1] R n R n K(R n ) be a Carathéodory multifunction and let M and N be positive constants. If (2.1), (2.5) and (2.11) hold then (1.1), (1.3) has at least one solution in H 2 ([0, 1]; R n ). Proof In view of Lemma 2.7, the proof is virtually identical to that of Theorem 3.3. Theorem 3.5 Let F : [0, 1] R n R n K(R n ) be a Carathéodory multifunction and let M and N be positive constants. If (2.1), (2.3) and (2.12) hold then (1.1), (1.4) has at least one solution in H 2 ([0, 1]; R n ). Proof The proof is similar to that of Theorem 3.3. We obtain an R > 0 defined by R := max{max{m, C / α }, N} + 1 so that all of the conditions of Theorem 1.4 are satisfied. The existence of a solution to (1.1), (1.4) now follows. Theorem 3.6 Let F : [0, 1] R n R n K(R n ) be a Carathéodory multifunction and let M and N be positive constants. If (2.1), (2.5) and (2.12) hold then (1.1), (1.4) has at least one solution in H 2 ([0, 1]; R n ). 13

14 Proof In view of Lemma 2.7, the proof is virtually identical to that of Theorem 3.4. For Carathéodory single-valued functions F = {f}, (1.1) reduces to x = f(t, x, x ), for a.e. t [0, 1], (3.5) and the following new existence results follow as corollaries to Theorems The proofs involve applying the respective theorems to (1.1) for the special case F (t, x, x ) = {f(t, x, x )}. Corollary 3.7 Let f : [0, 1] R n R n R n be a Carathéodory function and let M and N be positive constants. If (2.2), (2.7) and (2.10) hold then (3.5), (1.2) has at least one solution in H 2 ([0, 1]; R n ). Corollary 3.8 Let f : [0, 1] R n R n R n be a Carathéodory function and let M and N be positive constants. If (2.2), (2.8) and (2.10) hold then (3.5), (1.2) has at least one solution in H 2 ([0, 1]; R n ). Corollary 3.9 Let f : [0, 1] R n R n R n be a Carathéodory function and let M and N be positive constants. If (2.2), (2.7) and (2.11) hold then (3.5), (1.3) has at least one solution in H 2 ([0, 1]; R n ). Corollary 3.10 Let f : [0, 1] R n R n R n be a Carathéodory function and let M and N be positive constants. If (2.2), (2.8) and (2.11) hold then (3.5), (1.3) has at least one solution in H 2 ([0, 1]; R n ). Corollary 3.11 Let f : [0, 1] R n R n R n be a Carathéodory function and let M and N be positive constants. If (2.2), (2.7) and (2.12) hold then (3.5), (1.4) has at least one solution in H 2 ([0, 1]; R n ). Corollary 3.12 Let f : [0, 1] R n R n R n be a Carathéodory function and let M and N be positive constants. If (2.2), (2.8) and (2.12) hold then (3.5), (1.4) has at least one solution in H 2 ([0, 1]; R n ). 14

15 4 An Example In this final section we highlight the applicability of the new theorems to an example. Example 4.1 Let x = (x 1, x 2 ) and p = (p 1, p 2 ). Consider (1.1), (1.2) for n = 2, where F (t, x, p) = F (t, x 1, x 2, p 1, p 2 ) x 1 (0) x 2 (0) x 1 e x 2p 2 + κx 2 1p p 1 = κ [1,2], t [0, 1], x 2 e x 2p 2 + κx 2 2p p 2 x 1 1(0) 2 x 1 (1) x 1(1) 2 =, + 2 x 2(0) x 2 (1) x 2(1) 1 2 = There is no growth condition applicable to F and thus the theorems of [10] do not apply. We will apply Theorem 3.1. Firstly, for x M, with M to be chosen below, and x, p = 0, consider, for all f(t, x, p) F (t, x, p) x, f(t, x, p) = x 2 1e x 2p 2 + κ(x 1 p 1 ) 3 + x 1 p 1 + x 2 2e x 2p 2 + κ(x 2 p 2 ) 3 + x 2 p 2 = e x 2p 2 [x x 2 2] (since x 1 p 1 = x 2 p 2 ) > 0 for any positive choice of M. For convenience, choose M = 1, thus (2.1) holds. Now, for x 1, p = N = 2 we have p, f(t, x, p) = p 1 x 1 e x 2p 2 + κx 2 1p p p 2 x 2 e x 2p 2 + κx 2 2p p e x 2p 2 [p 1 x 1 + p 2 x 2 ] 2 > 0 for x 1, p = 2, thus (2.3) holds. It is easy to see that (2.10) holds for our choice of M = 1 and N = 2 and for the given boundary conditions. Thus Theorem 3.2 is applicable and the BVP has a solution. 15

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