Journal of Nonlinear Mathematical Physics Volume 12, Supplement 2 (2005), 36–45 SIDE VI Second Order Dynamic Inclusions Martin BOHNER a and Christopher C TISDELL b a University of Missouri–Rolla, Department of Mathematics, Rolla, MO 65401, U.S.A. E-mail: bohner@umr.edu b University of New South Wales, School of Mathematics, Sydney, NSW 2052, Australia1 E-mail: cct@maths.unsw.edu.au This article is a part of the special issue titled “Symmetries and Integrability of Difference Equations (SIDE VI)” Abstract The theory of dynamic inclusions on a time scale is introduced, hence accommodating the special cases of differential inclusions and difference inclusions. Fixed point theory for set-valued upper semicontinuous maps, Green’s functions, and upper and lower solutions are used to establish existence results for solutions of second order dynamic inclusions. 1 Introduction Let T̃ be a time scale such that 0, T ∈ T̃ for some T > 0 (for the definition of a time scale and related notation see Section 2), put T := [0, T ] ∩ T̃, and consider dynamic inclusions Z T g(t, s)F (s, y σ (s))∆s, t ∈ T, (1.1) y(t) ∈ 0 where F : T × R → CK(R) is a set-valued map and g : T × T → R is a single-valued continuous map (CK(R) denotes the set of nonempty, closed, and convex subsets of R). In Section 3 some general existence principles for inclusions (1.1) are derived by using fixed point theory discussed in [1]. In Section 4 we present a specific function g such that y is a solution of (1.1) if and only if y is a solution of the second order dynamic inclusion y ∆∆ (t) ∈ F (t, y σ (t)), t ∈ Tκ , y(0) = y(σ(T )) = 0 (1.2) (for the time scales specific notation we refer again to Section 2). Using this equivalence and the existence principles for (1.1) from Section 3, we then proceed to establish an existence result for the second order dynamic inclusion (1.2). We also offer another existence result for the dynamic inclusion (1.2) based on the notion of upper and lower solutions on time scales (see [4]). Finally, in Section 5, we present some results for time scales possessing a differentiable forward jump operator. Our investigations follow closely the arguments given by Agarwal, O’Regan, and Lakshmikantham [2] and Stehlik and Tisdell [9] for discrete second order inclusions. For more on boundary value problems on time scales we refer the reader to [7, 8, 10]. Copyright c 2004 by M Bohner and C C Tisdell 1 C C Tisdell gratefully acknowledges the financial support of the Australian Research Council’s Discovery Projects (DP0450752) Second Order Dynamic Inclusions 2 37 Preliminaries In this section we present some definitions and elementary results connected to the time scales calculus. For further study we refer the reader to the monographs [5, 6]. A time scale T is an arbitrary nonempty closed subset of the real numbers R. On T we define the forward and backward jump operators by σ(t) := inf {s ∈ T : s > t} and ρ(t) := sup {s ∈ T : s < t} for t ∈ T. A point t ∈ T with t > inf T is said to be left-dense if ρ(t) = t and right-dense if σ(t) = t, left-scattered if ρ(t) < t and right-scattered if σ(t) > t. The set [a, b] ∩ T with a, b ∈ T is abbreviated by [a, b], and we also shall use the notation [a, b]κ := [a, b] \ (ρ(b), b]. Next, the graininess function µ is defined by µ(t) := σ(t) − t for t ∈ T. For a function f : T → R the (delta) derivative f ∆ (t) at t ∈ T is defined to be the number (provided it exists) with the property such that for every ε > 0 there exists a neighbourhood U of t with f (σ(t)) − f (s) − f ∆ (t)(σ(t) − s) ≤ ε |σ(t) − s| for all s ∈ U. A useful formula is f σ = f + µf ∆ , where f σ := f ◦ σ. (2.1) We will use the product rule for the derivative of the product f g of two differentiable functions f and g (f g)∆ = f ∆ g + f σ g∆ = f g∆ + f ∆ gσ . (2.2) For a, b ∈ T and a differentiable function f , the Cauchy integral of f ∆ is defined by Z b f ∆ (t)∆t = f (b) − f (a). (2.3) a Note that in the case T = R we have σ(t) = t, µ(t) ≡ 0, f ∆ (t) = f ′ (t), and in the case T = Z we have σ(t) = t + 1, µ(t) ≡ 1, f ∆ (t) = ∆f (t) = f (t + 1) − f (t). Another important time scale is T = {q k : k ∈ N} with q > 1, for which σ(t) = qt, µ(t) = (q − 1)t, f ∆ (t) = f (qt) − f (t) , (q − 1)t and this time scale gives rise to so-called q-difference equations. 38 3 M Bohner and C C Tisdell Fixed Point Results A map G : R → CK(R) is called upper semicontinuous provided {uk }k∈N , {vk }k∈N ⊂ R with uk → u, vk → v (k → ∞) and vk ∈ G(uk ) for all k ∈ N always implies v ∈ G(u). Throughout this paper we assume that  F : T × R → CK(R) is such that  F (t, ·) is upper semicontinuous for all t ∈ T (3.1)  F(y) 6= ∅ for all y ∈ C(T), where C(T) abbreviates the set of all continuous functions y : T → R, and F(u) := {v ∈ C(T) : v(t) ∈ F (t, uσ (t)) for all t ∈ T} . The existence principles presented in the remaining part of this section rely on the following two fixed point results which are extracted from [1]. Theorem 1. Let E be a Banach space and C ∈ CK(E). If G : C → CK(C) is upper semicontinuous and compact, then G has a fixed point in C. Theorem 2. Let E be a Banach space. Assume U ⊂ E is open, and let 0 ∈ U . If G : U → CK(E) is upper semicontinuous and compact, then either G has a fixed point in U or there exists u ∈ ∂U and λ ∈ (0, 1) such that u ∈ λG(u). Now we prove the following two existence principles which shall be used to establish existence results for second order dynamic inclusions in Section 4. For a function y ∈ C(T) we put kyk = maxt∈T |y(t)|. We also use the notation |F (t, u)| = supv∈F (t,u) |v|. Theorem 3. Assume (3.1) and suppose that g : T × T → R is continuous. If for all r > 0 there exists a nonnegative function hr ∈ C(T) with |F (t, u)| ≤ hr (t) for all t∈T and all |u| ≤ r and if there exists a constant M with kyk = 6 M for all solutions y of Z T y(t) ∈ λ g(t, s)F (s, y σ (s))∆s, t ∈ T 0 for all λ ∈ (0, 1), then (1.1) has a solution. Proof. We define a linear and continuous operator T : C(T) → C(T) by Z T g(t, s)y(s)∆s, t ∈ T (T y)(t) = 0 so that (3.3) is equivalent to the fixed point problem y ∈ λ(T ◦ F)(y), where F : C(T) → CK(C(T)) is defined in (3.1). We also define E = C(T), U = {u ∈ E : kuk < M } . (3.2) (3.3) Second Order Dynamic Inclusions 39 Now we will apply Theorem 2 to the function T ◦ F by showing that T ◦ F : U → CK(E) is upper semicontinuous and compact. (3.4) Assume there exists u ∈ ∂U and λ ∈ (0, 1) with u ∈ λ(T ◦ F)(u). Then kuk = M , and therefore the second possibility given in Theorem 2 is ruled out. Hence, if (3.4) is true, then the first possibility given in Theorem 2 holds so that T ◦ F has a fixed point in U , which completes the proof. Hence it only remains to show (3.4). We first show that T ◦ F : U → CK(E) is upper semicontinuous. Let {uk }k∈N , {wk }k∈N ⊂ R such that uk → u0 , wk → w0 (k → ∞), and wk ∈ (T ◦ F)(uk ) for all k ∈ N. Thus there exists vk ∈ F(uk ) with wk = T vk . Since uk ∈ U for all k ∈ N, (3.2) implies by [3, page 262] that there exists a compact set Ω ⊂ E with {vk }k∈N ⊂ Ω. Therefore there exists a convergent subsequence {vkν }ν∈N of {vk }k∈N , say vkν → v0 as ν → ∞. Now vkν → v0 and ukν → u0 as ν→∞ and vkν (t) ∈ F (t, uσkν (t)) for all t ∈ T. Thus, since F (t, ·) is upper semicontinuous for all t ∈ T, we may conclude v0 (t) ∈ F (t, uσ0 (t)) for all t ∈ T and therefore v0 ∈ F(u0 ). Since vkν → v0 as ν → ∞ and T : E → E is continuous, we see that wkν = T vkν → T v0 as ν → ∞, and hence w0 = T v0 ∈ (T ◦ F)(u0 ). Therefore T ◦ F : U → CK(E) is upper semicontinuous. As it is also compact by the Arzelà–Ascoli theorem [3, Chapter 17], (3.4) holds and the proof is complete.  Theorem 4. Assume (3.1) and suppose that g : T × T → R is continuous. If there exists a nonnegative function h ∈ C(T) with |F (t, u)| ≤ h(t) for all t∈T and all u ∈ R, then (1.1) has a solution. Proof. If T and F are as in Theorem 3, then (1.1) is equivalent to the fixed point problem y ∈ (T ◦ F)(y). Now we can show as in Theorem 3 that T ◦ F : E → CK(E) is upper semicontinuous and compact, and hence the claim follows by using Theorem 1.  4 Existence Results for Dynamic Inclusions We first prove the following equivalence. Theorem 5. Assume (3.1). Define a function g : T × T → R by ( tσ(s) if t ≤ s σ(T ) − t g(t, s) = tσ(s) if t ≥ σ(s). σ(T ) − σ(s) Then y solves (1.1) iff y solves (1.2). (4.1) 40 M Bohner and C C Tisdell Proof. First assume y solves (1.1). Then there exists a function τ ∈ F(y) such that Z T g(t, s)τ (s)∆s y(t) = 0 Z T Z t g(t, s)τ (s)∆s g(t, s)τ (s)∆s + = 0  t  Z t Z t tσ(s) tσ(s) = − σ(s) τ (s)∆s + t− τ (s)∆s σ(T ) σ(T ) 0 T Z t Z t Z t t τ (s)∆s. σ(s)τ (s)∆s + t σ(s)τ (s)∆s − = σ(T ) 0 T 0 We use the product rule (2.2) and the definition of the integral (2.3) to find 1 y (t) = σ(T ) ∆ t Z σ(s)τ (s)∆s + 0 Z t τ (s)∆s T so that y ∆∆ (t) = τ (t) ∈ F (t, y σ (t)) Clearly, y(0) = 0 and Z Z T σ(s)τ (s)∆s − y(σ(T )) = = T σ(s)τ (s)∆s − 0 = 0 σ(T ) σ(s)τ (s)∆s + σ(T ) 0 0 Z t ∈ Tκ . for all Z 0 σ(T ) σ(s)τ (s)∆s + Z Z σ(T ) τ (s)∆s T σ(T ) σ(s)τ (s)∆s T so that y solves (1.2). Conversely, assume now that y solves (1.2). Then Z T Z T σ g(t, s)y ∆∆ (s)∆s g(t, s)F (s, y (s))∆s ∋ 0 0 Z T Z t Z t t ∆∆ ∆∆ = σ(s)y (s)∆s − σ(s)y (s)∆s + t y ∆∆ (s)∆s σ(T ) 0 0 T


t = T y ∆ (T ) − y(T ) + y(0) − ty ∆ (t) − y(t) + y(0) + t y ∆ (t) − y ∆ (T ) σ(T ) t tT ∆ y (T ) − y(T ) + y(t) − ty ∆ (t) = σ(T ) σ(T ) tµ(T ) ∆ t = − y (T ) − y(T ) + y(t) σ(T ) σ(T ) t t = (y(T ) − y(σ(T ))) − y(T ) + y(t) σ(T ) σ(T ) = y(t) so that y solves (1.1).  Using Theorem 5, we can now present our first existence result for second order dynamic inclusions. Second Order Dynamic Inclusions 41 Theorem 6. Assume (3.1). Suppose there exists a continuous and nondecreasing function Ψ : [0, ∞) → [0, ∞) with Ψ(u) > 0 for u > 0 and a function q : T → [0, ∞) such that |F (t, u)| ≤ q(t)Ψ(|u|) for all u ∈ R and t ∈ T. If Z T c > Q0 := max |g(t, s)|q(s)∆s, sup t∈T 0 c>0 Ψ(c) where g is defined by (4.1), then (1.2) has a solution. Proof. Suppose M > 0 satisfies M/Ψ(M ) > Q0 . Consider y ∆∆ (t) ∈ λF (t, y σ (t)), t ∈ Tκ , y(0) = y(σ(T )) = 0 with 0 < λ < 1. By Theorem 5, this is equivalent to (3.3). Let y be any solution of (3.3) for 0 < λ < 1. Then Z T Z T |g(t, s)|q(s)∆s ≤ Ψ (kyk) Q0 |g(t, s)|q(s)Ψ(|y(s)|)∆s ≤ Ψ (kyk) |y(t)| ≤ λ 0 0 so that kyk ≤ Q0 . Ψ (kyk) If kyk = M , then M/Ψ(M ) ≤ Q0 , contradicting the first line of this proof. Hence the statement follows from Theorem 3.  Our next existence result for second order dynamic inclusions uses upper and lower solutions defined as follows. Definition 1. A function α ∈ C(T) is called a lower solution of (1.2) if  F (t, ασ (t)) ∩ −∞, α∆∆ (t) 6= ∅, t ∈ Tκ , α(0) ≤ 0, α(σ(T )) ≤ 0. A function β ∈ C(T) is called an upper solution of (1.2) if 
F (t, β σ (t)) ∩ β ∆∆ (t), ∞ 6= ∅, t ∈ Tκ , β(0) ≥ 0, β(σ(T )) ≥ 0. Theorem 7. Assume (3.1). Suppose for all r > 0 there exists a nonnegative function hr ∈ C(T) with (3.2). If there exist lower and upper solutions α and β of (1.2) with α(t) ≤ β(t) for all t ∈ T, then (1.2) has a solution y with α(t) ≤ y(t) ≤ β(t) for all t ∈ T. Proof. We define  σ σ  α (t) if x < α (t) h(t, x) = β σ (t) if x > β σ (t)   x otherwise, ̺(x) = ( x x |x| if |x| ≤ 1 if |x| > 1,  σ σ  ̺(x − α (t)) if x < α (t) r(t, x) = ̺(x − β σ (t)) if x > β σ (t)   0 otherwise,   ∆∆  if x < ασ (t)  −∞, α (t)
Γ+ (t, x) = β ∆∆ (t), ∞ if x > β σ (t)   R otherwise, F+ (t, x) = F (t, h(t, x)) ∩ Γ+ (t, x), F+∗ (t, x) = F+ (t, x) + r(t, x). 42 M Bohner and C C Tisdell We apply Theorem 4 to the function F+∗ : T × R → CK(R). Clearly, Γ+ (t, ·) is upper semicountinuous for each t ∈ T and hence so is F+ (t, ·) and therefore F+∗ (t, ·). Now the modified problem y ∆∆ (t) ∈ F+∗ (t, y σ (t)), t ∈ Tκ , y(0) = y(σ(T )) = 0 (4.2) is equivalent by Theorem 5 to the problem y(t) ∈ Z 0 T g(t, s)F+∗ (s, y σ (s))∆s, t ∈ T, (4.3) where g is given in (4.1). Since |F+∗ (t, u)| ≤ |F+ (t, u)| + |r(t, u)| ≤ hkβk (t) + 1, Theorem 4 ensures that (4.3) has a solution y ∈ C(T). Hence by the above equivalence of (4.3) and (4.2), we conclude that there exists a solution y of (4.2). It remains to show that α(t) ≤ y(t) ≤ β(t) holds for all t ∈ T. Assume there exists m ∈ T with y(m) > β(m). Define u := y − β and let θ ∈ T be such that maxt∈T u(t) = u(θ). Therefore u(θ) > 0. By an argument given in [4] we may conclude that u∆∆ (ρ(θ)) ≤ 0 and σ(ρ(θ)) = θ. (4.4) But, since y ∆∆ (ρ(θ)) ∈ F+∗ (ρ(θ), y σ (ρ(θ))) = F+ (ρ(θ), y(θ)) + r(ρ(θ), y(θ)) = F+ (ρ(θ), y(θ)) + ̺(y(θ) − β(θ)), 
there exists w(θ) ∈ β ∆∆ (ρ(θ)), ∞ with y ∆∆ (ρ(θ)) = w(θ) + ̺(y(θ) − β(θ)) ≥ β ∆∆ (ρ(θ)) + ̺(y(θ) − β(θ)) so that u∆∆ (ρ(θ)) = y ∆∆ (ρ(θ) − β(θ)) ≥ ̺(y(θ) − β(θ)) > 0, contradicting (4.4). Hence y(t) ≤ β(t) for all t ∈ T, and a similar argument shows α(t) ≤ y(t) for all t ∈ T. The proof is complete.  5 Time Scales with Differentiable Forward Jump In this section we present an existence result for the second order dynamic inclusion problem (1.2) subject to the assumption that the time scale T has a differentiable forward jump operator σ : T → T. That is, we assume throughout this final section that σ is differentiable. (5.1) We start with some auxiliary results that are needed in the proof of our existence result. These results about time scales satisfying (5.1) are also interesting in their own right. Second Order Dynamic Inclusions 43 Lemma 1. Assume (5.1). If y : T → R is differentiable, then so is y σ : Tκ → R, and we have y σ∆ = σ ∆ y ∆σ . Proof. Using (2.1), we have y σ (t) = y(t) + µ(t)y ∆ (t) = y(t) − ty ∆ (t) + σ(t)y ∆ (t) so that, by the product rule (2.2), y σ∆ (t) = y ∆ (t) − σ(t)y ∆∆ (t) + σ(t)y ∆∆ (t) + σ ∆ (t)y ∆σ (t). This completes the proof.  Lemma 2. Assume (5.1). If y : T → R is twice differentiable, then so is y 2 : T → R, and we have
2
∆∆
2 y2 = 2y σ y ∆∆ + y ∆ + σ ∆ y ∆σ . Proof. First, (y 2 )∆ = y ∆ y + y ∆ y σ by the product rule (2.2). Applying the product rule again, we find
∆∆ y2 = y ∆∆ y σ + y ∆ y ∆ + y ∆∆ y σ + y ∆σ y σ∆ . Now the claim follows by using Lemma 1.  Theorem 8. Assume (3.1) and (5.1). If there exist constants L, K ≥ 0 with |F (t, u)| ≤ inf (Luw + K) w∈F (t,u) (t, u) ∈ T × R, for all (5.2) then any solution y of (1.2) satisfies kyk ≤ K max t∈T Z T |g(t, s)|∆s. 0 Proof. Suppose y solves (1.2). Then, by Theorem 5, y also solves (1.1), i.e., there exists τ ∈ F(y) such that Z T g(t, s)τ (s)∆s for all t ∈ T. y(t) = 0 Since τ ∈ F(y), we have for all s ∈ T |τ (s)| ≤ inf w∈F (s,y σ (s)) (Ly σ (s)w + K) ≤ Ly σ (s)y ∆∆ (s) + K ≤ L 2
∆∆ (s) + K, y 2 where we used Lemma 2 and the fact that σ is an increasing function. Thus Z T |g(t, s)||τ (s)|∆s |y(t)| ≤ 0   Z T L 2
∆∆ y (s) + K ∆s |g(t, s)| ≤ 2 0 Z T L 2 = − y (t) + K |g(t, s)|∆s 2 0 Z T ≤ K |g(t, s)|∆s 0 44 M Bohner and C C Tisdell for all t ∈ T, where we have used Z T 0 |g(t, s)| y 2
∆∆ (s)∆s = −y 2 (t). (5.3) It remains to prove formula (5.3). To this end, we use (4.1) to find Z t Z T Z T


∆∆ t 2 ∆ 2 ∆∆ σ(s) y σ(s) y 2 |g(t, s)| y (s)∆s = (s)∆s − (s)∆s σ(T ) 0 0 0 Z T
∆∆ y2 (s)∆s +t t   Z T Z t



t 2 ∆ 2 ∆ 2 ∆ 2 ∆ T y (T ) − y (s)∆s = t y (t) − y (s)∆s − σ(T ) 0 h
0 i
∆ ∆ +t y 2 (T ) − y 2 (t) i
∆
∆ t h T y 2 (T ) − y 2 (T ) + y 2 (0) + t y 2 (T ) = −y 2 (T ) + y 2 (0) − σ(T )
∆ 2 (T ) + y 2 (T ) µ(T ) y 2 = −y (t) + t σ(T ) 2 = −y (t). This completes the proof.  Theorem 9. If (3.1), (5.1), and (5.2) hold, then (1.2) has a solution. Proof. We may multiply both sides of the inequality in (5.2) by λ ∈ [0, 1] to find |λF (t, u)| ≤ inf (Luw + K) for all w∈λF (t,u) (t, u) ∈ T × R. Therefore Theorem 8 is applicable to y ∆∆ (t) ∈ λF (t, y σ (t)), t ∈ Tκ , y(0) = y(σ(T )) = 0, (5.4) and thus kyk ≤ R for all solutions y of (5.4), where R is defined by R := K max t∈T Z T |g(t, s)|∆s. 0 Now choose U to be the set U := {y ∈ C(T) : kyk < R + 1} . In view of the above argument, there cannot be any solution y of (5.4) with kyk = R + 1. Hence, the relevant operator arguments in the previous theorems are applicable. We have the necessary compactness and upper semicontinuity and thus, by Theorem 2, the boundary value problem (1.2) must have a solution. This concludes the proof.  Second Order Dynamic Inclusions 45 References [1] Agarwal R P, Meehan M, and O’Regan D, Fixed point theory and applications, Cambridge Tracts in Mathematics 141, Cambridge University Press, Cambridge, 2001. [2] Agarwal R P, O’Regan D, and Lakshmikantham V, Discrete second order inclusions, J. Difference Equ. Appl. 9 (2003), 879–885. [3] Agarwal R P, O’Regan D, and Wong P J Y, Positive solutions of differential, difference and integral equations, Kluwer Academic Publishers, Dordrecht, 1999. [4] Akın E, Boundary value problems for a differential equation on a measure chain, Panamer. Math. J. 10 (2000), 17–30. [5] Bohner M and Peterson A, Dynamic equations on time scales: an introduction with applications, Birkhäuser, Boston, 2001. [6] Bohner M and Peterson A, Advances in dynamic equations on time scales, Birkhäuser, Boston, 2003. [7] Henderson J, Peterson A, and Tisdell C C, On the existence and uniqueness of solutions to boundary value problems on time scales, Adv. Difference Equ. 2004 (2004), 93–109. [8] Henderson J and Tisdell C C, Topological transversality and boundary value problems on time scales, J. Math. Anal. Appl. 289 (2004), 110–125. [9] Stehlik P and Tisdell C C, On boundary value problems for second order discrete inclusions, Boundary Value Problems 1 (2005), in press. [10] Tisdell C C and Thompson H B, On the existence of solutions to boundary value problems on time scales, Dynam. Contin. Discrete Impuls. Systems (2005), in press.