Dynamic Boundary Value Problems of the Second

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These new conditions involve linear or quadratic growth constraints on f(t, p, q) in q. The new ...... Now for (t, p, q) ∈ CR7 we have. |f(t, p, q)| ≤ 2b +1+ ... (2.1), (1.3) with R = R9 and N = N2 + 1, where N2 is defined in the proof of. Theorem 3.2.
Dynamic Boundary Value Problems of the Second-Order: Bernstein-Nagumo Conditions and Solvability Johnny Henderson1 and Christopher C Tisdell2 1

Department of Mathematics Baylor University Waco, TX 76798-7328, USA E-mail: Johnny [email protected] 2

School of Mathematics The University of New South Wales, Sydney NSW 2052, Australia E-mail: [email protected] Research supported by The Australian Research Council’s Discovery Projects (DP0450752) Abstract: This article analyzes the solvability of second-order, nonlinear dynamic boundary value problems (BVPs) on time scales. New Bernstein-Nagumo conditions are developed that guarantee an a priori bound on the delta derivative of potential solutions to the BVPs under consideration. Topological methods are then employed to gain solvability. Key words: Bernstein-Nagumo conditions, boundary value problem, system of equations, time scale, dynamic equation, existence of solutions AMS Subject Classification. 39A10, 39A99. Running Head: Bernstein-Nagumo conditions Corresponding Author: C C Tisdell

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1. Introduction In the study of the existence of solutions to boundary value problems (BVPs), major advancements have been made due to the introduction, development and application of so-called “Bernstein-Nagumo” conditions [4], [18]. In particular, Bernstein-Nagumo conditions have allowed the treatment of BVPs featuring differential equations of the type (1.1)

x00 = f (t, x, x0 ),

t ∈ [a, b].

The significant point here is that the right-hand side of (1.1) may depend on x0 . The dependency of the right-hand side on x0 is naturally seen in many physical phenomena and we refer the reader to [21] and [7] for some nice examples. Bernstein-Nagumo conditions for (1.1) are sufficient conditions, involving differential inequalities, that guarantee an a priori bound on the first derivative x0 of potential solutions x, with the bound on x0 being in terms of an a priori bound on x. Topological ideas are then usually used to gain solvability. Bernstein-Nagumo conditions have also been of significance in the solvability of BVPs involving second-order finite-difference equations. These types of “discrete” BVPs arise as discrete approximations to “continuous” BVPs involving ordinary differential equations. For example, [11], [17], [24], [25], [26] formulated “discrete” Bernstein-Nagumo conditions so that the first finite-difference of solutions to the discrete BVP are bounded a priori, with the bound being independent of the stepsize and in terms of a bound on potential solutions to the discrete BVP. Particular interest in these ideas is seen through the elimination of “spurious solutions” [20] to the discrete BVP, and the convergence of solutions of the discrete BVP to solutions of the continuous BVP as the step-size tends to zero. In this paper we are interested in Bernstein-Nagumo conditions and the existence of solutions to the second-order dynamic equation (1.2)

y ∆∆ = f (t, y σ , y ∆ ),

t ∈ [a, b]T ;

subject to the boundary conditions (1.3)

y(a) = A,

y(σ 2 (b)) = B;

where: f : [a, b]T × Rd → Rd , d ≥ 1; and A, B ∈ Rd . Equation (1.2) subject to (1.3) is called a boundary value problem (BVP) where t comes from a so-called “time scale” T. The field of dynamic equations on time scales provides a natural framework for: (1) establishing new insight into the theories of non-classical difference equations; (2) forming novel knowledge about “differential-difference” equations; (3) advancing, in their own right, each of the theories of differential equations and (classical) difference equations. Recently, some Bernstein-Nagumo conditions for dynamic BVPs on time scales were formulated in [16] and [3]. In this article, some alternate Bernstein-Nagumo

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conditions are established, with the new results extending and complimenting those in [16] and [3] in a significant way. The extensions are demonstrated through the use of examples. The paper is organized as follows. In Section 2 a general existence result is presented for (1.2), (1.3). The result provides a natural motivation for the obtention of a priori bounds on solutions and greatly minimizes the proofs of the new results in the following sections. The main tool used here is Leray-Schauder topological degree. In Section 3 some new Bernstein-Nagumo conditions are presented for (1.2), (1.3). These new conditions involve linear or quadratic growth constraints on kf (t, p, q)k in kqk. The new results are stated for systems of BVPs on time scales, however we remark that the ideas are new even for scalar BVPs on time scales. In Section 4 we establish some new results that guarantee a priori bounds on solutions to (1.2), (1.3). The ideas are extensions from the case y ∆∆ = f (t, y σ ). In Sections 5 and 6, the abstract existence theorem of Section 2 is combined with the a priori bound results of Sections 3 and 4 to give a number of new existence results for (1.2), (1.3). Examples to highlight the theory and application of the new ideas are presented throughout the paper. For more on dynamic equations on time scales we refer to [5]. To see a discussion of recent and future applications of dynamic equations on time scales see the featured front page article in New Scientist [22]. For recent related articles on solvability of dynamic BVPs on time scales, please see [12], [10], [19], [9], [23], [13], [2], [6]. To understand the concept of time scales and the above notation, some definitions are useful. Definition 1.1. Define the forward (backward) jump operator σ(t) at t for t < sup T (respectively ρ(t) at t for t > inf T) by σ(t) = inf{τ > t : τ ∈ T},

(ρ(t) = sup{τ < t : τ ∈ T}, ) for all t ∈ T.

Also define σ(sup T) = sup T, if sup T < ∞, and ρ(inf T) = inf T, if inf T > −∞. For simplicity and clarity denote σ 2 (t) = σ(σ(t)) and y σ (t) = y(σ(t)). Define the graininess function µ : T → R by µ(t) = σ(t) − t. Throughout this work the assumption is made that T has the topology that it inherits from the standard topology on the real numbers R. Also assume throughout that a < b are points in T and define the time scale interval [a, b]T := {t ∈ T : a ≤ t ≤ b}. The jump operators σ and ρ allow the classification of points in a time scale in the following way: If σ(t) > t then call the point t right-scattered; while if ρ(t) < t then we say t is left-scattered. If t < sup T and σ(t) = t then call the point t right-dense;

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while if t > inf T and ρ(t) = t then we say t is left-dense. If T has a left-scattered maximum at m then define Tκ = T − {m}. Otherwise Tκ = T. We next define the so-called delta derivative. The novice could skip this definition and for the scalar case look at the results stated in Theorem 1.1. In particular in part (2) of Theorem 1.1 we see what the delta derivative is at right–scattered points and in part (3) of Theorem 1.1 we see that at right–dense points the derivative is similar to the definition given in calculus. Definition 1.2. Fix t ∈ Tκ and let y : T → Rd . Define y ∆ (t) to be the vector (if it exists) with the property that given ² > 0 there is a neighbourhood U of t such that, for all s ∈ U and each i = 1, . . . , d, |[yi (σ(t)) − yi (s)] − yi∆ (t)[σ(t) − s]| ≤ ²|σ(t) − s|. Call y ∆ (t) the (delta) derivative of y(t) at t. Definition 1.3. If Y ∆ (t) = y(t) then define the integral by Z t y(s)∆s = Y (t) − Y (a). a

Theorem 1.1. [14] Assume that y : T → Rd and let t ∈ Tκ . (1) If y is differentiable at t, then y is continuous at t. (2) If y is continuous at t and t is right-scattered, then y is differentiable at t with y(σ(t)) − y(t) y ∆ (t) = . σ(t) − t (3) If y is differentiable and t is right-dense, then y(t) − y(s) . s→t t−s (4) If y is differentiable at t, then y(σ(t)) = y(t) + µ(t)y ∆ (t). y ∆ (t) = lim

Next we define the important concept of right–dense continuity. An important fact concerning right–dense continuity is that every right-dense continuous function has a delta antiderivative [5, Theorem 1.74]. This implies that the delta definite integral of any right–dense continuous function exists. Definition 1.4. We say that y : T → R is right-dense continuous (and write y ∈ Crd (T; Rd )) provided y is continuous at every right-dense point t ∈ T, and lims→t− y(s) exists and is finite at every left-dense point t ∈ T. In this paper we will be interested in so-called “regular” time scales which we define as follows: Definition 1.5. We say a time scale T is regular provided either T = R or T is an isolated time scale (i.e., all points in T are isolated).

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In addition to R and hZ := {0, ±h, ±2h, ±3h, · · · }, h > 0, there are many other regular time scales, e.g. T = q N0 , q > 1 ([5, Example 1.41]) which is important in the theory of orthogonal and q-difference equations, and the harmonic Pn polynomials 1 numbers T = {tn = k=1 k , n = 1, 2, 3, · · · } ([5, Example 1.45]). For numerous other examples see Bohner and Peterson [5]. If T is an isolated time scale and f : T → R, then one can easily show that f σ∆ (t) =

(1.4)

µ(σ(t)) ∆σ f (t). µ(t)

As a consequence of (1.4), for all regular time scales, we have ¡ ¢∆∆ ≥ 2hxσ (t), x∆∆ (t)i + kx∆ (t)k2 , (1.5) kx(t)k2 which will be useful in several of the proofs in this paper. We next define S to be the set of all functions y : T → Rd given by S = {y : y ∈ C([a, σ 2 (b)]T ; Rd ), y ∆ ∈ C([a, σ(b)]T ; Rd ) and y ∆∆ ∈ Crd ([a, b]T ; Rd )}. A solution to (1.2) is a function y ∈ S which satisfies (1.2) for each t ∈ [a, b]. A widely used technique in the theory of BVPs on time scales involves reformulating the BVP (1.2), (1.3) as an equivalent delta integral equation. In particular the BVP (1.2), (1.3) is equivalent to the delta integral equation Z σ(b) (1.6) y(t) = G(t, s)f (s, y σ (s), y ∆ (s))∆s + φ(t), t ∈ [a, σ 2 (b)]T ; a

where

( G(t, s) =

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(b)−t) − (σ(s)−a)(σ , σ(s) ≤ t; σ 2 (b)−a 2

(b)−σ(s)) , t ≤ s; − (t−a)(σ σ 2 (b)−a

is the Green’s function (see [5, p.171]) for the BVP y ∆∆ = 0,

y(a) = 0,

y(σ 2 (b)) = 0;

and Aσ 2 (b) − Ba + (B − A)t φ(t) = . σ 2 (b) − a One of our tools used to gain a priori bounds and existence in this work is the use of dynamic inequalities on f . To minimize notation in the statement of the theorems, the following notation will be used. Let U be a non-negative constant and let V be a positive constant. Define the sets CU and DV by: (1.7) CU := {(t, p, q) ∈ [a, b]T × Rd × Rd : kpk ≤ U }; (1.8) DV

:= {(t, p, q) ∈ [a, b]T × Rd × Rd : kpk = V, −2hp, qi ≤ µ(t)kqk2 }.

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2. General Existence In this section an abstract existence result is presented for (1.2), (1.3). This new result emphasizes the natural search for a priori bounds on solutions to BVPs, which will be conducted in the following sections. Theorem 2.1. Let R and N be positive constants in R and let f : [a, b]T ×Rd ×Rd → Rd , d ≥ 1, be continuous. Consider the family of BVPs: (2.1) (2.2)

y ∆∆ = λf (t, y σ , y ∆ ), y(a) = λA,

t ∈ [a, b]T , λ ∈ [0, 1]; y(σ 2 (b)) = λB.

If all potential solutions to (2.1), (2.2) that satisfy: (2.3) (2.4)

ky(t)k ≤ R, ky ∆ (t)k ≤ N,

for t ∈ [a, σ 2 (b)]T ; for t ∈ [a, σ(b)]T ;

ky(t)k < R, ky ∆ (t)k < N,

for t ∈ [a, σ 2 (b)]T ; for t ∈ [a, σ(b)]T ;

also satisfy: (2.5) (2.6)

with R and N independent of λ, then the BVP (1.2), (1.3) has at least one solution. Proof Let Q := {y : y ∈ C([a, σ 2 (b)]T ; Rd ), y ∆ ∈ C([a, σ(b)]T ; Rd )}. In view of (1.6), we want to show that there exists at least one y ∈ Q with y satisfying (1.6). This solution with then naturally be in S. Define the operator T : Q → C([a, σ 2 (b)]T ; Rd ) by Z σ(b) T y(t) := G(t, s)f (s, y σ (s), y ∆ (s))∆s + φ(t), t ∈ [a, σ 2 (b)]T . a

Thus we want to show the existence of at least one y ∈ Q such that (2.7)

y = T y.

Consider the family of problems associated with (2.7), namely, (2.8)

H(y, λ) := y − λT y = 0,

λ ∈ [0, 1].

Note that (2.8) is equivalent to the family of BVPs (2.1), (2.2). Now consider Ω ⊂ Q with Ω := {y ∈ Q : sup ky(t)k < R, sup ky ∆ (t)k < N }. t∈[a,σ 2 (b)]T

t∈[a,σ(b)]T

¯ → C([a, σ (b)]T ; R ) is continuous and completely continuous Now, since T : Ω ¯ × [0, 1] → C([a, σ 2 (b)]T ; Rd ) is a compact (because f is continuous) then H : Ω mapping. 2

d

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¯ to (2.8) must By the assumptions of the theorem, all possible solutions y ∈ Ω satisfy y ∈ Ω and thus H(y, λ) 6= 0,

∀y ∈ ∂Ω and λ ∈ [0, 1].

Hence the following Leray-Schauder degrees are defined and a homotopy principle applies: dLS (H(y, λ), Ω, 0) = dLS (H(y, 1), Ω, 0) = dLS (H(y, 0), Ω, 0) = 1 since 0 ∈ Ω. By the non-zero property of Leray-Schauder degree, (2.8) has at least one solution in Ω for all λ ∈ [0, 1] and hence (1.2), (1.3) has at least one solution. ¤ 3. Nagumo Conditions In this section some new Bernstein-Nagumo conditions are presented for the dynamic BVP (1.2), (1.3). The first result allows linear growth on kf (t, p, q)k in kqk. Theorem 3.1. Let β1 , K1 and R1 be non-negative constants in R and let f satisfy: kf (t, p, q)k ≤ β1 kqk + K1 , for (t, p, q) ∈ CR1 ; with β1 (σ(b) − a) < 1.

(3.1) (3.2)

If y is a solution to (1.2), (1.3) that satisfies ky(t)k ≤ R1 for t ∈ [a, σ 2 (b)]T then ky ∆ (t)k ≤ N1 for t ∈ [a, σ(b)]T , where N1 is a constant involving kAk, kBk, σ 2 (b)−a, σ(b) − a and K1 . Proof Let y be a solution to (1.2), (1.3) that satisfies ky(t)k ≤ R1 for t ∈ [a, σ 2 (b)]T . Taking the delta derivative in both sides of (1.6) and then applying norms, we have, for each t ∈ [a, σ 2 (b)]T , ¶ Z tµ σ(s) − a ∆ kf (s, y σ (s), y ∆ (s)k∆s ky (t)k ≤ 2 (b) − a σ a ¶ Z σ(b) µ 2 σ (b) − σ(s) + kf (s, y σ (s), y ∆ (s)k∆s 2 σ (b) − a t kAk + kBk + 2 σ (b) − a Z σ(b) kAk + kBk . ≤ (β1 ky ∆ (s)k + K1 )∆s + 2 σ (b) − a a So using this we obtain sup t∈[a,σ(b)]T

ky ∆ (t)k ≤ [σ(b) − a](β1

sup t∈[a,σ(b)]T

ky ∆ (t)k + K1 ) +

kAk + kBk σ 2 (b) − a

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and a rearrangement yields sup

ky ∆ (t)k ≤

t∈[a,σ(b)]T

K1 [σ(b) − a](σ 2 (b) − a] + kAk + kBk [1 − β1 (σ(b) − a)][σ 2 (b) − a]

:= N1 . ¤ Our second Bernstein-Nagumo condition is now presented. It allows quadratic growth of kf (t, p, q)k in kqk. Theorem 3.2. Let T be a regular time scale and let α2 , K2 and R2 be non-negative constants in R. Let f satisfy: £ ¤ (3.3) kf (t, p, q)k ≤ α2 2hp, f (t, p, q)i + kqk2 + K2 , for (t, p, q) ∈ CR2 ; (3.4)

with

2α2 R2 < σ 2 (b) − a.

If y is a solution to (1.2), (1.3) that satisfies ky(t)k ≤ R2 for t ∈ [a, σ 2 (b)]T then ky ∆ (t)k ≤ N2 for t ∈ [a, σ(b)]T , where N2 is a constant involving kAk, kBk, σ 2 (b)−a, σ(b) − a, α2 , K2 and R2 . Proof Let y be a solution to (1.2), (1.3) that satisfies ky(t)k ≤ R2 for t ∈ [a, σ 2 (b)]T . Proceeding as in the proof of Theorem 3.1 we have, for t ∈ [a, σ 2 (b)]T , ky ∆ (t)k Z t ª ¢ σ(s) − a ¡ © α2 2h(y σ (s), f (s, y σ (s), y ∆ (s))i + ky ∆ (s)k2 + K2 ∆s ≤ 2 a σ (b) − a Z σ(b) 2 ª ¢ σ (b) − σ(s) ¡ © σ σ ∆ ∆ 2 α 2h(y (s), f (s, y (s), y (s))i + ky (s)k + K ∆s + 2 2 σ 2 (b) − a t kAk + kBk + 2 . σ (b) − a Now let r(t) := ky(t)k2 for t ∈ [a, σ 2 (b)]T . See, by the product rule, that r∆ (t) = hy σ (t) + y(t), y ∆ (t)i so (3.5)

kr∆ (t)k ≤ 2R2 ky ∆ (t)k.

From (1.5) we see that Z t ¢ σ(s) − a ¡ ∆∆ ∆ ∆s ky (t)k ≤ α r (s) + K 2 2 2 a σ (b) − a Z σ(b) 2 ¢ σ (b) − σ(s) ¡ ∆∆ kAk + kBk + α2 r (s) + K2 ∆s + . 2 σ (b) − a σ 2 (b) − a t

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Using delta integration by parts, it is not difficult to show Z t (t − a)r∆ (t) − (r(t) − r(a)) σ(s) − a ∆∆ r (s)∆s = 2 σ 2 (b) − a a σ (b) − a (t − a)r∆ (t) + kAk2 ≤ , σ 2 (b) − a and Z σ(b) 2 σ (b) − σ(s) ∆∆ kBk2 − (σ 2 (b) − t)r∆ (t) r (s)∆s ≤ . σ 2 (b) − a σ 2 (b) − a t The above estimates, in conjunction with (3.5), now give for t ∈ [a, σ(b)]T , £ ¤ α2 2R2 ky ∆ (t)k + kAk2 + kBk2 kAk + kBk ∆ ky (t)k ≤ + K2 [σ(b) − a] + 2 2 σ (b) − a σ (b) − a and a rearrangement gives α2 (kAk2 + kBk2 ) + K2 (σ(b) − a)(σ 2 (b) − a) + kAk + kBk ∆ ky (t)k ≤ σ 2 (b) − a − 2α2 R2 := N2 for each t ∈ [a, σ(b)]T . We now give an example to highlight the previous result.

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Example 3.1. Let T be a regular time scale. Consider (1.2), (1.3) for d = 1 and f = f1 given by f1 (t, p, q) = pq 2 (t + 1) + 1,

t ∈ [0, b],

0, b ∈ T.

For this f1 , the conditions of Theorem 3.2 are satisfied for the choices α2 = R2 (b + 1) and K2 = 2R22 (b + 1) + 1 with 2R22 (b + 1) < σ 2 (b). Proof We want to show that (3.3) and (3.4) hold for the above choices of α2 and β2 . Fix R2 ≥ 0 such that 2R22 (b + 1) < σ 2 (b). It is easy to see that |f1 (t, p, q)| ≤ R2 q 2 (b + 1) + 1,

for (t, p, q) ∈ CR2 .

Now, consider, for (t, p, q) ∈ CR2 :

£ ¤ α2 (2pf1 (t, p, q) + q 2 ) + K2 = α2 2p2 (t + 1) + 1)q 2 + 2p + K2 ≥ α2 (q 2 + 2p) + K2 ≥ α2 (q 2 − 2R2 ) + K2 = R2 (b + 1)q 2 + 1

for the choices α2 = R2 (b + 1) and K2 = 2R22 (b + 1) + 1. Thus (3.3) holds. Note the ¤ assumption 2R22 (b + 1) < σ 2 (b) ensures (3.4) holds. If T in the above example contains at least one point that is right-scattered then the theorems in [3] do not apply to the above f1 , since condition (F1 ) in [3] is violated.

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A corollary to Theorem 3.2 now follows. The conditions may be easier to verify in practice than those of Theorem 3.2. Corollary 3.1. Let T be a regular time scale and let θ, K and M be non-negative constants in R. Let f satisfy: (3.6) (3.7)

with

kf (t, p, q)k ≤ θkqk2 + K, 2θM [1 + σ 2 (b) − a] < σ 2 (b) − a.

for (t, p, q) ∈ CM ;

If y is a solution to (1.2), (1.3) that satisfies ky(t)k ≤ M for t ∈ [a, σ 2 (b)]T then ky ∆ (t)k ≤ W for t ∈ [a, σ(b)]T , where W is a constant involving kAk, kBk, σ 2 (b)−a, σ(b) − a, θ, K and M . Proof We show that the conditions of the corollary imply those of Theorem 3.2. The case for θ = 0 is obvious, so from now on assume θ > 0. Let y be a solution to (1.2), (1.3) that satisfies ky(t)k ≤ M for t ∈ [a, σ 2 (b)]T . We have, for each t ∈ [a, b]T , 2hy σ (t), y ∆∆ (t)i + ky ∆ (t)k2 ≥ 2M (−ky ∆∆ (t)k) + ky ∆ (t)k2 ≥ (1 − 2M θ)ky ∆ (t)k2 − 2M K, using (3.6) ky ∆∆ (t)k − K ≥ (1 − 2M θ) − 2M K, using (3.6) θ and a rearrangement leads to (3.3) with α2 =

θ , 1 − 2M θ

K2 =

K 1 − 2M θ

where M = R2 . Note that if (3.7) holds then (3.4) holds. Thus all of the conditions of Theorem 3.2 are satisfied and the result follows. ¤. 4. More on A Priori Bounds In this section we obtain more a priori bounds on solutions to (1.2), (1.3). The ideas may be viewed as extensions from the case where f = f (t, y σ ). The following result allows linear growth of kf (t, p, q)k in kpk and kqk. Theorem 4.1. Let α3 , β3 and K3 be non-negative constants in R. If f satisfies: (4.1) kf (t, p, q)k ≤ α3 kpk + β3 kqk + K3 , for (t, p, q) ∈ [a, b]T × Rd × Rd ; (4.2) with β3 (σ(b) − a) < 1; α3 (σ(b) − a) (4.3) and < 1; 1 − β3 (σ(b) − a) then all solutions y to (1.2), (1.3) satisfy ky(t)k ≤ R3 for t ∈ [a, σ 2 (b)]T , where R3 is a constant involving kAk, kBk, σ 2 (b) − a, σ(b) − a and K3 .

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Proof Let y be a solution to (1.2), (1.3) and consider the equivalent equation (1.6). For each t ∈ [a, σ 2 (b)]T we have Z σ(b) ky(t)k ≤ |G(t, s)| kf (s, y σ (s), y ∆ (s))k∆s + kφ(t)k a

Z

σ(b)



¡ ¢ |G(t, s)| α3 ky σ (s)k + β3 ky ∆ (s)k + K3 ∆s + max{kAk, kBk}.

a

Now since Z

σ(b)

(4.4)

|G(t, s)|∆s ≤ [σ(b) − a][σ 2 (b) − a],

t ∈ [a, σ 2 (b)]T

a

it follows that (4.5)

sup t∈[a,σ 2 (b)]T

ky(t)k ≤ [σ(b) − a][σ 2 (b) − a](α3 + β3

sup

ky(t)k

t∈[a,σ 2 (b)]T

ky ∆ (t)k + K3 ) + max{kAk, kBk}.

sup t∈[a,σ(b)]T

Similarly as in the proof of Theorem 3.1, if we take the delta derivative in (1.6), then take norms and apply the assumption on f we obtain, (4.6)

sup

ky ∆ (t)k ≤ [σ(b) − a](α3

t∈[a,σ(b)]T

+ β3

sup

sup

ky(t)k

t∈[a,σ 2 (b)]T

ky ∆ (t)k + K3 ) +

t∈[a,σ(b)]T

kAk + kBk . σ 2 (b) − a

So a rearrangement in (4.6) leads to (4.7)

sup t∈[a,σ(b)]T



ky (t)k ≤

[σ(b) − a](α3 supt∈[a,σ2 (b)]T ky(t)k + K3 ) +

kAk+kBk σ 2 (b)−a

1 − β3 (σ(b) − a)

.

If we substitute (4.7) into (4.5) and rearrange then we obtain · ¸ α3 (σ(b) − a)(σ 2 (b) − a) sup ky(t)k 1 − (4.8) 1 − β3 (σ(b) − a) t∈[a,σ 2 (b)]T · µ ¶ ¸ β kAk + kBk 2 ≤ (σ(b) − a)(σ (b) − a) K+ 2 +K 1 − β3 (σ(b) − a) σ (b) − a + max{kAk, kBk} and an additional rearrangement in (4.8) gives the desired bound on y, which we will call R3 . ¤ The following two theorems involve maximum principles on time scales.

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Theorem 4.2. Let R4 be a positive constant in R. If f is scalar-valued and satisfies: (4.9) (4.10) (4.11)

f (t, −R4 , q) < 0, for t ∈ [a, b]T , q ≤ 0; f (t, R4 , q) > 0, for t ∈ [a, b]T , q ≥ 0; with R4 > max{|A|, |B|};

then, for d = 1, all solutions y to (1.2), (1.3) that satisfy |y(t)| ≤ R4 for t ∈ [a, σ 2 (b)]T also satisfy |y(t)| < R4 for t ∈ [a, σ 2 (b)]T . Proof The proof is quite similar to that of [1] and so is only sketched. Let y be a solution to (1.2), (1.3) satisfying |y(t)| ≤ R4 for t ∈ [a, σ 2 (b)]T . We briefly show that y(t) < R4 for t ∈ [a, σ 2 (b)]T . The case showing y(t) > −R4 for t ∈ [a, σ 2 (b)]T is similar. Assume that there is some t0 ∈ [a, σ 2 (b)]T such that y(t0 ) = R4 . By assumption we have t0 ∈ (a, σ 2 (b))T . Arguing similarly to Akin we see that y σ (ρ(t0 )) = R4 with the maximum principle for time scales giving y ∆ (ρ(t0 )) ≥ 0, and 0 ≤ y ∆∆ (ρ(t0 )) = f (ρ(t0 ), R4 , y ∆ (ρ(t0 ))) < 0 and we reach a contradiction.

¤

Theorem 4.3. Let T be an regular time scale and let R5 be a positive constant in R. If f satisfies: 2hp, f (t, p, q)i + kqk2 > 0, with R5 > max{kAk, kBk};

(4.12) (4.13)

for (t, p, q) ∈ DR5 ;

then all solutions y to (1.2), (1.3) that satisfy ky(t)k ≤ R5 for t ∈ [a, σ 2 (b)]T also satisfy ky(t)k < R5 for t ∈ [a, σ 2 (b)]T . Proof Let y be a solution to (1.2), (1.3) satisfying ky(t)k2 ≤ R52 for t ∈ [a, σ 2 (b)]T . We briefly show that ky(t)k2 < R52 for t ∈ [a, σ 2 (b)]T . Assume that there is some t0 ∈ [a, σ 2 (b)]T such that ky(t0 )k2 = R52 . By assumption we have t0 ∈ (a, σ 2 (b))T . Arguing similarly to [2] we see that ky σ (ρ(t0 ))k2 = R52 with the maximum principle for time scales giving ¡ ¢∆ 0 ≤ ky(t)k2 t=ρ(t0 ) = 2hy(ρ(t0 )), y ∆ (ρ(t0 ))i + µ(ρ(t0 ))ky ∆ (ρ(t0 ))k2 and 0 ≥

¡

ky(t)k2

¢∆∆ t=ρ(t0 )

σ

≥ 2hy (ρ(t0 )), f (ρ(t0 ), y σ (ρ(t0 )), y ∆ (ρ(t0 ))i + ky ∆ (ρ(t0 ))k2 > 0 and we reach a contradiction. Finally, we give a result that allows quadratic growth of kf (t, p, q)k in kqk.

¤

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Theorem 4.4. Let T be a regular time scale and let α6 , K6 be non-negative constants in R. Let f satisfy £ ¤ (4.14) kf (t, p, q)k ≤ α6 2hp, f (t, p, q)i + kqk2 + K6 , for (t, p, q) ∈ [a, b]T × R2d . Then all solutions y to (1.2), (1.3) satisfy ky(t)k ≤ R6 for t ∈ [a, σ 2 (b)]T , where R6 is a constant involving kAk, kBk, σ 2 (b) − a, σ(b) − a, α6 and K6 . Proof Let y be a solution to (1.2), (1.3). Taking norms in (1.6) and then using the assumption on f , we obtain for each t ∈ [a, σ 2 (b)]T ky(t)k Z σ(b) ¡ £ ¤ ¢ ≤ |G(t, s)| α6 2hy σ (s), f (s, y σ (s), y ∆ (s))i + ky ∆ (s)k2 + K6 ∆s a

+ max{kAk, kBk}. If we let r(t) := ky(t)k2 for t ∈ [a, σ 2 (b)]T and use (1.5) then Z σ(b) ky(t)k ≤ |G(t, s)|(α6 r∆∆ (s) + K6 )∆s + max{kAk, kBk}. a

in a similar fashion to [6], using delta integration by parts and (4.4), we then obtain ky(t)k ≤ α6 γ 2 + γ + K6 [σ 2 (b) − a][σ(b) − a] := R6 where γ := max{kAk, kBk}.

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5. Existence for Scalar BVPs In this section an existence result is obtained for scalar BVPs on time scales. Theorem 5.1. Let β7 and K7 be non-negative constants in R and let R7 be a positive constant. If f is continuous, scalar-valued and satisfies: (5.1) (5.2) (5.3) (5.4) (5.5)

f (t, −R7 , q) < 0, for t ∈ [a, b]T , q ≤ 0; f (t, R7 , q) > 0, for t ∈ [a, b]T , q ≥ 0; with R7 > max{|A|, |B|}; |f (t, p, q)| ≤ β7 |q| + K7 , for (t, p, q) ∈ CR7 ; with β7 (σ(b) − a) < 1;

then, for d = 1, the scalar BVP (1.2), (1.3) has at least one solution. Proof We want to show that the conditions of Theorem 2.1 hold for some positive constants R and N . Consider (2.1), (2.2). Obviously, for λ = 0 we have the zero solution, so from now on assume λ ∈ (0, 1].

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If the conditions of our theorem hold then, for λ ∈ (0, 1], λf (t, −R7 , q) < 0, for t ∈ [a, b]T , q ≤ 0; λf (t, R7 , q) > 0, for t ∈ [a, b]T , q ≥ 0; with R7 > max{|λA|, |λB|}; |λf (t, p, q)| ≤ β7 |q| + K7 , for (t, p, q) ∈ CR7 ; with β7 (σ(b) − a) < 1; so that Theorems 3.1 and 4.2 are applicable to the family of BVPs (2.1), (2.2) for d = 1, with R4 = R7 , β7 = β1 and K7 = K1 . Note that these constants are independent of λ. Hence we may apply Theorem 2.1 with R = R7 and N = N1 + 1, where N1 is defined in the proof of Theorem 3.1, and the existence of at least one solution follows. ¤ An example is now presented to highlight the applicability of the previous theorem. Example 5.1. Consider the scalar BVP given by (5.6) (5.7)

y ∆∆ = t + [y σ ]3 + [y ∆ ]1/3 , y(0) = 0 = y(σ 2 (b)).

t ∈ [0, b]T ,

0, b ∈ T, σ(b) < 1;

We claim that the above BVP has at least one solution. Proof We want to show that the conditions of Theorem 5.1 hold. It is not difficult to see f (t, p, q) ≤ b + p3 < 0, for t ∈ [0, b]T , p < (−b)1/3 , q ≤ 0; so we may choose, say, −R7 = (−b − 1)1/3 . In addition f (t, p, q) ≥ p3 > 0, for t ∈ [0, b]T , p > 0, q ≥ 0; so R7 = −(−b − 1)1/3 will suffice. Now for (t, p, q) ∈ CR7 we have |f (t, p, q)| ≤ 2b + 1 + |q|1/3 ≤ β7 |q| + K7 ; for the choices β7 = 1 and K7 = 2b + 2. Note that the assumption σ(b) < 1 implies β7 (σ(b) − 0) < 1. Thus all of the assumptions of Theorem 5.1 hold and the existence of at least one solutions follows. ¤ The ideas of [16] are inapplicable to the above example due to their method requiring that the boundary conditions imply an a priori bound on y ∆ (a) or y ∆ (σ(b)). 6. Existence for Systems In this section a number of existence theorems are presented. The ideas combine the results of Sections 2, 3 and 4 in a natural way.

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Theorem 6.1. Let α8 , β8 and K8 be non-negative constants in R. If f is continuous and satisfies (6.1) (6.2) (6.3)

kf (t, p, q)k ≤ α8 kpk + β8 kqk + K8 , for (t, p, q) ∈ [a, b]T × R2d ; with β8 (σ(b) − a) < 1; α8 (σ(b) − a) < 1; and 1 − β8 (σ(b) − a)

then the BVP (1.2), (1.3) has at least one solution. Proof Consider the family (2.1), (2.2). We want to show that the conditions of Theorem 2.1 hold for some positive constants R and N . Consider (2.1), (2.2). If the assumptions of the theorem hold, then for each λ ∈ [0, 1] we have kλf (t, p, q)k ≤ α8 kpk + β8 kqk + K8 , for (t, p, q) ∈ [a, b]T × R2d and thus Theorem 4.1 is applicable to (2.1), (2.2) with the bound on solutions, R3 defined in the proof of Theorem 4.1, is independent of λ. Thus we have kλf (t, p, q)k ≤ β8 kqk + α8 + R3 + K8 , for (t, p, q) ∈ CR3 ; and 3.1 hold for (2.1), (2.2) with K1 = α8 + R3 + K8 and β1 = β8 . Hence Theorem 2.1 holds for positive constants R = R3 + 1 and N = N1 + 1, where N1 is defined in the proof of Theorem 3.1. The solvability of (1.2), (1.3) now follows. ¤ Theorem 6.2. Let T be a regular time scale. Let α9 and K9 and be non-negative constants in R and let R9 be a positive constant. If f is continuous and satisfies: (6.4) (6.5) (6.6) (6.7)

2hp, f (t, p, q)i + kqk2 > 0, with R9 > max{kAk, kBk}; £ ¤ kf (t, p, q)k ≤ α9 2hp, f (t, p, q)i + kqk2 + K9 , with 2α9 R9 < 1;

for (t, p, q) ∈ DR9 ; for (t, p, q) ∈ CR9 ;

then the BVP (1.2), (1.3) has at least one solution. Proof Consider the family (2.1), (2.2). We want to show that the conditions of Theorem 2.1 hold for some positive constants R and N . If the conditions of the theorem hold, then for each λ ∈ (0, 1], (6.8) 2hp, λf (t, p, q)i + kqk2 > 0, (6.9) with R9 > max{kλAk, kλBk}; £ ¤ (6.10) kλf (t, p, q)k ≤ α9 2hp, λf (t, p, q)i + kqk2 + K9 , (6.11) with 2α9 R9 < 1.

for (t, p, q) ∈ DR9 ; for (t, p, q) ∈ CR9 ;

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Hence Theorems 4.3 and 3.2 apply to (2.1), (1.3). Thus Theorem 2.1 applies to (2.1), (1.3) with R = R9 and N = N2 + 1, where N2 is defined in the proof of Theorem 3.2. The solvability of (1.2), (1.3) now follows. ¤ Theorem 6.3. Let T be a regular time scale. Let α10 , K10 , α11 , K11 be non-negative constants in R and let R11 be a positive constant. If f is continuous and satisfies: £ ¤ (6.12) kf (t, p, q)k ≤ α10 2hp, f (t, p, q)i + kqk2 + K10 , for (t, p, q) ∈ [a, b]T × R2d ; £ ¤ (6.13) kf (t, p, q)k ≤ α11 2hp, f (t, p, q)i + kqk2 + K11 , for (t, p, q) ∈ CR11 ; (6.14) with 2α11 R11 < 1; then the BVP (1.2), (1.3) has at least one solution. Proof We may apply Theorem 4.4 to the family (2.1), (2.2) with α10 = α6 , K10 = K6 and then apply Theorem 3.2 to (2.1), (2.2) with α11 = α2 , K11 = K2 . Then Theorem 2.1 applies to (2.1), (2.2) with R = R6 + 1 and N = N2 + 1. ¤ Theorem 6.4. Let T be a regular time scale. Let α12 and K12 be non-negative constants in R. Let γ := max{kAk, kBk}. If f is continuous and satisfies: £ ¤ (6.15) kf (t, p, q)k ≤ α12 2hp, f (t, p, q)i + kqk2 + K12 , for (t, p, q) ∈ [a, b]T × R2d ; ¡ ¢ (6.16) with 2α12 α12 γ 2 + γ + K12 [σ 2 (b) − a][σ(b) − a] < 1; then the BVP (1.2), (1.3) has at least one solution. Proof We may apply Theorem 4.4 to the family (2.1), (2.2) and then apply Theorem 3.2 to the family (2.1), (2.2). Existence to (1.2), (1.3) then follows from Theorem 2.1 with R = (α12 γ 2 + γ + K12 [σ 2 (b) − a][σ(b) − a]) + 1 and N = N2 + 1. ¤ Theorem 6.5. Let T be a regular time scale. Let β13 and R13 be non-negative constants in R. If f is continuous and satisfies: (6.17) (6.18) (6.19) (6.20)

2hp, f (t, p, q)i + kqk2 > 0, for (t, p, q) ∈ DR13 ; with R13 > max{kAk, kBk}; kf (t, p, q)k ≤ β13 kqk + K13 , for (t, p, q) ∈ CR13 ; with β13 (σ(b) − a) < 1;

then the BVP (1.2), (1.3) has at least one solution. Proof The assumptions (6.17) and (6.18) mean that Theorem 4.3 may be applied to the family (2.1), (2.2). The assumptions (6.19) and (6.20) imply that Theorem 3.1 may be applied to the family (2.1), (2.2). Thus, Theorem 2.1 holds for R = R13 and N = N1 + 1. ¤

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