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Hindawi Publishing Corporation Advances in Difference Equations Volume 2008, Article ID 796851, 13 pages doi:10.1155/2008/796851 Research Article Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations Ravi P. Agarwal, 1 Victoria Otero-Espinar, 2 Kanishka Perera, 1 and Dolores R. Vivero 2 1 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA 2 Departamento de An ´ alise Matem ´ atica, Facultade de Matem ´ aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain Correspondence should be addressed to Ravi P. Agarwal, agarwal@fit.edu Received 21 April 2008; Accepted 17 August 2008 Recommended by Paul Eloe This paper is devoted to using perturbation and variational techniques to derive some sufficient conditions for the existence of multiple positive solutions in the sense of distributions to a singular second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes those problems related to the negative exponent Emden-Fowler equation. Copyright q 2008 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The Emden-Fowler equation, u ΔΔ tqt u α  σt   0,t∈ 0, 1 T , 1.1 arises in the study of gas dynamics and fluids mechanics, and in the study of relativistic mechanics, nuclear physics, and chemically reacting system see, e.g., 1 and the references therein for the continuous model. The negative exponent Emden-Fowler equation α<0 has been used in modeling non-Newtonian fluids such as coal slurries 2. The physical interest lies in the existence of positive solutions. We are interested in a broad class of singular problem that includes those related with 1.1 and the more general equation u ΔΔ tqt u α  σt   g  t, u σ t  ,t∈ 0, 1 T . 1.2 Recently, existence theory for positive solutions of second-order boundary value problems on time scales has received much attention see, e.g., 3–6 for general case, 7 for the continuous case, and 8 for the discrete case. 2 Advances in Difference Equations In this paper, we consider the second-order dynamic equation with homogeneous Dirichlet boundary conditions: P ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −u ΔΔ tF  t, u σ t  , Δ-a.e.t∈  D κ  o , ut > 0,t∈ a, b T , ua0  ub, 1.3 where we say that a property holds for Δ-a.e. t ∈ A ⊂ T or Δ-a.e. on A ⊂ T, Δ-a.e., whenever there exists a set E ⊂ A with null Lebesgue Δ-measure such that this property holds for every t ∈ A \ E, T is an arbitrary time scale, subindex T means intersection to T, a, b ∈ T are such that a<ρb, D a, b T , D κ a, ρb T , D κ 2 a, ρ 2 b T , D o a, b T , D κ  o a, ρb T , and F : D × 0, ∞ → R is an L 1 Δ -Carath ´ eodory function on compact subintervals of 0, ∞, that is, it satisfies the following conditions. Ci For every x ∈ 0, ∞, F·,x is Δ-measurable in D o . ii For Δ-a.e. t ∈ D o , Ft, · ∈ C0, ∞. C c  For every x 1 ,x 2 ∈ 0, ∞ with x 1 ≤ x 2 , there exists m x 1 ,x 2  ∈ L 1 Δ D o  such that   Ft, x   ≤ m x 1 ,x 2  t for Δ-a.e.t∈ D o ,x∈  x 1 ,x 2  . 1.4 Moreover, in order to use variational techniques and critical point theory, we will assume that F satisfy the following condition. PM For every x ∈ 0, ∞, function P F : D × 0, ∞ → R defined for Δ-a.e. t ∈ D and all x ∈ 0, ∞,as P F t, x :  x 0 Ft, r dr, 1.5 satisfies that P F ·,x is Δ-measurable in D o . We consider the spaces C 1 0,rd  D κ  : C 1 rd  D κ  ∩ C 0 D, C 1 c,rd  D κ  : C 1 rd  D κ  ∩ C c D, 1.6 where C 1 rd D κ  is the set of all continuous functions on D such that they are Δ-differentiable on D κ and their Δ-derivatives are rd-continuous on D κ , C 0 D is the set of all continuous functions on D that vanish on the boundary of D,andC c D is the set of all continuous functions on D with compact support on a, b T . We denote as · CD the norm in CD,that is, the supremum norm. On the other hand, we consider the first-order Sobolev spaces H 1 Δ D :  v : D −→ R : v ∈ ACD,v Δ ∈ L 2 Δ  D o  , H  H 1 0,Δ D :  v : D −→ R : v ∈ H 1 Δ D,va0  vb  , 1.7 Ravi P. Agarwal et al. 3 where ACD is the set of all absolutely continuous functions on D. We denote as  t 2 t 1 fsΔs   t 1 ,t 2  T fsΔs for t 1 ,t 2 ∈ D, t 1 <t 2 ,f∈ L 1 Δ  t 1 ,t 2  T  . 1.8 The set H is endowed with the structure of Hilbert space together with the inner product ·, · H : H × H → R given for every v, w ∈ H × H by v, w H :  v Δ ,w Δ  L 2 Δ :  b a v Δ s · w Δ sΔs; 1.9 we denote as · H its induced norm. Moreover, we consider the sets H 0,loc : H 1 loc,Δ D ∩ C 0 D, H c,loc : H 1 loc,Δ D ∩ C c D, 1.10 where H 1 loc,Δ D is the set of all functions such that their restriction to every closed subinterval J of a, b T belong to the Sobolev space H 1 Δ J. We refer the reader to 9–11 for an introduction to several properties of Sobolev spaces and absolutely continuous functions on closed subintervals of an arbitrary time scale, and to 12 for a broad introduction to dynamic equations on time scales. Definition 1.1. u is said to be a solution in the sense of distributions to P if u ∈ H 0,loc , u>0 on a, b T , and equality  b a  u Δ s · ϕ Δ s − F  s, u σ s  · ϕ σ s  Δs  0 1.11 holds for all ϕ ∈ C 1 c,rd D κ . From the density properties of the first-order Sobolev spaces proved in 9, Seccion 3.2, we deduce that if u is solution in the sense of distributions, then, 1.11 holds for all ϕ ∈ H c,loc . This paper is devoted to prove the existence of multiple positive solutions to P by using perturbation and variational methods. This paper is organized as follows. In Section 2, we deduce sufficient conditions for the existence of solutions in the sense of distributions to P . Under certain hypotheses, we approximate solutions in the sense of distributions to problem P  by a sequence of weak solutions to weak problems. In Section 3, we derive some sufficient conditions for the existence of at least one or two positive solutions to P. These results generalize those given in 7 for T 0, 1, where problem P  is defined on the whole interval 0, 1 ∩ T and the authors assume that F ∈ C0, 1 × 0, ∞, R instead of C and PM.Thesufficient conditions for the existence of multiple positive solutions obtained in this paper are applied to a great class of bounded time scales such as finite union of disjoint closed intervals, some convergent sequences and their limit points, or Cantor sets among others. 4 Advances in Difference Equations 2. Approximation to P by weak problems In this section, we will deduce sufficient conditions for the existence of solutions in the sense of distributions to P, where F  f  g and f, g : D × 0, ∞ → R satisfy C and PM, f satisfies C c ,andg satisfies the following condition. C g  For every p ∈ 0, ∞, there exists M p ∈ L 1 Δ D o  such that   gt, x   ≤ M p t for Δ-a.e.t∈ D o ,x∈ 0,p. 2.1 Under these hypotheses, we will be able to approximate solutions in the sense of distributions to problem P  by a sequence of weak solutions to weak problems. First of all, we enunciate a useful property of absolutely continuous functions on Dwhose proof we omit because of its simplicity. Lemma 2.1. If v ∈ ACD,thenv ± : max{± v,0}∈ACD,  v   Δ − v Δ  ·  v   Δ ≤ 0,  v −  Δ  v Δ  ·  v −  Δ ≤ 0, 2.2 Δ-a.e. on D o . We fix {ε j } j≥1 a sequence of positive numbers strictly decreasing to zero; f or every j ≥ 1, we define f j : D × 0, ∞ → R as f j t, xf  t, max  x, ε j  for every t, x ∈ D × 0, ∞. 2.3 Note that f j satisfies C and C g ; consider the following modified weak problem  P j  ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −u ΔΔ tf j  t, u σ t   g  t, u σ t  , Δ-a.e.t∈  D κ  o , ut > 0,t∈ a, b T , ua0  ub. 2.4 Definition 2.2. u is said to be a weak solution to P j  if u ∈ H, u>0ona, b T , and equality  b a  u Δ s · ϕ Δ s −  f j  s, u σ s   g  s, u σ s  · ϕ σ s  Δs  0 2.5 holds for all ϕ ∈ C 1 0,rd D κ . u is said to be a weak lower solution to P j  if u ∈ Hu> 0ona, b T , and inequality  b a  u Δ s · ϕ Δ s −  f j  s, u σ s   g  s, u σ s  · ϕ σ s  Δs ≤ 0 2.6 holds for all ϕ ∈ C 1 0,rd D κ  such that ϕ ≥ 0onD. Ravi P. Agarwal et al. 5 The concept of weak upper solution to P j  is defined by reversing the previous inequality. We remark that the density properties of the first-order Sobolev spaces proved in 9, Seccion 3.2 allows to assert that relations in Definition 2.2 are valid for all ϕ ∈ H and for all ϕ ∈ H such that ϕ ≥ 0onD, respectively. By standard arguments, we can prove the following result. Proposition 2.3. Assume that f, g : D × 0, ∞ → R satisfy (C and (PM, f satisfies (C c , and g satisfies (C g . Then,ifforsomej ≥ 1 there exist u j and u j as a lower and an upper weak solution, respectively, to P j  such that u j ≤ u j on D,thenP j  has a weak solution u j ∈ u j , u j  : {v ∈ H : u j ≤ v ≤ u j on D}. Next, we will deduce the existence of one solution in the sense of distributions to P from the existence of a sequence of weak solutions to P j . In order to do this, we fix {a k } k≥1 , {b k } k≥1 ⊂ D two sequences such that {a k } k≥1 ⊂ a, a  b/2 T is strictly decreasing to a if a  σa, a k  a for all k ≥ 1ifa<σa and {b k } k≥1 ⊂ a  b/2,b T is strictly increasing to b if ρbb, b k  b for all k ≥ 1ifρb <b. We denote that D k :a k ,b k  T , k ≥ 1. Moreover, we fix {δ k } k≥1 a sequence of positive numbers strictly decreasing to zero such that  σ  a k  ,ρb k  T ⊂  a  δ k ,b− δ k  T ,δ k ≤ b − a 2 for k ≥ 1. 2.7 Proposition 2.4. Suppose that F  f  g and f, g : D × 0, ∞ → R satisfy (C and (PM, f satisfies (C c , and g satisfies (C g . Then, if for every j ≥ 1, u j ∈ H is a weak solution to P j  and ν δ : inf j≥1 min aδ,b−δ T ,u j > 0 ∀δ ∈  0, b − a 2  , 2.8 M : sup j≥1 max D u j < ∞, 2.9 then a subsequence of {u j } j≥1 converges pointwise in D to a solution in the sense of distributions u 1 to P. Proof. Let k ≥ 1 be arbitrary; we deduce, from 2.2, 2.7, 2.8,and2.9, that there exists a constant K k ≥ 0 such that for all j ≥ 1,  b k a k  u Δ j s  2 Δs   u Δ j  a k  2 · μ  a k    u Δ j  ρb k  2 · μ  ρ  b k    ρb k  σa k  u Δ j s ·   u j − ν δ k    Δ sΔs ≤ K k   u j ,  u j − ν δ k    H . 2.10 6 Advances in Difference Equations Therefore, for all j ≥ 1 so large that ε j <ν δ 1 ,asu j is a weak solution to P j , by taking ϕ 1 :u j − ν δ 1   ∈ H as the test function in 2.5,from2.9, C c  and C g , we can assert that there exists l ∈ L 1 Δ D o  such that  b 1 a 1  u Δ j s  2 Δs ≤ K 1   b a F  s, u σ j s  · ϕ σ 1 sΔs ≤ K 1  M  b a lsΔs, 2.11 that is, {u j } j≥1 is bounded in H 1 Δ D 1  and hence, there exists a subsequence {u 1 j } j≥1 which converges weakly in H 1 Δ D 1  and strongly in CD 1  to some u 1 ∈ H 1 Δ D 1 . For every k ≥ 1, by considering for each j ≥ 1 the weak solution to P k j u k j and by repeating the previous construction, we obtain a sequence {u k1 j } j≥1 which converges weakly in H 1 Δ D k1  and strongly in CD k1  to some u k1 ∈ H 1 Δ D k1  with {u k1 j } j≥1 ⊂ {u k j } j≥1 . By definition, we know that for all k ≥ 1, u k1 | D k  u k . Let u 1 : D → R be given by u 1 : u k on D k for all k ≥ 1andu 1 a : 0 : u 1 b so that u 1 > 0ona, b T , u 1 ∈ H 1 loc,Δ D ∩ Ca, b T , u 1 is continuous in every isolated point of the boundary of D,and{u k k } k≥1 converges pointwise in D to u 1 . We will show that u 1 ∈ C 0 D; we only have to prove that u 1 is continuous in every dense point of the boundary of D.Let0<ε<Mbe arbitrary, it follows from C c  and C g  that there exist m ε ∈ L 1 Δ D o  such that m ε ≥ 0onD o and Ft, x ≤ m ε t for Δ-a.e. t ∈ D o and all x ∈ ε, M;letϕ ε ∈ H be the weak solution to −ϕ ΔΔ ε tm ε t, Δ-a.e.t∈  D κ  o ,ϕ ε a0  ϕ ε b; 2.12 we know see 4 that ϕ ε > 0ona, b T . For all k ≥ 1 so large that ε k k <ε,sinceu k k and ϕ ε are weak solutions to some problems, by taking ϕ 2 u k k − ε − ϕ ε   ∈ H as the test function in their respective problems, we obtain  u k k , ϕ 2  H   b a F  s, u σ k k s  · ϕ σ 2 sΔs ≤  b a m ε s · ϕ σ 2 sΔs   ϕ ε , ϕ 2  H ; 2.13 thus, 2.2 yields to   ϕ 2   2 H ≤  u k k − ϕ ε , ϕ 2  H ≤ 0, 2.14 which implies that 0 ≤ u k k ≤ ε  ϕ ε on D and so 0 ≤ u 1 ≤ ε  ϕ ε on D. Thereby, the continuity of ϕ ε in every dense point of the boundary of D and the arbitrariness of ε guarantee that u 1 ∈ C 0 D. Finally, we will see that 1.11 holds for every test function ϕ ∈ C 1 c,rd D κ ;fixoneof them. For all k ≥ 1 so large that supp ϕ ⊂ a k ,b k  T and all j ≥ 1 so large that ε k j <ν δ k ,asu k j is a weak solution to P k j , by taking ϕ ∈ C 1 c,rd D κ  ⊂ C 1 0,rd D κ  as the test function in 2.5 and bearing in mind 2.7, we have  b k a k u Δ k j s · ϕ Δ sΔs   u k j ,ϕ  H   b k a k F  s, u σ k j s  · ϕ σ sΔs, 2.15 Ravi P. Agarwal et al. 7 whence it follows, by taking limits, that  b k a k  u k  Δ s · ϕ Δ s − F  s,  u k  σ s  · ϕ σ s  Δs  0, 2.16 which is equivalent because u 1 | D k  u k and ϕ  0  ϕ σ on D o \ D o k to  b a  u Δ 1 s · ϕ Δ s − F  s, u σ 1 s  · ϕ σ s  Δs  0, 2.17 and the proof is therefore complete. Propositions 2.3 and 2.4 lead to the following sufficient condition for the existence of at least one solution in the sense of distributions to problem P. Corollary 2.5. Let F  f  g be such that f,g : D × 0, ∞ → R satisfy (C and (PM, f satisfies (C c , and g satisfies (C g . Then, if for each j ≥ 1 there exist u j and u j a lower and an upper weak solution, respectively, to P j  such that u j ≤ u j on D and inf j≥1 min aδ, b−δ T u j > 0 ∀δ ∈  0, b − a 2  , sup j≥1 max D u j < ∞, 2.18 then P has a solution in the sense of distributions u 1 . Finally, fixed u 1 ∈ H 0,loc is a solution in the sense of distributions to P with F  f  g, we will derive the existence of a second solution in the sense of distributions to Pgreater than or equal to u 1 on D. For every k ≥ 1, consider the weak problem   P k  ⎧ ⎨ ⎩ −v ΔΔ tF  t,  u 1  v   σ t  − F  t, u σ 1 t  , Δ-a.e.t∈  D κ k  o , v  a k   0  v  b k  . 2.19 For every k ≥ 1, consider H k : H 1 0,Δ D k  as a subspace of H by defining it for every v ∈ H k as v  0onD \ D k and define the functional Φ k : H k ⊂ H → R for every v ∈ H k as Φ k v : 1 2   v   2 H −  b k a k G  s,  v   σ s  Δs, 2.20 where function G : D × 0, ∞ → R is defined for Δ-a.e. t ∈ D and all x ∈ 0, ∞ as Gt, x :  x 0  F  t, u σ 1 tr  − F  t, u σ 1 t  dr. 2.21 As a consequence of Lemma 2.1, we deduce that every weak solution to   P k  is nonnegative on D k and by reasoning as in 4,Section3, one can prove that Φ k is weakly lower semicontinuous, Φ k is continuously differentiable in H k , for every v, w ∈ H k , Φ  k vwv, w H −  b k a k  F  s,  u 1  v   σ s  − F  s, u σ 1 s  · w σ sΔs, 2.22 and weak solutions to   P k  match up to the critical points of Φ k . Next, we will assume the following condition. NI For Δ-a.e. t ∈ D o , ft, · is nonincreasing on 0, ∞. 8 Advances in Difference Equations Proposition 2.6. Suppose that F  f  g is such that f, g : D × 0, ∞ → R satisfy (C and ( PM, f satisfies (C c  and (NI, and g satisfies (C g . If {v k } k≥1 ⊂ H, v k ∈ H k is a bounded sequence in H such that inf k≥1 Φ k  v k  > 0, lim k→∞   Φ  k  v k    H ∗ k  0, 2.23 then {v k } k≥1 has a subsequence convergent pointwise in D to a nontrivial function v ∈ H such that v ≥ 0 in D and u 2 : u 1  v is a solution in the sense of distributions to P. Proof. Since {v k } k≥1 is bounded in H, it has a subsequence which converges weakly in H and strongly in C 0 D to some v ∈ H. For every k ≥ 1, by 2.2,weobtain   v − k   H ≤   Φ  k  v k    H ∗ k , 2.24 which implies, from 2.23,thatv ≥ 0onD and so u 2 : u 1  v>0ona, b T . In order to show t hat u 2 : u 1  v ∈ H 0,loc is a solution in the sense of distributions to P,fixϕ ∈ C 1 c,rd D k  arbitrary and choose k ≥ 1 so large that supp ϕ ⊂ a k ,b k  T , bearing in mind that u 1 is a solution in the sense of distributions to P, and the pass to the limit in 2.22 with v  v k and w  ϕ yields to 0   b a  v Δ s · ϕ Δ s −  F  s,  u 1  v  σ s  − F  s, u σ 1 s  · ϕ σ s  Δs   b a  u Δ 2 s · ϕ Δ s − F  s, u σ 2 s  · ϕ σ s  Δs; 2.25 thus, u 2 is a solution in the sense of distributions to P. Finally, we will see that v is not the trivial function; suppose that v  0onD. Condition NIensures that function G defined in 2.21 satisfies for every k ≥ 1andΔ-a.e. s ∈ D o , G  s,  v  k  σ s  ≥  f  s,  u 1  v  k  σ s  − f  s, u σ 1 s  ·  v  k  σ s   v  k  σ s 0  g  s, u σ 1 sr  − g  s, u σ 1 s  dr, 2.26 so that, by 2.20 and 2.22, we have, for every k ≥ 1, Φ k  v k  ≤ 1 2   v k   2 H −  v k ,v  k  H Φ  k  v k  v  k  −  b a  g  s,  u 1  v  k  σ s  − g  s, u σ 1 s  ·  v  k  σ sΔs   b a   v  k  σ s 0  g  s, u σ 1 sr  − g  s, u σ 1 s  dr  Δs; 2.27 moreover, as we know that v  k ≤ p on D for some p>0, it follows from C g  that there exists m ∈ L 1 Δ D o  such that Φ k  v k  ≤ 1 2    v − k   2 H −   v  k   2 H  Φ  k  v k  v  k   2  b a ms ·  v  k  σ sΔs ≤ 1 2   v − k   2 H    Φ  k  v k    H ∗ k ·   v  k   H  2  b a ms ·  v  k  σ sΔs, 2.28 Ravi P. Agarwal et al. 9 and hence, since {v  k } k≥1 is bounded in H and converges pointwise in D to the trivial function v, we deduce, from the second relation in 2.23 and 2.24, that lim k→∞ Φ k v k  ≤ 0 which contradicts the first relation in 2.23. Therefore, v is a nontrivial function. 3. Results on the existence and uniqueness of solutions In this section, we will derive the existence of solutions in the sense of distributions to P where F  f  g 0  ηg 1 , η ≥ 0 is a small parameter, and f, g 0 ,g 1 : D × 0, ∞ → R satisfy C, PM as well as the following conditions. H 1  There exists a constant x 0 ∈ 0, ∞ and a nontrivial function f 0 ∈ L 1 Δ D o  such that f 0 ≥ 0 Δ-a.e. on D o and ft, x ≥ f 0 t,g 0 t, x,g 1 t, x ≥ 0forΔ-a.e.t∈ D o ,x∈  0,x 0  . 3.1 H 2  For every p ∈ 0, ∞, there exist m p ∈ L 1 Δ D o  and K p ≥ 0 such that   ft, x   ≤ m p t for Δ-a.e.t∈ D o ,x∈ p, ∞,   g 1 t, x   ≤ K p for Δ-a.e.t∈ D o ,x∈ 0,p. 3.2 H 3  There are m 0 ∈ L 2 Δ D o  such that   g 0 t, x   ≤ λx  m 0 t for Δ-a.e.t∈ D o ,x∈ 0, ∞, 3.3 for some λ<λ 1 , where λ 1 is the smallest positive eigenvalue of problem −u ΔΔ tλu σ t,t∈ D κ 2 , ua0  ub. 3.4 3.1. Existence of one solution. Uniqueness Theorem 3.1. Suppose that f,g 0 ,g 1 : D × 0, ∞ → R satisfy (C,(PM, and ( H 1 –H 3 . Then, there exists a η 0 > 0 such that for every η ∈ 0,η 0 , problem P  with F  f  g 0  ηg 1 has a solution in the sense of distributions u 1 . Proof. Let η ≥ 0 be arbitrary; conditions H 1 –H 3  guarantee that g : g 0  ηg 1 satisfies C g . We will show that there exists a η 0 > 0 such that for every η ∈ 0,η 0 , hypotheses in Corollary 2.5 are satisfied. Let x 0 and f 0 be given in H 1 ,weknow,from4, Proposition 2.7, that we can choose ε ∈ 0, 1 so small that the weak solution u ∈ H to −u ΔΔ tεf 0 t, Δ-a.e.t∈  D κ  o ,ua0  ub, 3.5 satisfies that u > 0ona, b T and u ≤ x 0 on D. Let j ≥ 1 be so large that ε j <x 0 , we obtain, by H 1 ,that −u ΔΔ t ≤ f 0 t ≤ f j  t, u σ t   g  t, u σ t  , Δ-a.e.t∈ D o , 3.6 whence it follows that u is a weak lower solution to P j . 10 Advances in Difference Equations As a consequence of C, PM, and H 1 –H 3 , by reasoning as in 4, Theorem 4.2, we deduce that problem −u ΔΔ tf j  t, u σ t   g 0  t, u σ t   1, Δ-a.e.t∈  D κ  o , ut > 0,t∈ a, b T , ua0  ub 3.7 has some weak solution u j ∈ H which, from Lemma 2.1 and H 1 ,satisfiesthatu ≤ u j on D.Wewillseethat{ u j } j≥1 is bounded in C 0 D, by taking ϕ j :u j − x 0   ∈ H as the test function, we know from 2.2, H 2 , and H 3  that there exist m x 0 ∈ L 2 Δ D o  such that   ϕ j   2 H ≤  u j − x 0 ,ϕ j  H   b a  f j  s, u σ j s   g 0  s, u σ j s   1  · ϕ σ j sΔs ≤  b a  λ u σ j sm x 0 sm 0 s1  · ϕ σ j sΔs; 3.8 so that, it follows from the fact that the immersion from H into C 0 D is compact, see 9, Proposition 3.7, Wirtinger’s inequality 10, Corollary 3.2 and relation λ<λ 1 that {ϕ j } j≥1 is bounded in H and, hence, { u j } j≥1 is bounded in C 0 D. Thereby, condition H 2  allows to assert that there exists η 0 ≥ 0, such that for all η ∈ 0,η 0  − u ΔΔ j t ≥ f j  t, u σ j t   g 0  t, u σ j t   ηg 1  t, u σ j t  , Δ-a.e.t∈ D o , 3.9 holds, which implies that u j is a weak upper solution to P j . Therefore, for every j ≥ 1 so large, we have a lower and an upper solution to P j , respectively, such that 2.2 is satisfied and so, Corollary 2.5 guarantees that problem P has at least one solution in the sense of distributions u 1 . Theorem 3.2. If f : D × 0, ∞ → R satisfies (C,(C c , and (NI, then, P  with F  f has at most one solution in the sense of distributions. Proof. Suppose that P has two solutions in the sense of distributions u 1 ,u 2 ∈ H 0,loc .Letε>0 be arbitrary, take ϕ u 1 − u 2 − ε  ∈ H c,loc as the test function in 1.11,by2.2 and NI, we have ϕ 2 H ≤  u 1 − u 2 − ε, ϕ  H   b a  f  s, u σ 1 s  − f  s, u σ 2 s  · ϕ σ sΔs ≤ 0, 3.10 thus, u 1 ≤ u 2  ε on D. The arbitrariness of ε leads to u 1 ≤ u 2 on D and by interchanging u 1 and u 2 , we conclude that u 1  u 2 on D. Corollary 3.3. If f : D × 0, ∞ → R satisfies (C,(PM,(NI, and (H 1 -(H 2  with g 0  0  g 1 , then P with F  f has a unique solution in the sense of distributions. [...]... 124–130, 2006 6 R A Khan, J J Nieto, and V Otero-Espinar, “Existence and approximation of solution of three-point boundary value problems on time scale,” Journal of Difference Equations and Applications, vol 14, no 7, pp 723–736, 2008 7 R P Agarwal, K Perera, and D O’Regan, Positive solutions in the sense of distributions of singular boundary value problems,” Proceedings of the American Mathematical Society,... Perera, and D R Vivero, “Wirtinger’s inequalities on time scales, ” Canadian Mathematical Bulletin, vol 51, no 2, pp 161–171, 2008 11 A Cabada and D R Vivero, “Criterions for absolute continuity on time scales, ” Journal of Difference Equations and Applications, vol 11, no 11, pp 1013–1028, 2005 12 M Bohner and A Peterson, Dynamic Equations on Time Scales: An Introduction with Application, Birkh¨ user, Boston,... O’Regan, V Lakshmikantham, and S Leela, An upper and lower solution theory for singular Emden-Fowler equations,” Nonlinear Analysis: Real World Applications, vol 3, no 2, pp 275–291, 2002 3 R P Agarwal, V Otero-Espinar, K Perera, and D R Vivero, “Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods,” Journal of Mathematical Analysis and Applications,... the sense of distributions u1 , u2 such that u1 ≤ u2 on D and u2 − u1 ∈ H Proof Conditions H1 – H4 allow to suppose that for Δ-a.e t ∈ Do , f t, · is nonnegative, nonincreasing, and convex on 0, ∞ because these conditions can be obtained by simply replacing on D × x0 , ∞ f and g0 with f t, x0 and g0 t, x f t, x − f t, x0 , respectively Let u1 be a solution in the sense of distributions to P , its existence... Existence of two ordered solutions Next, by using Theorem 3.1 which ensures the existence of a solution in the sense of distributions to P , we will deduce, by applying Proposition 2.6, the existence of a second one greater than or equal to the first one on the whole interval D; in order to do this, we will assume that f, g0 , g1 : D × 0, ∞ → R satisfy C , PM , H1 – H3 , as well as the following conditions... Otero-Espinar, K Perera, and D R Vivero, Multiple positive solutions of singular Dirichlet problems on time scales via variational methods,” Nonlinear Analysis: Theory, Methods & Applications, vol 67, no 2, pp 368–381, 2007 5 Z Du and W Ge, “Existence of multiple positive solutions for a second-order Sturm-Liouville-like boundary value problem on a measure chain,” Acta Mathematicae Applicatae Sinica,... · is nonincreasing and convex on 0, x0 with x0 given in H1 H5 There are constants θ > 2, C1 , C2 ≥ 0 and x1 > 0 such that g1 t, x x 0< ≤ C1 xθ−1 C2 for Δ-a.e t ∈ Do , x ∈ 0, ∞ , 1 xg1 t, x θ g1 t, r dr ≤ 0 for Δ-a.e t ∈ Do , x ∈ x1 , ∞ 3.11 We will use the following variant of the mountain pass, see 13 Lemma 3.4 If Φ is a continuously differentiable functional defined on a Banach space H and there... guaranteed by Theorem 3.1, and let η > 0 be arbitrary; it is clear that F f g with g : g0 ηg1 satisfies hypothesis in Proposition 2.6; we will derive the existence of an η0 > 0 such that for every η ∈ 0, η0 , we are able to construct a sequence {vk }k≥1 ⊂ H in the conditions of Proposition 2.6 For every k ≥ 1 and v ∈ Hk , as a straight-forward consequence of NI , H3 , H5 , and the compact immersion from... 0 on a, b T , v1 H > R and Φ1 v1 < 0 and hence, since Φ1 0 class of paths in H1 joining 0 and v1 , it follows from 3.16 that c1 : inf max Φ1 v ≥ c0 > Φ1 0 , Φ1 v1 , 3.18 γ∈Γ1 v∈γ 0,1 hence, Lemma 3.4 establishes the existence of a sequence {vk }k≥1 ⊂ H1 such that lim Φ1 vk c1 , k→ ∞ vk lim 1 k→ ∞ Φ1 vk H 0 ∗ H1 3.19 Consequently, bearing in mind that H1 ⊂ Hk and Φk |H1 Φ1 for all k ≥ 1 and by removing... s, r dr; 3.23 s as a straight-forward consequence of the convexity of f and conditions H2 , H3 , H5 , and 3.17 , we deduce that there exist constants C7 > 0 and C8 , C9 ≥ 0 such that b a { vk σ HF s, vk σ s Δs ≥ C7 vk σ θ Lθ Δ − C8 vk σ 2 L2 Δ 1 − C9 3.24 Therefore, relations 3.20 , 3.21 , 3.22 , and 3.24 allow to assert that sequence }k≥1 is bounded in Lθ Do and so, as for every k ≥ 1, Δ 1 vk 2 2 . for the existence of solutions in the sense of distributions to P . Under certain hypotheses, we approximate solutions in the sense of distributions to problem P  by a sequence of weak solutions. 2.17 and the proof is therefore complete. Propositions 2.3 and 2.4 lead to the following sufficient condition for the existence of at least one solution in the sense of distributions to problem. of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations Ravi P. Agarwal, 1 Victoria Otero-Espinar, 2 Kanishka Perera, 1 and Dolores R. Vivero 2 1 Department of Mathematical