Boundary value problems associated with singular strongly nonlinear equations with functional terms

1 Introduction

Boundary value problems for highly nonlinear differential equations in the whole real line, even governed by nonlinear differential operators, have been widely investigated in the last decade. Such problems are involved in many applications in several fields, such as non-Newtonian fluid theory, diffusion of flows in porous media, nonlinear elasticity, theory of capillary surfaces and, more recently, the modeling of glaciology (see, e.g., [9, 20, 24]). Starting from the simplest types of ODEs governed by the p-Laplace operator Φ p ( z ) := | z | p − 2 z , that is,

many authors have proposed generalizations in various directions, in particular considering a more general nonlinear differential operator, a Φ-Laplacian type operator, which can be a generic homeomorphism, a singular or a non-surjective operator (see, e.g., [2, 3, 4, 12, 13, 14, 19]). We also refer the reader to the survey [11] and to the references therein included. Let us also mention equations with mixed differential operators, that is,

where a is a continuous positive function (see, e.g., [5, 8, 15, 18]). In the autonomous case, namely,

equation (1.1) also arises in some models, e.g. reaction-diffusion equations with non-constant diffusivity and porous media equations.

In some models intervening in the aforementioned applications, the dynamics may also depend an a functional argument, since it may present a delay or a non-local term (such as a convolution integral). For instance, in reaction-diffusion models, the reactive term may involve the whole domain. As far as we know, equations involving both the Φ-Laplacian operator and functional terms are less studied and understood due to technical difficulties, see [1, 21]. So, the main aim of this paper is to provide a quite general approach in order to treat functional differential equations governed by general nonlinear differential operators, covering various types of functional dependences (e.g., delayed ODEs and non local equations). We also allow a functional dependence of boundary conditions.

More in detail, in this paper we study the solvability (in a suitable sense) of the following general boundary value problem (BVP, in short):

where Φ : R → R , the so-called Φ-Laplacian operator, is a strictly increasing homeomorphism, k : I ⟶ R is a bounded non-negative function satisfying

f and ρ are Carathéodory functions, and G x , H a , H b are functional terms, i.e.,

• G : W 1 , 1 ( I ) → L ∞ ( I ) is a continuous operator which verifies suitable boundedness and monotonicity conditions;

• H a , H b : W 1 , 1 ( I ) → R are continuous and increasing operators.

The proposed framework is very general, since it contains, as particular cases, delayed and non-local differential equations; moreover, we point out that the function k(t) inside the differential operator may vanish on a set having zero Lebesgue measure. As a consequence, the differential equation in BVP (1.2) may be singular, and this requires an accurate choice of the space of the solutions (since they present a low regularity). In particular, we look for solutions in the Sobolev space W^1,1(I), and this justifies the choice of W^1,1(I) as the domain of the involved functional operators. However, as we show in our existence result, a possible higher regularity of the solutions is related to the rate of integrability of the function 1/k. More precisely, we shall prove that when 1 / k ∈ L θ ( w i t h 1 < θ ≤ ∞ ), then there exists a solution belonging to W^1,θ(I); in particular, when 1/k is continuous, we find C¹-solutions. Hence, from this point of view, our main result concerns both the existence and the regularity of the solutions.

In this framework, a typical approach to get existence results is given by the combination of fixed point techniques and the method of upper and lower solutions. A crucial tool which gives a priori bounds for the derivatives of the solutions is a Nagumo-type growth condition on the nonlinearity. Recently, in the paper [23] the authors obtained an existence result assuming a weak form of Wintner-Nagumo growth condition. The approach of [23] has been fruitfully extended to the context of singular equations: see [5, 6, 7, 8, 17]. In our main result (see Theorem 2.6 below) we assume the following weak Nagumo growth condition:

where μ ∈ L q ( I ) (for some q > 1 ) , ℓ ∈ L 1 ( I ) a n d ψ : ( 0 , ∞ ) → ( 0 , ∞ ) satisfies

This assumption allows to consider a very general operator Φ.

While we refer to Section 4 for some concrete examples illustrating the applicability of our results, here we limit ourselves to point out that our approach allows us to prove the solvability of, e.g.,

where Φ p ( z ) = | z | p − 2 z is the usual p-Laplace operator, ϑ 0 , δ are positive constants and the functional term G x = x τ is of delay-type, that is,

A brief plan of the paper is now in order.

1. In Section 2 we fix some preliminary definitions and we state our main existence result, namely Theorem 2.6.

2. In Section 3 we provide the proof of Theorem 2.6, which articulates into two steps: first, we perform a truncation argument and we introduce an auxiliary BVP to which suitable existence results do apply; then, we show that any solution of the ‘truncated’ problem is a solution of the original BVP. In doing this, we use in a crucial way the assumption of the existence of a well-ordered pair of lower and upper solutions of our problem.

3. In Section 4 we present some examples to which our Theorem 2.6 applies.

4. Finally, we close the paper with an Appendix containing the explicit proof of a technical Lemma, which in some previous papers was missing, and in other papers was either not complete or not correct.

2 Preliminaries and main results

Let a , b ∈ R satisfy a < b, and let I := [a, b]. As mentioned in the Introduction, throughout this paper we shall be concerned with BVPs of the following form

where Φ : R → R is a strictly increasing homeomorphism, f , ρ : I × R → R are Carathéodory functions, and k , G , H a , H b satisfy the following assumptions:

(H1) k : I → R is a nonnegative function satisfying

(H2) G : W 1 , 1 ( I ) → L ∞ ( I ) is continuous (with respect to the usual norms) and bounded when W^1,1(I) is thought of as a subspace of L^∞(I); this means, precisely, that for every r > 0 there exists η r > 0 such that

(H3) There exists a constant k ≥ 0 such that

(H4) H a , H b : W 1 , 1 ( I ) → R are continuous (with respect to the usual topologies) and monotone increasing, that is,

3 Proof of Theorem 2.6

The proof of Theorem 2.6 is rather technical and long; for this reason, after having introduced some constants and parameters used throughout, we shall proceed by establishing several claims. Roughly put, our approach consists of two steps.

STEP I: As a first step, by crucially exploiting the existence of a well-ordered pair of lower and upper solutions α, β for (2.1) (see, precisely, assumption (H5)), we perform a truncation argument and we introduce a new problem, say (P)_τ, to which some abstract results do apply.

STEP II: Then, we show that any solution of (P)_τ is actually a solution of (2.1). In doing this, we use again in a crucial way the fact that α and β are, respectively, a lower and an upper solution for (2.1).

We then begin by fixing some quantities which shall be used all over the proof.

First of all, we choose a real M > 0 in such a way that ∥ α ∥ L ∞ ( I ) ≤ M a n d ∥ β ∥ L ∞ ( I ) ≤ M . Moreover, using assumption (H2), we let η M > 0 be such that

With reference to assumption (H7), we then set

Now, since Φ is strictly increasing, we can choose N > 0 such that

accordingly, owing to (2.8), we fix L = L_M ≥ N > 0 in such a way that

and we consider the function y_L ∈ L¹(I) defined as:

Following the notation in Appendix A, we also define the truncating operators

Given any x ∈ W^1,1(I), we then consider the function F_x defined by

Finally, we consider the operators B a , B b : W 1 , 1 ( I ) → R defined as

Thanks to all these preliminaries, we can finally introduce the following BVP (which can be thought of as a truncated version of problem (2.1)):

We now proceed by following the steps described above.

1. In this first step we prove the following result: there exists (at least) one solution u ∈ W^1,1(I) of problem (3.8); this means, precisely, that

   • the map t ⟼ Φ(k(t) u'(t)) is in W^1,1(I) and

   Φ k ( t ) u ′ ( t ) ′ = F u ( t )  for a.e.  t ∈ I ;

   • u(a) = B_a[u] and u(b) = B_b[u].

   Furthermore, the following higher-regularity assertions hold:

   1. if 1 / k ∈ L ϑ ( I ) for some 1 < ϑ ≤ ∞ , one also has that x₀ ∈ W^1,ϑ(I);

   2. if k ∈ C ( I , R ) and k > 0 on I, then u ∈ C 1 ( I , R ) .

   CLAIM 1. There exists a non-negative function ψ ∈ L 1 ( I ) such that

   (3.9) F X ( t ) ≤ ψ ( t )  for a.e.  t ∈ I  and every  x ∈ W 1 , 1 ( I ) .

   First of all, by the choice of M and the very definition of T_x we have

   − M ≤ α ( t ) ≤ T X ( t ) ≤ β ( t ) ≤ M  for all  x ∈ W 1 , 1 ( I )  and any  t ∈ I ;

   as a consequence, owing to the choice of η M in (3.1), we get

   (3.10) G T x L ∞ ( I ) ≤ η M  for every  x ∈ W 1 , 1 ( I ) .

   Moreover, owing to the very definition of D, we also have

   (3.11) D T x ′ ( t ) ≤ y L ( t )  for any  x ∈ W 1 , 1 ( I )  and a.e.  t ∈ I .

   Gathering together (3.10) and (3.11), we deduce from assumption (H6) that there exists a non-negative function h = h_ηM,L ∈ L¹(I) such that

   (3.12) F X ( t ) ≤ f t , G T x ( t ) ρ t , D T x ′ ( t ) + π 2 ≤ h η M , y L ( t ) + π 2 ,

   and this estimate holds for every x ∈ W^1,1(I) and a.e. t ∈ I. Since

   ψ := h η M , y L + π / 2 ∈ L 1 ( I ) ,

   we conclude at once that F_x ∈ L¹(I) for every x ∈ W^1,1(I) (hence, F maps W^1,1(I) into L¹(I)) and that F satisfies estimate (3.9).

   CLAIM 2. F is continuous from W^1,1(I) into L¹(I).

   Let x₀ ∈ W^1,1(I) be fixed, and let { x n } n ⊆ W 1 , 1 ( I ) be a sequence converging to x₀ as n ⟶ ∞. Moreover, let {u_k := x_nk}_k be any sub-sequence of {x_n}_n.

   To demonstrate the continuity of F it suffices to show that, by choosing a further sub-sequence if necessary, one has

   (3.13) lim k → ∞ F u k = F x 0  in  L 1 ( I ) .

   First of all we observe that, since u_k ⟶ x₀ in W^1,1(I) as k ⟶ ∞, we have

   (3.14) lim k → ∞ u k ( t ) = x 0 ( t )  uniformly for  t ∈ I ;

   moreover, by Lemma A.1 we also have

   (3.15) lim k → ∞ T u k = T x 0  in  W 1 , 1 ( I ) .

   In particular, since (3.15ux00) implies that T u k ′ → T x 0 ′ i n L 1 ( I ) a s k → ∞ , by possibly choosing a sub-sequence we can assume that

   (3.16) lim k → ∞ T u k ′ ( t ) = T X 0 ′ ( t )  for a.e.  t ∈ I  . 

   Now, since G is continuous from W^1,1(I) to L^∞(I), from (3.15) we get

   (3.17) lim k → ∞ G T u k ( t ) = G T x 0 ( t )  for every  t ∈ I  . 

   Moreover, from (3.16) we get that

   (3.18) lim k → ∞ D T u k ′ ( t ) = D T x 0 ′ ( t )  for a.e.  t ∈ I .

   Gathering together (3.17), (3.18) and (3.14), we then obtain (remind that, by assumptions, f and ρ are Carathéodory functions on I × R)

   lim k → ∞ F u k ( t ) = lim k → ∞ − f t , G T u k ( t ) ρ t , D T u k ′ ( t ) + arctan u k ( t ) − T u k ( t ) = − f t , G T χ 0 ( t ) ρ t , D T x 0 ( t ) + arctan u 0 ( t ) − T u 0 ( t ) = F X 0 ( t )  for a.e.  t ∈ I .

   From this, a standard dominated-convergence based on (3.12) allows us to conclude that F_uk ⟶ F_x0 in L¹(I) as k ⟶ ∞, which is exactly the desired (3.13).

   CLAIM 3. B_a and B_b are continuous and bounded (from W 1 , 1 ( I ) t o R ) .

   As regards the continuity, since H a , H b are continuous from W^1,1(I) to R (by assumption (H4)) and since T is continuous on W^1,1(I) (by Lemma A.1), we deduce that B a = H a ∘ T a n d B b = H b ∘ T are continuous.

   As regards the boundedness, since H_a, H_b are monotone increasing (see (2.5)), for every fixed x ∈ W^1,1(I) we have

   H a [ α ] ≤ H a [ T x ] ≤ H a [ β ] and H b [ α ] ≤ H b [ T x ] ≤ H b [ β ]

   (remind that, by definition, α ≤ T x ≤ β for all x ∈ W^1,1(I)). From this, we deduce that B_a, B_b are globally bounded, and the claim is proved.

   Using the results established in the above claims, one can prove the existence of solutions for (3.8) (and the higher-regularity assertions (i)-(ii)) by arguing essentially as in [17, Lem. 2.1 and Thm. 2.2]. The key points are the following.

   • Thanks to Claim 3, it can be proved that for every x ∈ W^1,1(I) there exists a unique real number z = z x ∈ R such that

   B b [ x ] − B a [ x ] = ∫ a b 1 k ( t ) Φ − 1 ( z x + F x ( t ) ) d t ,

   where F x ( t ) := ∫ a t F x ( s ) d s . Moreover, the map x ⟼ z_x is bounded, i.e.,

   z x ≤ c 0  for every  χ ∈ W 1 , 1 ( I ) ,

   where c₀ > 0 is a universal constant which is independent of x.

   • The solutions of (3.8) are precisely the fixed points (in W^1,1(I)) of the operator A : W^1,1(I) ⟶ W^1,1(I) defined as follows:

   A x ( t ) := B a [ x ] + ∫ a t 1 k ( t ) Φ − 1 ( z x + F x ( t ) ) d t ( t ∈ I ) .

   • Using all the above claims, it can be proved that A is continuous, bounded and compact on W^1,1(I); thus, Schauder’s Fixed-Point theorem ensures that A possesses (at least) one fixed point x₀ ∈ W^1,1(I).

   • Finally, the higher-regularity assertions (i)-(ii) are straightforward consequences of the following simple observations:

   1. A ( W 1 , 1 ( I ) ) ⊆ W 1 , ϑ ( I ) i f 1 / k ∈ L ϑ ( I ) ( f o r s o m e 1 < ϑ ≤ ∞ ) ;

   2. A ( W 1 , 1 ( I ) ) ⊆ C 1 ( I , R ) i f k ∈ C ( I , R ) a n d k > 0 o n I .

   We proceed with the second step.

2. In this second step we establish the following result: if u ∈ W^1,1(I) is any solution of (3.8), then u is also a solution of (2.1).

   CLAIM 1. α(t) ≤ u(t) ≤ β(t) for every t ∈ I, so that

   T u ≡ u  and  G T u ≡ G u  on  I .

   We argue by contradiction and, to fix ideas, we assume that the (continuous) function v := u − α attains a strictly negative minimum on I.

   Since u solves (3.8), α is a lower solution of problem (2.1) and the operators H_a, H_b are monotone increasing (see assumption (H4)), we get

   u ( a ) = H a [ T u ] ≥ H a [ α ] ≥ α ( a ) and u ( b ) = H b [ T u ] ≥ H b [ α ] ≥ α ( b )

   (remind that, by definition, T_u ≥ α on I). As a consequence, it is possible to find three points t₁, t₂, θ ∈ int(I), with t₁ < θ < t₂, such that

   1. u(t_i)− α(t_i) = 0 for i = 1, 2;

   2. u(t) − α(t) < 0 for all t ∈ (t₁, t₂);

   3. u ( θ ) − α ( θ ) = min t ∈ I ( u ( t ) − α ( t ) ) < 0.

   In particular, from (2) we infer that T u ≡ α o n ( t 1 , t 2 ) , and thus

   D T u ′ ( t ) = D α ′ ( t ) = α ′ ( t )  for a.e.  t ∈ t 1 , t 2 .

   By using once again the fact that u solves (3.8), and since α is a lower solution of problem (2.1), from assumption (H3) we then obtain (for a.e. t ∈ (t₁, t₂))

   (3.19) Φ k ( t ) u ′ ( t ) ′ = − f t , G T u ( t ) ρ t , α ′ ( t ) + arctan ⁡ ( u ( t ) − α ( t ) ) < − f t , G T u ( t ) ρ t , α ′ ( t ) = − f t , G T u ( t ) + κ T u ( t ) ρ t , α ′ ( t ) + κ T u ( t ) ρ t , α ′ ( t )  (by (2.4), since  T u ≥ α  on  I  and  ρ ≥ 0  on  R ≤ − f t , 9 α ( t ) + κ α ( t ) ρ t , α ′ ( t ) + κ T u ( t ) ρ t , α ′ ( t ) since T u ≡ α  on  t 1 , t 2 = − f t , 9 α ( t ) ρ t , α ′ ( t ) ≤ Φ k ( t ) α ′ ( t ) ′

   We now consider the sets A₁, A₂ ⊆ I defined as follows:

   A 1 := t ∈ t 1 , θ : ∃ u ′ ( t ) , α ′ ( t )  and  u ′ ( t ) < α ′ ( t )  and  A 2 := t ∈ θ , t 2 : ∃ u ′ ( t ) , α ′ ( t )  and  u ′ ( t ) > α ′ ( t ) .

   Since u, α ∈ W^1,1(I) and since u < α on (t₁, t₂), it follows that both A₁ and A₂ have positive Lebesgue measure; as a consequence, there exist τ₁ ∈ A₁ and τ₂ ∈ A₂ such that (see also Remarks 2.2 and 2.5)

   1. k(τ_i)> 0 for i = 1, 2;

   2. K_u(τ_i) = k(τ_i) u′(τ_i) for i = 1, 2;

   3. K_α(τ_i) = k(τ_i) α'(τ_i) for i = 1, 2.

   By integrating both sides of (3.19) on [τ₁, θ], and using (b)-(c), we then get

   Φ ( K u ( θ ) ) − Φ ( k ( τ 1 ) u ′ ( τ 1 ) ) < Φ ( K α ( θ ) ) − Φ ( k ( τ 1 ) α ′ ( τ 1 ) ) .

   Since Φ is strictly increasing, by (a) and the choice of τ₁ we obtain

   (3.20) Φ ( K u ( θ ) ) − Φ ( K α ( θ ) ) < 0.

   On the other hand, by integrating both sides of inequality (3.19) on [θ, τ₂] (and using once again (b)-(c)), we derive that

   Φ ( k ( τ 2 ) u ′ ( τ 2 ) ) − Φ ( K u ( θ ) ) < Φ ( k ( τ 2 ) α ′ ( τ 2 ) ) − Φ ( K α ( θ ) ) ;

   Since Φ is strictly increasing, by (a) and the choice of τ₂ we obtain

   Φ ( K u ( θ ) ) − Φ ( K α ( θ ) ) > 0.

   This is clearly in contradiction with (3.20), and thus u(t) − α(t) ≥ 0 for every t ∈ I. By arguing exactly in the same way one can also prove that u(t) − β(t) ≤ 0 for every t ∈ I, and the claim is completely demonstrated.

   CLAIM 2. |u(t)| ≤ M and |G_u(t)| ≤ η_M for every t ∈ I.

   By statement (i) and the choice of M ≥ ∥ α ∥ L ∞ ( I ) , ∥ β ∥ L ∞ ( I ) , we get

   − M ≤ α ( t ) ≤ u ( t ) ≤ β ( t ) ≤ M  for every  t ∈ I ,

   and this proves that |u(t)| ≤ M for all t ∈ I. From this, by taking into account the choice of η_M in (3.1), we derive that |G_u(t)| ≤ η_M, as desired.

   CLAIM 3. If N > 0 is as in (3.2), then

   (3.21) min t ∈ I | K u ( t ) | ≤ N .

   By contradiction, let us assume that (3.21) does not hold; moreover, to fix ideas (and taking into account the continuity of K_u), let us suppose that

   (3.22) K u ( t ) > N  for every  t ∈ I .

   By integrating on I both sides of the above inequality, we obtain

   N ( b − a ) < ∫ a b K u ( t ) t = ∫ a b k ( t ) u ′ ( t ) d t = ( ★ ) ;

   from this, since (3.22) implies that u'(t) > N/k(t) for a.e. t ∈ I, we then get

   ( ⋆ ) ≤ ∥ k ∥ L ∞ ( I ) ∫ a b u ′ ( t ) d t = ∥ k ∥ L ∞ ( I ) ( u ( b ) − u ( a ) )  (by statement (ii) and the choice of  N ,  see  ( 3.2 ) ) ≤ ( 2 M ) ⋅ ∥ k ∥ L ∞ ( I ) < N ( b − a ) .

   This is clearly a contradiction, and thus min_I K_u ≤ N. By arguing exactly in the same way one can also show that sup_I K_u ≥ −N, and this proves (3.21).

   Claim 4. | K u ( t ) | ≤ L M f o r e v e r y t ∈ I .

   Arguing again by contradiction, we assume that there exists some point τ in I such that | K u ( τ ) | > L M ; moreover, to fix ideas, we suppose that K u ( τ ) > L M > 0.

   Since L_M > N (see (3.3)), by (3.21) (and the continuity of K_u) we deduce the existence of two points t₁, t₂ ∈ I, with (to fix ideas) t₁ < t₂, such that

   1. K u ( t 1 ) = N a n d K u ( t 2 ) = L M ;

   2. N < K u ( t ) < L M f o r a l l t ∈ ( t 1 , t 2 ) ⊆ I .

   In particular, from (b), (3.4) and the choice of N in (3.2) we derive that

   (3.23) k ( t ) u ′ ( t ) ≥ H M and 0 < u ′ ( t ) < L M k ( t ) ≤ y L ( t )

   for almost every t ∈ (t₁, t₂). Now, on account of (3.23) and of the very definition of D, we deduce that D u ′ ≡ u ′ on (t₁, t₂); as a consequence, since u is a solution of (3.8) and T_u ≡ u on I (by Claim 1.), we have (a.e. on (t₁, t₂))

   Φ K u ( t ) ′ = Φ k ( t ) u ′ ( t ) ′ =∣ f t , G u ( t ) ρ t , u ′ ( t )  (by  ( 2.9 ) ,  since  k ( t ) u ′ ( t ) ≥ H M  and  G u ≤ η M ,  see (ii)  ) ≤ ψ M Φ k ( t ) u ′ ( t ) ⋅ l M ( t ) + μ M ( t ) u ′ ( t ) q − 1 q = ψ M Φ K u ( t ) ⋅ l M ( t ) + μ M ( t ) u ′ ( t ) q − 1 q .

   In particular, since u0 > 0 a.e. on (t₁, t₂) (see (3.23)) and since

   Φ X u ( t ) > Φ ( N ) > 0  for any  t ∈ t 1 , t 2

   (by (b), the monotonicity of Φ and the choice of N in (3.2)), we obtain

   (3.24) | ( Φ ( K u ( t ) ) ) ′ | ≤ ψ M ( Φ ( K u ( t ) ) ) ⋅ ( l M ( t ) + μ M ( t ) ( u ′ ( t ) ) q − 1 q )

   for almost every t ∈ (t₁, t₂). Using this last inequality, we then get (remind that Φ ∘ K u is absolutely continuous, see Remark 2.5)

   ∫ ϕ ( N ) ϕ L M 1 ψ M d s = ∫ ϕ K u t 1 ϕ K u t 2 1 ψ M d s = ∫ t 1 t 2 ϕ K u ( t ) ′ ψ M ϕ K u ( t ) d t ≤ ∫ t 1 t 2 l M ( t ) + μ M ( t ) u ′ ( t ) q − 1 q d t ≤ l M L 1 ( I ) + ∫ t 1 t 2 μ M ( t ) u ′ ( t ) q − 1 q d t  (by Hölder's inequality, since  μ M ∈ L q ( I ) ≤ l M L 1 ( I ) + μ M L q ( I ) u t 2 − u t 1 q − 1 q (  since  | u ( t ) | ≤ M  for all  t ∈ I ,  see Claim  2 . ) ≤ l M L 1 ( I ) + μ M L q ( I ) ( 2 M ) q − 1 1 .

   This is in contradiction with the choice of L_M in (3.3), and thus K_u(t) ≤ L_M for every t ∈ I. By arguing exactly in the same way one can also show that K_u(t) ≥ −L_M for all t ∈ I, and the claim is completely proved.

   CLAIM 5. |u'(t)| ≤ L_M/k(t) for a.e. t ∈ I, so that

   D u ′ ≡ u ′ a . e . o n I .

   By Claim 4 and the very definition of K_u we get

   | u ′ ( t ) | = | K u ( t ) | k ( t ) ≤ L M k ( t )  for a.e.  t ∈ I ;

   as a consequence, since L_M/k(t) ≤ y_L(t) a.e. on I (see (3.4)), from the very definition of D in (3.5) we conclude that D_u0 ≡ u0 on I.

   Using the results established in the above claims, we can complete the proof of this step. Indeed, by Claim 1. we have T u ≡ u and G T u ≡ G u on I; moreover, by Claim 5. we know that D u ′ ≡ u ′ a.e. on I. Gathering together all these facts (and since u is a solution of (3.8)), for almost every t ∈ I we get

   ( ϕ ( K u ( t ) ) ) ′ = − f ( t , G T u ( t ) ) ρ ( t , D T u ′ ( t ) ) + arctan ⁡ ( u ( t ) − T u ( t ) ) = − f ( t , G u ( t ) ) ρ ( t , u ′ ( t ) ) ,

   = − f t , G u ( t ) ρ t , u ′ ( t ) ,

   and thus u solves the ODE (2.6). Furthermore, by (3.7) we have

   u ( a ) = B a [ u ] = H a [ T u ] = H a [ u ] and u ( b ) = B b [ u ] = H b [ T u ] = H b [ u ] ,

   and this proves that u is a solution of the BVP (2.1).

   Thanks to the results in Steps I and II, we are finally in a position to conclude the proof of Theorem 2.6. Indeed, by Step I we know that there exists (at least) one solution x₀ ∈ W^1,1(I) of the truncated BVP (3.8); on the other hand, we derive from Step II that x₀ is actually a solution of (2.1).

   To proceed further we observe that, owing to Claim 1. in Step II, we get that x₀ satisfies (2.10); moreover, the result in Step I ensures that

   • if 1 / k ∈ L ϑ ( I ) . for some 1 < ϑ ≤ ∞ then x 0 ∈ W 1 , ϑ ( I ) ;

   • if k ∈ C ( I , R ) and k > 0 on I, then x 0 ∈ C 1 ( I , R ) .

   Finally, by combining Claims 2. and 5. in Step II, we conclude that x₀ satisfies the ‘a-priori’ estimate (2.11), and the proof is complete.