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

We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type $$\big(\Phi(k(t)\,x'(t))\big)' + f(t,\mathcal{G}_x(t))\,\rho(t, x'(t)) = 0$$ on a compact interval $[a,b]$. These equations are quite general due to the presence of a strictly increasing homeomorphism $\Phi$, the so-called $\Phi$-Laplacian operator, of a nonnegative function $k$, which may vanish on a set of null measure, and moreover of a functional term $\mathcal{G}_x$. We look for solutions, in a suitable weak sense, which belong to the Sobolev space $W^{1,1}([a,b])$. Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments.


Introduction
Boundary value problems for highly nonlinear di erential equations in the whole real line, even governed by nonlinear di erential operators, have been widely investigated in the last decade. Such problems are involved in many applications in several elds, such as non-Newtonian uid theory, di usion of ows 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 Φp(z) := |z| p− z, that is, (Φp(x )) = f (t, x, x ), many authors have proposed generalizations in various directions, in particular considering a more general nonlinear di erential operator, a Φ-Laplacian type operator, which can be a generic homeomorphism, a sin-gular or a non-surjective operator (see, e.g., [2-4, 12-14, 19]). We also refer the reader to the survey [11] and to the references therein included. Let us also mention equations with mixed di erential operators, that is, where a is a continuous positive function (see, e.g., [5,8,15,18]). In the autonomous case, namely, a(t, x) ≡ a(x), equation (1.1) also arises in some models, e.g. reaction-di usion equations with non-constant di usivity 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-di usion 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 di culties, see [1,21]. So, the main aim of this paper is to provide a quite general approach in order to treat functional di erential equations governed by general nonlinear di erential 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): a.e. on I := [a, b], ( 1.2) 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 The proposed framework is very general, since it contains, as particular cases, delayed and non-local di erential equations; moreover, we point out that the function k(t) inside the di erential operator may vanish on a set having zero Lebesgue measure. As a consequence, the di erential 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 , (I), and this justi es the choice of W , (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 /k. More precisely, we shall prove that when /k ∈ L θ (with < θ ≤ ∞), then there exists a solution belonging to W ,θ (I); in particular, when /k is continuous, we nd C -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 xed 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 > ), ∈ L (I) and ψ : ( , ∞) → ( , ∞) satis es +∞ ds ψ(s) = +∞.
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 Φp(z) = |z| p− z is the usual p-Laplace operator, ϑ , δ are positive constants and the functional term Gx = xτ is of delay-type, that is, A brief plan of the paper is now in order.
In Section 2 we x some preliminary de nitions and we state our main existence result, namely Theorem 2.6. In Section 3 we provide the proof of Theorem 2.6, which articulates into two steps: rst, 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. In Section 4 we present some examples to which our Theorem 2.6 applies. 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.

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 x (t)) = a.e. on I, where Φ : R → R is a strictly increasing homeomorphism, f , ρ : I × R → R are Carathéodory functions, and k, G, Ha , H b satisfy the following assumptions: (H1) k : I → R is a nonnegative function satisfying k ∈ L ∞ (I) and /k ∈ L (I). We are now ready to state the main result of the paper.

(2.9)
Then, there exists a solution x ∈ W , (I) of problem (2.1), further satisfying Moreover, the following higher-regularity properties hold: (1) if /k ∈ L ϑ (I) for some < ϑ ≤ ∞, one also has that x ∈ W ,ϑ (I); (2) if k ∈ C(I, R) and k > on I, one also has that x ∈ C (I, R). Remark 2.7. It is worth highlighting that the existence of a well-ordered pair of lower and upper solutions α, β for (2.1) is far from being obvious (see, e.g., [11,22] and the reference therein for general results on this topic). Here, we limit ourselves to observe that, if ρ(t, ) = for every t ∈ I, then any constant function is both a lower and an upper solution for the ODE (2.6).
As a positive counterpart of the previous comment, we shall present in the next Section 4 a couple of examples of BVPs to which Theorem 2.6 applies.

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.
S I: As a rst 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.

S
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 xing some quantities which shall be used all over the proof.
First of all, we choose a real M > in such a way that α L ∞ (I) ≤ M and β L ∞ (I) ≤ M. Moreover, using assumption (H2), we let η M > be such that With reference to assumption (H7), we then set and we consider the function γ L ∈ L (I) de ned as: Following the notation in Appendix A, we also de ne the truncating operators Given any x ∈ W , (I), we then consider the function Fx de ned by Finally, we consider the operators Ba , B b : W , (I) → R de ned as Thanks to all these preliminaries, we can nally introduce the following BVP (which can be thought of as a truncated version of problem (2.1)): a.e. on I, We now proceed by following the steps described above.
Step I. In this rst step we prove the following result: there exists (at least) one solution u ∈ W , (I) of problem (3.8); this means, precisely, that Furthermore, the following higher-regularity assertions hold: (ii) if k ∈ C(I, R) and k > on I, then u ∈ C (I, R).
There exists a non-negative function ψ ∈ L (I) such that for a.e. t ∈ I and every x ∈ W , (I). (3.9) First of all, by the choice of M and the very de nition of Tx we have for all x ∈ W , (I) and any t ∈ I; as a consequence, owing to the choice of η M in (3.1), we get Moreover, owing to the very de nition of D, we also have for any x ∈ W , (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 and this estimate holds for every x ∈ W , (I) and a.e. t ∈ I. Since we conclude at once that Fx ∈ L (I) for every x ∈ W , (I) (hence, F maps W , (I) into L (I)) and that F satis es estimate (3.9). First of all we observe that, since u k → x in W , (I) as k → ∞, we have lim k→∞ u k (t) = x (t) uniformly for t ∈ I; (3.14) moreover, by Lemma A.1 we also have In particular, since (3.15) implies that T u k → T x in L (I) as k → ∞, by possibly choosing a sub-sequence we can assume that lim Moreover, from (3.16) we get that 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) From this, a standard dominated-convergence based on (3.12) allows us to conclude that Fu k → Fx in L (I) as k → ∞, which is exactly the desired (3.13). As regards the continuity, since Ha , H b are continuous from W , (I) to R (by assumption (H4)) and since T is continuous on W , (I) (by Lemma A.1), we deduce that Ba = Ha • T and As regards the boundedness, since Ha , H b are monotone increasing (see (2.5)), for every xed x ∈ W , (I) we have (remind that, by de nition, α ≤ Tx ≤ β for all x ∈ W , (I)). From this, we deduce that Ba , 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 [ where c > is a universal constant which is independent of x. • The solutions of (3.8) are precisely the xed points (in W , (I)) of the operator A : W , (I) → W , (I) de ned as follows: • Using all the above claims, it can be proved that A is continuous, bounded and compact on W , (I); thus, Schauder's Fixed-Point theorem ensures that A possesses (at least) one xed point x ∈ W , (I). • Finally, the higher-regularity assertions (i)-(ii) are straightforward consequences of the following simple observations: A(W , (I)) ⊆ W ,ϑ (I) if /k ∈ L ϑ (I) (for some < ϑ ≤ ∞); A(W , (I)) ⊆ C (I, R) if k ∈ C(I, R) and k > on I.
We proceed with the second step.
Step II. In this second step we establish the following result: if u ∈ W , (I) is any solution of (3.8), then u is also a solution of (2.1).

C 1. α(t) ≤ u(t) ≤ β(t) for every t ∈ I, so that
Tu ≡ u and G Tu ≡ Gu on I.
We argue by contradiction and, to x 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 Ha , H b are monotone increasing (see assumption (H4)), we get (remind that, by de nition, Tu ≥ α on I). As a consequence, it is possible to nd three points t , t , θ ∈ int(I), with t < θ < t , such that In particular, from (2) we infer that Tu ≡ α on (t , t ), and thus for a.e. t ∈ (t , t ).
Since u, α ∈ W , (I) and since u < α on (t , t ), it follows that both A and A have positive Lebesgue measure; as a consequence, there exist τ ∈ A and τ ∈ A such that (see also Remarks 2.2 and 2.5) By integrating both sides of (3.19) on [τ , θ], and using (b)-(c), we then get Since Φ is strictly increasing, by (a) and the choice of τ we obtain On the other hand, by integrating both sides of inequality (3.19) on [θ, τ ] (and using once again (b)-(c)), we derive that Since Φ is strictly increasing, by (a) and the choice of τ we obtain This is clearly in contradiction with (3.20), and thus u(t) − α(t) ≥ for every t ∈ I. By arguing exactly in the same way one can also prove that u(t) − β(t) ≤ for every t ∈ I, and the claim is completely demonstrated.   Arguing again by contradiction, we assume that there exists some point τ in I such that |Ku(τ)| > L M ; moreover, to x ideas, we suppose that Ku(τ) > L M > .
Since L M > N (see (3.3)), by (3.21) (and the continuity of Ku) we deduce the existence of two points t , t ∈ I, with (to x ideas) t < t , such that for almost every t ∈ (t , t ). Now, on account of (3.23) and of the very de nition of D, we deduce that D u ≡ u on (t , t ); as a consequence, since u is a solution of (3.8) and Tu ≡ u on I (by Claim 1.), we have (a.e. on (t , t ))

Φ(Ku(t)) = Φ(k(t) u (t)) = f (t, Gu(t)) ρ(t, u (t))
by (2.9), since k(t)u (t) ≥ H M and Gu ≤ η M , see (ii) In particular, since u > a.e. on (t , t ) (see (3.23)) and since Φ(Ku(t)) > Φ(N) > for any t ∈ (t , t ) (by (b), the monotonicity of Φ and the choice of N in (3.2)), we obtain for almost every t ∈ (t , t ). Using this last inequality, we then get (remind that Φ • Ku is absolutely continuous, see Remark 2.5)  By Claim 4 and the very de nition of Ku we get as a consequence, since L M /k(t) ≤ γ L (t) a.e. on I (see (3.4)), from the very de nition of D in (3.5) we conclude that D u ≡ u on I.
Using the results established in the above claims, we can complete the proof of this step. Indeed, by Claim 1. we have Tu ≡ u and G Tu ≡ Gu 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(Ku(t)) = −f (t, G Tu (t)) ρ(t, D T u (t)) + arctan u(t) − Tu(t) = −f (t, Gu(t)) ρ(t, u (t)), and thus u solves the ODE (2.6). Furthermore, by (3.7) we have

u(a) = Ba[u] = Ha[Tu] = Ha[u]
and and this proves that u is a solution of the BVP (2.1).
Thanks to the results in Steps I and II, we are nally in a position to conclude the proof of Theorem 2.6. Indeed, by Step I we know that there exists (at least) one solution x ∈ W , (I) of the truncated BVP (3.8); on the other hand, we derive from Step II that x is actually a solution of (2.1).
To proceed further we observe that, owing to Claim 1. in Step II, we get that x satis es (2.10); moreover, the result in Step I ensures that • if /k ∈ L ϑ (I) for some < ϑ ≤ ∞, then x ∈ W ,ϑ (I); • if k ∈ C(I, R) and k > on I, then x ∈ C (I, R). Finally, by combining Claims 2. and 5. in Step II, we conclude that x satis es the 'a-priori' estimate (2.11), and the proof is complete.
As a consequence, if we know that γ L ∈ L ϑ (I) (for some ϑ > ), (3.25) assumption (H6) can be replaced by the following weaker one: (H6)' for every R > and every non-negative function γ ∈ L ϑ (I) there exists a non-negative function h = h R,γ ∈ L (I) such that for a.e. t ∈ I, every z ∈ R with |z| ≤ R and every y ∈ L ϑ (I) such that |y(s)| ≤ γ(s) for a.e. s ∈ I

Some examples
In this last section of the paper we present some 'model BVPs' illustrating the applicability of our existence result in Theorem 2.6. We aim to show that all the assumptions of Theorem 2.6 are satis ed in this case, so that problem (4.1) possesses (at least) one solution x ∈ W , (I). We explicitly point out that, in view of the boundary conditions, x cannot be constant.
To begin with, we observe that assumption (H1) is trivially satis ed, since /k ∈ L ϑ (I) for all ϑ ∈ [ , ). (4.2) As regards assumptions (H2)-(H3), we rst notice that G is a continuous operator mapping W , (I) into C (I, R); as a consequence, owing to Remark 2.1, we know that G is continuous from W , (I) into L ∞ (I) (with the usual norms). Furthermore, if r > is any xed positive number, we have and thus (2.3) is satis ed with ηr := r . Finally, since a is non-decreasing and G is increasing (with respect to the point-wise order), it follows that is monotone increasing, so that (2.4) holds with κ = .
As regards assumption (H4), it is easy to check that H , H are continuous from W , (I) to R (remind that W , (I) is continuously embedded into C(I, R)); moreover, if x, y ∈ W , (I) are such that x ≤ y point-wise on I, then and so that H , H are also monotone increasing (w.r.t. to the point-wise order).
We now turn to prove the validity of assumptions (H5)-to-(H7). A (H6). Let R > be xed and let γ be a non-negative function belonging to L (I). Since, by assumption, a ∈ C(R, R), we have for a.e. t ∈ I, every z ∈ R with |z| ≤ R and every y ∈ L (I) such that |y(s)| ≤ γ(s) for a.e. s ∈ I.
A (H7). Let R > be arbitrarily xed. Since, by assumption, a ∈ C(R, R), we have the following estimate holding true for a.e. t ∈ I, every z ∈ [−R, R] and every y ∈ R. As a consequence, we conclude that estimate (2.9) is satis ed with the choice Since all the assumptions of Theorem 2.6 are ful lled, we can conclude that there exists (at least) one solution x ∈ W , (I) of problem (4.1), further satisfying − ≤ x (t) ≤ for every t ∈ I.

Example 4.2.
Let ϑ ∈ ( , ∞) be xed, and let τ ∈ ( , π). Moreover, let p, δ ∈ R be two positive real numbers satisfying the following relation Finally, let Φp(z) := |z| p− z be usual p-Laplace operator on R. We then consider the following BVP on where xτ is the delay-type function de ned as Problem We aim to show that all the assumptions of Theorem 2.6 are satis ed in this case, so that problem (4.4) possesses (at least) one solution x ∈ W , (I). We explicitly point out that, in view of the boundary conditions, x cannot be constant.
To begin with, we observe that assumption (H1) is trivially satis ed, since /k ∈ L ϑ (I) for all ϑ ∈ [ , ϑ ). (4.5) As regards assumptions (H2)-(H3), we rst notice that G is a well-de ned linear operator mapping W , (I) into L ∞ (I); as a consequence, since we get that G is continuous from W , (I) into L ∞ (R). Furthermore, if r > is any xed positive number, we also have and thus (2.3) is satis ed with ηr := r. Finally, since G is monotone increasing with respect to the point-wise order (as it is easy to check) and since f (t, z) = z, one straightforwardly derives that (2.4) holds with κ = .
As regards assumption (H4), it is easy to check that H , H π are continuous from W , (I) to R (remind that W , (I) is continuously embedded into C(I, R)); moreover, if x, y ∈ W , (I) are such that x ≤ y point-wise on I, then and H π [x] = π π (x(s) + ) ds ≤ π π (y(s) + ) ds = H π [y], so that H , H π are also monotone increasing (w.r.t. to the point-wise order).
We now turn to prove the validity of assumptions (H5)-to-(H7).  as a consequence, according to Remark 3.1, it su ces to demonstrate that assumption (H6) holds in the weaker form (H6)' (with ϑ = max{ , δ}). Let then R > be xed and let γ be a non-negative function belonging to the space L ϑ (I). Reminding that f (t, z) = z, we have the following computation for a.e. t ∈ I, every z ∈ R with |z| ≤ R and every y ∈ L (I) such that |y(s)| ≤ γ(s) for a.e. s ∈ I.
A (H7). Let R > be arbitrarily xed. Since k ∈ C(I, R) (and since f (t, z) = z), we have the following computation holding true for a.e. t ∈ I, every z ∈ [−R, R] and every y ∈ R with |k(t)y| ≥ . As a consequence, if we are able to demonstrate that t → R k(t) δ+ /q− ∈ L q (I), (4.6) we conclude that estimate (2.9) is satis ed with the choice In its turn, the needed (4.6) follows from (4.5) and from the fact that Since all the assumptions of Theorem 2.6 are ful lled, we can conclude that there exists (at least) one solution x ∈ W , (I) of problem (4.4), further satisfying ≤ x (t) ≤ for every t ∈ I.
Moreover, from (4.2) we deduce that x ∈ W ,ϑ (I) for all ϑ ∈ [ , ϑ ). We then consider the following BVP: x(s) · log + | √ t x (t)| = a.e. on I, x(− ) = , x( ) = . We aim to show that all the assumptions of Theorem 2.6 are satis ed in this case, so that problem (4.8) possesses (at least) one solution x ∈ W , (I). We explicitly point out that, in view of the boundary conditions, x cannot be constant.
To begin with, we observe that assumption (H1) is trivially satis ed, since /k ∈ L ϑ (I) for all ≤ ϑ ≤ ∞. (4.9) As regards assumptions (H2)-(H3), we rst notice that G is a well-de ned operator mapping W , (I) into L ∞ (I); as a consequence, since we have we immediately derive that G is continuous from W , (I) into L ∞ (I) (with the usual norms). Furthermore, if r > is any xed positive number, we have and thus (2.3) is satis ed with ηr := r. Finally, since G is monotone increasing with respect to the point-wise order and since f (t, z) = z, by arguing as in Example 4.2 we derive that (2.4) holds with κ = .
As regards assumption (H4), since H − and H are constant, it follows that these operators are continuous (from W , (I) to R) and monotone increasing (w.r.t. to the point-wise order).
We now turn to prove the validity of assumptions (H5)-to-(H7). x(s) · log + | √ t x (t)| = ; moreover, since H − ≡ and H ≡ , we immediately derive that α is a lower solution and β is an upper solution of problem (4.8).
Proof. We limit ourselves to prove only assertion (ii), since (A.1) is trivial and (i) is an immediate consequence of (A.1) and the well-known characterization of W , (I) in terms of absolutely continuous functions (see, e.g., [10]).
First of all we observe that, if we introduce the operators the operator T ω,ζ is the composition between m and M, that is, for all x ∈ W , (I).
As a consequence, to prove the lemma it su ces to show that both M and m are continuous on W , (I). Here we limit ourselves to demonstrate this fact only for the operator M, since the case of m goes along the same lines.
Let then x ∈ W , (I) be xed, and let {xn}n ⊆ W , (I) be a sequence converging to x as n → ∞ in To this end, we rst notice that, since y k → x in W , (I) as k → ∞, we also have that y k converges uniformly on I to x as k → ∞ (see, e.g., [ To this end, we rst x some notation which shall be useful in the sequel. Given any point t ∈ (a, b) and any ρ > , we set I(t , ρ) := [t − ρ, t + ρ]; moreover, given any function ξ ∈ W , (I), we de ne Notice that, since ξ ∈ W , (I), the set N ξ has zero Lebesgue measure.
We now start with the proof of (A.5). First of all, since y k → x in W , (I) as k → ∞, we clearly have that y k → x in L (I) (as k → ∞); as a consequence, it is possible to nd a non-negative function g ∈ L (I) and a set Z ⊆ I, with vanishing Lebesgue measure, such that (up to a sub-sequence) (i) y k (t) → x (t) as k → ∞ for every t ∈ I \ Z;