Boundary value problems for strongly nonlinear equations under a Wintner-Nagumo growth condition

AbstractWe study the following strongly nonlinear differential equation: 
 (a(t,x(t))Φ(x′(t)))′=f(t,x(t),x′(t)),a.e. in [0,T]$$\bigl(a \bigl(t,x(t) \bigr)\Phi\bigl(x'(t) \bigr) \bigr)'= f \bigl(t,x(t),x'(t) \bigr), \quad\text{a.e. in } [0,T] $$ subjected to various boundary conditions including, as particular cases, the classical Dirichlet, periodic, Neumann and Sturm-Liouville problems. We adopt the method of lower and upper solutions requiring a weak form of a Wintner-Nagumo growth condition.


Introduction
Boundary value problems involving φ-Laplacian-type operators have been intensively investigated (see, e.g., [1][2][3][4]), even for singular or non-surjective operators (see [5][6][7][8]). More recently, a certain interest has been devoted to differential operators involving also a nonlinear function of the state variable as in the following equation: where a is a continuous positive function, φ is a strictly increasing homeomorphism and f is a Carathéodory function. In this framework, results on the solvability of boundary value problems, both in compact intervals and on the whole real line, were established (see [9,10]). Finally, for results concerning differential inclusions or non-autonomous differential operators, see [11,12] and [13]. The usual technique in this context is generally based on the method of lower and upper solutions, combined with some Nagumo-type growth condition, which is needed to ensure an a priori bound for the derivatives of the solutions in order to apply a suitable fixed point result. In the above quoted papers, the presence of the nonlinear term a inside the differential operator influences the requirements about the differential operator , which was assumed to be homogeneous, or having at most linear growth at infinity.
Motivated by this, in the present paper we investigate the following general equation: involving a positive continuous function a, under growth condition (2). We show that this condition suffices to obtain existence results also for the general equation (3) and, mainly, it allows us to widen the class of operators . Indeed, we are now able to treat very general differential operators, not necessarily homogeneous, nor having polynomial growth, which are only required to be strictly increasing homeomorphisms. In this context, by using a different approach with respect to [4], we are able to prove existence results for solutions of (1) subjected to very general boundary value conditions including, as particular cases, Dirichlet, periodic, Sturm-Liouville and Neumann problems. Our results extend those in [4] both for the presence of the function a inside the differential operators and for the great generality of the structure on the boundary conditions. Finally, we also provide some examples of application of our results, in which the operator is not homogeneous and grows exponentially at infinity.

Preliminaries and auxiliary results
Let us consider the differential equation where a : I × R → R is a positive continuous function, : R → R is an increasing homeomorphism and f : I × J × R → R, with J ⊂ R an interval, is a Carathéodory function, that is, f (·, x, y) is measurable for every x, y ∈ J × R and f (t, ·, ·) is continuous for a.e. t ∈ I. In this context, a solution for equation (4) is a function x ∈ C 1 (I) with x(t) ∈ J for every t ∈ I such that the map t → a(t, x(t)) (x (t)) is absolutely continuous in I and (a(t, x(t)) (x (t))) = f (t, x(t), x (t)) for a.e. t ∈ I.
In what follows, we investigate the existence of solutions for equation (4) satisfying different boundary conditions. Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions, according to the following definition.
A lower solution [upper solution] for equation (4) is a function α ∈ C 1 (I) such that α(t) ∈ J , the map t → a(t, α(t)) (x (t)) is absolutely continuous in I and A well-ordered pair of upper and lower solutions in I for equation (4) is a pair α, β ∈ C 1 (I) where α is a lower solution, β is an upper solution and α(t) ≤ β(t) for every t ∈ I. For every pair of well-ordered lower and upper solutions, the functional interval [α, β] is defined as follows: Finally, we will adopt the following notations relatively to real numbers x, y: The fixed point technique which will be used is based on an existence result for the following functional differential boundary value problem: where ν 1 , ν 2 ∈ R are given constants, : R → R is an increasing homeomorphism, and Moreover, assume that there exist m, M > 0 such that Finally, suppose that there exists φ ∈ L 1 (I) such that Then, for every ν 1 , ν 2 ∈ R, there exists a function u ∈ C 1 (I) with A u · ( • u ) ∈ W 1,1 (I), a solution of problem (5).

Remark 1
In [8] the previous result was given in the context of research involving differential operators satisfying (0) = 0; however, as it is easy to check, this assumption is not necessary in order to prove Theorem 1.

The Dirichlet problem
The study of the solvability of (4) under various boundary conditions starts with the classical Dirichlet problem, that is, where c, d ∈ R are given. The following result states the existence of a solution for problem (D) belonging to the functional interval delimited by a well-ordered pair of lower and upper solutions.
Suppose that there exists a measurable function ψ : (0, +∞) → (0, +∞) such that +∞ ds and assume that there exist positive constants H, p ∈ (1, +∞] and positive functions ν ∈ L p (I), ∈ L 1 (I) such that for a.e. t ∈ I, every x ∈ J and |y| > H, with the convention p-1 p = 1 if p = +∞. Finally, assume that there exists a well-ordered pair α and β of lower and upper solutions in I for equation (4).
Then, for every c, d ∈ R such that α(0) ≤ c ≤ β(0), α(T) ≤ d ≤ β(T), problem (D) has a solution x c,d belonging to the functional interval [α, β], that is, Proof Let M > 0 be a constant such that α C(I) ≤ M, β C(I) ≤ M and put Since is an increasing homeomorphism, it is possible to choose a constant L > 0 such that Moreover, by (10) there exists a constant N > L such that and m 0 (N) Let us now introduce the truncation operator T : Let us consider the modified function f * : and the following auxiliary boundary value problem where q N (r) := (-N) ∨ [r ∧ N] for all r ∈ R.

Claim 1 There exists a solution to problem (19).
Let A : C 1 (I) → C(I) be the functional defined as Of course, A is well defined, bounded and, from the uniform continuity of a in I ×[-M, M], A is also uniformly continuous. Moreover, if B is a bounded subset of C 1 (I), i.e., there exists k > 0 such that x C 1 (I) ≤ k for all x ∈ B. Then, fixed > 0, by the uniform continuity of So, we deduce that |A x (t 1 ) -A x (t 2 )| < for all x ∈ B provided that |t 1t 2 | < η. So the functional A satisfies assumption (6).
a.e. on I.
Of course, F is continuous. Moreover, put we have that φ ∈ L 1 (I) by (9) and |F x (t)| ≤ φ(t) for almost all t ∈ I and all x ∈ C 1 (I) by (9) and (18). Therefore, F satisfies hypothesis (8). So, by applying Theorem 1, there exists a function u ∈ C 1 (I) such that, for all t ∈ I, Obviously, u is a solution of problem (19).
As we mentioned in Introduction, the Wintner-Nagumo condition (11) is weaker than similar growth conditions assumed in many papers related to boundary value problems of similar type, also with respect to results in which the differential operator does not contain the function a. Moreover, it allows us to widen the range of the differential operators we can consider that can be very general, not necessarily homogeneous, nor having polynomial growth. For instance, in the following example, we apply the existence result to an operator having exponential growth.
Example 1 Let us consider the following equation: it is immediate to verify that f is a Carathéodory function such that, whenever |x| ≤ r and |y| ≤ r, we have [-r,r] g(x) := γ r (t) with γ r ∈ L 1 ([0, T]), and assumption (9) and, for every y ∈ R, we get that assumptions (10) and (11)  The following example shows an equation governed by a differential operator of the type considered in [10] and [8], and with a right-hand side which satisfies condition (11), but not other Nagumo-type growth conditions as those assumed in the mentioned papers.
Example 2 Consider the following differential equation: With the same notations as in Example 1, we get that assumptions (9), (10) and (11) are satisfied and Theorem 2 can be applied. On the contrary, in order to apply the results in [10] there should exist a function θ such that |f (t, x, y)| = |y + 1| · |y| ≤ θ (|y|), with +∞ 1 θ(s) ds = +∞, which is not possible.

General nonlinear boundary conditions
We consider now more general boundary conditions in order to deal with periodic, Neumann, Sturm-Liouville boundary conditions for equation (4), using the result obtained in the previous section for Dirichlet problems, following the idea developed in [1]. To this aim, in what follows we adopt the notation (D c,d ) to denote the Dirichlet problem (D), when we need to emphasize the values of the boundary conditions.
The following lemma provides a compactness-type result for the solutions of equation (4) obtained by means of Theorem 2.

Lemma 1
Let assumption (9) be satisfied, and let α, β be a well-ordered pair of lower and upper solutions for equation (4). Then, for every pair of sequences (c n ) n and (d n ) n of real numbers satisfying c n ∈ [α(0), β(0)] and d n ∈ [α(T), β(T)] for every n ∈ N, and for every sequence (x n ) n of solutions of problem D c n ,d n , equibounded in C 1 and belonging to the functional interval [α, β], there exists a subsequence (x n k ) k such that x n k (t) → x 0 (t) uniformly in I for some solution x 0 of equation (4).
Proof Let (c n ) n , (d n ) n be a pair of sequences of real numbers satisfying α(0) ≤ c n ≤ β(0) and α(T) ≤ d n ≤ β(T) for every n ∈ N, and let (x n ) n be a sequence of solutions of problem D c n ,d n , equibounded in C 1 and belonging to the functional interval [α, β]. We can assume without restriction, possibly by passing to subsequences, that c n → c 0 , d n → d 0 . By the equiboundedness of the sequence (x n ) n we have |x n (t)| + |x n (t)| < r for all t ∈ I, for some r > 0. Therefore, by assumption (9) we get a t, x n (t) x n (t) = f t, x n (t), x n (t) ≤ γ r (t) for all n, a.e. t ∈ I.
So, the sequence of functions (a(t, x n (t)) (x n (t))) n is equicontinuous on I. Moreover, since the function 1/a(t, x) is uniformly continuous on I × [-r, r], we deduce that also the sequence (1/a(t, x n (t))) n is equicontinuous on I. Therefore, the sequence ( (x n (t))) n is equicontinuous on I, as it is the product of equicontinuous functions. Finally, by the continuity of the function -1 , we deduce that the sequence (x n ) n is uniformly continuous on I. Recalling that |x n (t)| ≤ r for every n ∈ N and a.e. t ∈ I, by the Ascoli-Arzelà theorem, we infer the existence of a subsequence (x n k ) k uniformly convergent to a continuous function y 0 , implying that also (x n k ) k uniformly converges to the function x 0 (t) = t 0 y 0 (s) ds. Moreover, notice that, for all t ∈ I, we have Hence, by the dominated convergence theorem we obtain a t, x 0 (t) x 0 (t) = a 0, x 0 (0) that is, x 0 is a solution of (4).
In order to handle various types of boundary condition, let us consider the following problem with very general nonlinear boundary conditions: where g : R 4 → R and h : R → R are continuous functions.

Theorem 3 Suppose that there exists a well-ordered pair α, β of lower and upper solutions for equation (4) such that
Then problem (22) admits a solution x, belonging to the functional interval [α, β], such that x C 1 ≤ , where is the constant given by Theorem 2 (see (12) Let c * := sup , and let us prove that c * ∈ . This is trivial if c * = α(0), whereas, if c * > α(0), let (c n ) n ⊂ be a sequence converging to c * from below. Put d n = h(c n ), by Theorem 2, for every n ∈ N, there exists a solution x n of problem (D c n ,d n ), belonging to the functional interval [α, β], such that x n satisfies (24) for every n ∈ N. So, we can apply Lemma 1 to infer the existence of a subsequence (x n k ) k of solutions with the properties x n k (t) → x * (t) for every t ∈ I for some solution x * of equation (4). Of course, c * := x * (0) and, by the continuity of h, we get that x * is a solution of problem (D c * ,h(c * ) ). Moreover, since c n k ∈ , we have g(x n k (0), x n k (T), x n k (0), x n k (T)) ≥ 0 for every k ∈ N and, by the continuity of the function g, we obtain So, taking account of (25), we get g(x * (0), x * (T), x * (0), x * (T)) = 0 and x * is a solution of problem (22).
Let us now treat the case c * < β(0). In order to do this, let us consider a decreasing sequence (c m ) m converging to c * . Of course, the functions x * and β can be considered as a well-ordered pair of lower and upper solutions for equation (4), with x * C(I) ≤ M and x * C(I) ≤ . By applying again Theorem 2, we deduce that, for every m ∈ N, there exists a solution ξ m of problem (D c m ,h(c m ) ) belonging to the functional interval [x * , β], again with ξ m C(I) ≤ M and ξ m C(I) ≤ for every m ∈ N. Hence, we can apply Lemma 1 again, obtaining the existence of a subsequence (ξ m k ) k of solutions such that for some solution ξ * of equation (4). By the continuity of h, the function ξ * is a solution of problem (D c * ,h(c * ) ). Moreover, since c m k > c * = max , we have c m / ∈ for every m ∈ N. Since ξ m C 1 (I) ≤ , necessarily we have g(ξ m (0), ξ m (T), ξ m (0), ξ m (T)) < 0. Thus, by the continuity of g, we have Nevertheless, being ξ * (t) ≥ x * (t) for every t ∈ I, with ξ * (0) = x * (0) and ξ * (t) = x * (T), we deduce that ξ * (0) ≥ x * (0) and ξ * (T) ≤ x * (T). So, by (23) and (25) we infer The general boundary conditions considered in problem (22) include, as a particular case, periodic boundary conditions, that is, the problem As an immediate consequence of Theorem 3, the following existence result holds. Let us consider now the following boundary value problem: where p, q : R 2 → R are continuous functions. The following existence result for problem (28) holds.
To this aim, put We have that α(T) ∈ ϒ, so ϒ = ∅. Let d * := sup ϒ, and let us prove that d * = max ϒ, that is, q(ξ * (T), ξ * (T)) ≥ 0 for some solution ξ * of problem (Q d * ). This is trivial if d * = α(T), whereas if d * > α(T), let (d n ) n ⊂ ϒ be an increasing sequence converging to d * , and let (ξ d n ) n be a sequence of solutions of problem (Q d n ) satisfying q(ξ d n (T), ξ d n (T)) ≥ 0. By virtue of Lemma 1 we get the existence of a subsequence (ξ d n k ) k such that for some solution ξ * of equation (4). By the continuity of p and q, we get ξ * is a solution of problem (Q d * ) such that q(ξ * (T), ξ * (T)) ≥ 0, hence d * ∈ ϒ.

Conclusions
We have proved an existence result for the following strongly nonlinear differential equation: Here the function a = a(t, x) is generic (it is only required to be continuous and positive) and mainly the differential operator is quite general, not necessarily homogeneous, nor having polynomial growth. Finally, we also present some examples of applications of these results, for which the results in the previous literature in the matter are not applicable.