Existence results for the problem (φ(u′))′=f(t,u,u′) with nonlinear boundary conditions1

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Introduction

In this paper we study existence results for the following problem, which we will refer by (P):[φ(u′(t))]′=f(t,u(t),u′(t))fora.e.t∈I=[a,b],0=g(u(a),u′(a),u′(b)),u(b)=h(u(a)).

The three following conditions hold:

  • (H1) f is a Carathéodory function, that is: f(t,·,·) is a continuous function for a.e. t∈I;f(·,u,v) is measurable for all (u,v)∈R2; and for every M>0 there exists a real-valued function ψψML1(I) such that|f(t,u,v)|≤ψ(t)for a.e. tI and for every (u,v)∈R2 with |u|≤M and |v|≤M.

  • φ:RR is continuous, increasing and φ(R)=R.

  • g:R3R is a continuous function, nondecreasing in the second variable and nonincreasing in the third one; h:RR is a continuous and nondecreasing function.

Problem (1.1) with different boundary conditions has been studied by different authors (see 2, 6 and references therein). Recently, Gao, Wang and Lin, following some ideas developed in [6], proved in [10] the existence of solution for the problem (1.1) with f a continuous function and considering Dirichlet and mixed boundary conditions, in the presence of lower and upper solutions. These results have been generalized in [9] to the case in which f is a Carathéodory function. On the other hand, existence results for linear boundary data were also presented by O’Regan in [7].

In [1], existence results for the Periodic and Neumann problems were proved. There we used the results given in [9]. However, the method of lower and upper solutions was indirectly applied.

In this paper we give results, in the spirit of [10], that generalize all of these existence theorems for linear boundary conditions by considering the conditions studied in [3] for the second-order problem u′′=f(t,u,u′), together with equations of the form (1.1). Here, the method of lower and upper solutions and the Nagumo condition, to obtain a priori bounds for the derivatives of the solutions, are applied, improving the results of 1, 9.

Now, let W1,1(I) be the Banach space of absolutely continuous functions on I. We say that yC1(I) is a lower solution for problem (P) if φy′∈W1,1(I) and satisfies (φ(y′(t)))′≥f(t,y(t),y′(t))fora.e.t∈I,g(y(a),y′(a),y′(b))≥0 and h(y(a))=y(b).

y is an upper solution of (P) if the reversed inequalities hold. If equalities hold, we say that y is a solution of (P).

Next, we define the Nagumo condition we are going to use. Note that the condition does not depend on the boundary data of the problem.


Definition 1.1. We say that f:R2R, a Carathéodory function, satisfies a Nagumo condition relative to the pair α and β, with α,β∈C(I),α≤β in I, if there exist functions k∈Lp(I),1≤p≤∞, and θ:[0,∞)→(0,∞) continuous, such that|f(t,u,v)|≤k(t)θ(|v|)fora.e.(t,u,v)∈Ω,where Ω={(t,u,v)∈I×R2:α(t)≤u≤β(t)}, and also thatφ(ν)−1(u)|(p−1)/pθ(|φ−1(u)|)du,−∞φ(−ν)−1(u)|(p−1)/pθ(|φ−1(u)|)du>μ(p−1)/p‖k‖p,beingμ=maxt∈Iβ(t)−mint∈Iα(t),ν=max{|α(a)−β(b)|,|α(b)−β(a)|}b−aand‖k‖p=supt∈I|k(t)|ifp=∞[∫ab|k(t)|pdt]1/pif1≤p<∞,where (p−1)/p≡1 for p=∞.

Section snippets

Main result

Following the ideas of [3], before introducing the main result of existence of solutions of problem (P), suppose that hypotheses (H1)–(H3) and the Nagumo condition relative to a lower solution α and an upper solution β,αβ are satisfied. Now, we start with the construction of the modified problem.

Firstly, we define p(t,x)=max{α(t),min{x,β(t)}} for all x∈R.

One can find the next result, with its proof, in [8].


Lemma 2.1. For each uC1(I) the next two properties hold:

  • 1.

    (d/dt)p(t,u(t)) exists for a.e. tI.

  • 2.

    If u

Final remarks

A more general result is obtained if we change the hypothesis (H2) by

  • There exist K<0<K+ such that φ:[K,K+]→R is a continuous and increasing function in [K,K+].

Note that the solutions (and also the lower and upper solutions) must be elements of the setC={v∈C1(I):K<v′<K+andφ∘v′∈W1,1(I)},so, in their definitions, this condition cannot be disregarded.

Instead of Definition 1.1 we must assume thatK<−ν≤ν<K+and there exist k∈Lp(I),1≤p≤∞ and θ:[0,∞)→(0,∞) continuous, such that|f(t,x,y)|≤k(t)θ(|y|)

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1

Research partially supported by DGICYT, project PB94-0610.

2

Research partially supported by the EEC project grant ERB CHRX-CT94-0555.

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