Construction of lower and upper solutions for first-order periodic problem

Boundary Value Problems, Oct 2015

In this paper, we construct nonconstant lower and upper solutions for the periodic boundary value problem x ′ + f ( t , x ) = e ( t ) , x ( 0 ) = x ( T ) and find their estimates. We prove the existence of positive solutions for the singular problem x ′ + g ( x ) = e ( t ) , x ( 0 ) = x ( T ) by using these results. MSC: 34B10, 34B18.

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Construction of lower and upper solutions for first-order periodic problem

Ma and Zhang Boundary Value Problems Construction of lower and upper solutions for first-order periodic problem Ruyun Ma 0 1 2 Lu Zhang 0 1 2 0 Lanzhou , 730070 , P.R. China 1 Northwest Normal University 2 Department of Mathematics In this paper, we construct nonconstant lower and upper solutions for the periodic boundary value problem x + f (t, x) = e(t), x(0) = x(T ) and find their estimates. We prove the existence of positive solutions for the singular problem x + g(x) = e(t), x(0) = x(T ) by using these results. solutions; periodic boundary value problem; singular problem; lower and upper 1 Introduction The method of lower and upper solutions is an elementary but powerful tool in the existence theory of solutions to initial value problems and for periodic boundary value problems, even in cases where no special structure is assumed on the nonlinearity. Starting with the pioneering work of Moretto [] for locally Lipschitzian ordinary differential equations, the method of upper and lower solutions for the periodic boundary value problem x + f (t, x) = , x() = x(T ) has been extended to the case of a continuous right-hand side f : [, T ] × R In Franco et al. [], the method of lower and upper solutions was applied to the periodic x = f (t, x), x() = x(T ), and they also obtained a similar result to Lemma . below. In order to apply these results, finding upper and lower solutions is very important. But the problem of construction of lower and upper solutions has been solved very rarely (see, e.g., [, ]). In this paper we fill this gap and present conditions ensuring the existence of nonconstant lower and upper solutions to the first-order periodic boundary value problem (.) and find their estimates. This enables us to prove the existence result for the periodic problem with strong singularity. © 2015 Ma and Zhang. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. In order to explain the main results of the paper, let us introduce some notation: C[, T ] stands for the set of functions continuous on [, T ]. For x ∈ C[, T ], we denote  T  x  = E = E± = If x ∈ C[, T ], then we write x+(t) = max{x(t), } and x–(t) = max{–x(t), }. For e ∈ C[, T ], we put and note that E = E+ – E–. Let us recall first some classical definitions and results. They are taken from []. Definition . A lower solution α (respectively, an upper solution β) of problem (.) is a function α ∈ C[, T ] such that The basic existence theorem of the method of upper and lower solutions for (.) can be stated as follows. Lemma . If problem (.) has a lower solution α and upper solution β such that α(t) ≤ β(t) for all t ∈ [, T ] (resp. β(t) ≤ α(t) for all t ∈ [, T ]), then problem (.) has at least one solution x such that α(t) ≤ x(t) ≤ β(t) for all t ∈ [, T ] (resp. β(t) ≤ x(t) ≤ α(t) for all t ∈ [, T ]). The paper is organized as follows. In Section  we develop a method to construct lower and upper solutions for (.). As the application, in Section , we establish an existence result for a nonlinear first-order periodic problem with strong singularity. 2 Construction of lower and upper solutions In this section we consider x + f (t, x) = e(t), x() = x(T ), where f : [, T ] × R → R and e : [, T ] → R are continuous. Let us consider an auxiliary boundary value problem R where δ ∈ C[, T ] satisfies δ¯ =  and ζ ∈ . f (t, x) ≤ c(t) for all t ∈ [, T ] and x ∈ B, B +  η  , where η(t) = φ–(t) – φ+(t) –+ and φ = c – e. If c¯ – e¯ ≤ , B ≤ β ≤ B +  η  in [, T ]. Taking x = + and ω (t) + φ+(t) – – φ–(t) + = , it is easy to see that β() = β(T ). Since | – t x[φ+(s) – – φ–(s) +] ds| = |  η(s) ds| ≤ t η , choosing ζ = B + η , we obtain which means that (.) holds. Now, using (.), it follows that + ≤ –, implying that β (t) = xω (t) = –x φ+(s) – – φ–(s) + ≤ –φ(t) = e(t) – c(t). From (.) and (.) we deduce that and the proof is completed. β (t) + f (t, β) ≤ β (t) + c(t) ≤ e(t) for all t ∈ [, T ], Using similar arguments, one can prove the following proposition. f (t, x) ≥ c(t) for all t ∈ [, T ] and x ∈ B, B +  η  , where η(t) = φ–(t) – φ+(t) –+ and φ = c – e. If c¯ – e¯ ≥ , B ≤ α ≤ B +  η  in [, T ]. Theorem . and Theorem . are simple examples of existence results which follow immediately from Lemma ., Propositions . and .. (–)i+f (t, x) ≤ (–)i+ci(t), (–)i+(c¯i – e¯) ≤  for all t ∈ [, T ], x ∈ [Bi, Bi +  ηi ] and all i ∈ {, }, Then problem (.) possesses a solution x such that Theorem . Assume that there are B, B ∈ R, c, c ∈ C[, T ] such that the assumptions of Theorem . (...truncated)


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Ruyun Ma, Lu Zhang. Construction of lower and upper solutions for first-order periodic problem, Boundary Value Problems, 2015, pp. 190, 2015, DOI: 10.1186/s13661-015-0457-7