Existence of solutions for some second order equations with nonhomogeneous boundary conditions and a function ϕ continuous on DOM(ϕ)⊂R

Introduction

 𝜙: ] -𝑑, 𝑑[→ ℝ (𝑑 ∈ ]0, +∞]) an increasing homeomorphism satisfying 𝜙(0) = 0,  𝑘: [0, 𝜉] × ℝ × ℝ → ℝ continuous.

Their key findings include: 1) For 𝑑 < +∞, problem (1) -( 2) admits at least one solution under Villari-type sign conditions on 𝑘, 𝑙 0 , 𝑙 𝜉 [4] . 2) For 𝑑 < +∞, existence is guaranteed if lower and upper solutions exist (ordered or not).

3) For 𝑑 = +∞, a solution exists when |𝑙 0 | is bounded, 𝑙 𝜉 is bounded above and 𝑘(𝑠, 𝑢, 𝑣) ≤ 𝜏 (𝑠) for some 𝜏 ∈ 𝐶([0, 𝜉]).

In 2015 and 2017, Eienne G. and Assohoun A. [5,6] extended somes results of Cristian B. and Jean M. to 𝐿 1 -Carathéodory function 𝑘. The paper is structured as follows:

In section 2, we some preliminary. In section 3, we prove that the problem (1) -( 2) admits at least one solution when 𝑙 0 is bounded from below, 𝑙 𝜉 is bounded from above and there exists 𝜏 ∈ 𝐿 1 (0, 𝜉) such that 𝑘(𝑠, 𝑢, 𝑣) ≥ 𝜏 (𝑠) for a.e. 𝑠 ∈ [0, 𝜉] and ∀(𝑢, 𝑣) ∈ ℝ 2 or when 𝑙 0 is bounded from above, 𝑙 𝜉 is bounded from below and there exists 𝜏 ∈ 𝐿 1 (0, 𝜉) satisfying 𝑓(𝑠, 𝑢, 𝑣) ≤ 𝜏(𝑠) for a.e. 𝑠 ∈ [0, 𝜉] and ∀(𝑢, 𝑣) ∈ ℝ 2 . After that, in section 4 we apply section 3's results to nonlinear beam equations. Finally, in section 5, we extend the study to the generalized problem (Φ(𝑣 ′ (𝑠))) ′ = 𝑘(𝑠, 𝑣 (𝑠), 𝑣 ′ (𝑠)), 𝑎. 𝑒. 𝑠 ∈ [0, 𝜉]

where Φ: 𝐷𝑜𝑚(Φ) ⊂ ℝ → ℝ is continuous and strictly increasing on [𝑎, 𝑏] ⊂ 𝐷𝑜𝑚(Φ), 𝑙 0 , 𝑙 𝜉: ℝ → ℝ are two continuous functions and 𝑘: [0, 𝜉] × ℝ × ℝ → ℝ a 𝐿 1 -Carathéodory function. 𝜙(𝛽 ′ (0)) ≤ 𝑙 0 (𝛽(0)) 𝑎𝑛𝑑 𝜙(𝛽 ′ (𝜉)) ≥ 𝑙 𝜉 (𝛽(𝜉)).

Preliminary

Theorem 2.1. The existence of a lower solution α and an upper solution β for (5) implies that the problem (5) has at least one solution.

Proof. See [5] and [6] . Theorem 2.2. The existence of a lower-solution 𝛼 and an upper-solution 𝛽 of ( 5) such that ∀ 𝑠 ∈ [0, 𝜉], 𝛼(𝑠) ≤ 𝛽 (𝑠), implies that the problem (5) admits at least one solution 𝑢 such that ∀ 𝑠 ∈ [0, 𝜉], 𝛼(𝑠) ≤ 𝑢(𝑠) ≤ 𝛽 (𝑠).

Existence result

To prove an analogous result for d = +∞, we will apply Theorem

Where k: [0, 𝜉] × ℝ × ℝ → ℝ is a 𝐿 1 -Carathéodory function, 𝑙 0 , 𝑙 𝜉: ℝ → ℝ are two continuous functions and 𝜙: ℝ → ℝ, an increasing homeomorphism such that 𝜙(0) = 0.

) and satisfies (10). We need the following results.

Lemma 3.1. Suppose that: a) There exists 𝑔 ∈ 𝐿 1 (0, 𝜉) such that 𝑘(𝑠, 𝑤, 𝑧) ≤𝑔(𝑠) for a.e. 𝑠 ∈ [0, 𝜉 ] and ∀(𝑤, 𝑧) ∈ ℝ 2 . b) There exists (𝜗 0 , 𝜗 𝜉 ) ∈ ℝ 2 such that ∀𝑠 ∈ [0, 𝑇], 𝑙 0 (𝑠) ≤ 𝜗 0 and 𝑙 𝜉 (𝑠) ≥ 𝜗 𝜉 .

If 𝑣 is a solution of (10), then ||𝑣′|| ∞ ≤ 𝑒 where

Proof. Let 𝑢 be a solution of (10). Then we have:

Lemma 3.2. Suppose that: a) There exists 𝑔 ∈ 𝐿 1 (0, 𝜉) such that 𝑘(𝑠, 𝑤, 𝑧) ≥ 𝑔(𝑠) for a.e. 𝑠 ∈ [0, 𝜉 ] and

b) There exists (𝜗 0 , 𝜗 𝜉 ) ∈ ℝ 2 such that ∀𝑠 ∈ [0, 𝜉], 𝑙 0 (𝑠) ≥ 𝜗 0 and 𝑙 𝜉 (𝑠) ≤ 𝜗 𝜉 .

If 𝑣 is a solution of (10), then ||𝑣′|| ∞ ≤ 𝑒, where Theorem 3.1. Suppose that: a) There exists 𝑔 ∈ 𝐿 1 (0, 𝜉) such that 𝑘(𝑠, 𝑤, 𝑧) ≤ 𝑔(𝑠) for a.e. 𝑠 ∈ [0, 𝜉 ] and ∀(𝑤, 𝑧) ∈ ℝ 2 . b) There exists (𝜗 0 , 𝜗 𝜉 ) ∈ ℝ 2 such that ∀𝑠 ∈ [0, 𝜉], 𝑙 0 (𝑠) ≤ 𝜗 0 and 𝑙 𝜉 (𝑠) ≥ 𝜗 𝜉 . c) The problem (10) admits a lower-solution 𝛼 and an upper-solution 𝛽.

It follows that problem (10) has at least one solution.

Then, using Theorem 2.2. we deduce that the problem (11) has a solution 𝑢 which is also a solution of problem (10) by Lemma 3.1. Theorem 3.2. Suppose that: a) There exists 𝑔 ∈ 𝐿 1 (0, 𝜉) such that 𝑘(𝑠, 𝑤, 𝑧) ≥ 𝑔(𝑠) for a.e. 𝑠 ∈ [0, 𝜉 ] and ∀(𝑤, 𝑧) ∈ ℝ 2 . b) There exists (𝜗 0 , 𝜗 𝜉 ) ∈ ℝ 2 such that ∀𝑠 ∈ [0, 𝜉], 𝑙 0 (𝑠) ≥ 𝜗 0 and 𝑙 𝜉 (𝑠) ≤ 𝜗 𝜉 . c) The problem (10) admits a lower-solution 𝛼 and an upper-solution 𝛽.

It follows that problem (10) has at least one solution.

Proof: The proof is similar to the proof Theorem 3.1. Remark 3.1. In contrast to Theorem 5 in [3] , |𝑔 0 | bounded is not necessary; 𝑔 0 bounded above or 𝑔 0 bounded below is sufficient when 𝑑 = +∞.

Application to some nonlinear beam equations

In 2006, P. Pablo A. and Pedro Pablo Cardenas A. prove in [7] that the problem u′′(s) + h(s, u(s), u′(s)) = 0, 0 < s < 𝜉, u′(0) = -l(u(0)), u′(T) = l(u(𝜉))

with ℎ: [0, 𝜉] × ℝ 2 → ℝ, and l: ℝ → ℝ continuous, admits at least one solution, if ℎ satisfies a Nagumo type condition and there exists an ordered couple of a lower and an upper solution of (12). The following result gives us some existence results without Nagumo type condition, when ℎ is a 𝐿 1 -Carathéodory function and no ordering is assumed between the lower and upper solutions.

Theorem 4.1. Suppose that: a) There exists 𝑔 ∈ 𝐿 1 (0, 𝜉) such that for a.e. 𝑠 ∈ [0, 𝜉 ] and ∀(𝑤, 𝑧) ∈ ℝ 2 , ℎ(𝑠, 𝑤, 𝑧) ≥ 𝑔(𝑠) (𝑟𝑒𝑠𝑝 ℎ(𝑠, 𝑤, 𝑧) ≤ 𝑔(𝑠)) . b) There exists 𝜗, ∈ ℝ such that ∀𝑠 ∈ [0, 𝜉], 𝑙(𝑠) ≥ 𝜗 and (resp 𝑙(𝑠) ≤ 𝜗). c) The problem (12) admits a lower-solution 𝛼 and an upper-solution 𝛽.

Then the problem (12) admits at least one solution.

Proof: Application of Theorem 3.1. (resp Theorem 3.2.) with 𝑘 = -ℎ, 𝑙 0 = -𝑙, 𝑙 𝜉 = 𝑙 and 𝜙(𝑥) = 𝑥.

Existence result for problem (3) -(4).

In this section we study the problem ( 3) -( 4), where Φ: 𝐷𝑜𝑚(Φ) ⊂ ℝ → ℝ is continuous and strictly increasing on [𝑎, 𝑏] ⊂ 𝐷𝑜𝑚(Φ), 𝑙 0 , 𝑙 𝜉: ℝ → ℝ are two continuous functions, and 𝑘: [0, 𝑇] × ℝ 2 → ℝ a 𝐿 1 -Carathéodory function. Let 𝑝 ∈ ℝ be such that Φ(0) + 𝑝 = 0. Let 𝑎′ = max{|𝑎|, |𝑏|} + 1.

Λ is an increasing homeomorphism such that Λ (0) = 0. Consider the functions 𝐺 0: ℝ → ℝ 𝑎𝑛𝑑 𝐺 𝜉: ℝ → ℝ given by 𝐺 0 (𝑠) = 𝑙 0 (𝑠) + 𝑝 𝑎𝑛𝑑 𝐺 𝜉 (𝑠) = 𝑙 𝜉 (𝑠) + 𝑝.

We have: Let 𝑟 ∈ ℝ be such that Φ(𝑏) + 𝑟 < 0. Let 𝑎 ′ = -𝑎 + 1.

Ψ is an increasing homeomorphism such that Ψ(0) = 0. Consider the functions 𝐺 0: ℝ → ℝ 𝑎𝑛𝑑 𝐺 𝜉: ℝ → ℝ given by 𝐺 0 (𝑠) = 𝑙 0 (𝑠) + 𝑟 𝑎𝑛𝑑 𝐺 𝜉 (𝑠) = 𝑙 𝜉 (𝑠) + 𝑟.

Consider the problem

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