Using the barrier strip argument, we obtain the existence of solutions for the nonlinear boundary value problem
$$
(\phi(u'))'=f(t,u,u'),\qquad u(0)=A,\qquad u'(1)=B,
$$
where $\phi$ is an increasing homeomorphism

Liu, Rui

Ma, Ruyun

Zhang, Lu

Electronic journal of qualitative theory of differential equations

Ruyun Ma

Lu Zhang

Rui Liu

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Existence results for some nonlinear problems with $\phi$-Laplacian

Ruikuan Liu

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Existence results for some nonlinear problems with  $\phi$-Laplacian

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Electronic Journal of Qualitative Theory of Differential Equations2015, No. 22, 1–7; http://www.math.u-szeged.hu/ejqtde/Existence results for nonlinear problems withφ-LaplacianRuyun Ma, Lu ZhangB and Ruikuan LiuDepartment of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. ChinaReceived 1 July 2014, appeared 4 May 2015Communicated by Ioannis K. PurnarasAbstract. Using the barrier strip argument, we obtain the existence of solutions for thenonlinear boundary value problem(φ(u′))′ = f (t, u, u′), u(0) = A, u′(1) = B,where φ is an increasing homeomorphism.Keywords: barrier strip, φ-Laplacian, topological transversality theorem, existence.2010 Mathematics Subject Classification: 34B15, 34B18, 34C12.1 IntroductionThe purpose of this article is to obtain some existence results for nonlinear problems of theform(φ(u′))′ = f (t, u, u′), t ∈ [0, 1], (1.1)u(0) = A, u′(1) = B, (1.2)where f : [0, 1] ×R2 → R is continuous and φ : (−∞,∞) → R is an increasing homeomor-phism. Problems of this form are called classical.A typical classical model is the well-known p-Laplacian equation(φp(u′))′ = f (t, u, u′), t ∈ [0, 1]where φp(s) := |s|p−2s (p > 1). Various two-point boundary value problems containing thisoperator have received a lot of attention lately with respect to existence of solutions, see forexample, [10, 13, 14, 16, 20] and the references therein. The key condition in those works relieson a growth restriction on f .In 1994, Kelevedjiev [18], using the topological transversality theorem and the barrier stripargument, studied the nonlinear non-homogeneous problemu′′ = f (t, u, u′), t ∈ [0, 1] (1.3)u(0) = A, u′(1) = B, (1.4)BCorresponding author. Email: zhl992934235@163.com2 R. Ma, L. Zhang and R. Liuand obtained the following theorem.Theorem A. Let f : [0, 1]×R2 → R be continuous, and let there exist constants Li, i = 1, 2, 3, 4,such that L2 > L1 ≥ B, L3 < L4 ≤ B,f (t, u, v) ≥ 0, (t, u, p) ∈ [0, 1]×R× [L1, L2], (1.5)andf (t, u, v) ≤ 0, (t, u, p) ∈ [0, 1]×R× [L3, L4]. (1.6)Then problem (1.3)–(1.4) has at least one solution.(1.5) and (1.6) are called barrier strip conditions. They control the behavior of u′ on [0, 1].Depending on the sign of f (t, u, u′) the curve of u′(t) on [0, 1] crosses the strips [0, 1]× [L1, L2]and [0, 1]× [L3, L4] not more than once. Therefore, the assumptions (1.5) and (1.6) are sufficientconditions to obtain an a priori bound for u(t) and u′(t).Very recently, Kelevedjiev and Tersian [19] generalized Theorem A to the following bound-ary value problem with p-Laplacian(φp(u′))′ = f (t, u, u′), u(0) = A, u′(1) = B;the existence of C2-solution is proved under barrier strip conditions similar to (1.5) and (1.6).This work raises the following question: “Can we replace the p-Laplacian operator by themore general increasing homeomorphism (φ(u′))′?”. In the present paper, answering thesequestions in the affirmative, we extend Theorem A to the case of φ-Laplacian. More precisely,we prove the following theorem.Theorem 1.1. Let φ : R → R be classical and an increasing homeomorphism. Assume that (1.5) and(1.6) are fulfilled. Then (1.1)–(1.2) has at least one solution in C1[0, 1].The rest of the paper is organized as follows. In Section 2, we state some notations andthe topological transversality theorem, which will be crucial in the proof of our main result.Section 3 is devoted to the preparation of a priori bounds for the possible solutions of a suitablefamily of problems, and finally we will prove the main result.For other results concerning the φ-Laplacian operator we refer the reader to [1–9,11,12,17].2 Notations and fixed point theoremAs usual, C[0, 1] is the Banach space of continuous functions defined on [0, 1] endowed withthe norm ‖ · ‖0, and C1[0, 1] is Banach space of continuously differentiable functions definedon [0, 1] endowed with the norm ‖u‖1 = max{‖u‖0, ‖u′‖0}.Let Y be a convex subset of Banach space E and U ⊂ Y be open in Y. Let L∂U(U, Y) be aset of compact maps from U to Y which are fixed points free on ∂U; here, as usual, U and ∂Uare the closure of U and the boundary of U in Y, respectively.A map F in L∂U(U, Y) is essential if every map G in L∂U(U, Y) such that G|∂U = F|∂U has afixed point in U. It is clear, in particular, every essential map has a fixed point in U.Theorem B (Topological transversality theorem, [15]). Let Y be a convex subset of a Banach spaceE and U ⊂ Y be open. Assume that(a) F, G : U → Y are compact maps;Existence results for nonlinear problems with φ-Laplacian 3(b) G ∈ L∂U(U, Y) is essential;(c) H(u,λ), λ ∈ [0, 1], is a compact homotopy joining F and G; i.e.,H(u, 1) = F(u), H(u, 0) = G(u);(d) H(u,λ), λ ∈ [0, 1], is fixed point free on ∂U.Then H(u,λ), λ ∈ [0, 1], has at least one fixed point in U and in particular there is a u0 ∈ U suchthat u0 = F(u0).Theorem C. Let l ∈ U be fixed and F ∈ L∂U(U, Y) be the constant map F(u) = l for u ∈ U, then Fis essential.3 Auxiliary results, proofs of the main resultsFor λ ∈ [0, 1], we consider the family of boundary value problems(φ(u′))′ = λ f (t, u, u′), t ∈ [0, 1], (3.1)u(0) = A, u′(1) = B, (3.2)Our first auxiliary result gives an a priori bound for the solutions of the problem (3.1)–(3.2).Lemma 3.1. Let φ be an increasing homeomorphism, (1.5) and (1.6) hold and u ∈ C1[0, 1] be a solutionof the problem (3.1)–(3.2). Then there exists a constant M independent of λ and u such that‖u‖1 < M.Proof. Suppose the setS0 = {t ∈ [0, 1] : L1 < u′(t) ≤ L2} and S1 = {t ∈ [0, 1] : L3 ≤ u′(t) < L4}are not empty. Let t0 ∈ S0, t1 ∈ S1 be fixed. Assume that there are t′0 ∈ (t0, 1] and t′1 ∈ (t1, 1]such thatu′(t′0) < u′(t0), u′(t′1) > u′(t1). (3.3)The continuity of u′(t) allows us to take t′0 and t′1 correspondingly from (t0, 1] ∩ S0 and(t1, 1] ∩ S1. On the other hand, from (1.5) and (1.6) we have respectively(φ(u′))′ = λ f (t, u, u′) ≥ 0 for t ∈ S0 and (φ(u′))′ = λ f (t, u, u′) ≤ 0 for t ∈ S1and, since φ is increasing, we obtain that u′(t) is monotone increasing for t ∈ S0 and monotonedecreasing for t ∈ S1. Thus,u′(t′0) ≥ u′(t0), u′(t′1) ≤ u′(t1).This contradicts (3.3). Consequentlyu′(t) ≥ u′(t0) for t ∈ (t0, 1], u′(t) ≤ u′(t1) for t ∈ (t1, 1],and in particularu′(1) ≥ u′(t0) > L1 ≥ B, u′(1) ≤ u′(t1) < L4 ≤ B.4 R. Ma, L. Zhang and R. LiuThe contradiction obtained shows that S0 and S1 are empty. Since u′ ∈ C[0, 1], thenL4 ≤ u′(t) ≤ L1 for t ∈ [0, 1], i.e.,|u′(t)| ≤ max{|L1|, |L4|}, t ∈ [0, 1].On the other hand for each t ∈ [0, 1], we haveu(t)− u(0) =∫ t0u′(s) ds,which provides|u(t)| ≤ M2 for t ∈ [0, 1],where M2 = M1 + |A|, M1 = max{|L1|, |L4|}.So, we obtained that each solution u of (3.1)–(3.2), satisfies‖u‖1 < M := max{M1, M2}+ 1,where M is independent of λ and u. This completes the proof of Lemma 3.1.Now, we introduce the set C1BC[0, 1] = {u ∈ C1[0, 1] : u(0) = A, u′(1) = B} and theoperator L : C1BC[0, 1]→ C[0, 1] defined byLu = (φ(u′))′.Introduce also the operator K : C[0, 1]→ C1BC[0, 1] defined byKv = A +∫ t0φ−1[ ∫ τ1v(s) ds + φ(B)]dτ.Lemma 3.2. The operator K : C[0, 1]→ C1BC[0, 1] is well-defined and continuous.Proof. It is clear that for each v ∈ C[0, 1], the function h(t) := ∫ t1 v(s) ds + φ(B) is continuousfor t ∈ [0, 1]. Since φ is an increasing homeomorphism, so φ−1(h(t)) is also continuous fort ∈ [0, 1].Thus (Kv)′(t) = φ−1(h(t)) is in C[0, 1]. Finally, it is easy to check that(Kv)(0) = A, (Kv)′(1) = B,which means (Kv)(t) ∈ C1BC[0, 1]. The continuity of K follows from the continuity ofA +∫ t0 φ−1[∫ τ1 v(s) ds + φ(B)] dτ on [0, 1].Lemma 3.3. The operator K is the inverse operator of L.Proof. Clearly, each function u ∈ C1BC[0, 1] has a unique v = Lu ∈ C[0, 1]. Also, each functionv ∈ C[0, 1] has a unique inverse image u ∈ C1BC[0, 1] of the formu = A +∫ t0φ−1[ ∫ τ1v(s) ds + φ(B)]dτ,which is the solution of the boundary value problem(φ(u′))′ = v(t), t ∈ [0, 1],u(0) = A, u′(1) = B.Existence results for nonlinear problems with φ-Laplacian 5So, the operator L is one-to-one. Further, to show that K is an invertible map, let Lu = v, i.e.,φ(u′)′ = v. ThenKv = K(Lu) = A +∫ t0φ−1[ ∫ τ1(φ(u′(s)))′ ds + φ(B)]dτ= A +∫ t0φ−1[φ(u′(τ))− φ(u′(1)) + φ(B)] dτ= A +∫ t0u′(τ) dτ= u(t).Proof of Theorem 1.1. At first, we introduce the setU ={u ∈ C1BC[0, 1] : ‖u‖1 < M}.According to Lemma 3.1, all solutions of problem (3.1)–(3.2) are interior points of U. Introducealso the mapN : C1[0, 1]→ C[0, 1], defined by (Nu)(t) = f (t, u(t), u′(t)),for t ∈ [0, 1] and u(t) ∈ U.Now, we consider the homotopyHλ : U × [0, 1]→ C1BC[0, 1],defined by H(u,λ) ≡ Hλ(u) = λKN(u) + (1− λ)l, where l = Bt+ A is the unique solution of(φ(u′))′ = 0, t ∈ [0, 1],u(0) = A, u′(1) = B.Since K and N are continuous, using the Arzelà–Ascoli theorem it is not difficult to see thatK is compact. Therefore the homotopy Hλ : U × [0, 1] → C1BC[0, 1] is compact. According toLemma 3.1, H(u,λ), λ ∈ [0, 1], is fixed point free on ∂U. Besides, H0(u) ≡ l, ∀u ∈ U; i.e., it isa constant map and so is essential, by Theorem C.So, by Theorem B we get the map H1(u) has a fixed point in U. It is easy to see that it is asolution of the boundary value problem (3.1)–(3.2) obtained for λ = 1 and, what is the same,of (1.1)–(1.2).Example 3.4. Consider the boundary value problem(φ(u′))′ = u′2 − 4u′ + 3, t ∈ [0, 1],u(0) = 0, u′(1) = B.where 2 < B < 3 and φ(s) = s31+s2 .It is not difficult to verify that φ is an increasing homeomorphism, f (t, u, u′) = u′2− 4u′+ 3,has two simple zeros 1 and 3.So, we can choose L1 = 4, L2 = 5, L3 = 54 , L4 =32 to see (1.5)–(1.6) hold and so theconsidered problem has at least one solution.6 R. Ma, L. Zhang and R. LiuAcknowledgementsThe authors are very grateful to the the anonymous referees for their valuable suggestions.This work was supported by NSFC (no. 11361054 and no. 11201378), SRFDP (no. 2012 6203110004),and GanSu provincial National Science Foundation of China (no. 1208RJZA258).References[1] L. J. Alías, B. Palmer, On the Gaussian curvature of maximal surfaces and the Calabi–Bernstein theorem, Bull. London Math. Soc. 33(2001), 454–458. MR1832557[2] A. Azzollini, Ground state solution for a problem with mean curvature operator inMinkowski space, J. Funct. Anal. 266(2014), 2086–2095. 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Existence results for nonlinear problems with φ-Laplacian

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Existence results for some nonlinear problems with $\phi$ -Laplacian

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Existence results for some nonlinear problems with $\phi$ -Laplacian