In this paper, we discuss the existence of nonnegative solutions to a fourth order singular boundary value problem at two points. Our result is based on a recent Birkhoff-Kellogg type fixed point theorem developed on translates of a cone on a Banach space.
Mathematics Subject Classification (2010): 47H10, 34B09, 34B16

BOUCHAL, Lydia

GEORGIEV, Svetlin Georgiev

MEBARKI, Karima

English

Studia Universitatis Babeş-Bolyai

Stud. Univ. Babeş-Bolyai Math. 69(2024), No. 4, 905–912DOI: 10.24193/subbmath.2024.4.14Nonnegative solutions for a class of fourth ordersingular eigenvalue problemsLydia Bouchal , Karima Mebarki and Svetlin Georgiev GeorgievAbstract. In this paper, we discuss the existence of nonnegative solutions to afourth order singular boundary value problem at two points. Our result is basedon a recent Birkhoff-Kellogg type fixed point theorem developed on translates ofa cone on a Banach space.Mathematics Subject Classification (2010): 47H10, 34B09, 34B16.Keywords: Fixed point, fourth-order boundary value problem, positive solution,cone.1. IntroductionIn the present paper, we investigate the following fourth order singular differen-tial equation with parameterv(4) = λg(t)f(v(t)), 0 < t < 1, (1.1)subject to the boundary conditionsv(0) = a1, v(1) = a2, v′′(0) = a3, v′′(1) = a4, (1.2)where aj ≥ 0, j ∈ {1, 2, 3, 4}, are given constants,(H1). f ∈ C([0,∞)),0 < A1 ≤ f(x) ≤ A2 +k∑j=0Bjxj , x ∈ [0,∞),A2 ≥ A1 > 0 and Bj ≥ 0, j ∈ {0, . . . , k}, k ∈ N0, are given constants.Received 30 January 2023; Accepted 01 March 2023.© Studia UBB MATHEMATICA. Published by Babeş-Bolyai UniversityThis work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives4.0 International License.906 Lydia Bouchal, Karima Mebarki and Svetlin Georgiev Georgiev(H2). g : (0, 1) → R+ is continuous and may be singular at t = 0 or/and t = 1 ,g 6≡ 0 on (0, 1) and∫ 10s(1− s)g(s)ds <∞.Fourth order two-point boundary value problems (BVPs for short) have beenreceived much attention by many authors due to their importance in physics. Usually,they are essential in describing a vast class of elastic deflections with several typesof boundary conditions such as whose ends are simply-supported at 0 and 1 (v(0) =v(1) = v′′(0) = v′′(1) = 0). A great number of research has been devoted to investigatethe existence of positive solutions to this class of problems, see [2, 1, 3, 4, 8, 9, 10,11, 12, 13] and the references therein. The authors in [2] discussed the existence,uniqueness and multiplicity of positive solutions to the following eigenvalue BVP bymeans of fixed point theorem and degree theoryv(4) = λf(t, (v(t)), 0 < t < 1, (1.3)v(0) = v(1) = v′′(0) = v′′(1) = 0, (1.4)where λ > 0 is a constant and f : [0, 1] × [0,∞) → [0,∞) is continuous. In [12] byapplying a Krasnosel’skii fixed point theorem of cone expansion and compressionthe author obtained the existence and multiplicity results of equation (1.3) withboundary conditions v(0) = v(1) = v′(0) = v′(1) = 0. In the literature, thereare few papers devoted to study fourth order singular eigenvalue problems. In thecase when aj = 0, j ∈ {1, 2, 3, 4}, the BVP (1.1)-(1.2) is investigated in [7] whenf ∈ C([0,∞)), f > 0 on [0,∞), f is nondecreasing on [0,∞) and there exist δ > 0,m ≥ 2 such that f(u) > δum, u ∈ [0,∞), and g ∈ C(0, 1), g > 0 on (0, 1) and0 <∫ 10s(1− s)g(s)ds <∞. In [7], Feng and Ge used the method of upper and lowersolutions and the fixed point index to discuss the existence of positive solutions.Our main result is as follows where we do not require any monotonicity assump-tions on f , and we do not assume that f is either superlinear or sublinear.Theorem 1.1. Suppose that (H1) and (H2) hold. Then there is a λ∗ > 0 such that theBVP (1.1)-(1.2) has at least one nonnegative solution for λ = λ∗.Note that our main result, in the particular case aj = 0, j ∈ {1, 2, 3, 4}, is validin the case when f is decreasing on [0,∞), while the corresponding result in [7] isnot valid. For instance, f(x) = 1 + 11+x2 , x ∈ [0,∞), satisfies (H1) for A1 = 1,A2 = 2, Bj = 0, j ∈ {0, . . . , k}, and f is decreasing on [0,∞), whereupon it does notsatisfy the conditions in [7]. Also, the conditions for g in [7] are more restrictive than(H2). For instance, g(t) =( 12−t)2t(1−t) , t ∈ (0, 1), satisfies (H2) and does not satisfy theconditions in [7] because g(12)= 0. Thus, we can consider the particular case of ourmain result, aj = 0, j ∈ {1, 2, 3, 4}, as a complementary result to the result in [7].The approach used in this paper is to rewrite the (BVP) (1.1)-(1.2) into a perturbedintegral equation of which we search for solutions in a suitable subset of a Banachspace by means of recent fixed point theorem of Birkhoff-Kellogg type developed byCalamai and Infante in [5]. Note that this fixed point theorem has been applied veryrecently to discuss the solvability of fourth order retarded equations in [6].A class of fourth order singular eigenvalue problems 907The paper is organized as follows. In Section 2, we give some auxiliary resultsneeded for the proof of our main result. In Section 3, we prove our main result. InSection 4, we give an example.2. Auxiliary resultsLet X be a real Banach space.Definition 2.1. A mapping F : Ω ⊂ X → X is said to be completely continuous if itis continuous and maps bounded sets into relatively compact sets.Definition 2.2. A closed, convex set K of X is said to be cone if1. αx ∈ K for any α ≥ 0 and for any x ∈ K,2. x,−x ∈ K implies x = 0.For a given y ∈ X, we consider the translate of a cone K, namelyKy = K + y = {x+ y : x ∈ K}.Given an open bounded subset D of X we denote DKy = D ∩ Ky, an open subset ofKy.Theorem 2.3. [5, Corollary 2.4] Let (X, ‖ ‖) be a real Banach space, K ⊂ X be acone, and D ⊂ X be an open bounded set with y ∈ DKy and DKy 6= Ky. Assume thatF : DKy → K is a completely continuous map and assume thatinfx∈∂DKy‖Fx‖ > 0.Then there exists x∗ ∈ ∂DKy and λ∗ ∈ (0,∞) such thatx∗ = y + λ∗F (x∗).Lety1(t) =(a1 +a46)(1− t) + a2t+a36(1− t)3 + a46(t3 − 1) + a36(t− 1), t ∈ [0, 1].We have0 ≤ y1(t) ≤ a1 + a2 + a3 + a4, t ∈ [0, 1],andy′1(t) = −a1 −a46+ a2 −12a3(1− t)2 +12a4t2 +a36, t ∈ [0, 1].y′′1 (t) = a3(1− t) + a4t, t ∈ [0, 1].Hence,y1(0) = a1, y1(1) = a2, y′′1 (0) = a3, y′′(1) = a4.Sety(t) = −y1(t), t ∈ [0, 1].Now, consider the BVPu(4) = λg(t)f(u(t)− y(t)), 0 < t < 1,u(0) = u(1) = u′′(0) = u′′(1) = 0,(2.1)908 Lydia Bouchal, Karima Mebarki and Svetlin Georgiev Georgievwhere f and g satisfy (H1) and (H2), respectively.Let X = C([0, 1]) be endowed with the norm ‖u‖ = maxt∈[0,1]|u(t)|. DefineK = {u ∈ X : u(t) ≥ 0, t ∈ [0, 1]}.Since 0 ≤∫ 10s(1− s)g(s)ds <∞, there exists a nonnegative constant C0 such that∫ 10s(1− s)g(s)ds = C0.Because g 6≡ 0 on (0, 1), there are C1 > 0, s0 ∈ (0, 1) and ε > 0 such that s0−ε, s0+ε ∈(0, 1) andg(s) ≥ C1, s ∈ (s0 − ε, s0 + ε).DefineG(t, s) = t(1− s)2s−s2−t26 , 0 ≤ t ≤ s ≤ 1,s(1− t) 2t−t2−s26 , 0 ≤ s ≤ t ≤ 1.We have0 ≤ G(t, s) ≤ 16s(1− s) ≤ 16, 0 ≤ t, s ≤ 1,Note that∫ 10G(s0 + ε, s)g(s)ds ≥∫ s0+εs0−εG(s0 + ε, s)g(s)ds≥ C1∫ s0+εs0−εG(s0 + ε, s)ds= C1∫ s0+εs0−εs(1− s0 − ε)2(s0 + ε)− (s0 + ε)2 − s26ds≥ 23C1ε(s0 − ε)2(1− s0 − ε)2> 0.For u ∈ X, define the operatorTu(t) =∫ 10G(t, s)g(s)f(u(s)− y(s))ds, t ∈ [0, 1].In [7], it is proved that any fixed point u ∈ X of the operator λT is a solution to theBVP (2.1). Fix C2 > a1 + a2 + a3 + a4 arbitrarily. DefineD = {u ∈ X : ‖u‖ < C2}.We have that D is an open bounded set in X, y ∈ D and DKy = D ∩Ky 6= Ky. Notethat for any u ∈ DKy , we haveu(t) = y(t) + z(t), t ∈ [0, 1],A class of fourth order singular eigenvalue problems 909for some z ∈ K, and so u(t)− y(t) = z(t) ≥ 0, t ∈ [0, 1], andf(u(t)− y(t)) ≤A2 + k∑j=0Bj(u(t)− y(t))j≤(A2 +k∑j=0Bj2j(|u(t)|j + |y1(t)|j))≤(A2 +k∑j=0Bj2j(Cj2 + (a1 + a2 + a3 + a4)j)), t ∈ [0, 1].2.1. Proof of the main resultSince f ∈ C([0,∞)) and g ∈ C(0, 1), we have that T : DKy → K is a continuousoperator. Next, for u ∈ DKy , we haveTu(t) =∫ 10G(t, s)g(s)f(u(s)− y(s))ds≤ 16A2 + k∑j=0Bj2j(Cj2 + (a1 + a2 + a3 + a4)j)∫ 10s(1− s)g(s) ds=16C0A2 + k∑j=0Bj2j(Cj2 + (a1 + a2 + a3 + a4)j) , t ∈ [0, 1],whereupon‖Tu‖ ≤ C0A2 + k∑j=0Bj2j(Cj2 + (a1 + a2 + a3 + a4)j) .Then, T (DKy ) is uniformly bounded. Moreover, for u ∈ DKy and t1, t2 ∈ [0, 1], t1 < t2,the Lebesgue dominated convergence theorem guarantees that|Tu(t1)− Tu(t2)|≤∫ 10|G(t1, s)−G(t2, s)|g(s)f(u(s)− y(s))ds ds≤A2 + k∑j=0Bj2j(Cj2 + (a1 + a2 + a3 + a4)j)∫ 10g(s)|G(t1, s)−G(t2, s)|ds→ 0, t1 → t2,Therefore, T (DKy ) is equicontinuous. According to the Arzelà-Ascoli compactnesscriterion, we conclude that the operator T : DKy → K is completely continuous.910 Lydia Bouchal, Karima Mebarki and Svetlin Georgiev GeorgievObserve that, for u ∈ ∂DKy ,maxt∈[0,1]|Tu(t)| ≥ Tu(s0 + ε) =∫ 10G(s0 + ε, s)g(s)f(u(s)− y(s))ds≥ A1∫ 10G(s0 + ε, s)g(s)ds≥ 23A1C1ε(s0 − ε)2(1− s0 − ε)2> 0.Consequentlyinfu∈∂DKy‖Tu‖ ≥ 23A1C1ε(s0 − ε)2(1− s0 − ε)2 > 0.Now, applying Theorem 2.3, we conclude that there are λ∗ ∈ (0,∞) and u∗ ∈ ∂DKysuch thatu∗(t) = y(t) + λ∗∫ 10G(t, s)g(s)f(u∗(s)− y(s))ds, t ∈ [0, 1].Letv∗(t) = u∗(t)− y(t), t ∈ [0, 1].Thenv∗(0) = u∗(0)− y(0) = a1,v∗(1) = u∗(1)− y(1) = a2,v∗′′(0) = u∗′′(0)− y′′(0) = a3,v∗′′(1) = u∗′′(1)− y′′(1) = a4andv∗(t) = λ∫ 10G(t, s)g(s)f(v∗(s))ds, t ∈ [0, 1],whereuponv∗(4)(t) = λg(t)f(v∗(t)), 0 < t < 1.Since u∗ ∈ ∂DKy , we have that u∗(t) = y(t) + z∗(t), t ∈ [0, 1], for some z∗ ∈ K, andthenv∗(t) = u∗(t)− y(t) = z∗(t) + y(t)− y(t) = z∗(t) ≥ 0, t ∈ [0, 1].A class of fourth order singular eigenvalue problems 9113. An exampleConsider the BVPu(4) = λ( 12−t)2t(1−t)(1 + 11+(u(t))2), t ∈ (0, 1),u(0) = 0, u(1) = 1, u′′(0) = 12 , u′′(1) = 1.(3.1)Heref(x) = 1 +11 + x2, x ∈ [0,∞), g(t) =(12 − t)2t (1− t), t ∈ (0, 1),anda0 = 0, a1 = 1, a2 =12, a3 = 1.By our main result, it follows that the BVP (3.1) has at least one nonnegative solution.References[1] Bai, Z., The method of lower and upper solutions for a bending of an elastic beamequation, J. Math. Anal. Appl., 248(2000), no. 1, 195-202.[2] Bai, Z., Wang, H., On positive solutions of some nonlinear fourth-order beam equations,J. Math. Anal. Appl., 270(2002), no. 2, 357-368.[3] Benslimane, S., Georgiev, S.G., Mebarki, K., Multiple nonnegative solutions for a ofclass fourth-order BVPs via a new topological approach, Adv. Theory Nonlinear Anal.Appl., 6(2022), no. 3, 390-404.[4] Bouchal, L., Georgiev, S.G., Mebarki, K., On the existence of solutions for a class ofnonlinear fully fourth order differential system, Proyecciones, 43(2024), no. 3, 613-629.[5] Calamai, A., Infante, G., An affine Birkhoff-Kellogg-type result in cones with applicationsto functional differential equations, Math. Methods Appl. Sci., (2022), 1-9.[6] Calamai, A., Infante, G., On fourth order retarded equations with functional boundaryconditions: A unified approach, Discrete Contin. Dyn. Syst. - S, Early access January2023.[7] Feng, M., Ge, W., Existence of positive solutions for singular eigenvalue problems, Elec-tron. J. Differential Equations, 2006(2006), no. 105, 1-9.[8] Han, G., Xu, Z., Multiple solutions of some nonlinear fourth-order beam equations, Non-linear Anal., 68(2008), no. 12, 3646-3656.[9] Liu, B., Positive solutions of fourth-order two point boundary value problems, Appl.Math. Comput., 148(2004), no. 2, 407-420.[10] Ma, R., Wang, H., On the existence of positive solutions of fourth-order ordinary differ-ential equations, Appl. Anal., 59(1995), no. 1-4, 225-231.[11] Pang, C., Dong, W., Wei, Z., Multiple solutions for fourth-order boundary value problem,J. Math. Anal. Appl., 314(2006), no. 2, 464-476.[12] Yao, Q., Positive solutions for eigenvalue problems of fourth-order elastic beam equa-tions, Appl. Math. Lett., 17(2004), no. 2, 237-243.[13] Zhu, Y., Weng, P., Multiple positive solutions for a fourth-order boundary value problem,Bol. Soc. Parana. Mat., 21(2003), no. 1-2, 9-19.912 Lydia Bouchal, Karima Mebarki and Svetlin Georgiev GeorgievLydia BouchalLaboratory of Applied Mathematics,Faculty of Exact Sciences,University of Bejaia, 06000 Bejaia, Algeriae-mail: lydia.bouchal@univ-bejaia.dzKarima MebarkiLaboratory of Applied Mathematics,Faculty of Exact Sciences,University of Bejaia, 06000 Bejaia, Algeriae-mail: karima.mebarki@univ-bejaia.dzSvetlin Georgiev GeorgievDepartment of Differential Equations,Faculty of Mathematics and Informatics,University of Sofia, Sofia, Bulgariae-mail: svetlingeorgiev1@gmail.com

Nonnegative solutions for a class of fourth order singular eigenvalue problems

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Nonnegative solutions for a class of fourth order singular eigenvalue problems

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