u =Δu = 0, on Ω, (1.2) in a non-empty bounded open set Ω R N (N > 1) with a sufficient smooth boundary Ω where,

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1 Khaleghi Moghadam Cogent Mathematics 6 : 6 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Existence of a non-trivial solution for fourth-order elastic beam equations involving Lipschitz non-linearity Mohsen Khaleghi Moghadam * Received: June 6 Accepted: August 6 Published: September 6 *Corresponding author: Mohsen Khaleghi Moghadam Department of Basic Sciences Sari Agricultural Sciences Natural Resources University 7 Sari Iran s: mohsen.haleghi@rocetmail. com m.haleghi@sanru.ac.ir Reviewing editor: Amar Debbouche Guelma University Algeria Additional information is available at the end of the article Abstract: In this paper we consider the existence of a non-trivial solution for a class of fourth-order elastic beam equations involving Lipschitz non-linearity with Navier boundary value condition. The technical approach is essentially based on a critical point theorem. As an application an example is presented. Subjects: Advanced Mathematics; Analysis - Mathematics; Applied Mathematics; Differential Calculus; Differential Equations; Mathematical Physics; Mathematics & Statistics; Pure Mathematics; Science Keywords: fourth-order equations; critical point; non-trivial solution mathematics subject classifications: J; B; E. Introduction The aim of this paper is to study the existence of non-trivial solution for the following boundary value problem: u iv + Au + Bu + gu =λf t u t [ ] u =u = u =u =. where A B are real constants f : [ ] R R is an L -Carathéodory function g: R R is a Lipschitz continuous function with Lipschitzian constant L g = λ > is a parameter. In recent years the fourth-order boundary value problems have been extensively considered by many authors for instance see Bai Wang Cabada Cid Sanchez 7 Grossinho Sanchez Tersian Liu Li 7a Peletier Troy Van der Vorst 99 references therein. These ind of problems arising in real-world phenomena play a fundamental role in different fields of research such as mechanical engineering control systems economics computer ABOUT THE AUTHOR Mohsen Khaleghi Moghadam is an assistant professor of Mathematics Department of Basic Sciences Sari Agricultural Sciences Natural Resources University Sari Iran. The author's ey research activities are: Nonlinear Analysis: Variational Principles Critical Point Theory Variational Inequalities; Partial Differential Equations: Semilinear Quasilinear Elliptic Boundary Value Problems. This paper relates to wider projects that the author will wor. PUBLIC INTEREST STATEMENT In the paper the author using variational methods established the existence of a non-trivial solution for a class of fourth-order elastic beam equations involving Lipschitz non-linearity with Navier boundary value condition. Using a critical points theorem the author has ensured the exact collections of the parameters in which the problem possesses at least a non-trivial solution has presented two examples to illustrate the results. 6 The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY. license. Page of

2 Khaleghi Moghadam Cogent Mathematics 6 : 6 science physics biology many others. For example fourth-order BVPs describe the deformations of an elastic beam in an equilibrium state whose both ends are simply supported. Also a non-linear fourth-order equation describes traveling waves in suspension bridges. So for this reason there are wide papers about these problems which authors have investigated by different methods such as fixed point theorems Liu & Li 7b lower upper solutions method Cabada et al. 7 critical point theory Bonanno Morse theory Han & Xu 7 mountain-pass theorem Gyulov & Morosanu that for more references we refer the reader to Afrouzi Heidarhani O Regan Bai Bonanno Di Bella Bonanno Di Bell Bonanno Di Bella O Regan Chai 7 Li 7 references therein. In Khalhali Heidarhani Razani. have studied the existence of infinitely many solutions. The authors in Heidarhani Ferrara Salari Caristi 6 established the existence multiplicity results by variational methods critical point theory for the following fourth-order Navier boundary value problem Δ px u = λf t u in Ω u =Δu = on Ω. in a non-empty bounded open set Ω R N N > with a sufficient smooth boundary Ω where Δ px u = Δ Δu px Δu is the px-biharmonic operator of fourth order with p C Ω λ [ f :Ω R R is an L -Carathéodory function. There are so many equations in engineering physics mathematics that are studied through Navier boundary conditions. For the use of Navier boundary conditions in the mathematical literature see Busuioc Ratiu the related references. Very recently some researchers have studied the existence multiplicity of solutions for impulsive fourth-order elastic beam equation; we refer the reader to Heidarhani Afrouzi Ferrara Moradi 6 Heidarhani Ferrara Khademloo 6 references therein. In the present paper based on a local minimum theorem Theorem. due to Bonanno we ensure an exact interval of parameters in which the problem. admits at least a non-trivial solution. We also refer the interested reader to the papers Bonanno Di Bella & O Regan ; Heidarhani a b ; Heidarhani et al. 6; Khaleghi Moghadam & Heidarhani in which Theorem. has been successfully employed to the existence of at least one nontrivial solution for some boundary value problems. Our main result Theorem. its consequence Theorem. ensure the existence of a nontrivial solution to problem.. Moreover when f has separable variables Theorem.6 points out some relevant consequences of the main result. As an example of our results a special case of Theorem. is presented here the proof of Theorem. comes in Remar.. Theorem. Let g: R R is a Lipschitz continuous function with Lipschizian constant L = π g =. Assume f : [ ] R R be a non-negative continuous function such that 6 f t sdsdt < π f t sdsdt;. Then for every λ Λ: = π π f t sdsdt π 6 f t sdsdt π f t sdsdt Page of

3 Khaleghi Moghadam Cogent Mathematics 6 : 6 the problem u iv + gu =λf t u u =u = u =u =. has at least one non-trivial generalized solution u such that < u u L [] π gsdsdt <. The rest of this paper is arranged as follows. In Section we recall some basic definitions the main tool Theorem. in Section we provide our main result that contains several theorems finally we illustrate the results by giving several examples as applications of our results.. Preliminaries First we here recall for the reader s convenience Bonanno Theorem. see also Bonanno Proposition. which is our main tool. For a given non-empty set X two functionals Φ Ψ: X R we define the following functions βr r = ρr r = inf v Φ ]r r [ sup Ψu Ψv u Φ ]r r [ r Φv Ψv sup Ψu u Φ sup ] r [ v Φ ]r r [ Φv r.. for all r r R with r < r. Theorem. Bonanno Theorem. Let X be a reflexive real Banach space Φ: X R a sequentially wealy lower semi-continuous coercive continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X Ψ: X R a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Put I λ =Φ λψ assume that there are r r R r < r such that βr r <ρr r. ρr r Then for each λ Λ=] [ there is u βr r λ Φ ]r r [ such that I λ u λ I λ u for all u Φ ]r r [ I u =. λ λ Let us introduce some notations that will be used later. Assume that A B be two real constants such that A max. π B π A π B < π for instance if A B then. holds. Also set A σ: = max π B π A π B π δ: = σ : = δ π 7 9 A + B.. Page of

4 Khaleghi Moghadam Cogent Mathematics 6 : 6 Clearly < <. Assume that the Lipschitsian constant L be such that L <δ π. Let X: = H [ ] H [ ] be the Sobolev space endowed with the usual norm. It is well-nown poincaré-type inequalities see for instance Peletier et al. 99 Lemma. u L [] π u L []. u L [] π u L [].6 for all u X. By..6 one can show that the following norm u X = u x A u x + B ux dx.7 is equivalent to the usual one in particular one has see for instance Proposition. in Bonanno Di Bella u L [] π u L [] π u L [] δπ u X. u πδ u X. Recall that function f :[ ] R R is said to be an L -Carathéodory function if the function t f t x is measurable for every x R the function x f t x is continuous for almost every t [ ] for every ρ > there exists a function l ρ L [ ] such that sup x ρ f t x l ρ t for almost every t [ ]. Put Ft ξ = ξ f t sds Gξ = ξ gsds for each t ξ [ ] R. By a wea solution of. we mean any u X such that.9 [u v Au v + Buv]dt + [gu λf t u]vdt =. for every v X. Moreover a function u:[ ] R is said to be a generalized solution to problem. if u C [ ] u AC[ ] u =u =u =u = u iv + Au + Bu + gu =λf t u for almost every t [ ]. If f g are continuous functions in [ ] R then each generalized solution u is a classical solution. The assumptions on f g imply that a wea solution to problem. is a generalized one see Bonanno & Di Bella Proposition.. Remar. If f t xx < gxx > for every t [ ] x then by.7. one can see that the problem. has only the trivial solution. Remar. If ut be a solution of problem. that the values of the parameters A B satisfy A B the non-linear functions gx ft x satisfy f t ux gut for very t [ ] then using Proposition. of Bonanno Di Bella ut for very t [ ]. Page of

5 Khaleghi Moghadam Cogent Mathematics 6 : 6 Main results We firstly introduce the functionals Φ Ψ: X R as follows Φu = u X Ψu = Ft utdt Gutdt.. for each u X. We need the following lemma in the proof of our main results. Lemma. The functional Φ: X R be a sequentially wealy lower continuous coercive continuously Gâteaux differentiable function on X with Φ u[v] = u v Au v + Buvdt guvdt v X. The functional Ψ: X R be a continuously Gâteaux differentiable function on X with Ψ u[v] = f t uvdt v X such that Φ : X X admits a continuous inverse X Ψ : X X is a compact operator... Proof Put Ju = Gudt for u X let u n u since J is sequentially wealy continuous functional on X sequentially wealy semi-continuity of X one has lim inf Φu =lim inf u n X n n lim Ju n n n u X Ju = Φu; hence Φ is a sequentially wealy lower continuous. To prove coercivity of Φ one has Φu u X L as u π δ X. Indeed by Lipschitz continuity g with the Lipschitsian constant L <δ π.6. one can conclude that Φu u X Gu dt u X We let u v X s ; then by the Mean Value Theorem for integrals u gs ds dt u L X u dt u L X π u u L X π δ u = X u X L as u π δ X. Ju + sv Ju s guvdt Gu + sv Gu guv s dt = gu + sηv gu v dt L s v L as s in which <η<. Hence J uv = guvdt for every v X. Also by a routine argument on the first term of Φu similar argument on the Ψu using Lebesgue Convergent Theorem one can follow.... Page of

6 Khaleghi Moghadam Cogent Mathematics 6 : 6 Now we show that Φ : X X admits a continuous inverse on X. Indeed we need to show that Φ is a Lipschitzian strongly monotone operator i.e. for every u v X there are two positive constants K L such that Φ u Φ v X L u v X.6 Φ u Φ v u v K u v X..7 So by Zeidler 9 Theorem 6.Ad Φ admits a Lipschitzian continuous inverse. By the Hölder s inequality the inequalities in. we have Φ u Φ v X = sup w X sup w X + sup w X in which L >. Also Φ u Φ v w u v w dt A u v w dt + B u v w dt gu gv w dt sup u v L [] w L [] w X + A u v L [] w + B u v w L [] L [] L [] + L sup u v L w [] L [] sup w X w X δ u v X w L [] + A δπ u v X w L [] + B + L δπ sup u v X w L [] sup w X w X + A δ π u v X w X + B δ π u v X w X δπ u v w X L [] δ u v w X X + L δ π sup u v X w X w X δ u v + A X δ π u v + B X δ π u v + L X δ π u v X u v X δ + A δ π + B δ π + L : = L δ π u v X Φ u Φ v u v u v L u X v L [] L u v δ π : = K u X v X where K > Because of L <δ π. It is clear that Ψ is a compact operator. Fourth-order differential equations such as. arise in the study of deflections of elastic beams on non-linear elastic foundations in engineering physical sciences. Therefore for its importance contribution to its area we consider the following theorem as a main result of this paper. Theorem. Let δ L be the real constants as defined above. Assume that there exist three non-negative constants c c d such that c = d c = d δ π +L δ π L [ ] [ ] i Ft ξ t ξ [ d ] Page 6 of

7 Khaleghi Moghadam Cogent Mathematics 6 : 6 ii max ξ c Ft ξdt + max ξ c Ft ξdt < Then for any λ Ft ddt. the problem. has at one least non-trivial generalized solution u X such that Proof Our goal is to use Theorem. to our problem. To this end tae Φ Ψ as given in.. respectively. From Lemma. we observe that the regularity assumptions on Φ Ψ are verified. Put Clearly v X; moreover it is easy to verify that v = A + B X d = δ π d 7 9 Also by similar argument one has Put r = d δ π L π d δ π L π Ft ddt max ξ c Ft ξdt d δ π L < π u X vt = 6 9 d t t t [ ] d t ] ] 6 d t t + t ] ]. 9 Φ v v X + δ π d δ π d + L + = d δ π + L π Φ v v X δ π d δ π d G vdt = c δ π L r π = d δ π +L π d δ π L = c δ π L π ; hence r < Φ v < r ; thus Taing.9. into account when Φu < r i i = one has max t [] ut c i i = ; hence π max ξ c Ft ξdt Gu dt < d δ π + L. π δ π d v dt δ π d L π δ v X = d δ π L π r < Φ v < r sup u Φ r i Ψu = G vdt δ π d + v + L π v v gs ds dt gs ds dt L v dt δ π d L π v L π δ v X sup r Φ v < π d δ π L <. Φ v r u Φ r i Ft utdt max Ft ξdt i =. ξ c i Ft ddt Page 7 of

8 Khaleghi Moghadam Cogent Mathematics 6 : 6 On the other h from i one has Ψ v = Ft vtdt Thus βr r Also by arguing before one has Ft vtdt = sup u Φ r r Ψu Ψ v r Φ v Taing ii into account we get βr r <ρr r. Ft ddt. max ξ c Ft ξdt Ft ddt r Φ v π < max Ft ξdt d δ π L ξ c Ψ v sup Ψu u Φ r ρr r Φ v r Ft ddt max ξ c Ft ξdt Φ v r π > Ft ddt d δ π L max Ft ξdt ξ c Ft ddt.. Hence Theorem. follows that for each λ d δ π L π Ft ddt max ξ c Ft ξdt d δ π L π max ξ c Ft ξdt the problem. has at least one non-trivial generalized solution u X such that r < u Gu dt < r. X Ft ddt As a simple consequence of Theorem. we point out the following corollary. Corollary. Let δ L be real constants as defined above. Assume that there exist two nonnegative constants d c with c = d δ π +L such that δ π L [ ] [ ] i Ft ξ t ξ [ d ] ii max Ft ξdt < δ π +L ξ c Ft ddt Then for every δ π +L λ Λ: = d δ π + L π Ft ddt d δ π L π max Ft ξdξ ξ c the problem. has at least one non-trivial generalized solution u such that < u X Gu dt < d δ π +L π. Ft ddt Page of

9 Khaleghi Moghadam Cogent Mathematics 6 : 6 In Corollary. taing Theorem. into account it is enough to tae c = c = c. Remar. Theorem. follows immediately from Theorem. taing into account that A = B = d = π. Example. Let g:r R be a Lipschitz continuous function with Lipschizian constant L = π g = f t s =f tf s where f s = n mins n n + s if s n= [n n + ] if s ] [ if t < f t = if t < if t. by simple calculations we obtain 6 6 f t sdsdt = f tdt f sds = f tdt π = f tdt = f t sdsdt = n= 6 = = f n+ tdt f tdt f tdt = + n= n π π f sds + n+ f sds π n= n n+ 6 f sds π f sds + f sds f sds = + + sds 6. = + π. Thus we ensure the inequality. holds. Hence owing to Theorem. for every λ Λ= π π where Λ ]79 679[ the problem u iv + gu =λf tf u; in [ ] u =u =; u =u = 6 π π Page 9 of

10 Khaleghi Moghadam Cogent Mathematics 6 : 6 has at least one non-trivial generalized solution u that < u X Gu π dt <. We now point out a consequence of Corollary. in which the function ft u has separable variables. Theorem.6 Let f :[ ] R be a non-negative non-zero essentially bounded function f :R R be a non-negative continuous function F ξ = ξ f xdx for every ξ R. Assume that there exists a positive constant d such that F d > C f F c where c = d δ π +L δ π L δ π + L f L C f = [] ; δ π + L f tdt then for every λ d δ π + L π F d f tdt the problem d δ π L π F c f F d L [] f tdt u iv + Au + Bu + gu =λf tf u u =u = u =u =. has at least one non-trivial generalized solution u such that < u X Gu dt < d δ π +L π. Finally we present an example to illustrate the results of Theorem.6. Example.7 Let A = 9 B =. Clearly δ = π 9.97 = π 9.9. Put L = π f s = f t = n mins n n + s if s n= [n n + ] if s ] [ if t < if t < if t 9π π + C f = 9π π + Put d =. By simple calculations we obtain c = F d = f sds = n= n 7. F c = f sds =. n+ + f sds =.7 n= n+ = π 9π + π 9[π 9π ] 7. Page of

11 Khaleghi Moghadam Cogent Mathematics 6 : 6 Hence owing to Theorem.6 for every λ ]9 76[ Λ d the problem u iv + 9u + u + gu =λf tf u; in [ ] u =u =; u =u = has at least one non-trivial generalized solution u that < u Gu X dt < 69.. Conclusion In the paper the author has established the existence of a non-trivial solution for a class of fourthorder elastic beam equations involving Lipschitz non-linearity with Navier boundary value condition. Using a critical points theorem the author has ensured the exact collections of the parameters in which the problem possesses at least a non-trivial solution on one-dimensional space. On N- dimensional spaces N > the same discussions can be used for future ideas based on the results of this paper. Also the existence of three solutions or infinitely many solutions can be used for future ideas based on the other tool theorems. Acnowledgements The author express his gratitude to referees for their useful suggestions. Funding This wor was supported by Sari agricultural sciences natural resources university [grant number -9-]. Author details Mohsen Khaleghi Moghadam s: mohsen.haleghi@rocetmail.com m.haleghi@ sanru.ac.ir Department of Basic Sciences Sari Agricultural Sciences Natural Resources University 7 Sari Iran. Citation information Cite this article as: Existence of a non-trivial solution for fourth-order elastic beam equations involving Lipschitz non-linearity Mohsen Khaleghi Moghadam Cogent Mathematics 6 : 6. References Afrouzi G. A. Heidarhani S. & O Regan D.. Existence of three solutions for a doubly eigenvalue fourthorder boundary value problem. Taiwanese Journal of Mathematics. Bai Z.. Positive solutions of some nonlocal fourthorder boundary value problem. Applied Mathematics Computation Bai Z. & Wang H.. On positive solutions of some nonlinear fourth-order beam equations. Journal of Mathematical Analysis Applications Bonanno G.. A critical point theorem via the Eel variational principle. Nonlinear Analysis Bonanno G. & Di Bella B.. A boundary value problem for fourth-order elastic beam equations. Journal of Mathematical Analysis Applications Bonanno G. & Di Bella B.. A fourth-order boundary value problem for a Sturm-Liouville type equation. Applied Mathematics Computation Bonanno G. Di Bella B. & O Regan D.. Non-trivial solutions for nonlinear fourth-order elastic beam equations. Computers Mathematics with Applications Busuioc A. & Ratiu T.. The second grade fluid averaged Euler equations with Navier-slip boundary conditions. Nonlinearity Cabada A. Cid J. A. & Sanchez L. 7. Positivity lower upper solutions for fourth order boundary value problems. Nonlinear Analysis Chai G. 7. Existence of positive solutions for fourthorder boundary value problem with variable parameters. Nonlinear Analysis Grossinho M. R. Sanchez L. & Tersian S. A.. On the solvability of a boundary value problem for a fourth-order ordinary differential equation. Applied Mathematics Letters 9. Gyulov T. & Morosanu G.. On a class of boundary value problems involving the p-biharmonic operator. Journal of Mathematical Analysis Applications Han G. & Xu Z. 7. Multiple solutions of some nonlinear fourth-order beam equations. Nonlinear Analysis Heidarhani S. a. Existence of solutions for a twopoint boundary-value problem of a fourth-order Sturm-Liouvillie type. Electronic Journal of Differential Equations. Heidarhani S. b. Non-trivial solutions for a class of p_... p_n-biharmonic systems with Navier boundary conditions. Annales Polonici Mathematici Heidarhani S.. Existence of non-trivial solutions for systems of n fourth order partial differential equations. Mathematica Slovaca Heidarhani S. Afrouzi G. A. Ferrara M. & Moradi S. 6. Variational approaches to impulsive elastic beam equations of Kirchhoff type. Complex Variables Elliptic Equations. doi:./ Heidarhani S. Ferrara M. & Khademloo S. 6. Nontrivial solutions for one-dimensional fourth-order Kirchhoff-type equations. Mediterranean Journal of Mathematics 7 6. Heidarhani S. Ferrara M. Salari A. & Caristi G. 6. Multiplicity results for px-biharmonic equations with Navier boundary. Complex Variables Elliptic Equations Khaleghi Moghadam M. & Heidarhani S.. Existence of a Non-trivial solution for nonlinear difference equations. Differential Equations & Applications 6 7. Page of

12 Khaleghi Moghadam Cogent Mathematics 6 : 6 Khalhali S. M. Heidarhani S. & Razani A.. Infinitely many solutions for a fourth-order boundary-value problem. Electronic Journal of Differential Equations 6. Li Y. 7. On the existence of positive solutions for the bending elastic beam equations. Applied Mathematics Computation 9 7. Liu X.-L. & Li W.-T. 7a. Existence multiplicity of solutions for fourth-order boundary values problems with three parameters. Mathematical Computer Modelling 6. Liu X.-L. & Li W.-T. 7b. Existence multiplicity of solutions for fourth-order boundary values problems with parameters. Journal of Mathematical Analysis Applications Peletier L. A. Troy W. C. & Van der Vorst R. C. A. M. 99. Stationary solutions of a fourth order nonlinear diffusion equation. V. V. Kurt Trans.. Differentsialnye Uravneniya Differential Equations. Zeidler E. 9. Nonlinear functional analysis its applications Vol. II/B. New Yor NY: Berlin-Heidelberg. 6 The Authors. This open access article is distributed under a Creative Commons Attribution CC-BY. license. You are free to: Share copy redistribute the material in any medium or format Adapt remix transform build upon the material for any purpose even commercially. The licensor cannot revoe these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit provide a lin to the license indicate if changes were made. You may do so in any reasonable manner but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Cogent Mathematics ISSN: - is published by Cogent OA part of Taylor & Francis Group. Publishing with Cogent OA ensures: Immediate universal access to your article on publication High visibility discoverability via the Cogent OA website as well as Taylor & Francis Online Download citation statistics for your article Rapid online publication Input from dialog with expert editors editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts waivers for authors in developing regions Submit your manuscript to a Cogent OA journal at Page of