Existence and location of solutions to fourth-order Lidstone coupled systems with dependence on odd derivatives
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Tusi Mathematical Research Group
ORIGINAL PAPER
Existence and location of solutions to fourth-order Lidstone coupled systems with dependence on odd derivatives Robert de Sousa1,2
•
Feliz Minho´s2,3
Received: 22 July 2020 / Accepted: 26 August 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract This paper addresses the existence and location results for coupled system with two fourth-order differential equations with dependence on all derivatives in nonlinearities and subject to Lidstone-type boundary conditions. To guarantee the existence and location of the solutions, we applied lower and upper solutions technique and degree theory. In this context, we highlight a new type of Nagumo condition to control the growth of the third derivatives and increases the number of applications, as well as a new type of definitions of upper and lower solutions for such coupled systems. Last section contains an application to a coupled system composed by two fourth order equations, which models the estimated bending of simply-supported beam with torsional solitons. Keywords Coupled nonlinear systems coupled lower and upper solutions Lidstone-type boundary value problems operator theory simply supported beams
Mathematics Subject Classification 34A34 34B10 34B15 37K40 47N20
Communicated by Julio Rossi. & Robert de Sousa [email protected] Feliz Minho´s [email protected] 1
Faculdade de Cieˆncias e Tecnologia, Nu´cleo de Matema´tica e Aplicac¸o˜es (NUMAT), Universidade de Cabo Verde, Campus de Palmarejo, 279 Praia, Cabo Verde
2
Centro de Investigac¸a˜o em Matema´tica e Aplicac¸o˜es (CIMA), Instituto de Investigac¸a˜o e Formac¸a˜o Avanc¸ada, Universidade de E´vora, Rua Roma˜o Ramalho, 59, 7000-671 E´vora, Portugal
3
Departamento de Matema´tica, Escola de Cieˆncias e Tecnologia, Centro de Investigac¸a˜o em Matema´tica e Aplicac¸o˜es (CIMA), Instituto de Investigac¸a˜o e Formac¸a˜o Avanc¸ada, Universidade de E´vora, Rua Roma˜o Ramalho, 59, 7000-671 E´vora, Portugal
R. de Sousa and F. Minhós
1 Introduction In this paper, we consider a fourth order coupled system composed by the nonlinear fully differential equations ( uð4Þ ðtÞ ¼ f ðt; uðtÞ; vðtÞ; u0 ðtÞ; v0 ðtÞ; u00 ðtÞ; v00 ðtÞ; u000 ðtÞ; v000 ðtÞÞ; t 20; 1½ ð1Þ vð4Þ ðtÞ ¼ hðt; uðtÞ; vðtÞ; u0 ðtÞ; v0 ðtÞ; u00 ðtÞ; v00 ðtÞ; u000 ðtÞ; v000 ðtÞÞ; t 20; 1½; where f ; h : ½0; 1 R8 ! R are continuous functions, with the Lidstone boundary conditions uð0Þ ¼ uð1Þ ¼ A1 ; u00 ð0Þ ¼ A2 ; u00 ð1Þ ¼ A3 ð2Þ vð0Þ ¼ vð1Þ ¼ B1 ; v00 ð0Þ ¼ B2 ; v00 ð1Þ ¼ B3 ; and Ai ; Bi 2 R, for i ¼ 1; 2; 3. These type of fourth-order equations are usually associate with beam theory, where the boundary conditions describe the different beam behaviors at the endpoints. In this context, fourth-order differential systems or equations are very relevant in engineering, especially in beam theory, electric circuits, elasticity theory, suspension bridges, and viscoelastic and inelastic flows, (see [2, 15, 19, 22, 27, 33] and
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