### Existence of Positive Solutions for a Pertubed Fourth-Order Equation

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**Authors:**M. R. H. TAVANI AND A. NAZARI

**DOI:**10.46793/KgJMat2104.623H

**Abstract:**

In this paper, a special type of fourth-order diﬀerential equations with a perturbed nonlinear term and some boundary conditions is considered which is very important in mechanical engineering. Therefore, the existence of a non-trivial solution for such equations is very important. Our goal is to ensure at least three weak solutions for a class of perturbed fourth-order problems by applying certain conditions to the functions that are available in the diﬀerential equation (problem (??)). Our approach is based on variational methods and critical point theory. In fact, using a fundamental theorem that is attributed to Bonanno, we get some important results. Finally, for some results, an example is presented.

**Keywords:**

Fourth-order equation, weak solution, critical point theory, variational methods.

**References:**

[1] G. Bonanno and P. Candito, Non-diﬀerentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Diﬀerential Equations 244 (2008), 3031–3059.

[2] G. Bonanno,A. Chinnì and S. Tersian, Existence results for a two point boundary value problem involving a fourth-order equation, Electron. J. Qual. Theory Diﬀer. Equ. 33 (2015), 1–9.

[3] G. Bonanno and S. A. Marano, On the structure of the critical set of non-diﬀerentiable functions with a weak compactness condition, Appl. Anal. 89 (2010), 1–10.

[4] A. Cabada and S. Tersian, Multiplicity of solutions of a two point boundary value problem for a fourth-order equation, Appl. Math. Comput. 24 (2011), 1599–1603.

[5] M. R. Grossinho and St. A. Tersian, The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation, Nonlinear Anal. 41(2000), 417–431.

[6] M. R. Heidari Tavani, Existence results for a perturbed fourth-order equation, J. Indones. Math. Soc. 23 (2017), 55–65.

[7] T. F. Ma, Positive solutions for a beam equation on a nonlinear elastic foundation, Math. Comput. Model. 39 (2004), 1195–1201.

[8] T. F. Ma and J. da Silva, Iterative solutions for a beam equation with nonlinear boundary conditions of third order, Appl. Math. Comput. 159 (2004), 11–18.

[9] S. Timoshenko, W. Weaver, Jr and D. H. Young, Vibrations Problems in Engineering, 5th Edition, John Wiley and Sons, New York, 1990.

[10] L. Yang, H. Chen and X. Yang, The multiplicity of solutions for fourth-order equations generated from a boundary condition, Appl. Math. Lett. 24 (2011), 1599–1603.