Existence of a non-trivial solution for fourth-order elastic beam equations involving Lipschitz non-linearity

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.

Keywords:

• fourth-order equations
• critical point
• non-trivial solution

AMS Subject Classifications:

• 35J35
• 34B15
• 58E05

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, and has presented two examples to illustrate the results.

1. Introduction

The aim of this paper is to study the existence of non-trivial solution for the following boundary value problem:(1.1) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{(iv)}+A{u}^{\prime }+Bu+g(u)=\mathit{\lambda }f(t,u),t\in [0,1],\hfill \\ u(0)=u(1)=0,\hfill \\ u^{″}(0)=u^{″}(1)=0,\hfill \end{array}\right.\end{array}$$(1.1)

where A, B are real constants, $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is an $L^1$-Carathéodory function, $g:\mathbb{R}\to \mathbb{R}$ is a Lipschitz continuous function with Lipschitzian constant $L\ge 0$ and $g(0)=0$, $\mathit{\lambda }>0$ is a parameter.

In recent years, the fourth-order boundary value problems have been extensively considered by many authors, for instance, see Bai and Wang (2002), Cabada, Cid, and Sanchez (2007), Grossinho, Sanchez, and Tersian (2005), Liu and Li (2007a), Peletier, Troy, and Van der Vorst (1995) and references therein. These kind of problems arising in real-world phenomena play a fundamental role in different fields of research, such as mechanical engineering, control systems, economics, computer science, physics, biology and 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, 2007b), lower and upper solutions method (Cabada et al., 2007), critical point theory (Bonanno, 2012), Morse theory (Han & Xu, 2007) and mountain-pass theorem (Gyulov & Morosanu, 2010) that for more references, we refer the reader to Afrouzi, Heidarkhani and O’Regan (2011), Bai (2010), Bonanno and Di Bella (2008), Bonanno and Di Bell (2010), Bonanno, Di Bella and O’Regan (2011), Chai (2007), Li (2007) and references therein. In Khalkhali, Heidarkhani and Razani (2012), (1.1) have studied the existence of infinitely many solutions. The authors in Heidarkhani, Ferrara, Salari and Caristi (2016) established the existence and multiplicity results by variational methods and critical point theory for the following fourth-order Navier boundary value problem(1.2) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathrm{\Delta }}_{p(x)}^{2}u=\mathit{\lambda }f(t,u),\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=\mathrm{\Delta }u=0,\hfill & \text{on}\mathit{\partial }\mathrm{\Omega },\hfill \end{array}\right.\end{array}$$(1.2)

in a non-empty bounded open set $\mathrm{\Omega }\subset {\mathbb{R}}^N(N>1)$ with a sufficient smooth boundary $\mathit{\partial }\mathrm{\Omega }$ where, ${\mathrm{\Delta }}_{p(x)}^{2}u={\mathrm{\Delta }(|\mathrm{\Delta }u|}^{p(x)-2}\mathrm{\Delta }u)$ is the p(x)-biharmonic operator of fourth order with $p(·)\in C^0(\mathrm{\Omega })$ and $\mathit{\lambda }\in [0,\infty )$ and $f:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ is an $L^2$-Carathéodory function.

There are so many equations in engineering, physics and mathematics that are studied through Navier boundary conditions. For the use of Navier boundary conditions in the mathematical literature, see Busuioc and Ratiu (2003) and the related references.

Very recently, some researchers have studied the existence and multiplicity of solutions for impulsive fourth-order elastic beam equation; we refer the reader to Heidarkhani, Afrouzi, Ferrara and Moradi (2016), Heidarkhani, Ferrara and Khademloo (2016), and references therein.

In the present paper, based on a local minimum theorem (Theorem 2.1) due to Bonanno (2012), we ensure an exact interval of parameters, in which the problem (1.1) admits at least a non-trivial solution. We also refer the interested reader to the papers (Bonanno, Di Bella, & O’Regan, 2011; Heidarkhani, 2012a; Heidarkhani, 2014; Heidarkhani, 2012b,2016; Khaleghi Moghadam & Heidarkhani, 2014) in which Theorem 2.1 has been successfully employed to the existence of at least one non-trivial solution for some boundary value problems.

Our main result, Theorem 3.2, and its consequence, Theorem 3.3, ensure the existence of a non-trivial solution to problem (1.1). Moreover, when f has separable variables, Theorem 3.6 points out some relevant consequences of the main result.

As an example of our results, a special case of Theorem 3.2 is presented here (the proof of Theorem 1.1 comes in Remark 3.4).

Theorem 1.1

Let $g:\mathbb{R}\to \mathbb{R}$ is a Lipschitz continuous function with Lipschizian constant $L=\frac{{\mathit{\pi }}^4}{5}$ and $g(0)=0$. Assume $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ be a non-negative and continuous function, such that(1.3) $$\begin{array}{c}\hfill {\int }_0^1{\int }_0^{16}f(t,s)dsdt<\frac{4}{3}{\int }_{\frac{3}{8}}^{\frac{5}{8}}{\int }_0^{\frac{3\mathit{\pi }}{2}}f(t,s)dsdt;\end{array}$$(1.3)

Then, for every$$\mathit{\lambda }\in \mathrm{\Lambda }:=\left]\frac{2048{\mathit{\pi }}^2}{5{\int }_{\frac{3}{8}}^{\frac{5}{8}}{\int }_0^{\frac{3\mathit{\pi }}{2}}f(t,s)dsdt},\frac{2048{\mathit{\pi }}^2}{15\left({\int }_0^1{\int }_0^{16}f(t,s)dsdt-{\int }_{\frac{3}{8}}^{\frac{5}{8}}{\int }_0^{\frac{3\mathit{\pi }}{2}}f(t,s)dsdt\right)}\right[,$$

the problem(1.4) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{(iv)}+g(u)=\mathit{\lambda }f(t,u),\hfill \\ u(0)=u(1)=0,\hfill \\ u^{″}(0)=u^{″}(1)=0,\hfill \end{array}\right.\end{array}$$(1.4)

has at least one non-trivial generalized solution $u_0$ such that $0<\frac{1}{2}{\Vert u_0^{″}\Vert }_{L^2([0,1])}^2-{\int }_0^1{\int }_0^{u_0}g(s)dsdt<\frac{2048{\mathit{\pi }}^2}{5}$.

The rest of this paper is arranged as follows. In Section 2, we recall some basic definitions and the main tool (Theorem 2.1) and in Section 3, we provide our main result that contains several theorems and finally, we illustrate the results by giving several examples as applications of our results.

2. Preliminaries

First, we here recall for the reader’s convenience (Bonanno, 2012, Theorem 5.1) (see also Bonanno, 2012, Proposition 2.1) which is our main tool. For a given non-empty set X, and two functionals $\mathrm{\Phi },\mathrm{\Psi }:X\to \mathbb{R}$, we define the following functions(2.1) $$\begin{array}{cc}\hfill \mathit{\beta }(r_1,r_2)& =\underset{v\in {\mathrm{\Phi }}^{-1}(]r_1,r_2[)}{inf}\frac{{sup}_{u\in {\mathrm{\Phi }}^{-1}(]r_1,r_2[)}\mathrm{\Psi }(u)-\mathrm{\Psi }(v)}{r_2-\mathrm{\Phi }(v)},\hfill \end{array}$$(2.1) (2.2) $$\begin{array}{cc}\hfill \mathit{\rho }(r_1,r_2)& =\underset{v\in {\mathrm{\Phi }}^{-1}(]r_1,r_2[)}{sup}\frac{\mathrm{\Psi }(v)-{sup}_{u\in {\mathrm{\Phi }}^{-1}(]-\infty ,r_1[)}\mathrm{\Psi }(u)}{\mathrm{\Phi }(v)-r_1},\hfill \end{array}$$(2.2)

for all $r_1,r_2\in \mathbb{R}$, with $r_1<r_2$.

Theorem 2.1

   Bonanno (2012, Theorem 5.1) Let X be a reflexive real Banach space, $\mathrm{\Phi }:X\to \mathbb{R}$ a sequentially weakly lower semi-continuous coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on $X^{\ast }$ and $\mathrm{\Psi }:X\to \mathbb{R}$ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Put $I_{\mathit{\lambda }}=\mathrm{\Phi }-\mathit{\lambda }\mathrm{\Psi }$ and assume that there are $r_1,r_2\in \mathbb{R}$, $r_1<r_2$, such that$$\mathit{\beta }(r_1,r_2)<\mathit{\rho }(r_1,r_2).$$

Then, for each $\mathit{\lambda }\in \mathrm{\Lambda }=]\frac{1}{\mathit{\rho }(r_1,r_2)},\frac{1}{\mathit{\beta }(r_1,r_2)}[$ there is $u_{0,\mathit{\lambda }}\in {\mathrm{\Phi }}^{-1}(]r_1,r_2[)$ such that $I_{\mathit{\lambda }}(u_{0,\mathit{\lambda }})\le I_{\mathit{\lambda }}(u)$ for all $u\in {\mathrm{\Phi }}^{-1}(]r_1,r_2[)$ and $I_{\mathit{\lambda }}^{\prime }(u_{0,\mathit{\lambda }})=0$.

Let us introduce some notations that will be used later. Assume that A and B be two real constants such that(2.3) $$\begin{array}{c}\hfill max\left\{\frac{A}{{\mathit{\pi }}^2},\frac{-B}{{\mathit{\pi }}^4},\frac{A}{{\mathit{\pi }}^2}-\frac{B}{{\mathit{\pi }}^4}\right\}<1,\end{array}$$(2.3) (for instance, if $A\le 0$ and $B\ge 0$, then (2.3) holds). Also, set$$\mathit{\sigma }:=max\left\{\frac{A}{{\mathit{\pi }}^2},\frac{-B}{{\mathit{\pi }}^4},\frac{A}{{\mathit{\pi }}^2}-\frac{B}{{\mathit{\pi }}^4},0\right\},$$$$\mathit{\delta }:=\sqrt{1-\mathit{\sigma }},$$

and(2.4) $$\begin{array}{c}\hfill k:=2{\mathit{\delta }}^2{\mathit{\pi }}^2{\left(\frac{2048}{27}-\frac{32}{9}A+\frac{13}{40}B\right)}^{-1}.\end{array}$$(2.4)

Clearly, $0<k<\frac{1}{2}$. Assume that the Lipschitsian constant L be such that $L<{\mathit{\delta }}^2{\mathit{\pi }}^4$. Let $X:=H^2([0,1])\cap H_0^1([0,1])$ be the Sobolev space endowed with the usual norm.

It is well-known poincaré-type inequalities (see, for instance, Peletier et al. (1995, Lemma 2.3))(2.5) $$\begin{array}{c}\hfill \Vert u^{\prime }{\Vert }_{L^2([0,1])}\le \frac{1}{\mathit{\pi }}{\Vert u^{″}\Vert }_{L^2([0,1])},\end{array}$$(2.5)

and(2.6) $$\begin{array}{c}\hfill {\Vert u\Vert }_{L^2([0,1])}\le \frac{1}{{\mathit{\pi }}^2}{\Vert u^{″}\Vert }_{L^2([0,1])},\end{array}$$(2.6)

for all $u\in X$. By (2.3)–(2.6), one can show that the following norm(2.7) $$\begin{array}{c}\hfill {\Vert u\Vert }_X={\left({\int }_0^1|u^{″}{(x)|}^2-A|u^{\prime }{(x)|}^2+B{|u(x)|}^{2}dx\right)}^{\frac{1}{2}},\end{array}$$(2.7)

is equivalent to the usual one and in particular one has (see, for instance Proposition 2.1 in Bonanno and Di Bella (2008))(2.8) $$\begin{array}{c}\hfill {\Vert u\Vert }_{L^2([0,1])}\le \frac{1}{\mathit{\pi }}\Vert u^{\prime }{\Vert }_{L^2([0,1])}\le \frac{1}{{\mathit{\pi }}^2}\Vert u^{″}{\Vert }_{L^2([0,1])}\le \frac{1}{\mathit{\delta }{\mathit{\pi }}^2}{\Vert u\Vert }_X,\end{array}$$(2.8)

and(2.9) $$\begin{array}{c}\hfill {\Vert u\Vert }_{\infty }\le \frac{1}{2\mathit{\pi }\mathit{\delta }}{\Vert u\Vert }_X.\end{array}$$(2.9)

Recall that function $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is said to be an $L^1$-Carathéodory function if the function $t\to f(t,x)$ is measurable for every $x\in \mathbb{R}$ and the function $x\to f(t,x)$ is continuous for almost every $t\in [0,1]$ and for every $\mathit{\rho }>0$, there exists a function $l_{\mathit{\rho }}\in L^1([0,1])$ such that $su{p}_{|x|\le \mathit{\rho }}|f(t,x)|\le l_{\mathit{\rho }}(t)$ for almost every $t\in [0,1]$.

Put $F(t,\mathit{\xi })={\int }_0^{\mathit{\xi }}f(t,s)ds$ and $G(\mathit{\xi })={\int }_0^{\mathit{\xi }}g(s)ds$ for each $(t,\mathit{\xi })\in [0,1]\times \mathbb{R}$. By a weak solution of (1.1) we mean any $u\in X$ such that(2.10) $$\begin{array}{c}\hfill {\int }_0^1[u^{″}{v}^{″}-A{u}^{\prime }v\prime +Buv]dt+{\int }_0^1[g(u)-\mathit{\lambda }f(t,u)]vdt=0,\end{array}$$(2.10)

for every $v\in X$. Moreover, a function $u:[0,1]\to \mathbb{R}$ is said to be a generalized solution to problem (1.1), if $u\in C^3[0,1]$, $u^{″\prime }\in AC([0,1])$, $u(0)=u(1)=u^{″}(0)=u^{″}(1)=0$ and $u^{(iv)}+A{u}^{″}+Bu+g(u)=\mathit{\lambda }f(t,u)$ for almost every $t\in [0,1]$. If f and g are continuous functions in $[0,1]\times \mathbb{R}$, then each generalized solution u is a classical solution. The assumptions on f and g imply that a weak solution to problem (1.1) is a generalized one (see Bonanno & Di Bella, 2008, Proposition 2.2).

Remark 2.2

If $f(t,x)x<0$ and $g(x)x>0$ for every $t\in [0,1]$ and $x\ne 0$, then by (2.7) and (2.10), one can see that the problem (1.1) has only the trivial solution.

Remark 2.3

If u(t) be a solution of problem (1.1) that the values of the parameters A and B satisfy $A^2-4B\ge 0$ and the non-linear functions g(x) and f(t, x) satisfy $f(t,u(x))\ge g(u(t))$ for very $t\in [0,1]$, then using Proposition 2.3 of Bonanno and Di Bella (2008), $u(t)\ge 0$ for very $t\in [0,1]$.

3. Main results

We firstly introduce the functionals $\mathrm{\Phi },\mathrm{\Psi }:X\to \mathbb{R}$ as follows(3.1) $$\begin{array}{c}\hfill \mathrm{\Phi }(u)=\frac{1}{2}{\Vert u\Vert }_X^2-{\int }_0^{1}G(u(t))dt,\end{array}$$(3.1)

and(3.2) $$\begin{array}{c}\hfill \mathrm{\Psi }(u)={\int }_0^{1}F(t,u(t))dt,\end{array}$$(3.2)

for each $u\in X$. We need the following lemma in the proof of our main results.

Lemma 3.1

The functional $\mathrm{\Phi }:X\to \mathbb{R}$ be a sequentially weakly lower continuous and coercive and continuously Gâteaux differentiable function on X with(3.3) $$\begin{array}{c}\hfill {\mathrm{\Phi }}^{\prime }(u)[v]={\int }_0^1{u}^{″}{v}^{″}-A{u}^{\prime }v\prime +Buvdt-{\int }_0^{1}g(u)vdt,\forall v\in X.\end{array}$$(3.3)

The functional $\mathrm{\Psi }:X\to \mathbb{R}$ be a continuously Gâteaux differentiable function on X with(3.4) $$\begin{array}{c}\hfill {\mathrm{\Psi }}^{\prime }(u)[v]={\int }_0^{1}f(t,u)vdt,\forall v\in X,\end{array}$$(3.4)

such that ${\mathrm{\Phi }}^{\prime }:X\to X^{\ast }$ admits a continuous inverse $X^{\ast }$ and ${\mathrm{\Psi }}^{\prime }:X\to X^{\ast }$ is a compact operator.

Proof

Put $J(u)={\int }_0^{1}G(u)dt$ for $u\in X$ and let $u_n\rightharpoonup u$ since J is sequentially weakly continuous functional on X and sequentially weakly semi-continuity of ${\Vert ·\Vert }_X$, one has$$\underset{n\to \infty }{lim}inf\mathrm{\Phi }(u_n)=\underset{n\to \infty }{lim}inf\frac{\Vert u_n{\Vert }_X^2}{2}-\underset{n\to \infty }{lim}J(u_n)\ge \frac{1}{2}{\Vert u\Vert }_X^2-J(u)=\mathrm{\Phi }(u);$$

hence, $\mathrm{\Phi }$ is a sequentially weakly lower continuous. To prove coercivity of $\mathrm{\Phi }$, one has(3.5) $$\begin{array}{c}\hfill \mathrm{\Phi }(u)\ge \frac{1}{2}{\Vert u\Vert }_X^2\left(1-\frac{L}{{\mathit{\pi }}^4{\mathit{\delta }}^2}\right)\to \infty ,\text{as}{\Vert u\Vert }_X\to \infty .\end{array}$$(3.5)

Indeed, by Lipschitz continuity g, with the Lipschitsian constant $L<{\mathit{\delta }}^2{\mathit{\pi }}^4$, (2.6) and (2.8), one can conclude that$$\begin{array}{cc}\hfill \mathrm{\Phi }(u)& \ge \frac{1}{2}{\Vert u\Vert }_X^2-{\int }_0^1|G(u)|dt\ge \frac{1}{2}{\Vert u\Vert }_X^2-{\int }_0^1\left({\int }_0^u|g(s)|ds\right)dt\hfill \\ \hfill & \ge \frac{1}{2}{\Vert u\Vert }_X^2-\frac{L}{2}{\int }_0^1{|u|}^{2}dt\ge \frac{1}{2}{\Vert u\Vert }_X^2-\frac{L}{2{\mathit{\pi }}^4}{\Vert u^{″}\Vert }_2^2\hfill \\ \hfill & \ge \frac{1}{2}{\Vert u\Vert }_X^2-\frac{L}{2{\mathit{\pi }}^4{\mathit{\delta }}^2}{\Vert u\Vert }_X^2=\frac{1}{2}{\Vert u\Vert }_X^2\left(1-\frac{L}{{\mathit{\pi }}^4{\mathit{\delta }}^2}\right)\to \infty ,\text{as}{\Vert u\Vert }_X\to \infty .\hfill \end{array}$$

We let $u,v\in X$ and $s\ne 0$; then, by the Mean Value Theorem for integrals$$\begin{array}{cc}\hfill \left|\frac{J(u+sv)-J(u)}{s}-{\int }_0^{1}g(u)vdt\right|& \le {\int }_0^1\left|\frac{G(u+sv)-G(u)}{s}-g(u)v\right|dt\hfill \\ \hfill & ={\int }_0^1\left|g(u+s\mathit{\eta }v)-g(u)\right|\left|v\right|dt\hfill \\ \hfill & \le {L|s|\Vert v\Vert }_{L^2}^2\to 0\text{as}s\to 0,\hfill \end{array}$$

in which $0<\mathit{\eta }<1$. Hence, $J^{\prime }(u)(v)={\int }_0^{1}g(u)vdt$ for every $v\in X$. Also, by a routine argument on the first term of $\mathrm{\Phi }(u)$, and similar argument on the $\mathrm{\Psi }(u)$ and using Lebesgue Convergent Theorem, one can follow (3.3) and (3.4).

Now, we show that ${\mathrm{\Phi }}^{\prime }:X\to X^{\ast }$ admits a continuous inverse on $X^{\ast }$. Indeed, we need to show that ${\mathrm{\Phi }}^{\prime }$ is a Lipschitzian and strongly monotone operator, i.e. for every $u,v\in X$, there are two positive constants $K_1$ and $L_1$ such that(3.6) $$\begin{array}{c}\hfill {|{\mathrm{\Phi }}^{\prime }(u)-{\mathrm{\Phi }}^{\prime }(v)|}_{X^{\ast }}\le L_1{\Vert u-v\Vert }_X,\end{array}$$(3.6)

and(3.7) $$\begin{array}{c}\hfill ⟨{\mathrm{\Phi }}^{\prime }(u)-{\mathrm{\Phi }}^{\prime }(v),u-v⟩\ge K_1{\Vert u-v\Vert }_X^2.\end{array}$$(3.7)

So by Zeidler (1985, Theorem 26.A(d)) ${\mathrm{\Phi }}^{\prime }$ admits a Lipschitzian continuous inverse.

By the Hölder’s inequality and the inequalities in (2.8), we have$$\begin{array}{cc}\hfill {\left|{\mathrm{\Phi }}^{\prime }(u)-{\mathrm{\Phi }}^{\prime }(v)\right|}_{X^{\ast }}& =\underset{{\Vert w\Vert }_X\le 1}{sup}\left|⟨{\mathrm{\Phi }}^{\prime }(u)-{\mathrm{\Phi }}^{\prime }(v),w⟩\right|\hfill \\ \hfill & \le \underset{{\Vert w\Vert }_X\le 1}{sup}\left\{{\int }_0^1|u^{″}-v^{″}||w^{″}|dt-A{\int }_0^1|u^{\prime }-v\prime ||w^{\prime }|dt+B{\int }_0^1|u-v||w|dt\right\}\hfill \\ \hfill & +\underset{{\Vert w\Vert }_X\le 1}{sup}\left\{{\int }_0^1|g(u)-g(v)||w|dt\right\}\le \underset{{\Vert w\Vert }_X\le 1}{sup}\left\{\Vert u^{″}-v^{″}{\Vert }_{L^2([0,1])}{\Vert w^{″}\Vert }_{L^2([0,1])}\right.\hfill \\ \hfill & \left.+|A|\Vert u^{\prime }{-v\prime \Vert }_{L^2([0,1])}\Vert w^{\prime }{\Vert }_{L^2([0,1])}{+|B|\Vert u-v\Vert }_{L^2([0,1])}{\Vert w\Vert }_{L^2([0,1])}\right\}\hfill \\ \hfill & +L\underset{{\Vert w\Vert }_X\le 1}{sup}\left\{{\Vert u-v\Vert }_{L^2([0,1])}{\Vert w\Vert }_{L^2([0,1])}\right\}\le \underset{{\Vert w\Vert }_X\le 1}{sup}\left\{\frac{1}{\mathit{\delta }}{\Vert u-v\Vert }_X{\Vert w^{″}\Vert }_{L^2([0,1])}\right.\hfill \\ \hfill & \left.+\frac{|A|}{\mathit{\delta }\mathit{\pi }}{\Vert u-v\Vert }_X\Vert w^{\prime }{\Vert }_{L^2([0,1])}+\frac{|B|}{\mathit{\delta }{\mathit{\pi }}^2}{\Vert u-v\Vert }_X{\Vert w\Vert }_{L^2([0,1])}\right\}\hfill \\ \hfill & +\frac{L}{\mathit{\delta }{\mathit{\pi }}^2}\underset{{\Vert w\Vert }_X\le 1}{sup}\left\{{\Vert u-v\Vert }_X{\Vert w\Vert }_{L^2([0,1])}\right\}\le \underset{{\Vert w\Vert }_X\le 1}{sup}\left\{\frac{1}{{\mathit{\delta }}^2}{\Vert u-v\Vert }_X{\Vert w\Vert }_X\right.\hfill \\ \hfill & \left.+\frac{|A|}{{\mathit{\delta }}^2{\mathit{\pi }}^2}{\Vert u-v\Vert }_X{\Vert w\Vert }_X+\frac{|B|}{{\mathit{\delta }}^2{\mathit{\pi }}^4}{\Vert u-v\Vert }_X{\Vert w\Vert }_X\right\}+\frac{L}{{\mathit{\delta }}^2{\mathit{\pi }}^4}\underset{{\Vert w\Vert }_X\le 1}{sup}\left\{{\Vert u-v\Vert }_X{\Vert w\Vert }_X\right\}\hfill \\ \hfill & \le \frac{1}{{\mathit{\delta }}^2}{\Vert u-v\Vert }_X+\frac{|A|}{{\mathit{\delta }}^2{\mathit{\pi }}^2}{\Vert u-v\Vert }_X+\frac{|B|}{{\mathit{\delta }}^2{\mathit{\pi }}^4}{\Vert u-v\Vert }_X+\frac{L}{{\mathit{\delta }}^2{\mathit{\pi }}^4}{\Vert u-v\Vert }_X,\hfill \\ \hfill & \le {\Vert u-v\Vert }_X\left\{\frac{1}{{\mathit{\delta }}^2}+\frac{|A|}{{\mathit{\delta }}^2{\mathit{\pi }}^2}+\frac{|B|}{{\mathit{\delta }}^2{\mathit{\pi }}^4}+\frac{L}{{\mathit{\delta }}^2{\mathit{\pi }}^4}\right\}:=L_1{\Vert u-v\Vert }_X,\hfill \end{array}$$

in which $L_1>0$. Also$$\begin{array}{cc}\hfill ⟨{\mathrm{\Phi }}^{\prime }(u)-{\mathrm{\Phi }}^{\prime }(v),u-v⟩& \ge {\Vert u-v\Vert }_X^2-L{\Vert u-v\Vert }_{L^2([0,1])}^2\hfill \\ \hfill & \ge \left(1-\frac{L}{{\mathit{\delta }}^2{\mathit{\pi }}^4}\right){\Vert u-v\Vert }_X^2:=K_1{\Vert u-v\Vert }_X^2,\hfill \end{array}$$

where $K_1>0$ (Because of $L<{\mathit{\delta }}^2{\mathit{\pi }}^4$). It is clear that ${\mathrm{\Psi }}^{\prime }$ is a compact operator. $\square$

Fourth-order differential equations such as (1.1) arise in the study of deflections of elastic beams on non-linear elastic foundations in engineering and physical sciences. Therefore, for its importance and contribution to its area, we consider the following theorem as a main result of this paper.

Theorem 3.2

Let k, $\mathit{\delta }$ and L be the real constants as defined above. Assume that there exist three non-negative constants $c_1$, $c_2$ and d such that $c_1=\frac{d}{\sqrt{k}}$, $c_2=\frac{d}{\sqrt{k}}\sqrt{\frac{2({\mathit{\delta }}^2{\mathit{\pi }}^4+L)}{{\mathit{\delta }}^2{\mathit{\pi }}^4-L}}$ and

(i)	$F(t,\mathit{\xi })\ge 0\forall (t,\mathit{\xi })\in \left(\left[0,\frac{3}{8}\right]\cup \left[\frac{5}{8},1\right]\times \left[0,d\right]\right)$
(ii)	${\int }_0^1{max}_{|\mathit{\xi }|\le c_1}F(t,\mathit{\xi })dt+{\int }_0^1{max}_{|\mathit{\xi }|\le c_2}F(t,\mathit{\xi })dt<2{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt.$

Then, for any$$\mathit{\lambda }\in \left]\frac{\frac{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^{2}k}}{{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt-{\int }_0^1{max}_{|\mathit{\xi }|\le c_1}F(t,\mathit{\xi })dt},\frac{\frac{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^{2}k}}{{\int }_0^1{max}_{|\mathit{\xi }|\le c_2}F(t,\mathit{\xi })dt-{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt}\right[,$$

the problem (1.1) has at one least non-trivial generalized solution $u_0\in X$ such that$$\frac{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^{2}k}<\frac{1}{2}{\Vert u_0\Vert }_X^2-{\int }_0^{1}G(u_0)dt<\frac{4d^2({\mathit{\delta }}^2{\mathit{\pi }}^4+L)}{{\mathit{\pi }}^{2}k}.$$

Proof

Our goal is to use Theorem 2.1 to our problem. To this end, take $\mathrm{\Phi }$ and $\mathrm{\Psi }$ as given in (3.1) and (3.2), respectively. From Lemma 3.1, we observe that the regularity assumptions on $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are verified. Put$$\begin{array}{c}\hfill \overline{v}(t)=\left\{\begin{array}{cc}-\frac{64}{9}d\left(t^2-\frac{3}{4}t\right),\hfill & t\in [0,\frac{3}{8}],\hfill \\ d,\hfill & t\in ]\frac{3}{8},\frac{5}{8}],\hfill \\ -\frac{64}{9}d\left(t^2-\frac{5}{4}t+\frac{1}{4}\right),\hfill & t\in ]\frac{5}{8},1].\hfill \end{array}\right.\end{array}$$

Clearly, $\overline{v}\in X$; moreover, it is easy to verify that

${\Vert \overline{v}\Vert }_X^2=2\left(\frac{2048}{27}-\frac{32}{9}A+\frac{13}{40}B\right)d^2=\frac{4{\mathit{\delta }}^2{\mathit{\pi }}^2}{k}{d}^2$ and$$\begin{array}{cc}\hfill \mathrm{\Phi }(\overline{v})& \le \frac{1}{2}{\Vert \overline{v}\Vert }_X^2+\left|{\int }_0^{1}G(\overline{v})dt\right|\le \frac{2{\mathit{\delta }}^2{\mathit{\pi }}^2{d}^2}{k}+{\int }_0^1\left({\int }_0^{\overline{v}}|g(s)|ds\right)dt\hfill \\ \hfill & \le \frac{2{\mathit{\delta }}^2{\mathit{\pi }}^2{d}^2}{k}+\frac{L}{2}{\int }_0^1{|\overline{v}|}^{2}dt\le \frac{2{\mathit{\delta }}^2{\mathit{\pi }}^2{d}^2}{k}+\frac{L}{2{\mathit{\pi }}^4}{\Vert {\overline{v}}^{″}\Vert }_2^2\hfill \\ \hfill & \le \frac{2{\mathit{\delta }}^2{\mathit{\pi }}^2{d}^2}{k}+\frac{L}{2{\mathit{\pi }}^4{\mathit{\delta }}^2}{\Vert \overline{v}\Vert }_X^2\hfill \\ \hfill & =\frac{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4+L)}{{\mathit{\pi }}^{2}k}\hfill \end{array}$$

Also by similar argument, one has$$\begin{array}{cc}\hfill \mathrm{\Phi }(\overline{v})& \ge \frac{1}{2}{\Vert \overline{v}\Vert }_X^2-\left|{\int }_0^{1}G(\overline{v})dt\right|\ge \frac{2{\mathit{\delta }}^2{\mathit{\pi }}^2{d}^2}{k}-{\int }_0^1\left({\int }_0^{\overline{v}}|g(s)|ds\right)dt\hfill \\ \hfill & \ge \frac{2{\mathit{\delta }}^2{\mathit{\pi }}^2{d}^2}{k}-\frac{L}{2}{\int }_0^1{|\overline{v}|}^{2}dt\ge \frac{2{\mathit{\delta }}^2{\mathit{\pi }}^2{d}^2}{k}-\frac{L}{2{\mathit{\pi }}^4}{\Vert {\overline{v}}^{″}\Vert }_2^2\hfill \\ \hfill & \ge \frac{2{\mathit{\delta }}^2{\mathit{\pi }}^2{d}^2}{k}-\frac{L}{2{\mathit{\pi }}^4{\mathit{\delta }}^2}{\Vert \overline{v}\Vert }_X^2\hfill \\ \hfill & =\frac{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^{2}k}\hfill \end{array}$$

Put $r_1=\frac{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^{2}k}=\frac{2c_1^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^2}$ and $r_2=\frac{4d^2({\mathit{\delta }}^2{\mathit{\pi }}^4+L)}{{\mathit{\pi }}^{2}k}=\frac{2c_2^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^2}$; hence, $r_1<\mathrm{\Phi }(\overline{v})<\frac{r_2}{2}$; thus,$$r_1<\mathrm{\Phi }(\overline{v})<r_2,\text{and}\frac{1}{r_2-\mathrm{\Phi }(\overline{v})}<\frac{{\mathit{\pi }}^{2}k}{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}<\frac{1}{\mathrm{\Phi }(\overline{v})-r_1}.$$

Taking (2.9) and (3.5) into account, when $\mathrm{\Phi }(u)<r_i$, $i=1,2$, one has ${max}_{t\in [0,1]}|u(t)|\le c_i$, $i=1,2$; hence,$$\underset{u\in {\mathrm{\Phi }}^{-1}(-\infty ,r_i)}{sup}\mathrm{\Psi }(u)=\underset{u\in {\mathrm{\Phi }}^{-1}(-\infty ,r_i)}{sup}{\int }_0^{1}F(t,u(t))dt\le {\int }_0^1\underset{|\mathit{\xi }|\le c_i}{max}F(t,\mathit{\xi })dt,i=1,2.$$

On the other hand, from (i), one has$$\mathrm{\Psi }(\overline{v})={\int }_0^{1}F(t,\overline{v}(t))dt\ge {\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,\overline{v}(t))dt={\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt.$$

Thus,$$\begin{array}{cc}\hfill 0\le \mathit{\beta }(r_1,r_2)& \le \frac{{sup}_{u\in {\mathrm{\Phi }}^{-1}(r_1,r_2)}\mathrm{\Psi }(u)-\mathrm{\Psi }(\overline{v})}{r_2-\mathrm{\Phi }(\overline{v})}\hfill \\ \hfill & \le \frac{{\int }_0^1{max}_{|\mathit{\xi }|\le c_2}F(t,\mathit{\xi })dt-{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt}{r_2-\mathrm{\Phi }(\overline{v})}\hfill \\ \hfill & <\frac{{\mathit{\pi }}^{2}k}{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}\left\{{\int }_0^1\underset{|\mathit{\xi }|\le c_2}{max}F(t,\mathit{\xi })dt-{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt\right\}.\hfill \end{array}$$

Also by arguing before, one has$$\begin{array}{cc}\hfill \mathit{\rho }(r_1,r_2)& \ge \frac{\mathrm{\Psi }(\overline{v})-{sup}_{u\in {\mathrm{\Phi }}^{-1}(-\infty ,r_1)}\mathrm{\Psi }(u)}{\mathrm{\Phi }(\overline{v})-r_1}\hfill \\ \hfill & \ge \frac{{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt-{\int }_0^1{max}_{|\mathit{\xi }|\le c_1}F(t,\mathit{\xi })dt}{\mathrm{\Phi }(\overline{v})-r_1}\hfill \\ \hfill & >\frac{{\mathit{\pi }}^{2}k}{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}\left\{{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt-{\int }_0^1\underset{|\mathit{\xi }|\le c_1}{max}F(t,\mathit{\xi })dt\right\}.\hfill \end{array}$$

Taking (ii) into account, we get $\mathit{\beta }(r_1,r_2)<\mathit{\rho }(r_1,r_2)$.

Hence, Theorem 2.1 follows that for each$$\mathit{\lambda }\in \left]\frac{\frac{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^{2}k}}{{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt-{\int }_0^1{max}_{|\mathit{\xi }|\le c_1}F(t,\mathit{\xi })dt},\frac{\frac{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^{2}k}}{{\int }_0^1{max}_{|\mathit{\xi }|\le c_2}F(t,\mathit{\xi })dt-{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt}\right[,$$

the problem (1.1) has at least one non-trivial generalized solution $u_0\in X$ such that $r_1<\frac{1}{2}{\Vert u_0\Vert }_X^2-{\int }_0^{1}G(u_0)dt<r_2$.

$\square$

As a simple consequence of Theorem 2.1, we point out the following corollary.

Corollary 3.3

Let k, $\mathit{\delta }$ and L be real constants as defined above. Assume that there exist two non-negative constants d and c with $c=\frac{d}{\sqrt{k}}\sqrt{\frac{2({\mathit{\delta }}^2{\mathit{\pi }}^4+L)}{{\mathit{\delta }}^2{\mathit{\pi }}^4-L}}$ such that

(i)’	$F(t,\mathit{\xi })\ge 0\forall (t,\mathit{\xi })\in \left(\left[0,\frac{3}{8}\right]\cup \left[\frac{5}{8},1\right]\times \left[0,d\right]\right)$
(ii)’	${\int }_0^1{max}_{|\mathit{\xi }|\le c}F(t,\mathit{\xi })dt<\left(\frac{3{\mathit{\delta }}^2{\mathit{\pi }}^4+L}{2\left({\mathit{\delta }}^2{\mathit{\pi }}^4+L\right)}\right){\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt$

Then, for every$$\mathit{\lambda }\in \mathrm{\Lambda }:=\left]\frac{4d^2({\mathit{\delta }}^2{\mathit{\pi }}^4+L)}{{\mathit{\pi }}^{2}k{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt},\frac{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^{2}k\left\{{\int }_0^1{max}_{|\mathit{\xi }|\le c}F(t,\mathit{\xi })d\mathit{\xi }-{\int }_{\frac{3}{8}}^{\frac{5}{8}}F(t,d)dt\right\}}\right[$$

the problem (1.1) has at least one non-trivial generalized solution $u_0$ such that $0<\frac{1}{2}{\Vert u_0\Vert }_X^2-{\int }_0^{1}G(u_0)dt<\frac{4d^2({\mathit{\delta }}^2{\mathit{\pi }}^4+L)}{k{\mathit{\pi }}^2}$.

In Corollary 3.3, taking Theorem 3.2 into account, it is enough to take $c_1=0$ and $c_2=c$.

Remark 3.4

Theorem 1.1 follows immediately from Theorem 3.3 taking into account that $A=B=0$ and $d=\frac{3\mathit{\pi }}{2}$.

Example 3.5

Let $g:\mathbb{R}\to \mathbb{R}$ be a Lipschitz continuous function with Lipschizian constant $L=\frac{{\mathit{\pi }}^4}{5}$ and $g(0)=0$, and $f(t,s)=f_0(t)f_1(s)$ where,$$\begin{array}{c}\hfill f_1(s)=\left\{\begin{array}{cc}{2}^{1-n}min\{s-n,n+1-s\},\hfill & \text{if}s\in \bigcup _{n=0}^{\infty }[n,n+1],\hfill \\ 0,\hfill & \text{if}s\in ]-\infty ,0[,\hfill \end{array}\right.\end{array}$$

and$$\begin{array}{c}\hfill f_0(t)=\left\{\begin{array}{cc}{10}^{-3},\hfill & \text{if}0\le t<\frac{3}{8},\hfill \\ 1,\hfill & \text{if}\frac{3}{8}\le t<\frac{5}{8},\hfill \\ {10}^{-3},\hfill & \text{if}\frac{5}{8}\le t\le 1.\hfill \end{array}\right.\end{array}$$

and, by simple calculations, we obtain$$\begin{array}{cc}\hfill {\int }_0^1{\int }_0^{16}f(t,s)dsdt& ={\int }_0^1{f}_0(t)dt{\int }_0^{16}{f}_1(s)ds={\int }_0^1{f}_0(t)dt\sum _{n=0}^{15}{\int }_n^{n+1}{f}_1(s)ds\hfill \\ \hfill & ={\int }_0^1{f}_0(t)dt\sum _{n=0}^{15}\frac{1}{2^{n+1}}={\int }_0^1{f}_0(t)dt\left(1-\frac{1}{2^{16}}\right)\hfill \\ \hfill & =\frac{1003}{4000}\left(1-\frac{1}{2^{16}}\right)\simeq 0/251\hfill \end{array}$$$$\begin{array}{cc}\hfill \frac{4}{3}{\int }_{\frac{3}{8}}^{\frac{5}{8}}{\int }_0^{\frac{3\mathit{\pi }}{2}}f(t,s)dsdt& =\frac{4}{3}{\int }_{\frac{3}{8}}^{\frac{5}{8}}{f}_0(t)dt\left({\int }_0^4{f}_1(s)ds+{\int }_4^{\frac{3\mathit{\pi }}{2}}{f}_1(s)ds\right)\hfill \\ \hfill & =\frac{4}{3}{\int }_{\frac{3}{8}}^{\frac{5}{8}}{f}_0(t)dt\left(\sum _{n=0}^3{\int }_n^{n+1}{f}_1(s)ds+{\int }_4^{\frac{3\mathit{\pi }}{2}}{f}_1(s)ds\right)\hfill \\ \hfill & =\frac{1}{3}\left(1-\frac{1}{2^4}+{\int }_4^{\frac{3\mathit{\pi }}{2}}{f}_1(s)ds\right)\hfill \\ \hfill & =\frac{1}{3}\left(1-\frac{1}{2^4}+\frac{1}{2^6}+{\int }_{4.5}^{\frac{3\mathit{\pi }}{2}}{2}^{-3}(5-s)ds\right)\hfill \\ \hfill & =\frac{1}{3}\left(1-\frac{1}{2^4}+\frac{1}{2^5}-\frac{1}{2^4}{\left(5-\frac{3\mathit{\pi }}{2}\right)}^2\right)\simeq 0.321\hfill \end{array}$$

Thus, we ensure the inequality (1.3) holds. Hence, owing to Theorem 1.1, for every$$\mathit{\lambda }\in \mathrm{\Lambda }=\left]\frac{2048{\mathit{\pi }}^2}{\frac{5}{4}\left(1-\frac{1}{2^5}-\frac{1}{2^4}{\left(5-\frac{3\mathit{\pi }}{2}\right)}^2\right)},\frac{2048{\mathit{\pi }}^2}{15\left\{\frac{1003}{4000}\left(1-\frac{1}{2^{16}}\right)-\frac{1}{4}\left(1-\frac{1}{2^5}-\frac{1}{2^4}{\left(5-\frac{3\mathit{\pi }}{2}\right)}^2\right)\right\}}\right[$$

where, $\mathrm{\Lambda }\subset ]15759,136789[$, the problem$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{(iv)}+g(u)=\mathit{\lambda }{f}_0(t)f_1(u);\text{in}[0,1],\hfill \\ u(0)=u(1)=0;\hfill \\ u^{″}(0)=u^{″}(1)=0,\hfill \end{array}\right.\end{array}$$

has at least one non-trivial generalized solution $u_0$ that $0<\Vert u_0{\Vert }_X^2-2{\int }_0^{1}G(u_0)dt<\frac{2048{\mathit{\pi }}^2}{5}$.

We now point out a consequence of Corollary 3.3, in which the function f(t, u) has separable variables.

Theorem 3.6

Let $f_0:[0,1]\to \mathbb{R}$ be a non-negative, non-zero and essentially bounded function and $f_1:\mathbb{R}\to \mathbb{R}$ be a non-negative and continuous function and $F_1(\mathit{\xi })={\int }_0^{\mathit{\xi }}{f}_1(x)dx$ for every $\mathit{\xi }\in \mathbb{R}$. Assume that there exists a positive constant d such that $F_1(d)>C_{f_0}{F}_1(c)$ where $c=\frac{d}{\sqrt{k}}\sqrt{\frac{2({\mathit{\delta }}^2{\mathit{\pi }}^4+L)}{{\mathit{\delta }}^2{\mathit{\pi }}^4-L}}$ and$$C_{f_0}=\left(\frac{2{\mathit{\delta }}^2{\mathit{\pi }}^4+2L}{3{\mathit{\delta }}^2{\mathit{\pi }}^4+L}\right)\frac{\Vert f_0{\Vert }_{L^1([0,1])}}{{\int }_{\frac{3}{8}}^{\frac{5}{8}}{f}_0(t)dt};$$

then, for every$$\mathit{\lambda }\in \left]\frac{4d^2({\mathit{\delta }}^2{\mathit{\pi }}^4+L)}{{\mathit{\pi }}^{2}k{F}_1(d){\int }_{\frac{3}{8}}^{\frac{5}{8}}{f}_0(t)dt},\frac{2d^2({\mathit{\delta }}^2{\mathit{\pi }}^4-L)}{{\mathit{\pi }}^{2}k\left\{F_1(c){\Vert f_0\Vert }_{L^1([0,1])}-F_1(d){\int }_{\frac{3}{8}}^{\frac{5}{8}}{f}_0(t)dt\right\}}\right[,$$

the problem(3.8) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{(iv)}+A{u}^{″}+Bu+g(u)=\mathit{\lambda }{f}_0(t)f_1(u)\hfill \\ u(0)=u(1)=0\hfill \\ u^{″}(0)=u^{″}(1)=0\hfill \end{array}\right.\end{array}$$(3.8)

has at least one non-trivial generalized solution $u_0$ such that $0<\frac{1}{2}{\Vert u_0\Vert }_X^2-{\int }_0^{1}G(u_0)dt<\frac{4d^2({\mathit{\delta }}^2{\mathit{\pi }}^4+L)}{k{\mathit{\pi }}^2}$.

Finally, we present an example to illustrate the results of Theorem 3.6.

Example 3.7

Let $A=9$ and $B=2$. Clearly, $\mathit{\delta }=\frac{\sqrt{{\mathit{\pi }}^2-9}}{\mathit{\pi }}\simeq 0.297$ and $k=\frac{1080({\mathit{\pi }}^2-9)}{24031}\simeq 0.0391$. Put $L=1$ and$$\begin{array}{c}\hfill f_1(s)=\left\{\begin{array}{c}{2}^{1-n}min\{s-n,n+1-s\},\text{if}s\in \bigcup _{n=0}^{\infty }[n,n+1],\hfill \\ 0,\text{if}s\in ]-\infty ,0[,\hfill \end{array}\right.\end{array}$$

and$$\begin{array}{c}\hfill f_0(t)=\left\{\begin{array}{c}1,\text{if}0\le t<\frac{3}{8},\hfill \\ {10}^{-3},\text{if}\frac{3}{8}\le t<\frac{5}{8},\hfill \\ 1,\text{if}\frac{5}{8}\le t\le 1,\hfill \end{array}\right.\end{array}$$

and$$C_{f_0}=\left(\frac{2\left(1-\frac{9}{{\mathit{\pi }}^2}\right){\mathit{\pi }}^4+2}{3\left(1-\frac{9}{{\mathit{\pi }}^2}\right){\mathit{\pi }}^4+1}\right)\frac{\frac{3}{4}+\frac{{10}^3}{4}}{\frac{{10}^3}{4}}=\simeq 0.714$$

Put $d=25$. By simple calculations, we obtain $c=25\sqrt{\frac{24031({\mathit{\pi }}^4-9{\mathit{\pi }}^2+1)}{540({\mathit{\pi }}^2-9)[{\mathit{\pi }}^4-9{\mathit{\pi }}^2-1]}}\simeq 178.84$ and$$F_1(d)={\int }_0^{25}{f}_1(s)ds=\sum _{n=0}^{24}{\int }_n^{n+1}{f}_1(s)ds=\sum _{n=0}^{24}\frac{1}{2^{n+1}}=1-\frac{1}{2^{25}},$$$$F_1(c)={\int }_0^{178.84}{f}_1(s)ds=1.$$

Hence, owing to Theorem 3.6, for every $\mathit{\lambda }\in ]249,32763[\subset {\mathrm{\Lambda }}_d$, the problem$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{(iv)}+9u^{″}+2u+g(u)=\mathit{\lambda }{f}_0(t)f_1(u);\text{in}[0,1],\hfill \\ u(0)=u(1)=0;\hfill \\ u^{″}(0)=u^{″}(1)=0,\hfill \end{array}\right.\end{array}$$

has at least one non-trivial generalized solution $u_0$ that $0<\Vert u_0{\Vert }_X^2-2{\int }_0^{1}G(u_0)dt<62109$.

4. Conclusion

In the paper, the author has 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 on one-dimensional space. On N-dimensional spaces $N>1$, 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.

Acknowledgements

The author express his gratitude to referees for their useful suggestions.

Additional information

Funding

This work was supported by Sari agricultural sciences and natural resources university [grant number 01-1395-04]

Notes on contributors

Mohsen Khaleghi Moghadam

Mohsen Khaleghi Moghadam is an assistant professor of Mathematics, Department of Basic Sciences, Sari Agricultural Sciences and Natural Resources University, Sari, Iran.

The author's key research activities are: (1) Nonlinear Analysis: Variational Principles, Critical Point Theory, Variational Inequalities; (2) Partial Differential Equations: Semilinear and Quasilinear, Elliptic Boundary Value Problems.

This paper relates to wider projects that the author will work.