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Existence and uniqueness of solutions to the nonlinear boundary value problem for fourthorder differential equations with all derivatives
Journal of Inequalities and Applications volume 2023, Article number: 23 (2023)
Abstract
This article addresses the existence and uniqueness of solution for fully fourthorder differential equations modeling beams on elastic foundations with nonlinear boundary conditions. The proof will rely on Perov’s fixed point theorem in complete generalized metric spaces to overcome the problems due to the presence of all lowerorder derivatives in the nonlinearity. Finally, some illustrating examples of the theory are presented.
1 Introduction
Recently, fourthorder ordinary differential equations for boundary value problems, which frequently appear in various academic and engineering fields, e.g., chemistry, physics, and dynamical system control, have attracted vast attention and have been broadly studied, especially on the existence and multiplicity of solution under different boundary conditions [1–29]. For example, Li [14] considered the existence of positive solutions to fully equations for cantilever beams, which was determined by the fixed point index theory for cones and sublinear or superlinear growth behaviors of the nonlinearity, through
where f is continuous. In [29], the authors obtained new conclusion on the uniqueness of solutions to problem (1.1) with the order reduction method and the linear operator theory. In [1], Almuthaybiri and Tisdell considered the following nonlinear fourthorder differential equation with linear boundary conditions (LBC) for continuous \(f: [0, 1]\times \mathbb{R}^{4}\rightarrow \mathbb{R}\):
where constants of \(a,c,e,h \in \mathbb{R}\) are known. They sharpen traditional uniqueness results by employing Rus’s contraction mapping theorem where it is accepted that the mapping is contraction according to the metric δ when the space \(C^{3}[0,1]\) is complete for the other metric d, where
We refer readers to the previous studies on cantilever beam equations [6, 15, 24] and the references therein.
In [18], Ma considered the following beam equation with thirdorder nonlinear boundary conditions (NLBC):
where f and g denote continuous functions. Problem (1.2) can model the deformation of an elastic beam on elastic bearings, g expresses the influence of vertical force on the right elastic bearing (\(t=1\)). Using the contraction principle and some restrictive conditions, the authors proved that the solution of problem (1.2) exists and is unique. The existence and multiplicity for such a problem have also been studied by using critical point theory [9, 10, 19]. For problem (1.2), the nonlinear part does not consist of any derivative terms. Therefore, it is an interesting problem to discuss the existence and uniqueness of solution when the nonlinearity depends on every derivative up to order three and the boundary conditions are nonlinear.
Based on the aforementioned work, the present paper aims to propose an existence and uniqueness theorem for fourthorder differential equations having all derivatives for boundary value problems:
with either NLBC
or LBC
where \(f\in C([0,1]\times \mathbb{R}^{4}, \mathbb{R})\) and \(g\in C(\mathbb{R}, \mathbb{R})\) are real functions. This paper has two significant features: the nonlinearity function f contains all derivatives of the unknown function with NLBC; the establishment of the existence and uniqueness of solutions for problems (1.3), (1.4) and (1.3), (1.5) is mainly based on Perov’s fixed point theorem (PFPT) and norm \(\\cdot \_{e}\).
This manuscript is structured as follows. Section 2 provides several essential lemmas associated with this work. Section 3 presents the main results obtained for problems (1.3), (1.4) and (1.3), (1.5).
The following assumptions are made in this paper:
\((H_{1})\): There exist four constants \(p_{i}>0\) such that, for all \(u_{i}, v_{i} \in \mathbb{R}\), \(i=1,2,3,4\),
\((H_{2})\): By \((H_{1})\), we obtain the following matrix:
where \(L_{g}\) is a constant and g is a Lipschitz function, and we suppose that its spectral radius \(\rho (M)<1\).
\((H_{3})\): By \((H_{1})\), we obtain the matrix
and we suppose that its spectral radius \(\rho (M_{1})<1\).
2 Preliminaries
Let the norm of Banach space \(E=C[0,1]\) be \(\u(t)\_{\infty}=\max_{0 \leq t \leq 1}u(t)\).
Lemma 2.1
([18])
Given \(h\in E\), there exists a unique solution for the linear problem
and the solution can be expressed by
with Green’s function G given by
Based on Lemma 2.1, we see that u is the solution of problem (1.3), (1.4) if and only if u is the solution of the formula
Differentiating equation (2.1) yields
where
Similarly, we can obtain
and
where
Therefore, we suppose that there exists a solution \(u_{1}(t)\) for problem (1.3), (1.4) that can be rewritten as follows:
where \(u_{2}\), \(u_{3}\), \(u_{4}\) are the first, second, and thirdorder derivatives of \(u_{1}\), respectively. This is a fixed point problem in \(E^{4}\) for a completely continuous operator
\(T_{i}v\in E\) (\(i=1,2,3,4\)) with \(v(t)=(v_{1}(t),v_{2}(t),v_{3}(t),v_{4}(t))\) are defined by the following integrals:
respectively.
Thus, it is straightforward to conclude that, for \(t,s\in [0,1]\),
and
Moreover, by a simple calculation, we obtain
Similarly, we obtain
and
To compute the fixed point T defined in equation (2.2), we review several conclusions and concepts from vectorvalued metric spaces and PFPT. Let \(d:E\times E\rightarrow R^{n}\) be a function, where E is a nonempty set. Then, if the following features are satisfied for all \(u,v,w\in E\):

(1)
\(d(u,v)=0\) if and only if \(u=v\), and \(d(u,v) \geq 0\) otherwise;

(2)
\(d(u,v)=d(v,u) \);

(3)
\(d(u,v)\leq d(u,w)+d(w,v)\),
d can be regarded as a vector metric on E. The meaning of “≤” is the natural componentwise order relation of \(\mathbb{R}^{n}\). Let \(\alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})^{T}\), \(\beta =(\beta _{1},\beta _{2},\ldots ,\beta _{n})^{T}\), and \(\alpha ,\beta \in \mathbb{R}^{n}\), “\(\alpha \leq \beta \)” means \(\alpha _{i} \leq \beta _{i}\) for \(i=1,2,\ldots ,n\).
Under these circumstances, \((E,d)\) is a generalized metric space, which has similar features, e.g., the completeness of the space, and the Cauchyness and convergence of a sequence, with usual metric spaces.
Theorem 2.2
Given a vector metric d and a complete generalized metric space \((E,d)\), let \(N:E\rightarrow E\), then
for every \(u,v\in E\), and square matrix M belongs to the set of all norder square matrices with positive elements, \(\mathcal{M}_{n\times n}(\mathbb{R}_{+})\). If \(\rho (M)<1\), then there is a unique fixed point \(u^{*}\) for N and
for every \(v\in E\) and \(k\geq 1\).
Theorem 2.3
([30])
Let δ be another vector metric on E, \((E,d)\) be a complete generalized metric space. If \(N:E\rightarrow E\) is continuous with respect to d and

(i)
There exists \(U\in \mathcal{M}_{n\times n}(\mathbb{R}_{+})\) such that \(d(Nu,Nv)\leq U\cdot \delta (u,v)\) for all \(u,v\in E\);

(ii)
There exists \(M\in \mathcal{M}_{n\times n}(\mathbb{R}_{+})\) with \(\rho (M)<1\) such that \(\delta (N(u),N(v))\leq M\delta (u,v)\) for all \(u,v\in E\).
Then N has a unique fixed point in E.
Remark 2.4
We refer readers to [22] for the properties of a nonnegative matrix with a spectral radius smaller than one.
3 Results
Theorem 3.1
Problem (1.3), (1.4) has a unique solution if \((H_{1})\) and \((H_{2})\) hold.
Proof
By (2.3) and (2.5), for every \(v=(v_{1},v_{2},v_{3},v_{4})\in E^{4}\), we have
which shows that \(T_{1}\) maps \(E^{4}\) into \(E_{1}\), a vector subspace of E. It is a Banach space given by
and the norm of \(E_{1}\) \(\u\_{e_{1}}=\inf \{\lambda :u(t)\leq \lambda e_{1}(t),t\in [0,1] \}\) with \(e_{1}(t)=t^{2}\).
Similarly, by (2.4) and (2.5), for every \(v\in E^{4}\), we obtain
and \(T_{2}\) maps \(E^{4}\) into the Banach space \((E_{2},\\cdot \_{e_{2}})\), where
and the norm \(\u\_{e_{2}}=\inf \{\lambda :u(t)\leq \lambda e_{2}(t),t\in [0,1] \}\) with \(e_{2}(t)=t\).
Hence, we conclude that \(T(E^{4})\subset X:=E_{1}\times E_{2}\times E^{2}\). Therefore, only the existence of a unique fixed point v for T in X has to be demonstrated to get the uniqueness solutions for problem (1.3), (1.4).
For \(w=(w_{1},w_{2},w_{3}, w_{4})\), \(v=(v_{1},v_{2},v_{3},v_{4}) \in X\), we define
Obviously, X is a complete generalized Banach metric space equipped with a vectorvalued metric d.
To apply Theorem 2.2, one should prove that, for all \(v, w \in X\), T satisfies
and some nonnegative matrix M with \(\rho (M)<1\). To this end, let \(w, v \in X\) be any elements of X. By (2.6), (2.7), (2.8), and \((H_{1})\), we get
This result yields that
Now, the definition of norm \(\\cdot \_{e_{1}}\) implies that
Similarly, we obtain
and
We can put the above four inequalities under the following vector inequality by using vectorvalued metric on X:
By \((H_{2})\), there is a unique solution to problem (1.3), (1.4), which follows from Theorem 2.2. □
From the previous argument, we know that the basic complete generalized Banach metric space used in obtaining Theorem 3.1 is X, not \(E^{4}\). If we consider problem (1.3), (1.4) in \(E^{4}\) by using an appropriate vectorvalued metric on \(E^{4}\) and PFPT, the result of Theorem 3.1 remains true, except that \((H_{2})\) is replaced with \((H_{3})\).
Theorem 3.2
Problem (1.3), (1.4) has a unique solution if \((H_{1})\) and \((H_{3})\) hold.
Proof
PFPT is applied to the complete generalized Banach metric space \(E^{4}\) having the following vectorvalued metric for \(w,v \in E^{4}\):
Note that
and
Similar to proving Theorem 3.1, we obtain
for \(\forall w,v\in E^{4}\). Thus, Perov’s fixed point theorem can be applied. □
Example 3.3
We consider problem (1.3), (1.4) with
and
According to the mean value theorem, \(b\) is a constant and g is a Lipschitz function. Taking \(a=\frac{1}{3}\) and \(b=1\), we deduce that \(\rho (M)=0.9629\) and \(\rho (M_{1})=1.2538\). The hypothesis of Theorem 3.1 can therefore be satisfied. But, if we take \(a=\frac{1}{2}\) and \(b=\frac{1}{4}\), then we deduce that \(\rho (M)=1.0291\) and \(\rho (M_{1})=0.9813\). The hypotheses of Theorem 3.2 are satisfied in this case.
If the function g in the nonlinear boundary condition is equal to zero, problem (1.3), (1.4) reduces to problem (1.3), (1.5).
Theorem 3.4
If \((H_{1})\) holds, then problem (1.3), (1.5) has a unique solution providing \(\rho (M)<1\) with \(L_{g}=0\), or \(\rho (M_{1})<1\) with \(L_{g}=0\).
The above theorems present a method for the existence and uniqueness of solution for fourthorder differential equations with all derivatives. For the nonlinear boundary, there are few studies at present, and the theories need to be further studied. For the linear boundary, by contrast to [1], the condition of Theorem 3.4 is not optimal. The results in [1] offer an advancement over traditional approaches based on the Rus fixed point theorem and two metrics. It is notable that the method used in [1] seems to be invalid for (1.3), (1.4) because the nonlinear boundary conditions are dominated by the function of \(u(1)\). Now we present the same existence and uniqueness results as [1] by utilizing Perov’s fixed point theorem with two metrics. Let p, q be constants. They satisfy \(\frac{1}{p}+\frac{1}{q}=1\) with \(p>1\) and \(q<1\). Define positive constants \(\beta _{i}\), \(c_{i}\), and \(\gamma _{i}\) for \(i=1,2,3,4\) by
and
Theorem 3.5
If \((H_{1})\) holds, then problem (1.3), (1.5) has a unique solution providing that \(\sum_{i=1}^{4}\gamma _{i}p_{i}<1\).
Proof
The main idea of this result is adopted from [1]. For \(w,v\in E^{4}\), we let
where \(\w_{1}\_{p}= ( \int _{0}^{1} w_{1}(t)^{p}\,dt )^{ \frac{1}{p}}\). Clearly, \(\delta (\cdot ,\cdot )\) is a vectorvalued metric in E; however, it is not complete. From the definition of two metrics \(d_{1}\) and δ on \(E^{4}\), we have
Note that problem (1.3), (1.5) having a unique solution means that the operator \(S:E^{4}\rightarrow E^{4}\) having a unique fixed point is proved, where S is given by
and for \(v(t)=(v_{1}(t),v_{2}(t),v_{3}(t),v_{4}(t))\), we define \(S_{i}v\in E\) (\(i=1,2,3,4\)) by
respectively. For \(w,v\in E^{4}\), by \((H_{1})\) and Hölder’s inequality, we obtain
With the help of norm \(\\cdot \_{\infty}\), we obtain
Combining the above inequalities, we have
This together with (3.1) indicates that condition (1) of Theorem 2.3 can be met and S is continuous on \(E^{4}\) with respect to metric \(d_{1}\).
By (3.2), for \(w,v\in E^{4}\) and \(i=1,2,3,4\), we have
Putting the above inequality over i under the following vector inequality:
Take \(P=(p_{1}, p_{2}, p_{3}, p_{4})\), \(\Gamma =(\gamma _{1},\gamma _{2},\gamma _{3},\gamma _{4})\in \mathbb{R}^{4}\). Then \(M_{2}=P^{T}\Gamma \). So, the rank of matrix \(M_{2}\) is equal to 1 and \(\rho (M_{2})=P\Gamma ^{T}=\sum_{i=1}^{4}\gamma _{i}p_{i}\). Thus, S is contractive on \(E^{4}\) with respect to the metric δ. Consequently, problem (1.3), (1.5) has a unique solution following Theorem 2.3. □
We all know that the Banach fixed point theorem and its generalization play a crucial role in the theory with unique fixed point properties. Surprisingly, the Banach fixed point theorem is equivalent to some of its generalization such as Perov’s fixed point theorem [31]. More surprisingly, all the uniqueness results can be obtained by the Banach fixed point theorem from Theorem 17.5 in [32]. However, in some cases, the introduction of the generalizations of the Banach fixed point will make the study more effective and convenient in a way which may overcome some difficulties encountered in the differential equation and differential system. Those difficulties contain the choice of a complete metric such that an operator T or \(T^{m}\) for some m is a strict contraction. For example, we consider problem (1.3), (1.5), as shown in [31], take
Then \((E^{4}, D)\) is a metric space. Note that \(\sum_{i=1}^{4}\gamma _{i}p_{i}<1\). The operator S may not be contractive with respect to metric D since the operator S such that
For problem (1.3), (1.4), if we take
then we obtain that
which shows that \(T:X\rightarrow X\) may not be contractive with respective to metric \(D_{1}\) even if \(\rho (M)<1\) holds. But, as shown in [31], there must be an integer \(n_{0} > 0\) so that for \(n\geq n_{0}\), operator \(S^{n}\) (or \(T^{n}\)) is contractive and the constant \(n_{0}\) depends heavily on the spectral radius of matrix \(M_{2}\) (or M).
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Acknowledgements
We acknowledge anonymous referees for their valuable suggestions.
Funding
This project was supported by the National Natural Science Foundation of China (11571207), the Shandong Natural Science Foundation (ZR2018MA011).
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YC: conceptualization, supervision, software, writingreview and editing; HC: writingoriginal draft. YC and HC read and approved this final manuscript.
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Chen, H., Cui, Y. Existence and uniqueness of solutions to the nonlinear boundary value problem for fourthorder differential equations with all derivatives. J Inequal Appl 2023, 23 (2023). https://doi.org/10.1186/s13660022029079
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DOI: https://doi.org/10.1186/s13660022029079
MSC
 34A34
 34B18
 46B45
Keywords
 Fully fourthorder differential equation
 Existence and uniqueness
 Perov’s fixed point theorem