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Existence of positive solutions for a singular thirdorder twopoint boundary value problem on the halfline
Boundary Value Problems volume 2022, Article number: 48 (2022)
Abstract
In this paper, we consider the following singular thirdorder twopoint boundary value problem on the halfline of the form
where \(\phi \in C[0,+\infty )\), \(f\in C([0,+\infty )\times (0,+\infty )\times \mathbb{R}^{{2}},\mathbb{R})\) may be singular at \(x=0\), and \(a_{1}\), \(b_{1}\) are positive constants. Using the Leray–Schauder nonlinear alternative and the diagonalization method together with the truncation function technique, we obtain the existence and qualitative properties of positive solutions for the problem. As applications, an example is given to illustrate our result.
1 Introduction
In this paper, we study the existence and qualitative properties of positive solutions to the singular thirdorder twopoint boundary value problem on the halfline of the form
where \(\phi \in C[0,+\infty )\) with \(\phi (t)>0\) for \(t\in (0,+\infty )\), \(f\in C([0,+\infty )\times (0,+\infty )\times \mathbb{R}^{{2}}, \mathbb{R})\) may be singular at \(x=0\), and \(a_{1}\), \(b_{1}\) are positive constants with \(a_{1}< b_{1}\).
Thirdorder differential equations on an infinite interval arise from many physical phenomena, such as free convection problems in boundary layer theory, and the draining or coating fluid flow problems [7, 17, 19, 20]. Hence the thirdorder boundary value problems on the infinite interval have been extensively studied. For more details on nonsingular thirdorder boundary value problems on the infinite interval, see, for instance, [1, 3, 4, 8, 9, 13, 15, 18, 21, 22] and the references therein. For singular thirdorder boundary value problems on the infinite interval, we refer the reader to [2, 5–7, 10, 12, 14, 16, 19, 20] and the references therein.
In recent years, Benbaziz and Djebali [5] considered the following singular thirdorder multipoint boundary value problem on the halfline:
where \(\alpha _{i}\geq 0\) (\(i=1,2,\ldots ,n_{1}\)) with \(\sum_{i=1}^{n_{1}}\alpha _{i}<1\), \(0<\xi _{1}<\xi _{2}<\cdots <\xi _{n_{1}}<+\infty \), \(\beta _{i}\geq 0\) (\(i=1,2,\ldots ,n_{2}\)) with \(\sum_{i=1}^{n_{2}}\beta _{i}<1\), \(0<\eta _{1}<\eta _{2}<\cdots <\xi _{n_{2}}<+\infty \). The nonlinearity \(f\in C((0,+\infty )\times [0,+\infty )\times [0,+\infty ),[0,+ \infty ))\) satisfies upper and lowerhomogeneity conditions in the space variables x, y and may be singular at time variable \(t=0\). The authors presented sufficient conditions which guarantee the existence of positive solutions to problem (1.2) by using the Krasnosel’skii fixed point theorem on cone compression and expansion of norm type. In [6], Benmezaï and Sedkaoui considered the following singular thirdorder twopoint boundary value problem on the halfline:
where κ is a positive constant, \(\phi \in L^{1} (0,+\infty )\) is nonnegative and does not vanish identically on \((0,+\infty )\), the function \(f: \mathbb{R}^{+} \times (0,+\infty )\times (0,+\infty ) \rightarrow \mathbb{R}^{+}\) is continuous and may be singular at the space variable and its derivative. They provided sufficient conditions for the existence of a positive solution to problem (1.3) by employing the Krasnosel’skii fixed point theorem on cone compression and expansion of norm type.
It is worthy to note that none of the nonlinearity f in works concerned with the singular thirdorder boundary value problems on the halfline we mentioned above involves the variables \(x''\). Up to now, we have not found the works that studied the fully nonlinear case of which f contains explicitly t and every derivative of x up to order two.
Motivated and inspired by the above works and [2], in this paper we present sufficient conditions for the existence of positive solutions to problem (1.1) and study the qualitative properties of positive solutions. Our main tool is the Leray–Schauder nonlinear alternative and the diagonalization method together with the truncation function technique.
The rest of this paper is organized as follows. In Sect. 2, we first discuss the existence of positive solutions for singular thirdorder boundary value problems on the finite interval by the Leray–Schauder nonlinear alternative, and then we investigate the existence of positive solutions to problem (1.1) by using the diagonalization method together with the truncation function technique. In Sect. 3, as application, we give an example to illustrate our result.
2 Main results
At first, we present some lemmas, which will be useful in the proof of our main results.
Lemma 2.1
([11])
Assume that Ω is a relatively open subset of a convex set C in a Banach space E. Let \(T:\overline{\Omega}\rightarrow C\) be a compact map and \(p\in \Omega \). Then either

(1)
T has a fixed point in Ω̅; or

(2)
there are \(x\in \partial \Omega \) and \(\lambda \in (0,1)\) such that \(x=(1\lambda )p+\lambda Tx\).
Lemma 2.2
Let \(b>0\) and suppose that \(f:[0,b]\times \mathbb{R}^{3}\rightarrow \mathbb{R}\) and \(\phi :[0,b]\to \mathbb{R}\) are continuous. In addition, assume that there is a constant \(M>\max \{a_{0}+\frac{b}{2}(a_{1}+b_{1}),b_{1}\}\), independent of λ, with
for any solution \(x\in C^{2}[0,b]\cap C^{3}(0,b)\) to
for each \(\lambda \in (0,1)\). Then problem (2.1)_{1} has at least one solution \(x\in C^{2}[0,b]\cap C^{3}(0,b)\) with \(\x\\leq M\).
Proof
Consider the Banach space \(E=C^{2}[0,b]\) with the norm
Take the convex subset \(C=E\) and the open set \(\Omega =\{x\in C:\x\< M\}\). Let us define the operator \(T:\overline{\Omega}\rightarrow E\) by
where
By the Arzelà–Ascoli theorem, we can easily prove that T is a compact operator, and \(x\in C^{2}[0,b]\cap C^{3}(0,b)\) is the solution to problem (2.1)_{1} if and only if \(x\in C^{2}[0,b]\) is a fixed point of operator T. Let \(\omega (t)=a_{0}+a_{1}t+\frac{b_{1}a_{1}}{2b}t^{2}\). Then \(\omega \in C\) with \(\\omega \=\max \{a_{0}+\frac{b}{2}(b_{1}+a_{1}),b_{1}\}< M\). Hence \(\omega \in \Omega \). Noticing that the solvability of problem (2.1)_{λ} is equivalent to the solvability of the operator equation \(x=(1\lambda )\omega +\lambda Tx\), it follows from the assumption that
Hence from Lemma 2.1, T has a fixed point on Ω̅, and thus problem (2.1)_{1} has at least one solution \(x\in \overline{\Omega}\). This completes the proof of the lemma. □
We now discuss the solvability of problem (1.1) by using the diagonalization method together with the truncation function technique.
Theorem 2.1
Suppose that \(f:[0,+\infty )\times (0,+\infty )\times \mathbb{R}^{{2}}\rightarrow \mathbb{R}\) is continuous, \(\phi \in C[0,+\infty )\) with \(\phi (t)>0\) for \(t\in (0,+\infty )\), and \(\phi (t)\) is nondecreasing on \((0,+\infty )\). In addition, assume that the following conditions hold:

(i)
\(zf(t,x,y,z)\geq 0\) for any \((t,x,y,z)\in [0,+\infty )\times (0,+\infty )\times \mathbb{R}^{{2}}\);

(ii)
There exist a function \(h\in C([0,+\infty ),(0,+\infty ))\) and a nondecreasing function \(g\in C((0,+\infty ),(0,+\infty ))\) such that
$$ f(t,x,y,z)\leq g(x)h(z), \quad (t,x,y,z)\in [0,+\infty )\times (0,+ \infty )\times [a_{1},b_{1}]\times [0,+\infty ), $$where
$$ \int _{0}^{+\infty}\frac{{\mathrm{{d}}}u}{h(u)}> \int _{0}^{b_{1}a_{1}} \frac{{\mathrm{{d}}}u}{h(u)}+ \frac{\phi (1)}{a_{1}} \int _{0}^{1+b_{1}}g(u){\, \mathrm{{d}}}u; $$ 
(iii)
For any constant \(K>0\), there exist a continuous function \(\Psi _{K}(t)\) defined on \([0,+\infty )\), which is positive and nondecreasing on \((0,+\infty )\), and a constant \(r\in [1,2)\) such that
$$ f(t,x,y,z)\geq \Psi _{K}(t)z^{r},\quad (t,x,y,z)\in [0,+ \infty ) \times (0,+\infty )\times [a_{1},b_{1}]\times [0,K]. $$
Then problem (1.1) has a convex and monotonically increasing positive solution \(x\in C^{2}[0,+\infty )\cap C^{3}(0,+\infty )\).
Proof
We shall complete the proof in two steps.
Step 1. We show that, for each \(n\in \mathbb{N}\), the singular boundary value problem on the finite interval
has a solution \(x\in C^{2}[0,n]\cap C^{3}(0,n]\). For this, consider the nonsingular problem
where \(m\in \mathbb{N}\). In order to prove that problem (2.3) has a solution, consider the modified problems
where \(\lambda \in (0,1)\) and
Obviously, \(f^{{\star}}\in C([0,+\infty )\times \mathbb{R}^{3},\mathbb{R})\) and
Let \(x\in C^{3}[0,n]\) be any solution of (2.4)_{λ}. We now assert that
Indeed, for any \(\eta \in [0,n)\) and \(\eta < t\leq n\), we have
and so
If \(x''(0)=0\), then (2.6) with \(\eta =0\) implies \(x''(t)=0\) for \(t\in [0,n]\), which contradicts \(x'(0)=a_{1}< b_{1}=x'(n)\). Thus \(x''(0)\neq 0\). Now we have two cases to consider:
Case 1. \(x''(t)\neq 0\) for \(t\in [0,n)\). If \(x''(t)< 0\) for \(t\in [0,n)\), then \(a_{1}=x'(0)>x'(n)=b_{1}\), which is a contradiction. Thus \(x''(t)>0\) for \(t\in [0,n)\).
Case 2. There exists \(\delta \in (0,n)\) with \(x''(t)\neq 0\) for \(t\in [0,\delta )\) and \(x''(\delta )=0\). Now (2.6) with \(\eta =\delta \) implies \(x''(t)=0\) for \(t\in [\delta ,n]\), and so \(x'(t)\equiv b_{1}\) on \([\delta ,n]\). If \(x''(t)<0\) for \(t\in [0,\delta )\), then \(b_{1}=x'(\delta )< x'(0)=a_{1}\), which is a contradiction. Hence \(x''(t)>0\) for \(t\in [0,\delta )\).
In summary, (2.5) is true. Hence \(x'''(t)\leq 0\) for \(t\in [0,n]\). Meanwhile,
and thus
Obviously, \(\max_{t\in [0, n]}x''(t)=x''(0)\). In addition, according to the differential mean value theorem, there exists \(\xi \in (0,1)\) such that
Now, from (2.5), (2.7), (2.8) and assumption (ii), we have
that is,
Integrating from 0 to ξ, we obtain
Consequently,
Let
Then from assumption (iii) and (2.9) we know that
Notice that \(x'''(t)\leq 0\) for \(t\in [0,n]\), it follows from (2.5) that
This together with (2.7) and (2.8) implies that
Therefore from Lemma 2.2, problem (2.4)_{1} [and consequently problem (2.3)] has a solution \(v_{m}\in C^{3}[0,n]\) with
Therefore from (2.10) and assumption (ii) it follows that
Notice that assumption (iii) guarantees that there are a continuous function \(\Psi _{V}(t)\), which is positive and nondecreasing on \((0,+\infty )\), and a constant \(r\in [1,2)\) such that
and so
Now we assert that
where
Indeed, we have two cases to consider:
Case 1. \(r=1\). Integrating (2.12) from t to n, we can obtain
and thus
Solving the above inequality, we have
Case 2. \(1< r<2\). Note that either \(v''_{m}(t)>0\) for \(t\in [0,n)\) or there exists \(\delta \in (0,n)\) such that \(v''_{m}(t)>0\) for \(t\in [0,\delta )\) and \(v''_{m}(t)=0\) for \(t\in [\delta ,n]\). Hence, there exists \(\delta \in (0,n]\) such that \(v''_{m}(t)>0\) for \(t\in [0,\delta )\). Multiplying (2.12) by \(({v''_{m}}(t))^{1r}\) and integrating from t to δ (note that ϕ and \(\Psi _{V}\) are nondecreasing on \([0,+\infty )\)), we have
Consequently,
Integrating from 0 to t, we obtain
and so
Therefore
In summary, (2.13) holds.
Notice that (2.10), (2.11), (2.13) and the Arzelà–Ascoli theorem guarantee that there exist a subsequence \(\mathbb{S}\) of \(\mathbb{N}\) and a function \(x_{n}\in C^{2}[0,n]\) such that \(v^{(j)}_{m}(t)\to x^{(j)}_{n}(t)\) (\(j=0,1,2\)) uniformly on \([0,n]\) as \(m\rightarrow \infty\) (\(m\in \mathbb{S}\)), and
Also note that
Let \(m\rightarrow \infty\) (\(m\in \mathbb{S}\)), then by the Lebesgue dominated convergence theorem (note (2.11) and assumption (ii)), we have
Consequently, \(x_{n}\in C^{2}[0,n]\cap C^{3}(0,n]\) is a solution of (2.2) and satisfies
Step 2. We prove the existence of solutions to problem (2.2) by using the diagonalization method. To do this, for \(n\geq 1\) an integer, we let
Then from (2.14) it follows that
Hence, by the Arzelà–Ascoli theorem, there exist a subsequence \(\mathbb{N}^{*}_{1}\) of \(\mathbb{N}\) and a function \(z_{1}(t)\in C^{2}[0,1]\) with \(u^{(j)}_{n}(t)\to z^{(j)}_{1}(t)\) (\(j=0,1,2\)) uniformly on \([0,1]\) as \(n\rightarrow \infty\) (\(n\in \mathbb{N}^{*}_{1}\)). Also \(a_{1}t\leq z_{1}(t)\leq b_{1}+1\), \(\Theta _{r}(t)\leq z_{1}'(t)\leq b_{1}\), \(0\leq z_{1}''(t)\leq V\), \(t\in [0,1]\) and \(z_{1}(0)=0\), \(z'_{1}(0)=a_{1}\). Let \(\mathbb{N}_{1}=\mathbb{N}^{*}_{1}\setminus \{1\}\). Also notice that
the Arzelà–Ascoli theorem guarantees the existence of a subsequence \(\mathbb{N}^{*}_{2}\) of \(\mathbb{N}_{1}\) and a function \(z_{2}(t)\in C^{2}[0,2]\) with \(u^{(j)}_{n}(t)\to z^{(j)}_{2}(t)\) (\(j=0,1,2\)) uniformly on \([0,2]\) as \(n\rightarrow \infty\) (\(n\in \mathbb{N}^{*}_{2}\)). Note that \(z_{2}(t)=z_{1}(t)\) on \([0,1]\) since \(\mathbb{N}^{*}_{2}\subset \mathbb{N}_{1}\). Also, \(a_{1}t\leq z_{2}(t)\leq 2b_{1}+1\), \(\Theta _{r}(t)\leq z_{2}'(t)\leq b_{1}\), \(0\leq z_{2}''(t)\leq V\), \(t\in [0,2]\) and \(z_{2}(0)=0\), \(z'_{2}(0)=a_{1}\). Let \(\mathbb{N}_{2}=\mathbb{N}^{*}_{2}\setminus \{2\}\) and proceed inductively to obtain for \(k=1,2,\ldots \) a subsequence \(\mathbb{N}_{k}\subset \mathbb{N}\) with \(\mathbb{N}_{k}\subset \mathbb{N}_{k1}\) and a function \(z_{k}(t)\in C^{2}[0,k]\) such that \(u^{(j)}_{n}(t)\to z^{(j)}_{k}(t)\) (\(j=0,1,2\)) uniformly on \([0,k]\) as \(n\rightarrow \infty\) (\(n\in \mathbb{N}_{k}\)). Also note that \(z_{k}(t)=z_{k1}(t)\) for \(t\in [0,k1]\) and \(a_{1}t\leq z_{k}(t)\leq kb_{1}+1\), \(\Theta _{r}(t)\leq z_{k}'(t)\leq b_{1}\), \(0\leq z_{k}''(t)\leq V\), \(t\in [0,k]\), \(z_{k}(0)=0\), \(z'_{k}(0)=a_{1}\).
We now define a function \(x(t)\) as follows. For any fixed \(t\in [0,+\infty )\), take \(k\in \mathbb{N}\) such that \(k\geq t\). Let \(x(t)=z_{k}(t)\). Then \(x(t)\) is well defined on \([0,+\infty )\), and \(x\in C^{2}[0,+\infty )\). In addition, we have
and
Arbitrarily, take \(k\in \mathbb{N}\). Note that \(\forall t\in (0,k]\) and \(\forall n\in{\mathbb{N}_{k}}\), we have
Let \(n\rightarrow \infty\) (\(n\in \mathbb{N}_{k}\)), from the Lebesgue dominated convergence theorem, it follows that
that is,
Thus \(x\in C^{3}(0,k]\) and
Since \(k\in \mathbb{N}\) is arbitrary, we have \(x\in C^{2}[0,+\infty )\cap C^{3}(0,+\infty )\) and
Notice that \(\Theta _{r}(t)\rightarrow b_{1}\) (\(t \rightarrow +\infty \)) and \(\Theta _{r}(t)\leq x'(t)\leq b_{1}\), \(t\in [0,+\infty )\), then \(x'(+\infty )=b_{1}\). In summary, \(x(t)\) is a convex and monotonically increasing positive solution of problem (1.1). This completes the proof of the theorem. □
3 An example
In this section, we give an example to illustrate our main result.
Example 3.1
Consider the singular thirdorder twopoint boundary value problems on the halfline
where \(\beta \in (0,1)\).
Let \(\phi (t)=t\), \(t\in (0,+\infty )\), and
Then \(f\in C([0,+\infty )\times (0,+\infty )\times \mathbb{R}^{{2}}, \mathbb{R})\), \(\phi \in C[0,+\infty )\) is positive and nondecreasing on \((0,+\infty )\). Obviously, condition (i) in Theorem 2.1 is satisfied. We now check conditions (ii) and (iii) of Theorem 2.1. To do this, let
Then \(g\in C((0,+\infty )(0,+\infty ))\), \(h\in C([0,+\infty ), (0,+\infty ))\), and \(\forall (t,x,y,z)\in [0,+\infty )\times (0,+\infty )\times (0,1) \times [0,+\infty )\), we have
In addition, it is clear that
and by the Cauchy inequality we have
Thus condition (ii) holds.
Finally, for any constant \(K>0\), take \(\Psi _{K}(t)\equiv 1/\sqrt{K}\) on \([0,+\infty )\) and \(r=1\). Then \(\forall (t,x,y,z)\in [0,+\infty )\times (0,+\infty )\times (0,1] \times [0,K]\), we have
that is, condition (iii) holds.
In summary, all the conditions in Theorem 2.1 are satisfied. Therefore, problem (3.1) has at least one convex, strictly increasing positive solution.
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The authors would like to thank the referees for their comments and suggestions.
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The three authors of this paper come from School of Mathematics and Statistics in Beihua University.
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This work was supported by the Natural Science Foundation of Jilin Province (Grant No. 20210101156JC).
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This work was carried out in collaboration between the three authors. MP designed the study and guided the research. YB and LW performed the analysis and wrote the first draft of the manuscript. YB, LW, and MP managed the analysis of the study. The three authors read and approved the final manuscript.
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Bao, Y., Wang, L. & Pei, M. Existence of positive solutions for a singular thirdorder twopoint boundary value problem on the halfline. Bound Value Probl 2022, 48 (2022). https://doi.org/10.1186/s13661022016300
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DOI: https://doi.org/10.1186/s13661022016300