On the Solvability of Nonlinear Third-Order Two-Point Boundary Value Problems

Abstract

Under barrier strips type assumptions we study the existence of $C^3[0,1]$—solutions to various two-point boundary value problems for the equation $x^{‴}=f(t,x,x^{\prime },x^{″}).$ We give also some results guaranteeing positive or non-negative, monotone, convex or concave solutions.

1. Introduction

In this paper, we study the solvability of boundary value problems (BVPs) for the differential equation

$$x^{‴}=f(t,x,x^{\prime },x^{″}),t\in (0,1),$$

(1)

with some of the boundary conditions

$$\begin{array}{c}\hfill x\left(0\right)=A,x^{\prime }\left(1\right)=B,x^{″}\left(1\right)=C,\end{array}$$

(2)

$$\begin{array}{c}\hfill x\left(0\right)=A,x^{\prime }\left(0\right)=B,x^{″}\left(1\right)=C,\end{array}$$

(3)

$$\begin{array}{c}\hfill x\left(0\right)=A,x\left(1\right)=B,x^{″}\left(1\right)=C,\end{array}$$

(4)

$$\begin{array}{c}\hfill x\left(0\right)=A,x^{\prime }\left(0\right)=B,x^{\prime }\left(1\right)=C,\end{array}$$

(5)

$$\begin{array}{c}\hfill x\left(1\right)=A,x^{\prime }\left(0\right)=B,x^{\prime }\left(1\right)=C,\end{array}$$

(6)

where $f:[0,1]\times D_x\times D_p\times D_q\to \mathbb{R},$ $D_x,D_p,D_q\subseteq \mathbb{R},$ and $A,B,C\in \mathbb{R}.$

The solvability of BVPs for third-order differential equations has been investigated by many authors. Here, we will cite papers devoted to two-point BVPs which are mostly with some of the above boundary conditions; in each of these works $A,B,C=0.$ Such problems for equations of the form

$$\begin{array}{c}\hfill x^{‴}=f(t,x),t\in (0,1),\end{array}$$

have been studied by H. Li et al. [1], S. Li [2] (the problem may be singular at $t=0$ and/or $t=1$), Z. Liu et al. [3,4], X. Lin and Z. Zhao [5], S. Smirnov [6], Q. Yao and Y. Feng [7]. Moreover, the boundary conditions in References [2,3] are (3), in Reference [4] they are (4), in References [1,5,7] they are (5), and in Reference [6] are

$$x\left(0\right)=x\left(1\right)=0,x^{\prime }\left(0\right)=C.$$

Y. Feng [8] and Y. Feng and S. Liu [9] have considered the equation

$$x^{‴}=f(t,x,x^{\prime }),t\in (0,1),$$

with (6) and (5), respectively. Y. Feng [10] and R. Ma and Y. Lu [11] have studied the equations

$$f(t,x,x^{\prime },x^{‴})=0and{x}^{‴}+M{x}^{″}+f(t,x)=0,t\in (0,1).$$

with (5). BVPs for the equation

$$x^{‴}=f(t,x,x^{\prime },x^{″}),t\in (0,1),$$

have been investigated by A. Granas et al. [12], B. Hopkins and N. Kosmatov [13], Y. Li and Y. Li [14]; the boundary conditions in [12] are (5), these in Reference [13] are (2) and (3), and in Reference [14]—(2).

Results guaranteeing positive or non-negative solutions can be found in References [2,3,4,7,8,9,10,11,13,14], and results that guarantee negative or non-positive ones in References [7,9,10]. The existence of monotone solutions has been studied in References [3,7,9].

As a rule, the main nonlinearity is defined and continuous on a set such that each dependent variable changes in a left- and/or a right-unbounded set; in Reference [13] it is a Carathéodory function on an unbounded set. Besides, the main nonlinearity is monotone with respect to some of the variables in References [1,5], does not change its sign in References [2,3,4,14] and satisfies Nagumo type growth conditions in Reference [14]. Maximum principles have been used in References [8,10], Green’s functions in References [1,2,4,5], and upper and lower solutions in References [1,7,8,9,10,11].

Here, we use a different tool—barrier strips which allow the right side of the equation to be defined and continuous on a bounded subset of its domain and to change its sign.

To prove our existence results we apply a basic existence theorem whose formulation requires the introduction of the BVP

$$\begin{array}{c}\hfill x^{‴}+a\left(t\right)x^{″}+b\left(t\right)x^{\prime }+c\left(t\right)x=f(t,x,x^{\prime },x^{″}),t\in (0,1),\end{array}$$

(7)

$$\begin{array}{c}\hfill V_i\left(x\right)=r_i,i=1,2,3(i=\overline{1,3}forshort),\end{array}$$

(8)

where $a,b,c\in C([0,1],\mathbb{R}),f:[0,1]\times D_x\times D_p\times D_q\to \mathbb{R},$

$$V_i\left(x\right)=\sum _{j=0}^2[a_{ij}{x}^{\left(j\right)}\left(0\right)+b_{ij}{x}^{\left(j\right)}\left(1\right)],i=\overline{1,3},$$

with constants $a_{ij}$ and $b_{ij}$ such that ${\sum }_{j=0}^2(a_{ij}^2+b_{ij}^2)>0,i=\overline{1,3},$ and $r_i\in \mathbb{R},$ $i=\overline{1,3}.$ Next, consider the family of BVPs for

$$x^{‴}+a\left(t\right)x^{″}+b\left(t\right)x^{\prime }+c\left(t\right)x=g(t,x,x^{\prime },x^{″},\lambda ),t\in (0,1),\lambda \in [0,1]\hspace{1em}\hspace{1em}{\left(7\right)}_{\lambda }$$

with boundary conditions (8), where g is a scalar function defined $[0,1]\times D_x\times D_p\times D_q\times [0,1],$ and $a,b,c$ are as above. Finally, $BC$ denotes the set of functions satisfying boundary conditions (8), and ${BC}_0$ denotes the set of functions satisfying the homogeneous boundary conditions $V_i\left(x\right)=0,i=\overline{1,3}.$ Besides, let $C_{BC}^3[0,1]=C^3[0,1]\cap BC$ and $C_{B{C}_0}^3[0,1]=C^3[0,1]\cap {BC}_0.$

The proofs of our existence results are based on the following theorem. It is a variant of Reference [12] (Chapter I, Theorem 5.1 and Chapter V, Theorem 1.2). Its proof can be found in Reference [15]; see also the similar result in Reference [16] (Theorem 4).

Lemma 1.

Suppose:

(i) Problem (7)${}_0$, (8) has a unique solution $x_0\in C^3[0,1].$

(ii) Problems (7), (8) and (7)${}_1$, (8) are equivalent.

(iii) The map ${\mathbf{L}}_h:C_{{BC}_0}^3[0,1]\to C[0,1]$ is one-to-one: here,

$${\mathbf{L}}_{h}x=x^{‴}+a\left(t\right)x^{″}+b\left(t\right)x^{\prime }+c\left(t\right)x.$$

(iv) Each solution $x\in C^3[0,1]$ to family (7)${}_{\lambda }$, (8) satisfies the bounds

$$m_i\le x^{\left(i\right)}\le M_i for t\in [0,1],i=\overline{0,3},$$

where the constants $-\infty <m_i,M_i<\infty ,i=\overline{0,3},$ are independent of λ and x.

(v) There is a sufficiently small $\sigma >0$ such that

$$[m_0-\sigma ,M_0+\sigma ]\subseteq D_x,[m_1-\sigma ,M_1+\sigma ]\subseteq D_p,[m_2-\sigma ,M_2+\sigma ]\subseteq D_q,$$

and $g(t,x,p,q,\lambda )$ is continuous for $(t,x,p,q,\lambda )\in [0,1]\times J\times [0,1]$ where $J=[m_0-\sigma ,M_0+\sigma ]\times [m_1-\sigma ,M_1+\sigma ]\times [m_2-\sigma ,M_2+\sigma ];$ $m_i,M_i,$ $i=\overline{0,3},$ are as in (iv).

Then boundary value problem (7), (8) has at least one solution in $C^3[0,1].$

For us, the equation from (7)${}_{\lambda }$ has the form

$$x^{‴}=\lambda f(t,x,x^{\prime },x^{″}).\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\left(1\right)}_{\lambda }$$

Preparing the application of Lemma 1, we impose conditions which ensure the a priori bounds from (iv) for the eventual $C^3[0,1]$ - solutions of the families of BVPs for (7)${}_{\lambda },\lambda \in [0,1],$ with one of the boundary conditions (k), $k=\overline{2,6}.$

So, we will say that for some of the BVPs (1), (k), $k=\overline{2,6},$ the conditions (H${}_{\mathbf{1}}$) and (H${}_{\mathbf{2}}$) hold for a $K\in \mathbb{R}$ (it will be specified later for each problem) if:

(H${}_{\mathbf{1}}$) 

There are constants $F_i^{\prime },L_i^{\prime },i=1,2,$ such that

$$F_2^{\prime }<F_1^{\prime }\le K\le L_1^{\prime }<L_2^{\prime },[F_2^{\prime },L_2^{\prime }]\subseteq D_q,$$

$$\begin{array}{c}\hfill f(t,x,p,q)\ge 0\mathrm{for}(t,x,p,q)\in [0,1]\times D_x\times D_p\times [L_1^{\prime },L_2^{\prime }],\end{array}$$

(9)

$$\begin{array}{c}\hfill f(t,x,p,q)\le 0\mathrm{for}(t,x,p,q)\in [0,1]\times D_x\times D_p\times [F_2^{\prime },F_1^{\prime }].\end{array}$$

(10)

(H${}_{\mathbf{2}}$) 

There are constants $F_i,L_i,i=1,2,$ such that

$$F_2<F_1\le K\le L_1<L_2,[F_2,L_2]\subseteq D_q,$$

$$f(t,x,p,q)\le 0for(t,x,p,q)\in [0,1]\times D_x\times D_p\times [L_1,L_2],$$

$$f(t,x,p,q)\ge 0for(t,x,p,q)\in [0,1]\times D_x\times D_p\times [F_2,F_1].$$

Besides, we will say that for some of the BVPs (1), (k), $k=\overline{2,6}$, the condition (H${}_{\mathbf{3}}$) holds for constants $m_i\le M_i,i=\overline{0,2},$ (they also will be specified later for each problem) if:

(H${}_{\mathbf{3}}$) 

$[m_0-\sigma ,M_0+\sigma ]\subseteq D_x,$$[m_1-\sigma ,M_1+\sigma ]\subseteq D_p,$$[m_2-\sigma ,M_2+\sigma ]\subseteq D_q$ and $f(t,x,p,q)$ is continuous on the set $[0,1]\times J,$ where J is as in $\left(v\right)$ of Lemma 1, and $\sigma >0$ is sufficiently small.

In fact, the present paper supplements P. Kelevedjiev and T. Todorov [15] where only conditions (H${}_{\mathbf{2}}$) and (H${}_{\mathbf{3}}$) have been used for studying the solvability of various BVPs for (1) with other boundary conditions. Here, (H${}_{\mathbf{1}}$) is also needed. Now, only (H${}_{\mathbf{1}}$) guarantees the a priori bounds for $x^{″}\left(t\right),x^{\prime }\left(t\right)$ and $x\left(t\right)$, in this order, for each eventual solution $x\in C^3[0,1]$ to the families (1)${}_{\lambda },\left(k\right),k=\overline{2,4},$ and (H${}_{\mathbf{1}}$) and (H${}_{\mathbf{2}}$) together guarantee these bounds for the families (1)${}_{\lambda },$ (k), $k=5,6.$ As in Reference [15], (H${}_{\mathbf{3}}$) gives the bounds for $x^{‴}\left(t\right).$

The auxiliary results which guarantee a priori bounds are given in Section 2, and the existence theorems are in Section 3. The ability to use (H${}_{\mathbf{1}}$) and (H${}_{\mathbf{2}}$) for studying the existence of solutions with important properties is shown in Appendix A. Examples are given in Section 4.

2. Auxiliary Results

This part ensures a priori bounds for the eventual $C^3[0,1]$-solutions of each family (1)${}_{\lambda },$ $\left(k\right),k=\overline{2,6},$ that is, it ensures the constants $m_i,M_i,i=\overline{0,2},$ from (iv) of Lemma 1 and (H${}_{\mathbf{3}}$).

Lemma 2.

Let $x\in C^3[a,b]$ be a solution to (1)${}_{\lambda }.$ Suppose (H${}_{\mathbf{1}}$) holds with $[0,1]$ replaced by $[a,b]$ and $K=x^{″}\left(b\right).$ Then

$$F_1^{\prime }\le x^{″}\left(t\right)\le L_1^{\prime } on [a,b].$$

Proof. 

By contradiction, assume that $x^{″}\left(t\right)>L_1^{\prime }$ for some $t\in [a,b).$ This means that the set

$$S_{+}=\{t\in [a,b]:L_1^{\prime }<x^{″}\left(t\right)\le L_2^{\prime }\}$$

is not empty because $x^{″}\left(t\right)$ is continuous on $[a,b]$ and $x^{″}\left(b\right)\le L_1^{\prime }.$ Besides, there is a $\gamma \in S_{+}$ such that

$$x^{‴}\left(\gamma \right)<0.$$

As $x\left(t\right)$ is a $C^3[a,b]$—solution to (1)${}_{\lambda },$

$$x^{‴}\left(\gamma \right)=\lambda f(\gamma ,x\left(\gamma \right),x^{\prime }\left(\gamma \right),x^{″}\left(\gamma \right)).$$

But, $(\gamma ,x\left(\gamma \right),x^{\prime }\left(\gamma \right),x^{″}\left(\gamma \right))\in S_{+}\times D_x\times D_p\times (L_1^{\prime },L_2^{\prime }]$ and (9) imply

$$x^{‴}\left(\gamma \right)\ge 0,$$

a contradiction. Consequently,

$$x^{″}\left(t\right)\le L_1^{\prime } \mathrm{for} t\in [a,b].$$

Along similar lines, assuming on the contrary that the set

$$S_{-}=\{t\in [a,b]:F_2^{\prime }\le x^{″}\left(t\right)<F_1^{\prime }\}$$

is not empty and using (10), we achieve a contradiction which implies that

$$F_1^{\prime }\le x^{″}\left(t\right)\mathrm{for}t\in [a,b].$$

 □

The proof of the next assertion is virtually the same as that of Lemma 2 and is omitted; it can be found in [15].

Lemma 3.

Let $x\in C^3[a,b]$ be a solution to (1)${}_{\lambda }.$ Suppose (H${}_{\mathbf{2}}$) holds with $[0,1]$ replaced by $[a,b]$ and $K=x^{″}\left(a\right).$ Then

$$F_1\le x^{″}\left(t\right)\le L_1 on [a,b].$$

Let us recall, conditions of type (H${}_{\mathbf{1}}$) and (H${}_{\mathbf{2}}$) are called barrier strips, see P. Kelevedjiev [17]. As can we see from Lemmas 2 and 3 they control the behavior of $x^{″}\left(t\right)$ on $[a,b],$ depending on the sign of $f(t,x,x^{\prime },x^{″})$ the curve of $x^{″}\left(t\right)$ on $[a,b]$ crosses the strips $[a,b]\times [L_1^{\prime },L_2^{\prime }],$ $[a,b]\times [L_1,L_2],$ $[a,b]\times [F_2^{\prime },F_1^{\prime }]$ and $[a,b]\times [F_2,F_1]$ not more than once. This property ensures the a priori bounds for $x^{″}\left(t\right).$

Lemma 4.

Let(H${}_{\mathbf{1}}$)hold for $K=C$. Then every solution $x\in C^3[0,1]$ to (1)${}_{\lambda },$ (2) or (1)${}_{\lambda },$ (3) satisfies the bounds

$$\left|x\left(t\right)\right|\le \left|A\right|+\left|B\right|+max\left\{\right|F_1^{\prime }|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$|x^{\prime }\left(t\right)|\le |B|+max\{|F_1^{\prime }|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$\begin{array}{c}\hfill F_1^{\prime }\le x^{″}\left(t\right)\le L_1^{\prime },t\in [0,1].\end{array}$$

(11)

Proof. 

Let first $x\left(t\right)$ be a solution to (1)${}_{\lambda },$ (2). Using Lemma 2 we conclude that (11) is true. Then, according to the mean value theorem, for each $t\in [0,1)$ there is a $\xi \in (t,1)$ such that

$$x^{\prime }\left(1\right)-x^{\prime }\left(t\right)=x^{″}\left(\xi \right)(1-t),$$

which together with (11) gives the bound for $|x^{\prime }\left(t\right)|.$ Again from the mean value theorem for each $t\in (0,1]$ there is an $\eta \in (0,t)$ with the property

$$x\left(t\right)-x\left(0\right)=x^{\prime }\left(\eta \right)t,$$

which yields the bound for $\left|x\right(t\left)\right|.$ The assertion follows similarly for (1)${}_{\lambda },$ (3). □

Lemma 5.

Let(H${}_{\mathbf{1}}$)hold for $K=C$. Then every solution $x\in C^3[0,1]$ to (1)${}_{\lambda },$ (4) satisfies the bounds

$$\left|x\left(t\right)\right|\le \left|A\right|+|B-A|+max\left\{\right|F_1^{\prime }|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$|x^{\prime }\left(t\right)|\le |B-A|+max\{|F_1^{\prime }|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$F_1^{\prime }\le x^{″}\left(t\right)\le L_1^{\prime },t\in [0,1].$$

Proof. 

By Lemma 2, $F_1^{\prime }\le x^{″}\left(t\right)\le L_1^{\prime }$ on $[0,1].$ Clearly, there is a $\mu \in (0,1)$ for which $x^{\prime }\left(\mu \right)=B-A.$ Further, for each $t\in [0,\mu )$ there is a $\xi \in (t,\mu )$ such that

$$x^{\prime }\left(\mu \right)-x^{\prime }\left(t\right)=x^{″}\left(\xi \right)(\mu -t),$$

from where, using the obtained bounds for $x^{″}\left(t\right)$, we get

$$|x^{\prime }\left(t\right)|\le |B-A|+max\{|F_1^{\prime }|,|L_1^{\prime }\left|\right\},t\in [0,\mu ].$$

We can proceed analogously to see that the same bound is valid for $t\in [\mu ,1].$ Finally, for each $t\in (0,1]$ there is an $\eta \in (0,t)$ such that

$$x\left(t\right)-x\left(0\right)=x^{\prime }\left(\eta \right)t,$$

which together with the obtained bound for$|x^{\prime }\left(t\right)|$ yields the bound for $\left|x\right(t\left)\right|.$. □

Lemma 6.

Let(H${}_{\mathbf{1}}$)and(H${}_{\mathbf{2}}$)hold for $K=C-B$. Then every solution $x\in C^3[0,1]$ to (1)${}_{\lambda },$ (2) or (1)${}_{\lambda },$ (3) satisfies the bounds

$$\left|x\left(t\right)\right|\le \left|A\right|+\left|B\right|+max\left\{\right|F_1|,|L_1|,|F_1^{\prime }|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$|x^{\prime }\left(t\right)|\le |B|+max\{|F_1|,|L_1|,|F_1^{\prime }|,|L_1^{\prime }\left|\right\},t\in [0,1],$$

$$min\{F_1,F_1^{\prime }\}\le x^{″}\left(t\right)\le max\{L_1,L_1^{\prime }\},t\in [0,1].$$

Proof. 

Let $x\left(t\right)$ be a solution to (1)${}_{\lambda },$ (5); the proof is similar for (1)${}_{\lambda },$ (6). We know there is a $\nu \in (0,1)$ for which $x^{″}\left(\nu \right)=C-B.$ Then, applying Lemmas 2 and 3 on the intervals $[0,\nu ]$ and $[\nu ,1],$ respectively, we get

$$F_1^{\prime }\le x^{″}\left(t\right)\le L_1^{\prime } \mathrm{on} [0,\nu ] \mathrm{and} F_1\le x^{″}\left(t\right)\le L_1 \mathrm{on} [\nu ,1]$$

and so the bounds for $x^{″}\left(t\right)$ follow. Further, as in the proof of Lemma 4 we establish consecutively the bounds for $|x^{\prime }\left(t\right)|$ and $\left|x\right(t\left)\right|.$. □

3. Existence Results

Theorem 1.

Let(H${}_{\mathbf{1}}$)hold for $K=C$ and(H${}_{\mathbf{3}}$)hold for

$$M_0=\left|A\right|+\left|B\right|+max\left\{\right|F_1^{\prime }{|,|}^{\prime }{L}_1\left|\right\},m_0=-M_0,$$

$$M_1=\left|B\right|+max\left\{\right|F_1^{\prime }|,|L_1^{\prime }\left|\right\},m_1=-M_1,m_2=F_1^{\prime },M_2=L_1^{\prime }.$$

Then each of BVPs (1), (2) and (1), (3) has at least one solution in $C^3[0,1].$

Proof. 

We will establish that the assertion is true for problem (1), (2) after checking that the hypotheses of Lemma 1 are fulfilled; it follows similarly and for (1), (3). We easily check that (i) holds for (1)${}_0,$ (2). Clearly, BVP (1), (2) is equivalent to BVP (1)${}_1,$ (2) and so (ii) is satisfied. Since now ${\mathbf{L}}_h=x^{‴},$(iii) also holds. Next, according to Lemma 4, for each solution $x\in C^3[0,1]$ to (1)${}_{\lambda }$, (2) we have

$$m_i\le x^{\left(i\right)}\left(t\right)\le M_i,t\in [0,1],i=0,1,2.$$

Now use that f is continuous on $[0,1]\times J$ to conclude that there are constants $m_3$ and $M_3$ such that

$$m_3\le \lambda f(t,x,p,q)\le M_3 \mathrm{for} \lambda \in [0,1] \mathrm{and} (t,x,p,q)\in [0,1]\times J,$$

which together with $(x\left(t\right),x^{\prime }\left(t\right),x^{″}\left(t\right))\in J$ for $t\in [0,1]$ and Equation (1)${}_{\lambda }$ implies

$$m_3\le x^{‴}\left(t\right)\le M_3,t\in [0,1].$$

These observations imply that (iv) holds, too. Finally, the continuity of f on the set J gives (v) and so the assertion is true by Lemma 1. □

Theorem 2.

Let(H${}_{\mathbf{1}}$)hold for $K=C$ and(H${}_{\mathbf{3}}$)hold for

$$M_0=\left|A\right|+|B-A|+max\left\{\right|F_1^{\prime }|,|L_1^{\prime }\left|\right\},m_0=-M_0,$$

$$M_1=|B-A|+max\left\{\right|F_1^{\prime }|,|L_1^{\prime }\left|\right\},m_1=-M_1,m_2=F_1^{\prime },M_2=L_1^{\prime }.$$

Then BVP (1), (4) has at least one solution in $C^3[0,1].$

Proof. 

It follows the lines of the proof of Theorem 1. Now the bounds

$$m_i\le x^{\left(i\right)}\left(t\right)\le M_i,t\in [0,1],i=0,1,2,$$

for each solution $x\in C^3[0,1]$ to a (1)${}_{\lambda }$, (4) follow from Lemma 5. □

Theorem 3.

Let(H${}_{\mathbf{1}}$)and(H${}_{\mathbf{2}}$)hold for $K=C-B$ and(H${}_{\mathbf{3}}$)hold for

$$M_0=\left|A\right|+\left|B\right|+max\left\{\right|F_1|,|L_1|,|F_1^{\prime }|,|L_1^{\prime }\left|\right\},m_0=-M_0,$$

$$M_1=\left|B\right|+max\left\{\right|F_1|,|L_1|,|F_1^{\prime }|,|L_1^{\prime }\left|\right\},m_1=-M_1,$$

$$m_2=min\{F_1,F_1^{\prime }\},M_2=max\{L_1,L_1^{\prime }\}.$$

Then each of BVPs (1), (5) and (1), (6) has at least one solution in $C^3[0,1].$

Proof. 

Arguments similar to those in the proof of Theorem 1 yield the assertion. Now the bounds

$$m_i\le x^{\left(i\right)}\left(t\right)\le M_i,t\in [0,1],i=0,1,2,$$

for each solution $x\in C^3[0,1]$ to (1)${}_{\lambda }$, (5) and (1)${}_{\lambda }$, (6) follow from Lemma 6. □

4. Examples

Through several examples we will illustrate the application of the obtained results.

Example 1.

Consider the BVPs for the equation

$$x^{‴}\left(t\right)=exp(x^{″}-3)+5x^{″}(x^{\prime 2}+1)-tsinx,t\in (0,1),$$

with boundary conditions (2) or (3).

For $F_2^{\prime }=-\left|C\right|-2,$ $F_1^{\prime }=-\left|C\right|-1,$ $L_1^{\prime }=max\left\{\right|C|,3\}+1,$ $L_2^{\prime }=max\left\{\right|C|,3\}+2$ and $\sigma =0.1,$ for example, each of these problems has a solution in $C^3[0,1]$ by Theorem 1.

Example 2.

Consider the BVP

$$x^{‴}\left(t\right)=\phi (t,x,x^{\prime })\left(lg\left((x^{″}+50)(60-x^{″})\right)-3\right),t\in (0,1),$$

$$x\left(0\right)=5,x^{\prime }\left(0\right)=10,x^{\prime }\left(1\right)=40,$$

where $\phi :[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ is continuous and does not change its sign.

If $\phi (t,x,p)\ge 0$ on $[0,1]\times {\mathbb{R}}^2,$ the assumptions of Theorem 3 are satisfied for $F_2=-36,$ $F_1=-35,F_2^{\prime }=-46,F_1^{\prime }=-45,L_1^{\prime }=40,L_2^{\prime }=41,L_1=55,L_2=56$ and $\sigma =0.01,$ for example, and if $\phi (t,x,p)\le 0$ on $[0,1]\times {\mathbb{R}}^2,$ they are satisfied for $F_2^{\prime }=-36,F_1^{\prime }=-35,F_2=-46,F_1=-45,$ $L_1=40,L_2=41,L_1^{\prime }=55,$ $L_2^{\prime }=56$ and $\sigma =0.01,$ for example; it is clear, $K=30.$ Thus, the considered problem has at least one solution in $C^3[0,1].$ Let us note, here $D_q=(-50,60).$

Example 3.

Consider the BVP

$$x^{‴}\left(t\right)=\frac{t(x^{″}+8)(x^{″}+3)\sqrt{625-x^{\prime 2}}}{\sqrt{900-x^2}\sqrt{100-x^{\prime \prime 2}}},t\in (0,1),$$

$$x\left(0\right)=9,x\left(1\right)=1,x^{″}\left(1\right)=-4.$$

For $F_2^{\prime }=-6,F_1^{\prime }=-5,L_1^{\prime }=-3,L_2^{\prime }=-2$ and $\sigma =0.1,$ for example, this problem has a positive, decreasing, concave solution in $C^3[0,1]$ by Theorem A1; notice, here $D_x,D_p$ and $D_q$ are bounded.