Existence of solutions to fourth-order differential equations with deviating arguments

In this paper, we consider fourth-order differential equations on a half-line with deviating arguments of the form \(u^{(4)}(t)+q(t)f(t,[u(t)],[u'(t)],[u''(t)],u'''(t))=0\), \(0< t<{+\infty}\), with the boundary conditions \(u(0)=A\), \(u'(0)=B\), \(u''(t)-au'''(t)=\theta(t)\), \(-\tau\leq{t}\leq 0\); \(u'''(+\infty)=C\). We present sufficient conditions for the existence of a solution between a pair of lower and upper solutions by using Schäuder’s fixed point theorem. Also, we establish the existence of three solutions between two pairs of lower and upper solutions by using topological degree theory. An important feature of our existence criteria is that the obtained solutions may be unbounded. We illustrate the importance of our results through two simple examples.

In recent years considerable attention has been focused on the existence of solutions to boundary value problems involving differential equations with deviating arguments (DEDA) [1–16]. While most of these works deal with problems on finite intervals and the literature is satisfactory, study of infinite interval problems has been just initiated in [2, 13–17]. This study compare to boundary value problems for second and higher order ordinary differential equations over infinite intervals (and their wide variety of applications to real world problems) [18–24] is far from complete, and needs attention. To fill some of this gap, in this paper we shall provide existence criteria for fourth-order differential equations with deviating arguments of the form

where

and \(q:(0,+\infty)\rightarrow{(0,+\infty)}\), \(f:[0,+\infty)\times{\mathbb{R}}^{n+1}\times{\mathbb{R}}^{n+1}\times{\mathbb{R}}^{n+1}\times\mathbb{R}\rightarrow{\mathbb{R}}\) are continuous, and \(\tau_{j,i}\): \([0,+\infty)\rightarrow{(0,+\infty)}\) are continuous for all \(j=0,1,2\), \(i=1,2,\ldots,n\). In what follows we shall always assume that \(\lim_{t\rightarrow{+\infty}}(t-\tau_{j,i}(t))=+\infty\), \(j=0,1,2\), \(i=1,2,\ldots,n\). We define the positive real number τ as

and seek the solutions of (1.1) which satisfy the boundary conditions

where \(\theta\in\mathcal{C}[-\tau,0]\), \(A,B\in\mathbb{R}\), \(a,C\geq 0\), and \(u'''(+\infty)= \lim_{t\rightarrow{+\infty}}u'''(t)\). For this, following as in several above works, and inspired by the contributions in [25–30], we shall employ the method of upper and lower solutions.

The plan of this paper is as follows: In Section 2, we state some definitions and lemmas which are needed to prove the main results. In Section 3, we show that in the presence of a pair of upper and lower solutions the problem (1.1)-(1.2) has at least one solution. Also in this section, we establish that in the presence of two pairs of upper and lower solutions the problem (1.1)-(1.2) has at least three solutions. Finally, in Section 4, we illustrate two examples which show the importance of our results.

In this section we introduce some necessary definitions, lemmas, and preliminary results that will be used in main results which give the existence of solutions of the problem (1.1)-(1.2). First, we construct the Green’s function for the linear boundary value problem

Lemma 2.1

Let \(e\in{L^{1}[0,+\infty)}\). Then the solution \(u\in\mathcal{C}^{3}[-\tau,+\infty) \cap\mathcal {C}^{4}(0,+\infty)\) of the problem (2.1) can be expressed as

where

and

Proof

Since \(e\in{L^{1}[0,+\infty)}\), we can integrate (2.1) from t to +∞, and use \(u'''(+\infty)=C\), to get

Integrating the above equation on \([0,t]\), and applying Fubini’s theorem and using \(u''(0)-au'''(0)=\theta(0)\), we obtain

Again integrating (2.4) twice on \([0,t]\), and applying Fubini’s theorem and using \(u(0)=A\) and \(u'(0)=B\), we find

for \(t\in[0,+\infty)\). Now we consider the following third order linear differential equation:

If the above equation is rearranged, we have

and solving this linear equation on \([t,0]\), we find

Next, integrating (2.5) twice on \([t,0]\), and applying Fubini’s theorem and using the following boundary conditions:

we obtain

for \(t\in[-\tau,0]\). This completes the proof of the lemma. □

Remark 2.2

\(G(t,s)\) defined in (2.3) is the Green’s function of the BVP

Lemma 2.3

The Green’s function \(G(t,s)\) has the following properties:

1. (1)

   \(G(t,s)\) is twice continuously differentiable on \([0,+\infty )\times[0,+\infty)\) and

   $$\frac{\partial^{3}G(t,s)}{\partial{t^{3}}} \Big|_{t=s^{+}}-\frac{\partial ^{3}G(t,s)}{\partial{t^{3}}} \Big|_{t=s^{-}}=-1; $$
2. (2)

   \(\frac{\partial^{i}G(t,s)}{\partial{t^{i}}}\geq 0\), \(\forall(t,s)\in[0,+\infty)\times[0,+\infty)\), for \(i=0,1,2,3\);

3. (3)

   \(\sup_{t\in[0,+\infty)}\frac{G(t,s)}{1+t^{3}}\leq (\frac{a\sqrt[3]{4}+1}{6} )\), \(\sup_{t\in[0,+\infty)} ( \frac{1}{1+t^{2}}\frac {\partial G(t,s)}{\partial{t}} )\leq (\frac{a+1}{2} )\), \(\sup_{t\in[0,+\infty)} (\frac{1}{1+t}\frac{\partial ^{2}G(t,s)}{\partial{t^{2}}} )\leq(a+1)\), \(\sup_{t\in[0,+\infty)}\frac{\partial^{3}G(t,s)}{\partial {t^{3}}}\leq1\).

Proof

(1) and (2) are obvious. Here we shall prove the first inequality of (3). We note that for all integers k and l

For \(s\leq{t}\), we have

and for \(s\geq{t}\)

The other parts can be proved similarly. □

We consider the space X defined by

with the norm

where

It is clear that \((X,\|\cdot\|)\) is a Banach space. Next we define the mapping \(T:X\rightarrow\mathcal{C}^{3}[-\tau,+\infty )\cap\mathcal{C}^{4}(0,+\infty)\) by

where

and

Lemma 2.4

The mapping \(T:X\rightarrow\mathcal{C}^{3}[-\tau,+\infty)\cap\mathcal {C}^{4}(0,+\infty)\) in (2.6) has the following properties:

1. (1)

   \(Tu(0)=A\), \((Tu)'(0)=B\), \((Tu)''(t)-a(Tu)'''(t)=\theta(t)\) for \(t\in [-\tau,0]\),

2. (2)

   \(Tu(t)\) is three-times continuously differentiable on \(t\in[-\tau ,+\infty)\),

3. (3)

   \((Tu)^{(4)}(t)=-q(t)f(t,[u(t)],[u'(t)],[u''(t)],u'''(t))\), \(t\in (0,+\infty)\),

4. (4)

   fixed points of T are solutions of BVP (1.1)-(1.2).

When applying the Schäuder fixed point theorem to prove the existence result, it is necessary to show that the operator \(T_{1}\) (defined later) is completely continuous. For this, we need the following modified version of the Arzela-Ascoli lemma (see [18, 20]).

Lemma 2.5

\(M\subset{X}\) is relatively compact if the following conditions hold:

1. (1)

   all functions belonging to M are uniformly bounded,

2. (2)

   all functions belonging to M are equi-continuous on any compact sub-interval of \([-\tau,+\infty)\),

3. (3)

   all functions from M are equi-convergent at infinity, that is, for any \(\epsilon>0\), there exists a \(T=T(\epsilon)>0\) such that, for all \(t\geq T\) and any \(u\in{M}\),

   $$\begin{aligned}& \biggl|\frac{u(t)}{1+t^{3}}-\lim_{t\rightarrow+\infty}\frac{u(t)}{1+t^{3}} \biggr| < \epsilon,\qquad \biggl|\frac{u'(t)}{1+t^{2}}-\lim_{t\rightarrow+\infty }\frac{u'(t)}{1+t^{2}} \biggr|< \epsilon,\\& \biggl|\frac{u''(t)}{1+t}-\lim_{t\rightarrow+\infty}\frac {u''(t)}{1+t} \biggr|< \epsilon \quad\textit{and} \quad \bigl|u'''(t)-\lim_{t\rightarrow+\infty}u'''(t)\bigr|< \epsilon. \end{aligned}$$

Definition 2.6

A function \(\alpha\in X\cap{\mathcal {C}^{4}(0,+\infty)}\) is called a lower solution of (1.1)-(1.2) provided

Similarly, a function \(\beta\in X\cap{\mathcal{C}^{4}(0,+\infty)}\) is called an upper solution of (1.1)-(1.2) provided

Also, we say \(\alpha(\beta)\) is a strict lower solution (strict upper solution) for problem (1.1)-(1.2) if all the above inequalities are strict.

Remark 2.7

If

then on integrating (2.12) and using the boundary restrictions in Definition 2.6, we find that \(\alpha'(t)\leq\beta'(t)\), \(\alpha (t)\leq\beta(t)\) for all \(t\in[0,+\infty)\) and \(\beta'(t)\leq\alpha '(t)\), \(\alpha(t)\leq\beta(t)\) for all \(t\in[-\tau,0)\).

Definition 2.8

Let α, β be lower and upper solutions for the problem (1.1)-(1.2) satisfying

A continuous function f is said to satisfy Nagumo’s condition with respect to the pair of functions α, β if there exist positive functions φ and \(h\in{\mathcal{C}[0,+\infty)}\) such that

for all \(t\in[0,+\infty)\), and \((x_{0},\ldots,x_{n})\in{[[\alpha(t)],[\beta(t)]]}\), \(y_{i}(t-\tau_{1,i}(t))\in[\alpha'(t-\tau_{1,i}(t)),\beta'(t-\tau _{1,i}(t))]\) if \(t-\tau_{1,i}(t)>0\), \(y_{i}(t-\tau_{1,i}(t))\in[\beta'(t-\tau_{1,i}(t)),\alpha'(t-\tau _{1,i}(t))]\) if \(t-\tau_{1,i}(t)\leq0\), \(0\leq i\leq n\), \(\tau_{1,0}=0\), \((z_{0},\ldots,z_{n})\in[[\alpha''(t)], [\beta''(t)]] \), \(w\in\mathbb{R}\), and

In this section we state and prove our existence results. We begin with the following lemma.

Lemma 3.1

Suppose the following conditions hold.

(H₁):

BVP (1.1)-(1.2) has a pair of lower and upper solutions α, β satisfying

$$\begin{aligned} \alpha''(t)\leq\beta''(t),\quad \textit{for }t\in[-\tau,+\infty), \end{aligned}$$

and f satisfies Nagumo’s condition with respect to the pair of functions α, β.

(H₂):

There exists a constant \(\gamma>1\) such that

$$\begin{aligned} \sup_{0\leq{t}< +\infty}(1+t)^{\gamma}q(t)\varphi (t)< +\infty, \end{aligned}$$

where φ is the function in Nagumo’s condition of f.

and

satisfies \(\|u\|^{3}_{\infty}<{R}\).

Proof

We can choose \(R>\eta\) such that

and

where C is the nonhomogeneous boundary value, and

Let u be a solution of the differential equation (1.1) satisfying (3.1) and (3.2). If \(|u'''(t)|< R\), for all \(t\in[0,+\infty)\), there is nothing to prove. If this is not true, there exists a \(t_{0}\in[0,+\infty)\) such that \(|u'''(t_{0})|\geq R\). Since \(\lim_{t\rightarrow+\infty} u'''(t)=C < R\), there exists a \(T>0\) such that

Let \(t_{1}=\inf\{t\leq T: |u'''(s)|< R\mbox{ for all }s\in[t,+\infty)\}\). Then \(|u'''(t_{1})|=R\) and \(|u'''(t)|< R\) for all \(t>t_{1}\) and there exists a \(t_{2}\) such that \(|u'''(t)|\geq R\) for \(t\in[t_{2},t_{1}]\). So we have two cases to consider \(u'''(t_{1})=R\) and \(u'''(t)\geq R\) for \(t\in [t_{2},t_{1}]\), or \(u'''(t_{1})=-R\) and \(u'''(t)\leq-R\) for \(t\in[t_{2},t_{1}]\). We assume that \(u'''(t_{1})=R\) and \(u'''(t)\geq R\) for \(t\in[t_{2},t_{1}]\), then we have

which is a contradiction. In the case \(u'''(t_{1})=-R\) and \(u'''(t)\leq -R\) for \(t\in[t_{2},t_{1}]\), we obtain a similar contradiction. Thus, \(|u'''(t)|< R\) for all \(t\in[0,+\infty)\). From the boundary condition (1.2) we also have

for all \(t\in[-\tau,0]\). Therefore, \(|u'''(t)|< R\) for \(t\in[-\tau,0)\). To sum up, we have \(\|u\|^{3}_{\infty}<{R}\). □

Theorem 3.2

Suppose conditions (H₁) and (H₂) hold. Suppose further that

(H₃):

For any fixed \(t\in[0,+\infty)\), \(y_{i},z_{i},w\in{\mathbb{R}}\), \(i=0,\ldots,n\), when

$$\begin{aligned} &\alpha\bigl(t-\tau_{0,i}(t)\bigr)\leq{x_{i}}\leq\beta \bigl(t-\tau_{0,i}(t)\bigr), \quad i=0,1,\ldots,n,\\ &f\bigl(t,x_{0},x_{1},\ldots,\alpha\bigl(t- \tau_{0,i}(t)\bigr),\ldots,x_{n},y_{0}, \ldots,y_{n},z_{0},\ldots,z_{n},w\bigr) \\ &\quad\leq {f}(t,x_{0},x_{1},\ldots,x_{i}, \ldots,x_{n},y_{0},\ldots,y_{n},z_{0}, \ldots,z_{n},w) \\ &\quad\leq{f}\bigl(t,x_{0},x_{1},\ldots,\beta\bigl(t-\tau _{0,i}(t)\bigr),\ldots,x_{n},y_{0}, \ldots,y_{n},z_{0},\ldots,z_{n},w\bigr). \end{aligned}$$

(H₄):

For any fixed \(t\in[0,+\infty)\), \(x_{i},z_{i},w\in{\mathbb{R}}\), \(i=0,\ldots,n\) when

$$\alpha'\bigl(t-\tau_{1,i}(t)\bigr)\leq{y_{i}} \leq\beta'\bigl(t-\tau_{1,i}(t)\bigr),\quad t- \tau_{1,i}(t)>0, $$

or when

$$\begin{aligned}& \beta'\bigl(t-\tau_{1,i}(t)\bigr)\leq{y_{i}} \leq\alpha'\bigl(t-\tau_{1,i}(t)\bigr), \quad t- \tau_{1,i}(t)\leq0, i=0,1,\ldots,n, \\& f\bigl(t,x_{0},\ldots,x_{n},y_{0},\ldots, \alpha'\bigl(t-\tau_{1,i}(t)\bigr), \ldots,y_{n},z_{0}, \ldots,z_{n},w\bigr) \\& \quad\leq{f}(t,x_{0},\ldots,x_{n},y_{0}, \ldots,y_{i},\ldots,y_{n},z_{0}, \ldots,z_{n},w) \\& \quad\leq{f}\bigl(t,x_{0},\ldots,x_{n},y_{0}, \ldots,\beta'\bigl(t-\tau _{1,i}(t)\bigr), \ldots,y_{n},z_{0},\ldots,z_{n},w\bigr). \end{aligned}$$

(H₅):

For any fixed \(t\in[0,+\infty)\), \(x_{i},y_{i},w\in{\mathbb{R}}\), \(i=0,\ldots,n\) when

$$\begin{aligned}& \alpha''\bigl(t-\tau_{2,i}(t)\bigr) \leq{z_{i}}\leq\beta''\bigl(t- \tau_{2,i}(t)\bigr), \quad i=0,1,\ldots,n, \\& f\bigl(t,x_{0},\ldots,x_{n},y_{0}, \ldots,y_{n},z_{0},\ldots,\alpha '' \bigl(t-\tau_{2,i}(t)\bigr),\ldots,z_{n},w\bigr)\\& \quad\leq {f}(t,x_{0},\ldots,x_{n},y_{0}, \ldots,y_{n},z_{0},\ldots,z_{i}, \ldots,z_{n},w)\\& \quad\leq{f}\bigl(t,x_{0},\ldots,x_{n},y_{0}, \ldots,y_{n},z_{0},\ldots,\beta'' \bigl(t-\tau _{2,i}(t)\bigr),\ldots,z_{n},w\bigr), \end{aligned}$$

where \(\tau_{0,0}=\tau_{1,0}=\tau_{2,0}=0\).

(H₆):

$$\int_{0}^{\infty}\max\{s,1\}q(s)\,ds< +\infty,\qquad \int_{0}^{\infty}\max\{s,1\}q(s)\varphi(s)\,ds< +\infty. $$

Proof

Let R be a positive number as in Lemma 3.1 and define the auxiliary functions,

as follows:

where, for \(i=1,2,\ldots,n\),

if \(t-\tau_{1,i}(t)>0\),

if \(t-\tau_{1,i}(t)\leq0\),

and

We consider the modified boundary value problem

with the boundary conditions (1.2). We will show that the problem (3.3)-(1.2) has at least one solution u in X. Now for \(u\in X\), we define two operators \(\widetilde{T}_{1}\), \(T_{1}\) by

and

where \(l(t)\) is as in (2.7) and

We want to show that the operator \(T_{1}\) is completely continuous. We split the proof in the following parts:

(1) \(T_{1}:X\rightarrow{X}\) is well defined. Obviously, for any \(u\in{X}\) by direct calculation, we have

and for \(t\in(0,+\infty)\),

which show that \(T_{1}u(t)\in{C^{3}[-\tau,+\infty)}\). Further, we have

where \(H= \max_{0\leq{s}\leq\sup_{t\in[0,+\infty )}|u'''(t)|}h(s)\). Now from (3.5) it follows that

which implies

Next since

we also have

By Lebesgue’s dominated convergent theorem, L’Hopital’s rule, and (3.6), (3.7), we obtain

that is, \(\lim_{t\rightarrow+\infty}\frac{(\widetilde {T}_{1}u)(t)}{1+t^{3}}=0\), and

which implies that \(\sup_{t\in[0,+\infty)}\frac {|(T_{1}u)(t)|}{1+t^{3}}<+\infty\). Similarly, we have

that is, \(\lim_{t\rightarrow+\infty}\frac{(\widetilde {T}_{1}u)'(t)}{1+t^{2}}=0\) and \(\lim_{t\rightarrow+\infty}\frac{(\widetilde {T}_{1}u)'(t)}{1+t^{2}}= \lim_{t\rightarrow+\infty}\frac{l'(t)}{1+t^{2}}+\lim_{t\rightarrow +\infty}\frac{(T_{1}u)'(t)}{1+t^{2}}=\frac{C}{2}\), which implies that \(\sup_{t\in[0,+\infty)}\frac {|(T_{1}u)'(t)|}{1+t^{2}}<+\infty\),

that is, \(\lim_{t\rightarrow+\infty}\frac{(\widetilde {T}_{1}u)''(t)}{1+t}=0\) and \(\lim_{t\rightarrow+\infty}\frac{(T_{1}u)''(t)}{1+t}= \lim_{t\rightarrow+\infty}\frac{l''(t)}{1+t}+\lim_{t\rightarrow+\infty }\frac{(\widetilde{T}_{1}u)''(t)}{1+t}=C\), which implies that \(\sup_{t\in[0,+\infty)}\frac {|(T_{1}u)''(t)|}{1+t}<+\infty\), and by (3.5)

then

Therefore \(T_{1}u\in{X}\).

(2) \(T_{1}:X\rightarrow{X}\) is continuous. For any convergent sequence \(u_{m}\rightarrow{u}\) in X, we have

Thus the continuity of \(F_{3}\) implies that

Since \(u'''_{m}(t)\rightarrow{u'''(t)}\), \(\sigma=\sup\{s_{1}:s_{1}=\sup_{t\in[0,+\infty)}|u_{m}'''(t)|, m\in{N}\}< +\infty\).

Let \(H_{1}= \max_{0 \leq s\leq\max\{\sup_{t\in[0,+\infty )}|u'''(t)|, \sigma\}}h(s)\). Then we have

Thus, we find

that is,

as \(m\rightarrow+\infty\).

that is,

as \(m\rightarrow+\infty\).

that is,

as \(m\rightarrow+\infty\).

that is,

as \(m\rightarrow+\infty\).

that is,

as \(m\rightarrow +\infty\).

that is,

as \(m\rightarrow+\infty\).

To show \(\|T_{1}u_{m}-T_{1}u\|^{3}_{\infty}\rightarrow0\), as \(m\rightarrow+\infty\), we need the following:

and

Hence, it follows that

that is,

as \(m\rightarrow+\infty\).

Combining (3.8)-(3.14), we find \(\|(T_{1}u_{m})-(T_{1}u)\| \rightarrow0\), as \(m\rightarrow+\infty\); so \(T_{1}:X\rightarrow{X}\) is continuous.

(3) \(T_{1}:X\rightarrow{X}\) is compact. The operator \(T_{1}\) is compact if \(T_{1}\) maps bounded subsets of X into relatively compact sets. Let K be any bounded subset of X, then \(r_{3}= \sup_{0\leq s\leq\{\sup_{t\in[0,+\infty )}|u'''(t)|,u\in K\}} h(s)<+\infty\). For any \(u\in{K}\), we have the following:

which implies that

where

Therefore, \(T_{1}K\) is uniformly bounded. We also know that \(\psi_{1}(t)\) and \(\psi'_{1}(t)\) are continuous on \([-\tau ,0]\). Thus in view of \([-\tau,0]\) compact, \(\psi_{1}(t)\) and \(\psi'_{1}(t)\) are also uniformly continuous. Thus it follows that for \(t_{1},t_{2}\in [-\tau,0]\),

further since

is continuous on \([-\tau,0]\), we find

Next, for \(t_{1},{t_{2}}\in[0,\varepsilon]\) with \(\varepsilon>0\) a constant, we have

and

Thus, \(T_{1}K\) is equi-continuous. Finally, we will show that \(T_{1}K\) is equi-convergent at infinity. In fact, when \(t>0\) we have

and

Hence all conditions of Lemma 2.5 are fulfilled, so \(T_{1}K\) is relatively compact. Therefore, \(T_{1}:X\rightarrow{X}\) is completely continuous.

(4) \(T_{1}:X\rightarrow{X}\) has at least one fixed point. Let \(\Omega=\{u\in{X},\|u\|\leq N\}\) where

For any \(u\in\Omega\), it is easy to see that \(\|T_{1}u\|\leq\Omega\), and thus \(T_{1}\Omega\subset\Omega\). The Schäuder fixed point theorem now guarantees that the operator \(T_{1}\) has at least one fixed point in Ω, which is a solution of BVP (3.3)-(1.2). Now we shall show that this solution u satisfies the inequalities (3.1) and (3.2) which in view of the definitions of \(F_{3}\), \(F_{2}\), \(F_{1}\), and \(F_{0}\) will imply that u is in fact a solution of (1.1)-(1.2). For this, we only prove that \({u''(t)}\leq\beta''(t)\), \(t\in[-\tau ,+\infty)\). A similar argument can be used to prove \({\alpha''(t)\leq u''(t)}\), \(t\in[-\tau,+\infty)\). If not true, we set \(\omega(t)=u''(t)-\beta''(t)\), then there exists \(t^{*}\in[-\tau,+\infty)\) such that \(\omega(t^{*})=\sup_{-\tau\leq{t}<+\infty}\omega(t)>0\). Obviously, if \(t^{*}=-\tau\) then \(\omega'(t^{*})\leq0\), and if \(t^{*}\in(-\tau,0]\) then \(\omega'(t^{*})=0\). However, from the boundary condition, we have \(\omega'(t^{*})= \frac{1}{a}\omega(t^{*})>0\), which gives a contraction. If \(t^{*}\in(0,+\infty)\), then we have

By the definition of auxiliary functions and \(R>\sup_{t\in[0,+\infty )}|\beta'''(t)|\), we have

Now if \(u''(t^{*}-\tau_{2,1}(t^{*}))>\beta''(t^{*}-\tau_{2,1}(t^{*}))\) from the definition of \(\widetilde{z}_{1}\) it follows that

and if \(u''(t^{*}-\tau_{2,1}(t^{*}))\leq\beta''(t^{*}-\tau_{2,1}(t^{*}))\) from the condition (H₅), we have

Similarly, we consider the cases \(u''(t^{*}-\tau_{2,i}(t^{*}))>\beta''(t^{*}-\tau_{2,i}(t^{*}))\) or \(u''(t^{*}-\tau_{2,i}(t^{*}))\leq\beta''(t^{*}-\tau_{2,i}(t^{*}))\), \(i=2,3,\ldots,n\), and obtain the inequality

Next, if \(u'(t^{*})>\beta'(t^{*})\) from the definition of \(F_{1}\) it follows that

and if \(u'(t^{*})\leq\beta'(t^{*})\) from the definition of \(F_{1}\) and the condition (H₄) we have

If \(t^{*}-\tau_{1,1}(t^{*})>0\), while discussing the cases \(u'(t^{*}-\tau_{1,1}(t^{*}))>\beta'(t^{*}-\tau_{1,1}(t^{*}))\) we use the definition of \(\widetilde{y}_{1}\), and when discussing the cases \(u'(t^{*}-\tau _{1,1}(t^{*}))\leq\beta'(t^{*}-\tau_{1,1}(t^{*}))\) we use (H₄), and obtain

Similarly, if \(t^{*}-\tau_{1,1}(t^{*})\leq0\), while discussing the cases \(u'(t^{*}-\tau_{1,1}(t^{*}))\geq\beta'(t^{*}-\tau_{1,1}(t^{*}))\) we use (H₄), while when discussing the cases \(u'(t^{*}-\tau _{1,1}(t^{*}))<\beta'(t^{*}-\tau_{1,1}(t^{*}))\) we use the definition of \(\widetilde{y}_{1}\), to again find

Following exactly as above, using the definition of \(\widetilde{y}_{i}\) and (H₄), we consider the cases \(i=2,\ldots,n\) to finally obtain

Next, if \(u(t^{*})>\beta(t^{*})\) from the definition of \(F_{0}\) it follows that

and if \(u(t^{*})\leq\beta(t^{*})\) from the definition of \(F_{0}\) and the condition (H₃) we have

Similarly, we use the definition of \(\widetilde{x}_{i}\) and (H₃) while discussing the cases \(u(t^{*}-\tau_{0,i}(t^{*}))>\beta(t^{*}-\tau _{0,i}(t^{*}))\) or \(u(t^{*}-\tau_{0,i}(t^{*}))\leq\beta(t^{*}-\tau_{0,i}(t^{*}))\), \(i=1,\ldots,n\), to get

which implies that

which is a contractions.

If \(t^{*}=+\infty\) then \(\omega(+\infty)= \sup_{t\in[-\tau ,+\infty)}\omega(t)>0\). From the boundary conditions, we also have \(\omega'(+\infty)=u'''(+\infty)-\beta'''(+\infty)\leq0\). But this implies that \(\omega''(+\infty)\leq0\) and \(\omega'(+\infty )=0\). However, now following as above, we find \(\omega''(+\infty)>0\), which is a contradiction. Thus, \(u''(t)\leq\beta''(t)\), \(t\in[-\tau ,+\infty)\).

Consequently, we have

which on integration and using boundary conditions gives

and now a further integration leads to

Further, since all conditions of the Lemma 3.1 are satisfied, \(\|u\|^{3}_{\infty}< R\). Consequently, we have

and hence, u is a solution of (1.1)-(1.2). □

Theorem 3.3

Assume that there exist two pairs of upper and lower solutions \(\beta _{k}\), \(\alpha_{k}\), \(k=1,2\) of BVP (1.1)-(1.2), where \(\alpha_{2}\), \(\beta_{1}\) are strict and

and f satisfies Nagumo’s condition with respect to \(\alpha_{1}\), \(\beta_{2}\). Suppose further that conditions (H₂)-(H₆) hold with α and β replaced by \(\alpha_{1}\) and \(\beta_{2}\), respectively. Then the problem (1.1)-(1.2) has at least three solutions \(u_{1}\), \(u_{2}\), and \(u_{3}\) such that

Proof

First we define the truncated functions \(\widetilde {F}_{0}\), \(\widetilde{F}_{1}\), \(\widetilde{F}_{2}\), \(\widetilde{F}_{3}\) the same as \(F_{0}\), \(F_{1}\), \(F_{2}\), \(F_{3}\) in Theorem 3.2 with α, β replaced by \(\alpha_{1}\) and \(\beta_{2}\), respectively. Consider the modified differential equation

To show that (3.18)-(1.2) has at least three solutions, we define operators \(\widetilde{T}_{2}\), \(T_{2} \) as

and

where \(l(t)\) is as in (2.7) and

As in Theorem 3.2, \(T_{2}\) is completely continuous. By using the degree theory, we will show that \(T_{2}\) has at least three fixed points which are solutions of (3.18)-(1.2). We note that R in Lemma 3.1 instead of α, β now depends on \(\alpha_{1}\), \(\beta_{2}\). Set \(\Omega_{2}=\{u\in{X},\|u\|< N\}\) where N is as in (3.15) then for any \(u\in\overline{\Omega}_{2}\), it follows that \(\|T_{2}u\|< N\), thus \(T_{2}\overline{\Omega}_{2}\subset{\Omega_{2}}\), and so we have \(\deg(I-T_{2},\Omega_{2},0)=1\). Set

Since \(\alpha_{2}''\nleq\beta_{1}''\), \(\alpha^{\prime\prime}_{1}(t)\leq{\alpha ^{\prime\prime}_{2}(t)}\leq\beta^{\prime\prime}_{2}(t)\), \(\alpha^{\prime\prime}_{1}(t)\leq{\beta ^{\prime\prime}_{1}(t)}\leq\beta^{\prime\prime}_{2}(t)\), we find \(\Omega_{\alpha_{2}}\neq\emptyset\neq\Omega^{\beta_{1}}\), \(\overline{\Omega}_{\alpha_{2}}\cap\overline{\Omega^{\beta _{1}}}=\emptyset\), \(\Omega_{2}\backslash{\overline{\Omega_{\alpha_{2}}\cup\Omega^{\beta _{1}}}}\neq\emptyset\). Now since \(\alpha_{2}\), \(\beta_{1}\) are strict lower and upper solutions there is no solution in \(\partial\Omega_{\alpha_{2}}\cup\partial\Omega^{\beta_{1}}\). The additivity of degree implies that

We will show that \(\deg(I-T_{2},\Omega_{\alpha_{2}},0)=\deg(I-T_{2},\Omega^{\beta _{1}},0)=1\). For this, we define new operators \(\widetilde{T}_{3}:\overline{\Omega }_{{2}}\rightarrow\overline{\Omega}_{{2}}\) and \(T_{3}:\overline{\Omega }_{{2}}\rightarrow\overline{\Omega}_{{2}}\) as

and

where \(l(t)\) is as in (2.7) and

Here the functions \(\widehat{F}_{0}\), \(\widehat{F}_{1}\), \(\widehat{F}_{2}\), \(\widehat {F}_{3}\) are same as \(\widetilde{F}_{0}\), \(\widetilde{F}_{1}\), \(\widetilde {F}_{2}\), \(\widetilde{F}_{3}\) except that \(\alpha_{1}\) is replaced by \(\alpha_{2}\). Now similar to the proof of Theorem 3.2 we find that u is a fixed point of \(T_{3}\) only when \(\alpha''_{2}(t)\leq{u''(t)}\leq\beta''_{2}(t)\), \(t\in[-\tau,+\infty)\). Since the lower solution \(\alpha_{2}\) is strict, \(\alpha''_{2}(t)\neq u''(t)\), \(t\in(-\tau,+\infty)\). Therefore, \(u\in\Omega_{\alpha_{2}}\). Hence, it follows that

Also, \(T_{3}\overline{\Omega}_{2}\subset\Omega_{2}\), so that we have

Therefore,

Similarly, we have

and this leads to

Finally, using the properties of the degree, we conclude that \(T_{2}\) has at least three fixed points

which are the claimed solutions of the BVP (1.1)-(1.2). □

Example 4.1

Consider the fourth-order nonlinear differential equation on the half-line with deviating arguments

where

Clearly, (4.1) is a particular case of (1.1) with \(q(t)= \frac{1}{(1+t)^{3}}\),

It follows that

We consider (4.1) together with the following boundary conditions:

Comparing this with (1.2), we find \(\theta(t)= \frac{4}{3}\), \(a=\frac{1}{3}\), \(A=2\), \(B=0\), \(C=0\).

For (4.1)-(4.2) a direct substitution shows that

are upper and lower solutions such that \(\beta,\alpha\in X\cap{\mathcal {C}^{4}(0,+\infty)}\). Further, for these functions we have

We also note that when \(t\in[0,+\infty)\) and

the function f is continuous and satisfies Nagumo’s condition with respect to α and β, that is,

Hence we can take \(\varphi(t)=12\) and \(h(w)=(w+1)^{2}\). Now if \(1<\gamma \leq3\), then

and

and these imply that conditions (H₁), (H₂), and (H₆) are fulfilled. Now we will show that f satisfies conditions (H₃)-(H₅) of Theorem 3.2. For \(t\in[0,+\infty)\), \(y_{i},z_{i},w\in{\mathbb{R}}\), \(i=0,1\), when

since f is increasing with respect to \(x_{1}\),

for \(x_{i},z_{i},w\in{\mathbb{R}}\), \(i=0,1\), when

or

since f is decreasing on \([\beta'(t-\tau_{1,1}),\alpha'(t-\tau _{1,1}(t))]\) for \(t\in[0,1)\) and increasing on \([\alpha'(t-\tau _{1,1}(t)),\beta'(t-\tau_{1,1})]\) for \(t\in[1,+\infty)\) with respect to \(y_{1}\),

and for \(t\in[0,+\infty)\), \(x_{i},y_{i},w\in{\mathbb{R}}\), \(i=0,1\), when

since f is increasing with respect to \(z_{0}\),

also when

since f is increasing respect to \(z_{1}\),

Theorem 3.2 now ensures that the BVP (4.1)-(4.2) has at least one solution \(u(t)\) such that

and

also \(\|u\|^{3}_{\infty}<{R}\) where \(R>\sqrt{\exp(192)(1+\eta^{2})}\), \(\eta \geq4\), and \(\gamma=2\).

Example 4.2

Consider the fourth-order nonlinear differential equation on the half-line with deviating arguments

where

Clearly, (4.3) is a particular case of (1.1) with \(q(t)= \frac{1}{(1+t)^{3}}\),

It follows that

We consider (4.3) together with the following boundary conditions:

Comparing this with (1.2), we find \(\theta(t)= t-1\), \(a= \frac{1}{3}\), \(A= \frac{13}{2}\), \(B=4\), \(C= \frac{23}{28}\).

For (4.3)-(4.4) we take \(\alpha_{1}\), \(\alpha_{2}\) and \(\beta _{1}\), \(\beta_{2}\) as follows:

and

and by direct substitution verify that \(\alpha_{2},\beta_{1}\in X\cap {\mathcal{C}^{4}(0,+\infty)}\) are its strict lower and upper solutions, and \(\alpha_{1},\beta_{2}\in X\cap{\mathcal{C}^{4}(0,+\infty)}\) are lower and upper solutions, and satisfy the assumption (3.17).

We also verify that for every \(t\in[0,+\infty)\), \(w\in\mathbb{R}\),

we have

Hence the function f satisfies Nagumo’s condition with \(h(w)=(w+1)^{2}\) and \(\varphi(t)=41\). Now if \(1<\gamma\leq3\), then

and

and these imply that conditions (H₁), (H₂), and (H₆) are fulfilled. Now we shall show that f satisfies conditions (H₃)-(H₅) of Theorem 3.2. For \(t\in[0,+\infty)\), \(y_{i},z_{i},w\in{\mathbb{R}}\), \(i=0,1\), when

since f is increasing with respect to \(x_{1}\),

for \(x_{i},z_{i},w\in{\mathbb{R}}\), \(i=0,1\), when

or

Since f is decreasing on \([\beta'(t-\tau_{1,1}),\alpha'(t-\tau _{1,1}(t))]\) for \(t\in[0,{1}/{2})\) and increasing on \([\alpha'(t-\tau _{1,1}(t)),\beta'(t-\tau_{1,1})]\) for \(t\in[{1}/{2},+\infty)\) with respect to \(y_{1}\),

and for \(t\in[0,+\infty)\), \(x_{i},y_{i},w\in{\mathbb{R}}\), \(i=0,1\), when

since f is increasing with respect to \(z_{1}\),

This ensures that in Theorem 3.3 all assumptions (H₁)-(H₆) are fulfilled. Therefore, we conclude that the problem (4.3)-(4.4) has at least three solutions.

The authors are grateful for the referees’ careful reading and comments on this paper that led to the improvement of the original manuscript. This work was done when the third author was on academic leave, visiting Texas A&M University Kingsville, Department of Mathematics. He gratefully acknowledges the financial support of The Scientific and Technological Research Council of Turkey (TUBITAK).

Affiliations

1. Economics, Commercial and Management Sciences, Preparatory School of Oran, BP 65 CH 2 Achaba Hnifi, Technople de l’USTO, Bir El Djir, Algeria

   • Mostepha Naceri

2. Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX, 78363-8202, USA

   • Ravi P Agarwal
   •  & Erbil Çetin

3. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

   • Ravi P Agarwal

4. Department of Mathematics, Faculty of Science, Ege University, Bornova, Izmir, 35100, Turkey

   • Erbil Çetin

5. Mathematics Faculty of Science, Oran University, Es-Senia, BP1524, Algeria

   • El Haffaf Amir

Corresponding author

Correspondence to Ravi P Agarwal.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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