Skip to main content

Theory and Modern Applications

Existence of solutions for an ordinary second-order hybrid functional differential equation


In this paper, we study the existence of solutions for an initial value problem of an ordinary second-order hybrid functional differential equation (SHDE) using a fixed point theorem of Dhage. Example is given to illustrate the obtained result.

1 Introduction

In this paper, we consider the initial value problems of the second-order hybrid functional differential equation (in short SHDE):

$$ \textstyle\begin{cases} \frac{d^{2}}{dt^{2}} ( \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))} ) = g( t, x(\varphi _{3}(t))) \quad t\in J = [0,T], \\ x(0)=h(0,x(0)) \quad \text{and}\quad {x'(0)}=\frac{dh}{d t}| _{t=0}, \end{cases} $$

where \(f \in C(J\times \mathbb{R},\mathbb{R} \setminus \{0\}) \), \(g \in C(J\times \mathbb{R},\mathbb{R})\), \(h \in C(J\times R,R)\), and \(\varphi _{i} \in C(J)\) with \(\varphi _{i}(0)=0\), \(i=1,2,3\). By a solution of SHDE (1.1) we mean a function \(x \in C(J, \mathbb{R})\) such that

  1. (i)

    the function \(t \to \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))} \) is continuous for each \(x \in C(J,\mathbb{R})\) and

  2. (ii)

    x satisfies the equations in (1.1).

The importance of the investigations of hybrid differential equations lies in the fact that they include several dynamic systems as special cases. The consideration of hybrid differential equations is implicit in the works of Krasnoselskii [1] and extensively treated in several papers on hybrid differential equations with different perturbations. See [29] and [10] and the references therein. This class of hybrid differential equations includes the perturbations of original differential equations in different ways.

Here we study the existence of solutions for the initial value problem of second-order hybrid differential equation (1.1). Some remarks and an example to illustrate our results are given.

This paper is organized as follows: In Sect. 2, we recall some useful preliminaries. In Sect. 3, we prove an auxiliary theorem related to the linear variant of problem (1.1) and state sufficient conditions which guarantee the existence of solutions to problem (1.1). Also, conditions are added to our problem in order to obtain a new existence theorem, and an illustrative example is presented.

2 Preliminaries

In this section, we introduce some basic definitions and preliminary facts which we need in the sequel.

Definition 2.1


An algebra X is a vector space endowed with an internal composition law noted by

$$ (\cdot ) :\quad X \times X \rightarrow X,\quad \quad (x, y) \rightarrow x.y, $$

which is associative and bilinear. A normed algebra is an algebra endowed with a norm satisfying the following property:

For all \(x, y \in X \), we have

$$ \Vert x.y \Vert \leq \Vert x \Vert . \Vert y \Vert . $$

A complete normed algebra is called a Banach algebra.

Definition 2.2


Let X be a normed vector space. A mapping \(T : X \rightarrow X\) is called D-Lipschitzian if there exists a continuous and nondecreasing function \({\phi }:\mathbb{R}^{+}\to \mathbb{R}^{+}\) such that

$$ \Vert Tx - Ty \Vert \leq {\phi }\bigl( \Vert x - y \Vert \bigr) $$

for all \(x, y \in X \), where \({\phi }(0) = 0\).

Sometimes, we call the function ϕ to be a D-function of the mapping T on X. Obviously, every Lipschitzian mapping is D-Lipschitzian. Further, if \({\phi }(r)< r\), for \(r>0\), then T is called a nonlinear contraction on X. An important fixed point theorem that has been commonly used in the theory of nonlinear integral equations is a generalization of the Banach contraction mapping principle proved in [11].

Recently Dhage in [12] has proven a fixed point theorem involving three operators in a Banach algebra by blending the Banach fixed point theorem with Shauder’s fixed point principle.

Lemma 2.3


LetSbe a nonempty, closed convex, and bounded subset of a Banach algebraX, and let\(A,C : X \rightarrow X\)and\(B : S \rightarrow X\)be three operators such that:

  1. (a)

    AandCare Lipschitzian with Lipschitz constantsδandρ, respectively;

  2. (b)

    Bis compact and continuous;

  3. (c)

    \(x = AxBy + Cx \Rightarrow x \in S \)for all\(y \in S\);

  4. (d)

    \(\delta M + \rho < r\)for\(r > 0\)where\(M = \Vert B(S) \Vert \).

Then the operator equation\(AxBx+Cx=x\)has a solution in S.

3 Main results

In this section, we formulate our main result for SHDE (1.1) depending on the fixed point theorems due to Dhage [13].

Let \(X = C(J,\mathbb{R})\) of the vector of all real-valued continuous functions on \(J = [0,T]\). We equip the space X with the norm \(\Vert x \Vert = \sup_{t \in J } \vert x(t) \vert \). Clearly, \(C(J,\mathbb{R})\) is a complete normed algebra with respect to this supremum norm. Consider the following assumptions:


The functions \(f : J \times \mathbb{R} \rightarrow \mathbb{R} \backslash \{0\}\) and \(h : J \times \mathbb{R} \rightarrow \mathbb{R}\) are continuous, and there exist two functions \(k,L \in C(J,\mathbb{R}_{+})\), with norms \(\Vert k \Vert \) and \(\Vert L \Vert \) respectively, such that

$$\begin{aligned}& \bigl\vert h(t, x) - h(t, y) \bigr\vert \leq k(t) \vert x - y \vert , \\& \bigl\vert f(t, x) - f(t, y) \bigr\vert \leq L(t) \vert x - y \vert \end{aligned}$$

for all \(t \in J\) and \(x, y \in \mathbb{R}\).


\(g:J\times \mathbb{R}\rightarrow \mathbb{R}\). There exist a function \(p \in C(J,\mathbb{R}_{+})\) and a continuous nondecreasing function \(\varPsi : [0,\infty )\to (0,\infty )\) such that

$$ \bigl\vert g(t,x) \bigr\vert \leq { p(t)\varPsi \bigl( \vert x \vert \bigr)} , \quad \forall (t,x) \in J \times \mathbb{R}. $$

\(\varphi _{i} : J \to J\) are continuous functions with \(\varphi _{i}(0)=0\), \(i=1,2,3\).


There exists a number \(r > 0\) such that

$$\begin{aligned} r\geq \frac{H+ G \Vert p \Vert \varPsi (r) \frac{T^{2}}{2}}{1-( \Vert L \Vert \Vert p \Vert \varPsi (r) \frac{T^{2}}{2}+ \Vert k \Vert )}, \end{aligned}$$

where \(G=\sup_{t\in J } \vert f(t, 0) \vert \), \(H=\sup_{t\in J } \vert h(t, 0) \vert \), and

$$\begin{aligned} & \Vert L \Vert \Vert p \Vert \varPsi (r) \frac{T^{2}}{2}+ \Vert k \Vert < 1. \end{aligned}$$

Now, we shall prove the following lemma.

Lemma 3.1

Assume that hypotheses\((A_{1})-(A_{4})\)hold. Then a function\(x \in C(J, \mathbb{R} )\)is a solution of SHDE (1.1) if, and only if, it satisfies the following quadratic integral equation:

$$ x =h\bigl(t, x\bigl(\varphi _{1}(t)\bigr)\bigr) +f \bigl(t, x\bigl(\varphi _{2}(t)\bigr)\bigr) \int _{0}^{t} (t-s) g\bigl( s, x\bigl(\varphi _{3}(s)\bigr)\bigr) \,ds. $$


First, assume that x is a solution of SHDE (1.1), applying integration to both sides of (1.1) from 0 to t, we obtain

$$ \frac{d}{dt} \biggl( \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))} \biggr)- \frac{d}{dt} \biggl( \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))} \biggr)\bigg| _{t=0} = \int _{0}^{t} g\bigl( s, x\bigl(\varphi _{3}(s)\bigr)\bigr)\,ds. $$

On the other hand (due to the fact that \(f(0,x(0)) \neq 0 \) and \(\varphi _{i}(0)= 0\), \(i=1,2,3\)), we have

$$\begin{aligned} \begin{aligned} &\frac{d}{dt} \biggl( \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))} \biggr)\bigg| _{t=0} \\ &\quad = \frac{f(0,x(0)) (x^{\prime }(0)- \frac{d h}{d t}| _{t=0} )- (x(0)- h(0,x(0)) )\frac{d f}{d t} | _{t=0}}{f^{2}(0,x(0))}=0. \end{aligned} \end{aligned}$$


$$\begin{aligned}& x(0)= h\bigl(0,x(0)\bigr), \\& x^{\prime }(0)= \frac{d h}{d t}\bigg| _{t=0}, \end{aligned}$$

therefore, we get

$$\begin{aligned} \frac{d}{dt} \biggl( \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))} \biggr) = \int _{0}^{t} g\bigl( s, x\bigl(\varphi _{3}(s)\bigr)\bigr) \,ds. \end{aligned}$$

Again integrating both sides of (3.4) from 0 to t, we obtain

$$\begin{aligned} \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))}- \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))}\bigg| _{t=0} = \int _{0}^{t} (t-s) g\bigl( s, x\bigl(\varphi _{3}(s)\bigr)\bigr)\,ds, \end{aligned}$$

and we have

$$ \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))}\bigg| _{t=0} = \frac{x(0)- h(0,x(0))}{f(0,x(0))}=0. $$

Hence, Eq. (3.5) becomes

$$\begin{aligned} \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))}= \int _{0}^{t} (t-s) g\bigl( s, x\bigl(\varphi _{3}(s)\bigr)\bigr) \,ds, \end{aligned}$$


$$ x(t) =h\bigl(t, x\bigl(\varphi _{1}(t)\bigr)\bigr) +f\bigl(t, x\bigl( \varphi _{2}(t)\bigr)\bigr) \int _{0}^{t} (t-s) g\bigl( s, x\bigl(\varphi _{3}(s)\bigr)\bigr) \,ds. $$

Thus, Eq. (3.3) holds.

Conversely, assume that x satisfies Eq. (3.3). Then dividing by \(f(t, x(t))\) and making direct differentiation for both sides of Eq. (3.3), we obtain

$$\begin{aligned} \frac{d}{dt} \biggl( \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))} \biggr) = \int _{0}^{t} g\bigl( s, x\bigl(\varphi _{3}(s)\bigr)\bigr) \,ds. \end{aligned}$$

Then, again by direct differentiation, Eq. (1.1) is satisfied.

$$ \frac{d^{2}}{dt^{2}} \biggl( \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))} \biggr) = g\bigl( t, x\bigl(\varphi _{3}(t)\bigr)\bigr). $$

Again, substituting \(t = 0\) in Eq. (3.3) (due to the fact that \(f(0,x(0)) \neq 0 \) and \(\varphi _{i}(0)= 0\), \(i=1,2,3\)) yields

$$ \frac{x(0)-h(0, x(0))}{f(0,x(0))}\to 0 \quad \text{as } t\to 0, $$

hence \(x(0) = h(0, x(0))\), and

$$\begin{aligned} &\frac{d}{dt} \biggl( \frac{x(t)- h(t,x(\varphi _{1}(t)))}{f(t,x(\varphi _{2}(t)))} \biggr)\bigg| _{t=0} = 0, \\ & \frac{f(t,x(t))(x^{\prime }(t)- \frac{d h}{d t})-(x(t)- h(t,x(t)))\frac{d f}{d t}}{f^{2}(t,x(t))}\bigg| _{t=0}=0, \\ & f\bigl(0,x(0)\bigr) \biggl( x^{\prime }(0)- \frac{d h}{d t}\bigg| _{t=0} \biggr)- \bigl(x(0)- h\bigl(0,x(0)\bigr) \bigr) \frac{d f}{d t}\bigg| _{t=0}=0. \end{aligned}$$

Since we have proven \(x(0) = h(0, x(0))\), this yields \(x^{\prime }(0)= \frac{d h}{d t}| _{t=0}\). The proof is completed. □

3.1 Existence of solution

Now, our target is to prove the following existence theorems.

Theorem 3.2

Assume that hypotheses\((A_{1})-(A_{4})\)hold. Then SHDE (1.1) has at least one solution defined on J.


By Lemma 3.1, problem (1.1) is equivalent to the quadratic functional integral equation (3.3). Set \(X = C(J,\mathbb{R})\) and define a subset S of X as

$$ S :=\bigl\{ x \in X, \Vert x \Vert \leq r\bigr\} , $$

where r satisfies inequality (3.1).

Clearly S is a closed, convex, and bounded subset of the Banach space X.

Corresponding to the functions f, g, and h, we introduce the three operators \(A:X \to X\), \(B:S \to X\), and \(C:X \to X\) defined by

$$\begin{aligned}& (Ax) (t) = f\bigl(t, x\bigl(\varphi _{2}(t)\bigr) \bigr), \quad t\in J, \end{aligned}$$
$$\begin{aligned}& (Bx) (t) = \int _{0}^{t} (t-s) g\bigl( s, x\bigl(\varphi _{3}(t)\bigr)\bigr) \,ds, \quad t\in J, \end{aligned}$$
$$\begin{aligned}& (Cx) (t) = h\bigl(t,x\bigl(\varphi _{1}(t)\bigr)\bigr), \quad t\in J. \end{aligned}$$

Then the integral equation (3.3) can be rewritten as follows:

$$ x(t) = Ax(t)\cdot Bx(t)+ Cx(t), \quad t\in J. $$

We shall show that A, B, and C satisfy all the conditions of Lemma 2.3. This will be achieved in the following series of steps.

Step 1. To show that A and C are Lipschitzian on X, let \(x, y \in X\). So

$$\begin{aligned} \bigl\vert Ax(t) - Ay(t) \bigr\vert =& \bigl\vert f\bigl(t, x\bigl( \varphi _{2}(t)\bigr)\bigr) - f\bigl(t, y\bigl(\varphi _{2}(t) \bigr)\bigr) \bigr\vert \\ \leq & L(t) \bigl\vert x\bigl(\varphi _{2}(t)\bigr) - y\bigl( \varphi _{2}(t)\bigr) \bigr\vert \leq \Vert L \Vert \Vert x - y \Vert , \end{aligned}$$

which implies \(\Vert Ax -Ay \Vert \leq \Vert L \Vert \Vert x - y \Vert \) for all \(x, y \in X\). Therefore, A is a Lipschitzian on X with Lipschitz constant \(\Vert L \Vert \).

Similarly, for any \(x, y \in X\), we have

$$\begin{aligned} \bigl\vert Cx(t) - Cy(t) \bigr\vert =& \bigl\vert h\bigl(t, x\bigl( \varphi _{1}(t)\bigr)\bigr) - h\bigl(t, y\bigl(\varphi _{1}(t) \bigr)\bigr) \bigr\vert \\ \leq &k(t) \bigl\vert x\bigl(\varphi _{1}(t)\bigr) - y\bigl(\varphi _{1}(t)\bigr) \bigr\vert \leq \Vert k \Vert \Vert x - y \Vert . \end{aligned}$$


$$ \Vert Cx - Cy \Vert \leq \Vert k \Vert \Vert x - y \Vert . $$

This shows that C is a Lipschitz mapping on X with the Lipschitz constant \(\Vert k \Vert \).

Step 2. To prove that B is a compact and continuous operator on S into X.

First we show that B is continuous on X. Let \(\{x_{n}\}\) be a sequence in S converging to a point \(x \in S\). Then, by the Lebesgue dominated convergence theorem, let us assume that \(t \in J\), and since \(\varphi _{3}(t)\) is a continuous function and \(g(t,x(t))\) is continuous in x, then \(g(t,x_{n}(\varphi _{3}(t)))\) converges to \(g(t,x(\varphi _{3}(t)))\), (see assumption \((A_{2})\)). Applying the Lebesgue dominated convergence theorem, we get

$$\begin{aligned} \lim_{n \rightarrow \infty }Bx_{n}(t) =& \lim_{n \rightarrow \infty } \int _{0}^{t} (t-s) g(s,x_{n}\bigl( \varphi _{3}(s)\bigr) \,ds \\ =& \int _{0}^{t} (t-s) g(s,x\bigl(\varphi _{3}(s)\bigr) \,ds \\ =& Bx(t). \end{aligned}$$

Thus, \(Bx_{n} \rightarrow Bx\) as \(n \rightarrow \infty \) uniformly on \(\mathbb{R}_{+}\), and hence B is a continuous operator on S into S.

Now, we show that B is a compact operator on S. It is enough to show that \(B(S)\) is a uniformly bounded and equicontinuous set in X. To see this, let \(x \in S\) be arbitrary. Then, by hypothesis \((A_{2})\),

$$\begin{aligned} \bigl\vert Bx(t) \bigr\vert \leq & \int _{0}^{t} (t-s) \bigl\vert g(s,x\bigl(\varphi _{3}(s)\bigr) \bigr\vert \,ds \\ \leq & \int _{0}^{t} (t-s)p(t)\varPsi \bigl( \vert x \vert \bigr) ]\,ds \\ \leq & \Vert p \Vert \varPsi (r) \int _{0}^{t} (t-s)\,ds \\ \leq & \Vert p \Vert \varPsi (r) \frac{T^{2}}{2}=K \end{aligned}$$

for all \(t \in J\). Taking supremum over t,

$$ \bigl\Vert Bx(t) \bigr\Vert \leq K $$

for all \(x \in S\). This shows that B is uniformly bounded on S.

Now, we proceed to showing that \(B(S)\) is also an equicontinuous set in X. Let \(t_{1}, t_{2} \in J\), and \(x \in S\) (without loss of generality assume that \(t_{1}< t_{2}\)), then we have

$$\begin{aligned} &(Bx) (t_{2}) - (Bx) (t_{1}) \\ &\quad \leq \int _{0}^{t_{2}} (t_{2}-s)g\bigl(s,x\bigl( \varphi _{3}(s)\bigr)\bigr)\,ds - \int _{0}^{t_{1}} (t_{1}-s) g\bigl(s,x\bigl( \varphi _{3}(s)\bigr)\bigr) \,ds \\ &\quad \leq \int _{0}^{t_{1}} (t_{2}-s)g\bigl(s,x\bigl( \varphi _{3}(s)\bigr)\bigr)\,ds + \int _{t_{1}}^{t_{2}} (t_{2}-s)g\bigl(s,x\bigl( \varphi _{3}(s)\bigr)\bigr)\,ds \\ &\quad \quad{} - \int _{0}^{t_{1}} (t_{1}-s)g\bigl(s,x\bigl( \varphi _{3}(s)\bigr)\bigr) \,ds \\ &\quad \leq \int _{0}^{t_{1}} \bigl[(t_{2}-s)-(t_{1}-s) \bigr]g\bigl(s,x\bigl(\varphi _{3}(s)\bigr)\bigr)\,ds + \int _{t_{1}}^{t_{2}} (t_{2}-s)g\bigl(s,x\bigl( \varphi _{3}(s)\bigr)\bigr)\,ds \end{aligned}$$


$$\begin{aligned} & \bigl\vert (Bx) (t_{2}) - (Bx) (t_{1}) \bigr\vert \\ &\quad \leq \int _{0}^{t_{1}} (t_{2}-t_{1}) \bigl\vert g\bigl(s,x\bigl(\varphi _{3}(s)\bigr)\bigr) \bigr\vert \,ds + \int _{t_{1}}^{t_{2}} (t_{2}-s) \bigl\vert g \bigl(s,x\bigl(\varphi _{3}(s)\bigr)\bigr) \bigr\vert \,ds \\ &\quad \leq \int _{0}^{t_{1}} (t_{2}-t_{1}) p(t) \varPsi \bigl( \vert x \vert \bigr)\,ds + \int _{t_{1}}^{t_{2}} (t_{2}-s) p(t)\varPsi \bigl( \vert x \vert \bigr) \,ds \\ &\quad \leq \Vert p \Vert \varPsi (r)\biggl[T(t_{2}-t_{1}) + \int _{t_{1}}^{t_{2}} (t_{2}-s) \,ds\biggr] \\ &\quad \leq \Vert p \Vert \varPsi (r) \biggl[T(t_{2}-t_{1})+ \frac{(t_{2}-t_{1})^{2}}{2}\biggr], \end{aligned}$$


$$\begin{aligned} \bigl\vert (Bx) (t_{2}) - (Bx) (t_{1}) \bigr\vert &\leq \Vert p \Vert \varPsi (r) \biggl[T(t_{2}-t_{1})+ \frac{(t_{2}-t_{1})^{2}}{2}\biggr], \end{aligned}$$

which is independent of \(x \in S\). Hence, for \(\epsilon > 0\), there exists \(\delta > 0\) such that

$$ \vert t_{2} - t_{1} \vert < \delta \quad \implies \quad \bigl\vert (Bx) (t_{2}) - (Bx) (t_{1}) \bigr\vert < \epsilon $$

for all \(t_{2}, t_{1} \in J\) and for all \(x \in S\). This shows that \(B(S)\) is an equicontinuous set in X. Now, the set \(B(S)\) is a uniformly bounded and equicontinuous set in X, so it is compact by the Arzela–Ascoli theorem. As a result, B is a complete continuous operator on S.

Step 3. Hypothesis \((c)\) of Lemma 2.3 is satisfied. Let \(x \in X \) and \(y \in S \) be arbitrary elements such that \(x = AxBy + Cx\). Then we have

$$\begin{aligned} \bigl\vert x(t) \bigr\vert &\leq \bigl\vert Ax(t) \bigr\vert \bigl\vert By(t) \bigr\vert + \bigl\vert Cx(t) \bigr\vert \\ &\leq { \bigl\vert f\bigl(t, x\bigl(\varphi _{2}(t)\bigr)\bigr) \bigr\vert \int _{0}^{t} (t-s) \bigl\vert g\bigl(s,y\bigl( \varphi _{3}(s)\bigr)\bigr) \bigr\vert \,ds + \bigl\vert h\bigl(t, x \bigl(\varphi _{1}(t)\bigr)\bigr) \bigr\vert } \\ &\leq \bigl( \bigl\vert f \bigl(t, x\bigl(\varphi _{2}(t)\bigr) \bigr) - f (t, 0) \bigr\vert + \bigl\vert f (t, 0) \bigr\vert \bigr) \int _{0}^{t} (t-s) p(t)\varPsi \bigl( \vert y \vert \bigr) \,ds \\ &\quad{} +\bigl( \bigl\vert h \bigl(t,x\bigl(\varphi _{1}(t)\bigr) \bigr) - h (t, 0) \bigr\vert + \bigl\vert h (t, 0) \bigr\vert \bigr) \\ &\leq \bigl( \Vert L \Vert \bigl\vert x\bigl(\varphi _{2}(t) \bigr) \bigr\vert + G\bigr) \Vert p \Vert \varPsi (r) \int _{0}^{t} (t-s) \,ds+ \Vert k \Vert \bigl\vert x\bigl(\varphi _{3}(t)\bigr) \bigr\vert + H \\ &\leq \bigl( \Vert L \Vert r + G\bigr) \Vert p \Vert \varPsi (r) \frac{T^{2}}{2}+ \Vert k \Vert r + H. \end{aligned}$$


$$ \bigl\vert x(t) \bigr\vert \leq \bigl( \Vert L \Vert r + G\bigr) \Vert p \Vert \varPsi (r) \frac{T^{2}}{2}+ \Vert k \Vert r + H. $$

Taking supremum over t,

$$ \Vert x \Vert \leq \bigl( \Vert L \Vert r + G\bigr) \Vert p \Vert \varPsi (r) \frac{T^{2}}{2}+ \Vert k \Vert r+H. $$

Therefore, \(x \in S\).

Step 4. Finally we show that \(\delta M+ \rho <1 \), that is, \((d)\) of Lemma 2.3 holds.


$$\begin{aligned} M =& \bigl\Vert B(S) \bigr\Vert \\ =&\sup_{x\in S} \Bigl\{ \sup_{t\in J} \bigl\vert Bx(t) \bigr\vert \Bigr\} \\ \leq & \Vert p \Vert \varPsi (r) \frac{T^{2}}{2}, \end{aligned}$$

and by \((A_{4})\), we have

$$ \Vert L \Vert M+ \Vert k \Vert < 1 $$

with \(\delta = \Vert L \Vert \) and \(\rho = \Vert k \Vert \).

Thus all the conditions of Lemma 2.3 are satisfied, and hence the operator equation \(x = AxBx + Cx\) has a solution in S. In consequence, problem (1.1) has a solution on J. This completes the proof. □

3.2 Remarks and examples

  • If we replace conditions \((A_{2})\) and \((A_{4})\) with the following conditions:


    \(g:J \times \mathbb{R}\rightarrow \mathbb{R}\) satisfies the Caratheodory condition, i.e., g is measurable in t for any \(x\in {\mathbb{R}} \) and continuous in x for almost all \(t\in {[0,T]}\).

    There exist two positive real functions \(t \rightarrow a(t) \), \(t \rightarrow b(t) \) such that

    $$ \bigl\vert g(t,x) \bigr\vert \leq { a(t)+ b(t) \vert x \vert } , \quad \forall (t,x) \in J \times \mathbb{R}; $$

    There exists a number \(r > 0\) such that

    $$\begin{aligned} r\leq \frac{ \Vert L \Vert . \Vert a \Vert T^{2}+G \Vert b \Vert T^{2}+ \Vert k \Vert }{2 \Vert b \Vert . \Vert L \Vert T^{2}}, \end{aligned}$$

    where \(G=\sup_{t\in J } \vert f(t, 0) \vert \) and \(( \Vert L \Vert \Vert a \Vert +G \Vert b \Vert )T^{2}+ \Vert k \Vert <1\),

and Step 3 in the proof can be replaced with the following. Let \(x \in X \) and \(y \in S \) be arbitrary elements such that \(x = AxBy + Cx\). Then we have

$$\begin{aligned} \bigl\vert x(t) \bigr\vert &\leq \bigl\vert Ax(t) \bigr\vert \bigl\vert By(t) \bigr\vert + \bigl\vert Cx(t) \bigr\vert \\ &\leq { \bigl\vert f\bigl(t, x\bigl(\varphi _{2}(t)\bigr)\bigr) \bigr\vert \int _{0}^{t} (t-s) \bigl\vert g\bigl(s,y\bigl( \varphi _{3}(s)\bigr)\bigr) \bigr\vert \,ds + \bigl\vert h\bigl(t, x \bigl(\varphi _{1}(t)\bigr)\bigr) \bigr\vert } \\ &\leq \bigl( \bigl\vert f \bigl(t, x\bigl(\varphi _{2}(t)\bigr) \bigr) - f (t, 0) \bigr\vert + \bigl\vert f (t, 0) \bigr\vert \bigr) \int _{0}^{t} (t-s) (a(t)+ b(t) \bigl\vert x\bigl( \varphi _{3}(s) \bigr\vert \bigr) \,ds \\ &\quad{} +\bigl( \bigl\vert h \bigl(t,x\bigl(\varphi _{1}(t)\bigr) \bigr) - h (t, 0) \bigr\vert + \bigl\vert h (t, 0) \bigr\vert \bigr) \\ &\leq \bigl( \Vert L \Vert \bigl\vert x\bigl(\varphi _{2}(t) \bigr) \bigr\vert + G\bigr) \bigl( \Vert a \Vert + \Vert b \Vert r \bigr) \int _{0}^{t} (t-s) \,ds+ \Vert k \Vert \bigl\vert x\bigl(\varphi _{3}(t)\bigr) \bigr\vert + H \\ &\leq \bigl( \Vert L \Vert r + G\bigr) \bigl( \Vert a \Vert + \Vert b \Vert r \bigr) \frac{T^{2}}{2}+ \Vert k \Vert r + H. \end{aligned}$$


$$ \bigl\vert x(t) \bigr\vert \leq \bigl( \Vert L \Vert r + G\bigr) \bigl( \Vert a \Vert + \Vert b \Vert r \bigr) T^{2}+ \Vert k \Vert r + H. $$

Taking supremum over t,

$$ \Vert x \Vert \leq \bigl( \Vert L \Vert r + G\bigr) \bigl( \Vert a \Vert + \Vert b \Vert r \bigr) T^{2}+ \Vert k \Vert r+H. $$

From this estimate we show that the operator A maps the set S into itself with

$$ r= \frac{ \Vert L \Vert . \Vert a \Vert T^{2}+G \Vert b \Vert T^{2}+ \Vert k \Vert }{2 \Vert b \Vert . \Vert L \Vert T^{2}}. $$
  • Our results can be obtained using the technique of measures of noncompactness in the Banach algebras (under assumptions \((A_{1})-(A_{4})\)) and a fixed point theorem for the product of two operators verifying a Dhage type condition as follows in [2].


Consider the second-order functional differential equation

$$ \textstyle\begin{cases} \frac{d^{2}}{dt^{2}} ( \frac{x(t)- \frac{\cos \pi t+2t^{2}}{1+5t^{2}} \vert x(t) \vert }{ (\frac{ \vert x(t) \vert +1}{ \vert x(t) \vert +2} )\frac{7-e^{t}}{2\sqrt{25-t^{2}}}+\frac{2-t}{10}} ) = \frac{(t-1)^{2}+3}{35(13-t^{2})}(7 \vert x(t) \vert +15), \quad t\in J=[0,1], \\ x(0)=h(0,x(0))\quad \text{and}\quad {x'(0)}=\frac{dh}{d t}| _{t=0},\end{cases} $$


$$\begin{aligned}& f\bigl(t,x(t)\bigr)= \biggl(\frac{ \vert x(t) \vert +1}{ \vert x(t) \vert +2} \biggr) \frac{7-e^{t}}{2\sqrt{25-t^{2}}}+ \frac{2-t}{10}, \\& \bigl\vert f\bigl(t,x(t)\bigr)-f\bigl(t,y(t)\bigr) \bigr\vert \leq \biggl( \frac{7-e^{t}}{2\sqrt{25-t^{2}}} \biggr) \vert x-y \vert , \\& h\bigl(t,x(t)\bigr)=\frac{\cos \pi t+2t^{2}}{1+5t^{2}} \bigl\vert x(t) \bigr\vert , \\& \bigl\vert h\bigl(t,x(t)\bigr)-h\bigl(t,y(t)\bigr) \bigr\vert \leq \biggl( \frac{1+2t^{2}}{1+5t^{2}} \biggr) \vert x-y \vert , \end{aligned}$$


$$ g\bigl(t,x(t)\bigr)= \frac{(t-1)^{2}+3}{35(13-t^{2})}\bigl(7 \bigl\vert x(t) \bigr\vert +15\bigr)= \biggl( \frac{(t-1)^{2}+3}{13-t^{2}} \biggr) \biggl( \frac{ \vert x \vert }{5}+ \frac{3}{7} \biggr). $$

Take \(p(t)=\frac{(t-1)^{2}+3}{13-t^{2}}\) and \(\varPsi (x)=\frac{ \vert x \vert }{5}+\frac{3}{7}\).

We can easily verify that \(x(0)=h(0,x(0))\), \(x'(0)=\frac{dh}{dt}| _{t=0} \).

\(\Vert k \Vert =1/2\), \(\Vert p \Vert =4/13\), \(\Vert L \Vert =7/10\), and \(G=1\).

For condition \(\Vert L \Vert \Vert p \Vert \varPsi (r) \frac{T^{2}}{2}+k< 1\) is satisfied, r should be chosen \(r<21.07\).


  1. Krasnoselskij, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations (1964)

    Google Scholar 

  2. Dhage, B.C.: Attractivity and positivity results for nonlinear functional integral equations via measure of noncompactness. Differ. Equ. Appl. 2(3), 299–318 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Dhage, B.C.: On α-condensing mappings in Banach algebras. Math. Stud. 63(1), 146–152 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Dhage, B.C.: Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations. Differ. Equ. Appl. 2(4), 465–486 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Herzallah, M.A.E., Baleanu, D.: On fractional order hybrid differential equations. Abstr. Appl. Anal. 2014, Article ID 389386 (2014)

    Article  MathSciNet  Google Scholar 

  6. Melliani, S., Hilal, K., Hannabou, M.: Existence results in the theory of hybrid fractional integro-differential equations. J. Univers. Math. 1(2), 166–179 (2018)

    Google Scholar 

  7. Zhao, Y., Sun, S., Han, Z., Li, Q.: Theory of fractional hybrid differential equations. Comput. Math. Appl. 62(3), 1312–1324 (2011)

    Article  MathSciNet  Google Scholar 

  8. Dhage, B.C.: Basic results in the theory of hybrid differential equations with linear perturbations of second type. Tamkang J. Math. 44(2), 171–186 (2013)

    Article  MathSciNet  Google Scholar 

  9. Dhage, B.C.: Basic results in the theory of hybrid differential equations with mixed perturbations of second type. Funct. Differ. Equ. 19(1–2), 87–106 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Zheng, L., Zhang, X.: Modeling and Analysis of Modern Fluid Problems (2017)

    MATH  Google Scholar 

  11. Dhage, B.C.: A nonlinear alternative in Banach algebras with applications to functional differential equations. Nonlinear Funct. Anal. Appl. 8, 563–575 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Dhage, B.C.: On some variants of Schauder’s fixed point principle and applications to nonlinear integral equations. J. Math. Phys. Sci. 25, 603–611 (1988)

    MathSciNet  MATH  Google Scholar 

  13. Dhage, B.C.: A fixed point theorem in Banach algebras involving three operators with applications. Kyungpook Math. J. 44(1), 145–155 (2004)

    MathSciNet  MATH  Google Scholar 

Download references


The authors wish to express their gratitude to the anonymous referees for their valuable comments and suggestions, which allowed to improve an early version of this work.

Availability of data and materials

Not applicable.


There is no source of funding.

Author information

Authors and Affiliations



All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Shorouk Al-Issa.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El-Sayed, A., Hashem, H. & Al-Issa, S. Existence of solutions for an ordinary second-order hybrid functional differential equation. Adv Differ Equ 2020, 296 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: