Hermite-Hadamard-Fejér type fractional inequalities relating to a convex harmonic function and a positive symmetric increasing function

1.   Introduction

Functional differential equations arise widely in many fields such as mathematical biology, economy, physics, or biology, see ^{[16,19,28,40]}. This explains the great interest in the qualitative properties of these kinds of equations. Oscillation phenomena appear in various models from real world applications; see, e.g., the papers ^[12,35,38] for models from mathematical biology where oscillation and/or delay actions may be formulated by means of cross-diffusion terms. As a part of this approach, the oscillation theory of this type of equation has been extensively developed, see ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]}. In particular, the oscillation criteria of first-order differential equations with deviating arguments have numerous applications in the study of higher-order functional differential equations (e.g., one can study the oscillatory behavior of higher-order functional differential equations by relating oscillation of these equations to that of associated first-order functional differential equations); see, e.g., the papers ^[13,36,37].

Recently, there has been great interest in studying the oscillation of all solutions of the first-order delay differential equation

and its dual advanced equation

where $b, c, \tau, \sigma \in C([t_0, \infty), [0, \infty))$ such that $\tau(t)\leq t$, $\underset{t\rightarrow \infty} {\lim}\ \tau(t) = \infty$, and $\sigma(t)\geq t$. In most of these works, the delay (advanced) function is assumed to be nondecreasing, see ^{[14,16,29,31,32,33,34,42,45]} and the references therein. As shown in ^[8], the oscillation character of Eq (1.1) with nonmontone delay, is not an easy extension to the oscillation problem for the nondecreasing delay case. Many authors ^{[1,3,5,6,7,8,9,10,11,15,20,25,27,39,43]} have developed and generalized the methods used to study the oscillation of equations (1.1) and (1.2) with monotone delays and to study this property for the nonmonotone case. Only a few works, however, dealt with the oscillation of equations (1.1) and (1.2) with oscillatory coefficients. For example, ^[16,44] studied the oscillation of Eq (1.1) where the delay function $\tau(t)$ is assumed to be nondecreasing and constant (i.e., $\tau(t) = t-\alpha, \ \alpha > 0$), respectively. Also, Kulenovic and Grammatikopoulos ^[29] studied the oscillation of a first-order nonlinear functional differential equation that contains both equations (1.1) and (1.2). The authors obtained liminf and limsup oscillation criteria for the case when the coefficient function does not need to be nonnegative. However, the delay (advanced) and the coefficient functions are assumed to be nondecreasing (for limsup conditions) and nonnegative on a sequence of intervals $\{(r_n, \, s_n)\}_{n\geq0}$ such that $\lim_{n\rightarrow \infty}\ \left(s_n-r_n\right) = \infty$ (for liminf conditions), respectively.

Our aim in this work is to obtain oscillation criteria for equations (1.1) and (1.2) where $b(t)$ and $c(t)$ are continuous functions on $[t_0, \infty)$. We relax the nonnegative restriction on the coefficient functions $b(t)$ and $c(t)$. To accomplish this goal, using the ideas of ^[27], we develop and enhance the work of Kwong ^[30]. This procedure leads to new sufficient oscillation criteria that improve and generalize those mentioned in ^[16,29,44].

2.   Main results

From now on, we assume that $b(t)$ and $c(t)$ are only continuous functions on $[t_0, \infty)$.

2.1.   Delay differential equations

Let $\Lambda(t)$ and $\Lambda_i(t)$, $t\geq t_0$, $i\in \mathbb{N}$ be defined as follows (see ^[27]):

Also, we define the function $g(t)$ and the sequence $\{Q_n(v, u)\}_{n = 0}^{\infty}$, $\tau(v) \leq u \leq v$, as follows:

and

The proofs of our main results are essentially based on the following lemma.

Lemma 2.1. Let $n \in \mathbb{N}_0$, $T^* > t_0$, $T\geq T^*$ and $x(t)$ be a solution of Eq (1.1) such that $x(t) > 0$ for all $t \geq T^*$. If $b(t)\geq0$ on $[T, \, T_1]$, $T_1\geq \Lambda_{n+2}(T)$, then

Proof. It follows from Eq (1.1) that $x'(t)\leq0$ on $[\Lambda_1(T), \, T_1]$. Therefore,

Dividing Eq (1.1) by $x(t)$ and integrating from $u$ to $v$, $\tau(v) \leq u \leq v$, we obtain

Since $x'(t)\leq0$ on $[\Lambda_1(T), \, T_1]$, we get

for $v \in [\Lambda_3(T), \, T_1]$ and consequently, for $u \leq \zeta \leq v$, we have

Substituting in (2.4), we get

Repeating this argument $n$ times, we obtain

The proof of the lemma is complete.

Let $\{T_k\}_{k\geq 0}$ be a sequence of real numbers such that $\lim_{k\rightarrow \infty}\ T_k = \infty$ and

Also, we define the sequence $\{\beta_n\}_{n\geq 1}$, $\beta_n > 1$ for all $n\in \mathbb{N}$ as follows:

Theorem 2.1. Let $n \in \mathbb{N}$ such that (2.5) and (2.6) are satisfied. If

then every solution of Eq (1.1) is oscillatory.

Proof. Assume, for the sake of contradiction, that $x(t)$ is an eventually positive solution of Eq (1.1). Then there exists a sufficiently large $T^* > t_0$ such that $x(t) > 0$ for $t > T^*$. Suppose that $T_{k_1}\in \{T_k\}_{k\geq 0}$ such that $T_{k_1} > T^*$. In view of (2.3), (2.5) and (2.6), it follows that

Then there exists $t_*\in(g(\Lambda_{n+4}(T_{k_1})), \Lambda_{n+4}(T_{k_1}))$ such that

Integrating Eq (1.1) from $t_*$ to $t$, we get

It is easy to see that

Substituting in (2.8), we have

Since $x'(t)\leq0$ on $[\Lambda_1 (T_{k_1}), \, \Lambda_{n+4}(T_{k_1})]$, it follows that

By (2.3) and $\zeta_1 \in [\Lambda_{n+2}(T_{k_1}), \Lambda_{n+3}(T_{k_1})]$ for $\tau(\zeta) < \zeta_1 < g(\zeta)$, $g(\Lambda_{n+4}(T_{k_1})) < \zeta < \Lambda_{n+4}(T_{k_1})$, we get

Dividing Eq (1.1) by $x(t)$ and integrating from $g(\Lambda_{n+4}(T_{k_1}))$ to $t_*$, we obtain

Using (2.9), we get

From this, (2.3) and (2.6), we get

It follows from (2.6) and (2.7) that

Combining this and (2.10) we get

that is

The proof of the theorem is complete.

Theorem 2.2. Let $n \in \mathbb{N}_0$ such that (2.5) is satisfied. If

then every solution of Eq (1.1) is oscillatory.

Proof. Let $x(t)$ be an eventually positive solution of Eq (1.1). Then the exists $T^* > t_0$ such that $x(t) > 0$ for all $t\geq T^*$. It is not difficult to prove that

where $T_{k_1}\in \{T_k\}$ such that $T_{k_1} > T^*$. Using (2.3), we get

Using the positivity of $x(\Lambda_{n+4}(T_{k_1}))$ we have

Then

which contradicts (2.11). The proof of the theorem is complete.

Remark 2.1.

● (1) It should be noted that the monotonicity of the delay function $\tau(t)$ is required in many previous works to study the oscillation of Eq (1.1) with oscillating coefficients; see, for example, ^[16,29,44]. In this work, the sequence $\{\Lambda_i(t)\}_{i\geq0}$ plays a central role in the derivation of our results. In fact the delay function $\tau(t)$ does not need to be monotone. Therefore, our results substantially improve and generalize [29, Theroems 6, 7] for Eq (1.1). Furthermore, using our approach, many previous oscillation studies for Eq (1.1) with monotone delays can be used to study the oscillation of Eq (1.1) with general delays (the delay does not need to be monotone) and oscillating coefficients.

(2) There are numerous lower bounds for the quotient $\frac{x(\tau(t))}{x(t)}$, where $x(t)$ is a positive solution of Eq (1.1) with a nonnegative continuous function $b(t)$, see ^{[6,16,22,24,42]}. For example, [22], Lemma 1] and Lemma 2.1 can be used instead of Lemma 2.1 to improve our results in the case where $b(t)$ is a nonnegative continuous function. In this case, the adjusted version of the results improves Theorems 2.1 and 2.2. Even in the case where $\tau(t)$ is nondecreasing, the improvement is substantial.

2.2.   Advanced differential equations

Similar results for the (dual) advanced differential equation (1.2) can be obtained easily. The details of the proofs are omitted since they are quite similar to Eq (1.1).

We will use the following notation:

where

Also, we define the sequence $\{R_n(u, v)\}_{n = 0}^{\infty}$, $v \leq u \leq \sigma(v)$ as follows:

In order to obtain the oscillation criteria for Eq (1.2) we need the following conditions:

Let the sequence $\{T_k\}_{k\geq 0}$ be a sequence of real numbers such that $\lim_{k\rightarrow \infty}\ T_k = \infty$ and

Also, we define the sequence $\{\gamma_n\}_{n\geq 1}$, $\gamma_n > 1$ for all $n\in \mathbb{N}$ as follows:

Theorem 2.3. Let $n \in \mathbb{N}$ such that (2.14) and (2.15) are satisfied. If

then every solution of Eq (1.2) is oscillatory.

Theorem 2.4. Let $n \in \mathbb{N}_0$ such that (2.14) is satisfied. If

then every solution of Eq (1.2) is oscillatory.

3.   Numerical examples

Example 3.1. Consider the delay differential equation

where $b\in C([1, \infty), \mathbb{R})$ such that

$\{r_k\}_{k\ge0}$ is a sequence of positive integers such that $r_{k+1} > r_{k}+\frac{215}{121}$ and $\lim_{k\rightarrow \infty}\ r_k = \infty$, and

In view of (2.1) and (2.2), it is easy to see that

and

respectively.

Letting $T_k = 3r_k$, $k \in \mathbb{N}_0$, so $\Lambda_5(T_k) = 3r_k+\frac{645}{121}$, and hence

It is obvious that $g(\Lambda_{5}(T_{k})) = 3r_k+\frac {46}{11}$ and

Therefore,

for $t\in \left[3r_k+\frac {46}{11}, \, 3r_k+\frac{645}{121}\right]$. Denote $\beta_2 = \exp \left(\eta \exp \left(\eta\right)\right) > 1$. Then

Also,

for all $\eta\geq 0.61$ and $k \in \mathbb{N}_0$ and

It is obvious that

In view of this, (3.2) and (3.3), all conditions of Theorem 2.1 with $n = 1$ are satisfied for all $\eta\geq 0.61$. Therefore all solutions of Eq (3.1) are oscillatory for $\eta\geq 0.61$.

However, if we assume that $a_k = 3r_k$ and $b_k = 3r_k+\frac{645}{121}$, then $b_k-a_k = \frac{645}{121} < \infty$. It follows that [29,Theorem 3] cannot be applied to Eq (3.1). Note also that since $\tau$ is not monotone, [29,Theorem 6] cannot be applied to this example.

Example 3.2. Consider the advanced differential equation

where $c\in C([0, \infty), \mathbb{R})$ such that

where $\alpha, \delta \geq0$ and $\{r_k\}_{k\ge0}$ is a sequence of positive integers such that $r_{k+1} > r_{k}+\frac{49}{18}$ and $\lim_{k\rightarrow \infty}\ r_k = \infty$, and

In view of (2.12) and (2.13), it follows that

and

respectively.

Clearly,

If we assume that $T_k = 4r_k+\frac{10}{3}$, $k \in \mathbb{N}_0$, then $\Omega_4(T_k) = 4r_k-\frac{68}{9}$. It follows from (3.5) that

Thus

Therefore all conditions of Theorem 2.4 with $n = 0$ are satisfied for all $\delta\geq \frac{9\pi}{ 2+27\, \pi}$, and hence Eq (3.4) is oscillatory for $\delta\geq \frac{9\pi}{ 2+27\, \pi}$.

Acknowledgments

The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number (IF-PSAU- 2022/01/22323).

Conflicts of interest

The authors declare that they have no competing of interests regarding the publication of this paper.