Generalized $ \Xi $-metric-like space and new fixed point results with an application

Hasanen A. Hammad; Maryam G. Alshehri; Hasanen A. Hammad; Maryam G. Alshehri

doi:10.3934/math.2023127

AIMS Mathematics

2023, Volume 8, Issue 2: 2453-2472. doi: 10.3934/math.2023127

Previous Article Next Article

Research article Special Issues

Generalized $\Xi$ -metric-like space and new fixed point results with an application

Hasanen A. Hammad ^{1,2
,
,},
Maryam G. Alshehri ³

1.
Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Buraydah 52571, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
3.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia

Received: 11 September 2022 Revised: 26 October 2022 Accepted: 31 October 2022 Published: 04 November 2022
MSC : 47H10, 54H25

This paper is devoted to generalizing $\Xi$ -metric spaces and $b$ - metric-like spaces to present the structure of generalized $\Xi$ -metric-like spaces. The topological properties of this space and examples to support it are being investigated. Moreover, as demonstrated in the previous literature, the concept of Lipschitz mappings is presented more generally and some results of fixed points are derived in the aforementioned space. Finally, some theoretical results have been implicated in the discussion of the existence and uniqueness of the solution to the Fredholm integral equation.

Keywords:

Citation: Hasanen A. Hammad, Maryam G. Alshehri. Generalized $\Xi$ -metric-like space and new fixed point results with an application[J]. AIMS Mathematics, 2023, 8(2): 2453-2472. doi: 10.3934/math.2023127

Related Papers:

[1]	Changdev P. Jadhav, Tanisha B. Dale, Vaijanath L. Chinchane, Asha B. Nale, Sabri T. M. Thabet, Imed Kedim, Miguel Vivas-Cortez . On solutions of fractional differential equations for the mechanical oscillations by using the Laplace transform. AIMS Mathematics, 2024, 9(11): 32629-32645. doi: 10.3934/math.20241562
[2]	Duoduo Zhao, Kai Zhou, Fengming Ye, Xin Xu . A class of time-varying differential equations for vibration research and application. AIMS Mathematics, 2024, 9(10): 28778-28791. doi: 10.3934/math.20241396
[3]	Takumi Washio, Akihiro Fujii, Toshiaki Hisada . On random force correction for large time steps in semi-implicitly discretized overdamped Langevin equations. AIMS Mathematics, 2024, 9(8): 20793-20810. doi: 10.3934/math.20241011
[4]	Omar Bazighifan, Areej A. Al-moneef, Ali Hasan Ali, Thangaraj Raja, Kamsing Nonlaopon, Taher A. Nofal . New oscillation solutions of impulsive conformable partial differential equations. AIMS Mathematics, 2022, 7(9): 16328-16348. doi: 10.3934/math.2022892
[5]	Amjid Ali, Teruya Minamoto, Rasool Shah, Kamsing Nonlaopon . A novel numerical method for solution of fractional partial differential equations involving the $\psi$ -Caputo fractional derivative. AIMS Mathematics, 2023, 8(1): 2137-2153. doi: 10.3934/math.2023110
[6]	Ahmad Qazza, Rania Saadeh, Emad Salah . Solving fractional partial differential equations via a new scheme. AIMS Mathematics, 2023, 8(3): 5318-5337. doi: 10.3934/math.2023267
[7]	Aslı Alkan, Halil Anaç . The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method. AIMS Mathematics, 2024, 9(9): 25333-25359. doi: 10.3934/math.20241237
[8]	Shaomin Wang, Cunji Yang, Guozhi Cha . On the variational principle and applications for a class of damped vibration systems with a small forcing term. AIMS Mathematics, 2023, 8(9): 22162-22177. doi: 10.3934/math.20231129
[9]	Armin Hadjian, Juan J. Nieto . Existence of solutions of Dirichlet problems for one dimensional fractional equations. AIMS Mathematics, 2022, 7(4): 6034-6049. doi: 10.3934/math.2022336
[10]	Paul Bosch, José M. Rodríguez, José M. Sigarreta . Oscillation results for a nonlinear fractional differential equation. AIMS Mathematics, 2023, 8(5): 12486-12505. doi: 10.3934/math.2023627

Abstract

1. Introduction

The theory of fractional calculus has played an important role in engineering and natural sciences. Currently, the concept of fractional calculus has been effectively used in many social, physical, signal, image processing, biological and engineering problems. Further, it has been realized that a fractional system provides a more accurate interpretation than the integer-order system in many real modeling problems. For more details, one can refer to ^{[1,2,3,4,5,6,7,8,9,10]}.

Oscillation phenomena take part in different models of real world applications; see for instance the papers ^{[11,12,13,14,15,16,17]} and the papers cited therein. More precisely, we refer the reader to the papers ^[18,19] on bio-mathematical models where oscillation and/or delay actions may be formulated by means of cross-diffusion terms. Recently and although it is rare, the study on the oscillation of fractional partial differential equations has attracted many researchers. In ^{[20,21,22,23]}, the researchers have established the requirements of the oscillation for certain kinds of fractional partial differential equations.

In ^[24], Luo et al. studied the oscillatory behavior of the fractional partial differential equation of the form

$\begin{align*} &D_{+, t}^{1+\alpha}u(y, t)+p( t)D_{+, t}^{\alpha}u(y, t)+q(y, t)\int_{0}^{ t}( t-\hbar)^{-\alpha}u(y, \hbar)d\hbar\\ & = a( t)\Delta u(y, t)+\sum\limits_{i = 1}^{m}a_{i}( t)\Delta u(y, t-\tau_{i}), \quad (y, t)\in Q\times \mathbb{R}_{+} = H \end{align*}$

subject to either of the following boundary conditions

$\begin{align*} \frac{\partial u(y, t)}{\partial \nu}+\beta(y, t)u(y, t)& = 0, \quad (y, t)\in \partial Q\times \mathbb{R}_{+}\, , \\ u(y, t)& = 0, \quad (y, t)\in \partial Q \times \mathbb{R}_{+}\, . \end{align*}$

They have obtained some sufficient conditions for the oscillation of all solutions of this kind of fractional partial differential equations by using the integral averaging technique and Riccati transformations.

On other hand in ^[25], Xu and Meng considered a fractional partial differential equation of the form

$\begin{align*} &D_{+, t}^{\alpha}(r( t)D_{+, t}^{\alpha}u(y, t))+p( t)D_{+, t}^{\alpha}u(y, t)+q(y, t)f(u(y, t))\\ & = a( t)\Delta u(y, t)+\sum\limits_{i = 1}^{m}b_{i}( t)\Delta u(y, t-\tau_{i}), \quad (y, t)\in Q\times \mathbb{R}_{+} = H \end{align*}$

with the Robin boundary condition

$\frac{\partial u(y, t)}{\partial N}+g(y, t)u(y, t) = 0, \quad (y, t)\in \partial Q\times \mathbb{R}_{+},$

they obtained some oscillation criteria using the integral averaging technique and Riccati transformations.

Prakash et al. ^[26] considered the oscillation of the fractional differential equation

$\begin{align*} \frac{\partial}{\partial t}(r( t)D_{+, t}^{\alpha}u(y, t))+q(y, t)f\biggl(\int_{0}^{ t}( t-v)^{-\alpha}u(y, v)dv\biggl) = a( t)\Delta u(y, t), (y, t)\in Q\times \mathbb{R}_{+} \end{align*}$

with the Neumann boundary condition

$\begin{align*} \frac{\partial u(y, t)}{\partial N} = 0, \quad (y, t)\in \partial Q\times \mathbb{R}_{+}, \end{align*}$

they obtained some oscillation criteria by using the integral averaging technique and Riccati transformations.

Furthermore in ^[27], Ma et al. considered the forced oscillation of the fractional partial differential equation with damping term of the form

$\begin{align*} \frac{\partial}{\partial t}(r( t)D_{+, t}^{\alpha}u(y, t))+p( t)D_{+, t}^{\alpha}u(y, t)+q(y, t)f(u(y, t)) = a( t)\Delta u(y, t)+\tilde{g}(y, t), \quad (y, t)\in Q\times \mathbb{R}_{+} \end{align*}$

with the boundary condition

$\frac{\partial u(y, t)}{\partial N}+\beta(y, t)u(y, t) = 0, \quad (y, t)\in \partial Q\times \mathbb{R}_{+}\, ,$

they obtained some oscillation criteria by using the integral averaging technique.

From the above mentioned literature, one can notice that the Riccati transformation method has been incorporated into the proof of the oscillation results. Unlike previous results, however, we study in this paper the forced oscillation of the fractional partial differential equation with the damping term of the form

$\begin{align} &\frac{\partial}{\partial t}\left(a( t)\frac{\partial}{\partial t}\left(r(t)g\left(D_{+, t}^{\alpha} u(y, t)\right)\right)\right)+p( t)\frac{\partial}{\partial t}\left(r( t)g(D_{+, t}^{\alpha} u(y, t))\right)\\ & = b( t)\Delta u(y, t)+\sum\limits_{i = 1}^{m} a_{i}( t)\Delta u(y, t-\tau_{i})\\ &\quad-q(y, t)\int_{0}^{ t}( t-\hbar)^{-\alpha}u(y, \hbar)d\hbar+f(y, t), \quad (y, t)\in Q \times \mathbb{R_{+}} = H \end{align}$

(1.1)

via the application of the integral averaging technique only. Equation (1.1) is presented under a high degree of generality providing a general platform for many particular cases. Here, $D^{\alpha}_{+, t}u(y, t)$ is the Riemann-Liouville fractional partial derivative of order $\alpha$ of $u, \alpha\in (0, 1),$ $\Delta$ is the Laplacian in $\mathbb{R}^{n}$ , i.e.,

$\Delta u(y, t) = \sum^{n}_{r = 1}\frac{\partial^{2}u(y, t)}{\partial y_{r}^{2}},$

$Q$ is a bounded domain of $\mathbb{R}^{n}$ with the piecewise smooth boundary $\partial Q$ and $\mathbb{R}_{+}: = (0, \infty)$ .

Further, we assume the Robin and Dirichlet boundary conditions

$\begin{equation} \frac{\partial u(y, t)}{\partial N}+\gamma (y, t)u(y, t) = 0, \quad (y, t)\in \partial Q \times \mathbb{R}_{+} \end{equation}$

(1.2)

and

$\begin{equation} u(y, t) = 0, \quad (y, t)\in \partial Q \times \mathbb{R}_{+}, \end{equation}$

(1.3)

where $N$ is the unit outward normal to $\partial Q$ and $\gamma(y, t) > 0$ is a continuous function on $\partial Q \times \mathbb{R}_{+}$ . The following conditions are assumed throughout:

$\bf (H_{1})$ $a(t)\in C^{1}([t_{0}, \infty); \mathbb{R}_{+})$ and $r(t)\in C^{2}([t_{0}, \infty); \mathbb{R}_{+})$ ;

$\bf (H_{2})$ $g(t)\in C^{2}(\mathbb{R}; \mathbb{R})$ is an increasing function and there exists a positive constant $k$ such that $\frac{y}{g(y)} = k > 0,$ $y g(y)\neq 0$ for $y\neq 0$ ;

$\bf (H_{3})$ $p(t)\in C([t_{0}, \infty); \mathbb{R})$ and $A(t) = \int_{ t_{0}}^{ t}\frac{p(\zeta)}{a(\zeta)}d\zeta$ ;

$\bf (H_{4})$ $b(t), a_{i}(t)\in C(\mathbb{R_{+}}; \mathbb{R_{+}})$ and $\tau_{i}$ are non-negative constants, $i\in I_{m} = \{1, 2, \ldots, m\}$ ;

$\bf (H_{5})$ $q(y, t)\in C(H; \mathbb{R}_{+})$ and $q(t) = \min_{ y\in Q}q(y, t)$ ;

$\bf (H_{6})$ $f(y, t)\in C(\bar{H}; \mathbb{R})$ .

By a solution of the problems (1.1) and (1.2) (or (1.1)–(1.3)), we mean a function $u(y, t)\in C^{2+\alpha}(\bar{Q}\times[0, \infty)),$ which satisfies (1.1) on $H$ and the boundary condition (1.2) (or (1.3)).

A solution $u(y, t)$ of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is non-oscillatory.

The rest of the paper is organized as follows. Some basic definitions and known lemmas are included in Section 2. In Sections 3 and 4, we study the oscillations of (1.1) and (1.2), and (1.1) and (1.3), respectively. Section 5 deals with some applications for the sake of showing the feasibility and effectiveness of our results. Lastly, we add a conclusion in Section 6.

2. Preliminaries

Before we start the main work, we present some basic lemmas and definitions which are applied in what follows.

Definition 1. ^[4] The Riemann-Liouville fractional integral of order $\alpha > 0$ of a function $y: \mathbb{R}_{+}\rightarrow \mathbb{R}$ on the half-axis $\mathbb{R}_{+}$ is defined by

$\begin{equation*} (I^{\alpha}_{+}y)( t): = \frac{1}{\Gamma(\alpha)}\int^{ t}_{0}( t-\vartheta)^{\alpha-1}y(\vartheta)d\vartheta, \quad t > 0 \end{equation*}$

provided the right-hand side is pointwise defined on $\mathbb{R}_{+}$ , where $\Gamma$ is the gamma function.

Definition 2. ^[4] The Riemann-Liouville fractional derivative of order $\alpha > 0$ of a function $y:\mathbb{R}_{+} \rightarrow \mathbb{R}$ on the half-axis $\mathbb{R}_{+}$ is defined by

$\begin{equation*} (D_{+}^{\alpha}y)( t): = \frac{d^{\lceil \alpha \rceil}}{d t^{\lceil \alpha \rceil}}\left(I_{+}^{\lceil \alpha \rceil -\alpha}y\right)( t), \quad t > 0 \end{equation*}$

provided the right-hand side is pointwise defined on $\mathbb{R}_{+}$ , where $\lceil \alpha \rceil$ is the ceiling function of $\alpha$ .

Definition 3. ^[4] The Riemann-Liouville fractional partial derivative of order $0 < \alpha < 1$ with respect to $t$ of a function $u(y, t)$ is defined by

$\begin{equation*} (D^{\alpha}_{+, t}u)(y, t): = \frac{1}{\Gamma(1-\alpha)}\frac{\partial}{\partial t}\int^{ t}_{0}( t-\vartheta)^{-\alpha}u(y, \vartheta)d\vartheta \end{equation*}$

provided the right-hand side is pointwise defined on $\mathbb{R}_{+}$ .

Lemma 1. ^[4] Let $y$ be a solution of (1.1) and

$\begin{equation*} L( t): = \int^{ t}_{0}( t-\vartheta)^{-\alpha}y(\vartheta)d\vartheta \end{equation*}$

for $\alpha \in (0, 1)$ and $t > 0.$ Then

$\begin{equation*} L^{\prime}( t) = \Gamma (1-\alpha)(D^{\alpha}_{+}y)( t). \end{equation*}$

Lemma 2. ^[4] Let $\alpha \ge 0, m\in\mathbb{N}$ and $D = \frac{d}{d t}$ . If the fractional derivatives $(D_{a+}^{\alpha}y)(t)$ and $(D_{a+}^{\alpha+m}y)(t)$ exist, then

$\begin{align*} \left(D^{m}D_{a+}^{\alpha}y\right)( t) = (D_{a+}^{\alpha+m}y)( t). \end{align*}$

Lemma 3. ^[4] If $\alpha\in(0, 1)$ , then

$\left(I_{a+}^{\alpha}D_{a+}^{\alpha}y\right)( t) = y( t)-\frac{y_{1-\alpha}(a)}{\Gamma(\alpha)}( t-a)^{\alpha-1},$

where $y_{1-\alpha}(t) = (I_{a+}^{1-\alpha}y)(t)$ .

Lemma 4. ^[5] The smallest eigenvalue $\beta_{0}$ of the Dirichlet problem

$\begin{align*} \Delta \omega(y)+\beta \omega(y)& = 0\mathit{\text{ in }}Q\\ \omega(y)& = 0 \mathit{\text{ on }} \partial Q \end{align*}$

is positive and the corresponding eigenfunction $\phi(y)$ is positive in $Q.$

3. Oscillation of (1.1) and (1.2)

In this section, we establish the oscillation criteria for (1.1) and (1.2).

Theorem 1. If $\bf {(H_{1})-(H_{6})}$ are valid, $\lim\limits_{ t\to\infty}I_{+}^{1-\alpha}U(0) = C_{0}$ and if

$\begin{equation} \liminf\limits_{ t\to\infty}\int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau \right]d\hbar < 0 \end{equation}$

(3.1)

and

$\begin{equation} \limsup\limits_{ t\to\infty}\int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau \right]d\hbar > 0 \end{equation}$

(3.2)

for some constants $C_{0}, C_{1}$ and $C_{2}$ with $F(t) = \int_{Q}f(y, t)dy$ , then all solutions of (1.1) and (1.2) are oscillatory.

Proof. If $u(y, t)$ is a non-oscillatory solution of (1.1) and (1.2) then there exists a $t_{0}\ge 0$ such that $u(y, t) > 0$ (or $u(y, t) < 0), t\ge t_{0}.$

Case $1$ . Let $u(y, t) > 0$ for $t\ge t_{0}.$ Integrating (1.1) over $Q,$ we get

$\begin{align} &\frac{d}{d t}\left(a( t)\frac{d}{d t}\left(r( t)g(D_{+}^{\alpha} U( t))\right)\right)+p( t)\frac{d}{d t}\left(r( t)g(D_{+}^{\alpha} U( t))\right)\\ & = b( t)\int_{Q}\Delta u(y, t)dy+\sum\limits_{i = 1}^{m} a_{i}( t)\int_{Q}\Delta u(y, t-\tau_{i})dy\\ &\quad-\int_{Q}\left(q(y, t)\int_{0}^{ t}( t-\hbar)^{-\alpha}u(y, \hbar)d\hbar\right)dy+\int_{Q}f(y, t)dy, \end{align}$

(3.3)

where $U(t) = \int_{Q}u(y, t)dy$ with $U(t) > 0.$ By (1.2) and Green's formula, we have

$\begin{equation} \int_{Q}\Delta u(y, t)dy = \int_{\partial Q}\frac{\partial u}{\partial N}d\zeta = -\int_{\partial Q} \gamma(y, t)u(y, t)d\zeta < 0 \end{equation}$

(3.4)

and

$\begin{equation} \int_{Q}\Delta u(y, t-\tau_{i})dy < 0. \end{equation}$

(3.5)

Also, by $\bf {(H_{5})},$ one can get

$\begin{align} \int_{Q}\left(q(y, t)\int_{0}^{ t}( t-\hbar)^{-\alpha}u(y, \hbar)d\hbar\right)dy&\ge q( t)\int_{0}^{ t}( t-\hbar)^{-\alpha}\left(\int_{Q}u(y, \hbar)dy\right)d\hbar\\ & = q( t)L( t), \end{align}$

(3.6)

where $L(t) = \int_{0}^{ t}(t-\hbar)^{-\alpha}U(\hbar)d\hbar.$ Because of the inequalities (3.4)–(3.6), (3.3) becomes

$\begin{align*} \frac{d}{d t}\left(a( t)\frac{d}{d t}\left(r( t)g(D_{+}^{\alpha} U( t))\right)\right)+p( t)\frac{d}{d t}\left(r( t)g(D_{+}^{\alpha} U( t))\right)\le -q( t)L( t)+F( t)\le F( t). \end{align*}$

Thus, we get

$\begin{align*} \left(e^{A( t)}a( t)\left(r( t)g(D_{+}^{\alpha}U( t))\right)^{\prime}\right)^{\prime} = e^{A( t)}\left(\left(a( t)\left(r( t)g(D_{+}^{\alpha} U( t))\right)^{\prime}\right)^{\prime}+p( t)\left(r( t)g(D_{+}^{\alpha} U( t))\right)^{\prime}\right)\nonumber \le e^{A(t)}F(t). \end{align*}$

Integrating the above inequality over $[t_{0}, t]$ , one can get

$\begin{equation*} e^{A( t)}a( t)\left(r( t)g(D_{+}^{\alpha}U( t))\right)^{\prime}\le\int_{ t_{0}}^{ t} e^{A(\zeta)}F(\zeta)d\zeta+C_{1}, \end{equation*}$

where

$C_{1} = e^{A( t_{0})}a( t_{0})\left(r( t_{0})g(D_{+}^{\alpha}U( t_{0}))\right)^{\prime}.$

Again integrating the above inequality over $[t_{0}, t]$ , we get

$\begin{equation*} r( t)g(D_{+}^{\alpha}U( t))\le C_{2}+\int_{ t_{0}}^{ t}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{ t}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau, \end{equation*}$

where $C_{2} = r(t_{0})g(D_{+}^{\alpha}U(t_{0})).$ Then using $\bf {(H_{5})}$ , we obtain

$\begin{equation*} \frac{D_{+}^{\alpha}U( t)}{k}\le \frac{C_{2}}{r( t)}+\frac{1}{r( t)}\int_{ t_{0}}^{ t}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\frac{1}{r( t)}\int_{ t_{0}}^{ t}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau. \end{equation*}$

Applying the Riemann-Liouville fractional integral operator of order $\alpha$ to the above inequality and using Lemma 3, we obtain

$\begin{align*} U( t)-\frac{I_{0}^{1-\alpha}U(0)}{\Gamma(\alpha)} t^{\alpha-1}&\le\frac{k}{\Gamma(\alpha)} \int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\biggl[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau\\ &\quad+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau\biggl]d\hbar\, . \end{align*}$

Then

$\begin{align*} \liminf\limits_{ t\to\infty}U( t)&\le\liminf\limits_{ t\to\infty}\frac{C_{0}}{\Gamma(\alpha)} t^{\alpha-1}+\liminf\limits_{ t\to\infty}\left\{\frac{k}{\Gamma(\alpha)} \int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau\right.\right.\\ &\quad\left.\left.+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau\right]d\hbar\right\}. \end{align*}$

Therefore, by our hypothesis, as given by (3.1), we get $\liminf_{ t\to\infty}U(t)\le 0$ . This leads to a contradiction to $U(t) > 0.$

Case $2$ . Let $u(y, t) < 0$ for $t\ge t_{0}.$ Just as in Case $1,$ we can obtain that (3.3) holds and $U(t) < 0.$ By (1.2) and Green's formula, we get

$\begin{equation} \int_{Q}\Delta u(y, t)dy = \int_{\partial Q}\frac{\partial u}{\partial N}d\zeta = -\int_{\partial Q} \gamma(y, t)u(y, t)d\zeta > 0 \end{equation}$

(3.7)

and

$\begin{equation} \int_{Q}\Delta u(y, t-\tau_{i})dy > 0. \end{equation}$

(3.8)

Also, by $\bf {(H_{5})},$ we have

$\begin{align} \int_{Q}\left(q(y, t)\int_{0}^{ t}( t-\hbar)^{-\alpha}u(y, \hbar)d\hbar\right)dy&\le q( t)\int_{0}^{ t}( t-\hbar)^{-\alpha}\left(\int_{Q}u(y, \hbar)dy\right)d\hbar\\ & = q( t)L( t). \end{align}$

(3.9)

Because of the inequalities (3.7)–(3.9), (3.3) becomes

$\begin{equation} \frac{d}{d t}\left(a( t)\frac{d}{d t}\left(r( t)g(D_{+}^{\alpha} U( t))\right)\right)+p( t)\frac{d}{d t}\left(r( t)g(D_{+}^{\alpha} U( t))\right)\ge -q( t)L( t)+F(t)\ge F(t), \end{equation}$

(3.10)

that is,

$\left(e^{A( t)}a( t)\left(r( t)g(D_{+}^{\alpha}U( t))\right)^{\prime}\right)^{\prime} =\\ e^{A( t)}\left(\left(a( t)\left(r( t)g(D_{+}^{\alpha} U( t))\right)^{\prime}\right)^{\prime}+p( t)\left(r( t)g(D_{+}^{\alpha} U( t))\right)^{\prime}\right)\nonumber \ge e^{A( t)}F( t).$

Integrating the above inequality over [ $t_{0}, t$ ], we have

$\begin{equation*} e^{A( t)}a( t)\left(r( t)g(D_{+}^{\alpha}U( t))\right)^{\prime}\ge\int_{ t_{0}}^{ t} e^{A(\zeta)}F(\zeta)d\zeta+C_{1}, \end{equation*}$

where

$C_{1} = e^{A( t_{0})}a( t_{0})\left(r( t_{0})g(D_{+}^{\alpha}U( t_{0}))\right)^{\prime}.$

Again integrating the above inequality over [ $t_{0}, t$ ], we obtain

$\begin{equation*} r( t)g(D_{+}^{\alpha}U( t))\ge C_{2}+\int_{ t_{0}}^{ t}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{ t}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau, \end{equation*}$

where $C_{2} = r(t_{0})g(D_{+}^{\alpha}U(t_{0})).$ Then using $\bf {(H_{5})}$ , we obtain

$\begin{align*} \frac{D_{+}^{\alpha}U( t)}{k}\ge \frac{C_{2}}{r( t)}+\frac{1}{r( t)}\int_{ t_{0}}^{ t}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\frac{1}{r( t)}\int_{ t_{0}}^{ t}\frac{1}{e^{A(\tau)}a(\tau)}\biggl(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F(\zeta)d\zeta\biggl)d\tau. \end{align*}$

Applying the Riemann-Liouville fractional integral operator of order $\alpha$ to the above inequality and using Lemma $3,$ we obtain

$\begin{align*} U( t)-\frac{I_{0}^{1-\alpha}U(0)}{\Gamma(\alpha)} t^{\alpha-1}\ge&\frac{k}{\Gamma(\alpha)} \int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\biggl[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau\\&+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\biggl(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F(\zeta)d\zeta\biggl)d\tau\biggl]d\hbar. \end{align*}$

Then

$\begin{align*} \limsup\limits_{ t\to\infty}U( t)&\ge\limsup\limits_{ t\to\infty}\frac{C_{0}}{\Gamma(\alpha)} t^{\alpha-1}+\limsup\limits_{ t\to\infty}\biggl\{\frac{k}{\Gamma(\alpha)} \int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\biggl[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau\\ &\quad+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\biggl(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F(\zeta)d\zeta\biggl)d\tau\biggl]d\hbar\biggl\}. \end{align*}$

Therefore, by our hypothesis given by (3.2), we get $\limsup_{ t\to\infty}U(t)\ge 0$ . This leads to a contradiction to $U(t) < 0.$

4. Oscillation of (1.1) and (1.3)

In this section, we establish the oscillation criteria for (1.1) and (1.3).

Theorem 2. If $\bf {(H_{1})-(H_{5})}$ are valid, $\lim\limits_{ t\to\infty}I_{+}^{1-\alpha}U_{1}(0) = A_{1}$ and if

$\begin{equation} \liminf\limits_{ t\to\infty}\int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\biggl[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\biggl(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F_{1}(\zeta)d\zeta\biggl)d\tau \biggl]d\hbar < 0 \end{equation}$

(4.1)

and

$\begin{equation} \limsup\limits_{ t\to\infty}\int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\biggl[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\biggl(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F_{1}(\zeta)d\zeta\biggl)d\tau \biggl]d\hbar > 0 \end{equation}$

(4.2)

for some constants $A_{1}, C_{1}$ and $C_{2}$ with

$F_{1}( t) = \int_{Q}f(y, t)\phi(y)dy\;\mathit{\text{and}}\; U_{1}( t) = \int_{Q}u(y, t)\phi(y)dy,$

then all solutions of $(1.1)$ and $(1.3)$ are oscillatory.

Proof. If $u(y, t)$ is a non-oscillatory solution of (1.1) and (1.3) then there exists a $t_{0}\ge 0$ such that $u(y, t) > 0$ (or $u(y, t) < 0$ ) for $t\ge t_{0}.$

Case $1$ . Let $u(y, t) > 0$ for $t\ge t_{0}.$ Multiplying $(1.1)$ by $\phi(y)$ and then integrating over $Q,$ we get

$\begin{align} &\int_{Q}\frac{\partial}{\partial t}\left(a( t)\frac{\partial}{\partial t}\left(r( t)g(D_{+, t}^{\alpha} u(y, t))\right)\right)\phi(y)dy+\int_{Q}p( t)\frac{\partial}{\partial t}\left(r( t)g(D_{+, t}^{\alpha} u(y, t))\right)\phi(y)dy\\ & = \int_{Q}b( t)\Delta u(y, t)\phi(y)dy+\int_{Q}\sum\limits_{i = 1}^{m} a_{i}( t)\Delta u(y, t-\tau_{i})\phi(y)dy\\ &\quad-\int_{Q}\left(q(y, t)\int_{0}^{ t}( t-\hbar)^{-\alpha}u(y, \hbar)d\hbar\right)\phi(y)dy+\int_{Q}f(y, t)\phi(y)dy. \end{align}$

(4.3)

By Lemma 4 and Green's formula, we have

$\begin{equation} \int_{Q}\Delta u(y, t)\phi(y)dy = \int_{Q}u(y, t)\Delta \phi(y)dy = -\beta_{0}\int_{Q}u(y, t)\phi(y)dy < 0 \end{equation}$

(4.4)

and

$\begin{equation} \int_{Q}\Delta u(y, t-\tau_{i})\phi(y)dy < 0. \end{equation}$

(4.5)

Also, by $\bf {(H_{5})},$ we get

$\begin{align} \int_{Q}\left(q(y, t)\int_{0}^{ t}( t-\hbar)^{-\alpha}u(y, \hbar)\phi(y)d\hbar\right)dy&\ge q( t)\int_{0}^{ t}( t-\hbar)^{-\alpha}\left(\int_{Q}u(y, \hbar)\phi(y)dy\right)d\hbar\\ & = q( t)L_{1}( t), \end{align}$

(4.6)

where

$L_{1}( t) = \int_{0}^{ t}( t-\hbar)^{-\alpha}U_{1}(\hbar)d\hbar > 0.$

Because of the inequalities (4.4)–(4.6), (4.3) becomes

$\begin{equation*} \frac{d}{d t}\left(a( t)\frac{d}{d t}\left(r( t)g(D_{+}^{\alpha} U_{1}( t))\right)\right)+p( t)\frac{d}{d t}\left(r( t)g(D_{+}^{\alpha} U_{1}( t))\right)\le -q( t)L_{1}( t)+F_{1}( t)\le F_{1}( t), \end{equation*}$

that is,

$\left(e^{A( t)}a( t)\left(r( t)g(D_{+}^{\alpha}U_{1}( t))\right)^{\prime}\right)^{\prime} = \\ e^{A( t)}\biggl[\biggl(a( t)\biggl(r( t)g(D_{+}^{\alpha} U_{1}( t))\biggl)^{\prime}\biggl)^{\prime}+p( t)\biggl(r( t)g(D_{+}^{\alpha} U_{1}( t))\biggl)^{\prime}\biggl]\le e^{A( t)}F_{1}( t).$

Integrating the above inequality over $[t_{0}, t]$ , we have

$\begin{equation*} e^{A( t)}a( t)\left(r( t)g(D_{+}^{\alpha}U_{1}( t))\right)^{\prime}\le\int_{ t_{0}}^{ t} e^{A(\zeta)}F_{1}(\zeta)d\zeta+C_{1}, \end{equation*}$

where

$C_{1} = e^{A( t_{0})}a( t_{0})\left(r( t_{0})g(D_{+}^{\alpha}U_{1}( t_{0}))\right)^{\prime}.$

Again integrating the above inequality over $[t_{0}, t]$ , we have

$\begin{equation*} r( t)g(D_{+}^{\alpha}U_{1}( t))\le C_{2}+\int_{ t_{0}}^{ t}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{ t}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F_{1}(\zeta)d\zeta\right)d\tau, \end{equation*}$

where $C_{2} = r(t_{0})g(D_{+}^{\alpha}U_{1}(t_{0})).$ Then using $\bf {(H_{5})}$ , we obtain

$\begin{equation*} \frac{D_{+}^{\alpha}U_{1}( t)}{k}\le \frac{C_{2}}{r( t)}+\frac{1}{r( t)}\int_{ t_{0}}^{ t}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\frac{1}{r( t)}\int_{ t_{0}}^{ t}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F_{1}(\zeta)d\zeta\right)d\tau. \end{equation*}$

Applying the Riemann-Liouville fractional integral operator of order $\alpha$ to the above inequality and using Lemma 3, we obtain

$\begin{align*} U_{1}( t)-\frac{I_{0}^{1-\alpha}U_{1}(0)}{\Gamma(\alpha)} t^{\alpha-1}&\le\frac{k}{\Gamma(\alpha)} \int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\biggl[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau\\ &\quad+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\biggl(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F_{1}(\zeta)d\zeta\biggl)d\tau\biggl]d\hbar. \end{align*}$

Then

$\begin{align*} \liminf\limits_{ t\to\infty}U_{1}( t)&\le\liminf\limits_{ t\to\infty}\frac{A_{1}}{\Gamma(\alpha)} t^{\alpha-1}+\liminf\limits_{ t\to\infty}\biggl\{\frac{k}{\Gamma(\alpha)} \int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\biggl[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau\\&+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\biggl(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F_{1}(\zeta)d\zeta\biggl)d\tau\biggl]d\hbar\biggl\}. \end{align*}$

Therefore, by our hypothesis given by (4.1), we get $\liminf_{ t\to\infty}U_{1}(t)\le 0$ . This leads to a contradiction to $U_{1}(t) > 0.$

Case $2$ . Let $u(y, t) < 0$ for $t\ge t_{0}.$ Multiplying $(1.1)$ by $\phi(y)$ and then integrating over $Q,$ one can get (4.3). Using Green's formula, we have

$\begin{equation} \int_{Q}\Delta u(y, t)\phi(y)dy = \int_{Q}u(y, t)\Delta \phi(y)dy = -\beta_{0}\int_{Q}u(y, t)\phi(y) dy > 0 \end{equation}$

(4.7)

and

$\begin{equation} \int_{Q}\Delta u(y, t-\tau_{i})\phi(y)dy > 0. \end{equation}$

(4.8)

Also, by $\bf {(H_{5})},$ we have

$\begin{align} \int_{Q}\biggl(q(y, t)\int_{0}^{ t}( t-\hbar)^{-\alpha}u(y, \hbar)d\hbar\biggl)\phi(y)dy&\le q( t)\int_{0}^{ t}( t-\hbar)^{-\alpha}\biggl(\int_{Q}u(y, \hbar)\phi(y)dy\biggl)d\hbar\\ & = q( t)L_{1}( t), \end{align}$

(4.9)

where

$L_{1}( t) = \int_{0}^{ t}( t-\hbar)^{-\alpha}U_{1}(\hbar)d\hbar < 0.$

Because of the inequalities (4.7)–(4.9), (4.3) becomes

$\begin{equation} \frac{d}{d t}\biggl(a( t)\frac{d}{d t}\biggl(r( t)g(D_{+}^{\alpha} U_{1}( t))\biggl)\biggl)+p( t)\frac{d}{d t}\biggl(r( t)g(D_{+}^{\alpha} U_{1}( t))\biggl)\ge -q( t)L_{1}( t)+F_{1}( t)\ge F_{1}( t), \end{equation}$

(4.10)

that is,

$\biggl(e^{A( t)}a( t)\biggl(r( t)g(D_{+}^{\alpha}U_{1}( t))\biggl)^{\prime}\biggl)^{\prime} = \\ e^{A( t)}\biggl(\biggl(a( t)\biggl(r( t)g(D_{+}^{\alpha} U_{1}( t))\biggl)^{\prime}\biggl)^{\prime}+p( t)\biggl(r( t)g(D_{+}^{\alpha} U_{1}( t))\biggl)^{\prime}\biggl) \ge e^{A( t)}F_{1}( t).$

Integrating the above inequality over $[t_{0}, t]$ , we get

$\begin{equation*} e^{A( t)}a( t)\left(r( t)g(D_{+}^{\alpha}U_{1}( t))\right)^{\prime}\ge\int_{ t_{0}}^{ t} e^{A(\zeta)}F_{1}(\zeta)d\zeta+C_{1}, \end{equation*}$

where

$C_{1} = e^{A( t_{0})}a( t_{0})\left(r( t_{0})g(D_{+}^{\alpha}U_{1}( t_{0}))\right)^{\prime}.$

Again integrating the above inequality over $[t_{0}, t]$ , we get

$\begin{equation*} r( t)g(D_{+}^{\alpha}U_{1}( t))\ge C_{2}+\int_{ t_{0}}^{ t}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{ t}\frac{1}{e^{A(\tau)}a(\tau)}\biggl(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F_{1}(\zeta)d\zeta\biggl)d\tau, \end{equation*}$

where $C_{2} = r(t_{0})g(D_{+}^{\alpha}U_{1}(t_{0})).$ Then using $\bf {(H_{5})}$ , we obtain

$\begin{equation*} \frac{D_{+}^{\alpha}U_{1}( t)}{k}\ge \frac{C_{2}}{r( t)}+\frac{1}{r( t)}\int_{ t_{0}}^{ t}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\frac{1}{r( t)}\int_{ t_{0}}^{ t}\frac{1}{e^{A(\tau)}a(\tau)}\biggl(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F_{1}(\zeta)d\zeta\biggl)d\tau. \end{equation*}$

Applying the Riemann-Liouville fractional integral operator of order $\alpha$ to the above inequality and using Lemma 3, we obtain

$\begin{align*} U_{1}( t)-\frac{I_{0}^{1-\alpha}U_{1}(0)}{\Gamma(\alpha)} t^{\alpha-1}&\ge\frac{k}{\Gamma(\alpha)} \int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau\right.\\ &\quad\left.+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F_{1}(\zeta)d\zeta\right)d\tau\right]d\hbar. \end{align*}$

Then

$\begin{align*} \limsup\limits_{ t\to\infty}U_{1}( t)&\ge\limsup\limits_{ t\to\infty}\frac{A_{1}}{\Gamma(\alpha)} t^{\alpha-1}+\limsup\limits_{ t\to\infty}\left\{\frac{k}{\Gamma(\alpha)} \int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau\right.\right.\\ &\quad\left.\left.+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau} e^{A(\zeta)}F_{1}(\zeta)d\zeta\right)d\tau\right]d\hbar\right\}. \end{align*}$

Therefore, by our hypothesis given by $(4.2)$ , we get $\limsup_{ t\to\infty}U_{1}(t)\ge 0$ . This leads to a contradiction to $U_{1}(t) < 0.$

5. Applications

In this section, we give two examples to illustrate our main results.

Example 1. Let us consider the fractional partial differential system

$\begin{align} D_{+, t}^{\frac{5}{2}}u(y, t)& = \frac{1}{\pi}\Delta u(y, t)+2 t\Delta u(y, t-1)\\ &\quad-\left(y^{2}+\frac{1}{ t^{2}}\right)\int_{0}^{ t}( t-\hbar)^{-\frac{1}{2}}u(y, \hbar)d\hbar+e^{2 t}\cos (t)\sin (y), \quad (y, t)\in (0, \pi)\times \mathbb{R_{+}} \end{align}$

(5.1)

with the condition

$\begin{align} u_{y}(0, t) = u_{y}(\pi, t) = 0. \end{align}$

(5.2)

In the above, $a(t) = 1, r(t) = 1, g(t) = t, \alpha = 1/2, p(t) = 0, b(t) = 1/\pi, m = 1, a_{1}(t) = 2 t, \tau_{1} = 1$ , $q(y, t) = (y^{2}+\frac{1}{ t^{2}}), f(y, t) = e^{2 t}\cos (t) \sin (y), Q = (0, \pi), q(t) = \min_{y\in(0, \pi)}q(y, t) = 1/ t^{2}$ and $t_{0} = 0$ .

Since $F(t) = \int_{0}^{\pi}e^{2 t}\cos (t) \sin (y) dy = 2e^{2 t}\cos (t)$ and $A(t) = 0,$ we have

$\begin{align*} &\int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau \right]d\hbar\\ & = C_{2}\int_{0}^{ t}( t-\hbar)^{-1/2}d\hbar+C_{1}\int_{0}^{ t}( t-\hbar)^{-1/2}\left(\int_{0}^{\hbar}d\tau\right)d\hbar\\ &\quad+\frac{2}{5}\int_{0}^{ t}( t-\hbar)^{-1/2}\left(2\int_{0}^{\hbar}e^{2\tau}\cos (\tau )d\tau+\int_{0}^{\hbar}e^{2\tau}\sin (\tau) d\tau-2\int_{0}^{\hbar}d\tau\right)d\hbar\\ & = (C_{2}-6/25)\int_{0}^{ t}( t-\hbar)^{-1/2}d\hbar+(C_{1}-4/5)\int_{0}^{ t}\hbar( t-\hbar)^{-1/2}d\hbar\\ &\quad+6/25\int_{0}^{ t}e^{2\hbar}( t-\hbar)^{-1/2}\cos (\hbar) d\hbar+8/25\int_{0}^{ t}e^{2\hbar}( t-\hbar)^{-1/2}\sin( \hbar) d\hbar. \end{align*}$

Fixing $y^{2} = t-\hbar$ , then

$\begin{align} &\int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau \right]d\hbar\\ & = 2(C_{2}-6/25)\sqrt{ t}+4/3(C_{1}-4/5) t^{3/2}\\ &\quad+4/25e^{2 t}\left(3\cos (t) \int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy+3\sin (t)\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy\right.\\ &\quad\left.+4\sin (t)\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy-4\cos (t) \int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy\right). \end{align}$

(5.3)

Pointing out that

$\begin{align*} |e^{-2y^{2}}\cos (y^{2})|\le e^{-2y^{2}}, \; |e^{-2y^{2}}\sin (y^{2})|\le e^{-2y^{2}}\;\text{ and }\;\lim\limits_{ t\to\infty}\int_{0}^{\sqrt{ t}}e^{-2y^{2}}dy = \frac{\sqrt{2\pi}}{4}, \end{align*}$

we can conclude that

$\begin{align*} \lim\limits_{ t\to\infty}\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy\quad\text{ and }\quad\lim\limits_{ t\to\infty}\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy \end{align*}$

are convergent. Thus, we have that

$\begin{align*} &\lim\limits_{t\to\infty}\left[\cos (t) \left(3\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy-4\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy\right)\right.\\ &\quad\left.+\sin t\left(3\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy+4\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy\right)\right] \end{align*}$

is convergent. Fixing

$\begin{align*} \lim\limits_{ t\to\infty}\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy = P, \quad\lim\limits_{ t\to\infty}\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy = Q \end{align*}$

and considering the sequence

$\begin{align*} t_{n} = \frac{3\pi}{2}+2n\pi-\arctan \left(\frac{3P-4Q}{3Q+4P}\right), \end{align*}$

we get

$\begin{align*} &\lim\limits_{n\to\infty}\left\{\cos (t_{n}) \left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy-4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy\right)\right.\\ &\quad\left.+\sin (t_{n})\left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy+4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy\right)\right\}\\ & = \sqrt{(3P-4Q)^{2}+(3Q+4P)^{2}}\sin\left(\frac{3\pi}{2}+2n\pi-\arctan \left(\frac{3P-4Q}{3Q+4P}\right)+\arctan \left(\frac{3P-4Q}{3Q+4P}\right)\right)\\ & = 5\sqrt{P^{2}+Q^{2}}\sin\left(\frac{3\pi}{2}+2n\pi\right)\\ & = -5\sqrt{P^{2}+Q^{2}}. \end{align*}$

Since $\lim\limits_{n\to\infty} t_{n} = \infty$ , from (5.3), we have

$\begin{align*} &\liminf\limits_{ t\to\infty}\int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau \right]d\hbar\\ & = \liminf\limits_{n\to\infty}\left\{2(C_{2}-6/25)\sqrt{ t_{n}}+4/3(e^{-1}C_{1}-4/5) t_{n}^{3/2}+4/25e^{2 t_{n}}\left[\cos (t_{n}) \left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy\right.\right.\right.\\ &\quad\left.\left.\left.-4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy\right)+\sin (t_{n})\left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy+4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy \right)\right]\right\}\\ & = -\infty < 0. \end{align*}$

Similarly, fixing

$\begin{align*} t_{n} = \frac{\pi}{2}+2n\pi-\arctan \left(\frac{3P-4Q}{3Q+4P}\right), \end{align*}$

we get

$\begin{align*} &\lim\limits_{n\to\infty}\left[\cos (t_{n}) \left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy-4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy\right)\right.\\ &\quad\left.+\sin (t_{n})\left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy+4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy\right)\right]\\ & = 5\sqrt{P^{2}+Q^{2}}. \end{align*}$

Thus, from (5.3), we can get

$\limsup\limits_{ t\to\infty}\int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau \right]d\hbar\\ = \limsup\limits_{n\to\infty}\\ \left\{2(C_{2}-6/25)\sqrt{ t_{n}}+4/3(e^{-1}C_{1}-4/5) t_{n}^{3/2}+4/25e^{2 t_{n}}\left[\cos (t_{n}) \left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy\right.\right.\right.\\ \quad\left.\left.\left.-4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy\right)+\sin (t_{n})\left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy+4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy\right) \right]\right\}\\ = \infty > 0.$

Therefore, by referring to Theorem 1, the solutions of (5.1) and (5.2) are oscillatory.

Example 2. Let us consider the fractional partial differential system

$\frac{1}{1875\sqrt{\pi}}D_{+, t}^{\frac{5}{2}}u(y, t) = \frac{1}{10^{5} t^{\frac{5}{2}}}\Delta u(y, t)+\\ \left(\frac{16}{75\times 10^{2}\pi t^{3}}-\frac{16e^{2 t}\cos (t)}{5\pi t^{3}}\right)\int_{0}^{ t}( t-\hbar)^{-\frac{1}{2}}u(y, \hbar)d\hbar\\ +e^{2 t}\cos (t)\cos(10y), \quad (y, t)\in (0, \pi)\times (0, 1.5)$

(5.4)

with the condition

$\begin{align} u(0, t) = u(\pi, t) = 0. \end{align}$

(5.5)

In the above, $a(t) = 1$ , $r(t) = \frac{1}{1875\sqrt{\pi}}$ , $g(t) = t$ , $\alpha = 1/2$ , $p(t) = 0$ , $b(t) = \frac{1}{2\times 10^{5} t^{\frac{5}{2}}}$ , $m = 1$ , $a_{1}(t) = \frac{1}{2\times 10^{5} t^{\frac{5}{2}}}$ , $\tau_{1} = 0$ , $q(y, t) =$ $\frac{-16}{75\times 10^{2}\pi t^{3}}+\frac{16e^{2 t}\cos (t)}{5\pi t^{3}}, f(y, t) =$ $e^{2 t}\cos (t) \sin (y), Q = (0, \pi), q(t) = \min_{y\in(0, \pi)}q(y, t) =$ $\frac{-16}{75\times 10^{2}\pi t^{3}}+\frac{16e^{2 t}\cos (t)}{5\pi t^{3}}$ and $t_{0} = 0$ . It is obvious that $\beta_{0} = 1$ and $\phi(y) = \sin (y).$ Since $F_{1}(t) = \int_{0}^{\pi}e^{2 t}\cos (t) \cos(10y) \sin (y) dy =$ $\frac{-2}{99}e^{2 t}\cos (t)$ and $A(t) = 0,$ we have

$\begin{align*} &\int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left(C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau \right)d\hbar\\ & = -\frac{3750\sqrt{\pi}}{99}\int_{0}^{ t}( t-\hbar)^{\alpha-1}\left(\int_{ t_{0}}^{\hbar}\left(\int_{ t_{0}}^{\tau}e^{2\zeta}\cos (\zeta) d\zeta\right)d\tau \right)d\hbar\\ & = \frac{50\sqrt{\pi}}{11}\int_{0}^{ t}( t-\hbar)^{-1/2}d\hbar+\frac{500\sqrt{\pi}}{33}\int_{0}^{ t}\hbar( t-\hbar)^{-1/2}d\hbar\\ &\quad-\frac{50\sqrt{\pi}}{33}\left[3\int_{0}^{ t}e^{2\hbar}( t-\hbar)^{-1/2}\cos (\hbar) d\hbar+4\int_{0}^{ t}e^{2\hbar}( t-\hbar)^{-1/2}\sin( \hbar) d\hbar\right]. \end{align*}$

Fixing $y^{2} = t-\hbar$ , then

$\int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau \right]d\hbar\\ = \frac{10^{2}\sqrt{\pi t}}{11}+\frac{2\times 10^{3}}{99} t^{3/2}-\frac{10^{2}\sqrt{\pi}}{33}e^{2 t}\left\{3\cos (t) \int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy\right.\\ \left.+3\sin (t)\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy+4\sin (t)\\ \int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy-4\cos (t) \int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy\right\}.$

(5.6)

Pointing out that

$|e^{-2y^{2}}\cos (y^{2})|\le e^{-2y^{2}}, \; |e^{-2y^{2}}\sin (y^{2})|\le e^{-2y^{2}}\;\text{ and }\; \lim\limits_{ t\to\infty}\int_{0}^{\sqrt{ t}}e^{-2y^{2}}dy = \frac{\sqrt{2\pi}}{4},$

we can conclude that

$\lim\limits_{ t\to\infty}\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy\quad\text{ and }\quad\lim\limits_{ t\to\infty}\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy$

are convergent. Thus, we have that

$\begin{align*} \lim\limits_{ t\to\infty}&\left[\cos (t) \left(3\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy-4\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy\right)\right.\\ &\quad\left.+\sin t\left(3\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy+4\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy\right)\right] \end{align*}$

is convergent. Fixing

$\lim\limits_{ t\to\infty}\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\cos (y^{2})dy = P\;\text{ and }\;\lim\limits_{ t\to\infty}\int_{0}^{\sqrt{ t}}e^{-2y^{2}}\sin (y^{2})dy = Q$

and considering the sequence

$t_{n} = \frac{3\pi}{2}+2n\pi-\arctan \biggl(\frac{3P-4Q}{3Q+4P}\biggl),$

we get

$\begin{align*} \lim\limits_{n\to\infty}&\left\{\cos (t_{n}) \left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy-4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy\right)\right.\\ &\quad\left.+\sin (t_{n})\left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy+4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy\right)\right\}\\ & = \sqrt{(3P-4Q)^{2}+(3Q+4P)^{2}}\sin\left(\frac{3\pi}{2}+2n\pi-\arctan \left(\frac{3P-4Q}{3Q+4P}\right)+\arctan \left(\frac{3P-4Q}{3Q+4P}\right)\right)\\ & = 5\sqrt{P^{2}+Q^{2}}\sin\left(\frac{3\pi}{2}+2n\pi\right)\\ & = -5\sqrt{P^{2}+Q^{2}}. \end{align*}$

Since $\lim\limits_{n\to\infty} t_{n} = \infty$ , from (5.6), we have

$\limsup\limits_{ t\to\infty} \int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau \right]d\hbar\\ = \limsup\limits_{n\to\infty}\left\{\frac{10^{2}\sqrt{\pi t_{n}}}{11}\sqrt{ t_{n}}+\frac{2\times 10^{3}}{99} t_{n}^{3/2}-\frac{10^{2}\sqrt{\pi}}{33}e^{2 t_{n}}\left[\cos (t_{n}) \left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy\right.\right.\right.\\ \quad\left.\left.\left.-4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy\right)+\sin (t_{n})\left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy+4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy \right)\right]\right\}\\ = \infty > 0.$

Similarly, fixing

$\begin{align*} t_{n} = \frac{\pi}{2}+2n\pi-\arctan \left(\frac{3P-4Q}{3Q+4P}\right), \end{align*}$

we get

Thus, from (5.6), we get

$\liminf\limits_{ t\to\infty} \int_{0}^{ t}\frac{( t-\hbar)^{\alpha-1}}{r(\hbar)}\left[C_{2}+\int_{ t_{0}}^{\hbar}\frac{C_{1}}{e^{A(\tau)}a(\tau)}d\tau+\int_{ t_{0}}^{\hbar}\frac{1}{e^{A(\tau)}a(\tau)}\left(\int_{ t_{0}}^{\tau}e^{A(\zeta)}F(\zeta)d\zeta\right)d\tau \right]d\hbar\\ = \liminf\limits_{n\to\infty}\left\{\frac{10^{2}\sqrt{\pi t_{n}}}{11}\sqrt{ t_{n}}+\frac{2\times 10^{3}}{99} t_{n}^{3/2}-\frac{10^{2}\sqrt{\pi}}{33}e^{2 t_{n}}\left[\cos (t_{n}) \left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy\right.\right.\right.\\ \quad\left.\left.\left.-4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy\right)+\sin (t_{n})\left(3\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\sin (y^{2})dy+4\int_{0}^{\sqrt{ t_{n}}}e^{-2y^{2}}\cos (y^{2})dy \right)\right]\right\}\\ = -\infty < 0.$

Therefore, by referring to Theorem 2, the solutions of (5.4) and (5.5) are oscillatory. In fact, $u(y, t) = t^{5/2}\cos(10y)$ is a solution of (5.4) and (5.5) and its oscillatory behavior is demonstrated in Figure 1.

Figure 1. Oscillatory behavior of

$u(y, t) = t^{5/2}\cos(10y)$ .

DownLoad: Full-Size Img PowerPoint

6. Conclusions

In this paper, we have obtained some new oscillation results for the fractional partial differential equation with damping and forcing terms under Robin and Dirichlet boundary conditions. The main results are proved by using only the integral averaging technique and without implementing the Riccati approach. Further, the obtained results are justified by some examples which can not be commented upon by using the previous results. Our results have been obtained for the general equation which may cover other particular cases.

Acknowledgments

A. Palanisamy was supported by the University Grants Commission (UGC-Ref. No.: 958/(CSIR-UGC NET JUNE 2018)), New Delhi, India and the third author was partially supported by the DST-FIST Scheme (Grant No. SR/FST/MST-115/2016), New Delhi, India. J. Alzabut is thankful to Prince Sultan University and Ostim Technical University for their endless support. The authors are grateful to the reviewers for their precious help in improving this manuscript.

Conflict of interest

The authors declare that they have no conflicts of interest.

References

[1]	S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, Fund. Math., 3 (1922), 133–181.
[2]	L. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468–1476. https://doi.org/10.1016/j.jmaa.2005.03.087 doi: 10.1016/j.jmaa.2005.03.087
[3]	L. Cirić, H. Lakzian, V. Rakočević, Fixed point theorems for $w$ -cone distance contraction mappings in $tvs$ -cone metric spaces, Fixed Point Theory Appl., 2012 (2012), 3. https://doi.org/10.1186/1687-1812-2012-3 doi: 10.1186/1687-1812-2012-3
[4]	W. S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal., 72 (2010), 2259–2261. https://doi.org/10.1016/j.na.2009.10.026 doi: 10.1016/j.na.2009.10.026
[5]	S. Janković, Z. Kadelburg, S. Radenović, On cone metric spaces: a survey, Nonlinear Anal., 74 (2011), 2591–2601. https://doi.org/10.1016/j.na.2010.12.014 doi: 10.1016/j.na.2010.12.014
[6]	H. A. Hammad, H. Aydi, C. Park, Fixed point approach for solving a system of Volterra integral equations and Lebesgue integral concept in $F_CM$ -spaces, AIMS Math., 7 (2021), 9003–9022. https://doi.org/10.3934/math.2022501 doi: 10.3934/math.2022501
[7]	H. A. Hammad, M. De la Sen, Application to Lipschitzian and integral systems via a quadruple coincidence point in fuzzy metric spaces, Mathematics, 10 (2022), 1905. https://doi.org/10.21203/rs.3.rs-976766/v1 doi: 10.21203/rs.3.rs-976766/v1
[8]	A. A. Rawashdeh, W. Shatanawi, M. Khandaqji, N. Shahzad, Normed ordered and $\Xi$ -metric spaces, Int. J. Math. Math. Sci., 2012 (2012), 272137. https://doi.org/10.1155/2012/272137 doi: 10.1155/2012/272137
[9]	C. Cevik, I. Altun, Vector metric spaces and some properties, Topol. Meth. Nonlinear Anal., 34 (2009), 375–382.
[10]	R. Cristescu, Order structures in normed vector spaces, Editura Ştiinţifică Enciclopedică, 1983.
[11]	J. Matkowski, Integrable solutions of functional equations, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1975.
[12]	R. Wegrzyk, Fixed point theorems for multifunctions and their applications to functional equations, Diss. Math., 201 (1982), 1–28.
[13]	Z. Pales, I. R. Petre, Iterative fixed point theorems in $\Xi$ -metric spaces, Acta. Math. Hungarica, 140 (2013), 134–144.
[14]	R. Wang, B. Jiang, H. Huang, Fixed point theorem for Hardy-Rogers type contraction mapping in $\Xi$ -metric spaces, Acta. Anal. Funct. Appl., 21 (2019), 362–368.
[15]	L. Cirić, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267–273. https://doi.org/10.2307/2040075 doi: 10.2307/2040075
[16]	A. Basile, M. Graziano, M. Papadaki, I. Polyrakis, Cones with semi-interior points and equilibrium, J. Math. Econ., 71 (2017), 36–48. https://doi.org/10.1016/j.jmateco.2017.03.002 doi: 10.1016/j.jmateco.2017.03.002
[17]	N. Mehmood, A. A. Rawashdeh, S. Radenović, New fixed point results for $\Xi$ -metric spaces, Positivity, 23 (2019), 1101–1111.
[18]	H. Huang, Topological properties of $\Xi$ -metric spaces withapplications to fixed point theory, Mathematics, 7 (2019), 1222. https://doi.org/10.3390/math7121222 doi: 10.3390/math7121222
[19]	R. A. Rashwan, H. A. Hammad, M. G. Mahmoud, Common fixed point results for weakly compatible mappings under implicit relations in complex valued g-metric spaces, Inf. Sci. Lett., 8 (2019), 111–119. http://dx.doi.org/10.18576/isl/080305 doi: 10.18576/isl/080305
[20]	B. Vetro, P. Vetro, Fixed point theorems for $\eta$ - $\wp$ contractive type mappings, Nonlinear Anal., 75 (2011), 2154–2165.
[21]	H. A. Hammad, H. Aydi, M. De la Sen, Analytical solution for differential and nonlinear integral equations via $F_{\varpi _{e}}$ -Suzuki contractions in modified $\varpi _{e}$ -metric-like spaces, J. Func. Spaces, 2021 (2021), 6128586. https://doi.org/10.1155/2021/6128586 doi: 10.1155/2021/6128586
[22]	M. A. Alghmandi, N. Hussain, P. Salimi, Fixed point and coupled fixed point theorems on $b$ -metric-like spaces, J. Inequal. Appl., 2013 (2013), 402. https://doi.org/10.1186/1029-242X-2013-402 doi: 10.1186/1029-242X-2013-402
[23]	N. Hussain, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Fixed points of contractive mappings in $b$ -metric-like spaces, Sci. World J., 2014 (2014), 471827. https://doi.org/10.1155/2014/471827 doi: 10.1155/2014/471827
[24]	H. Aydi, A. Felhi, S. Sahmim, Common fixed points via implicit contractions on $b$ -metric-like spaces, J. Nonlinear Sci. Appl., 10 (2017), 1524–1537. https://doi.org/10.22436/jnsa.010.04.20 doi: 10.22436/jnsa.010.04.20
[25]	H. K. Nashine, Z. Kadelburg, Existence of solutions of cantilever beam problem via $\alpha$ - $\beta$ - $FG$ -contractions in $b$ -metric-like spaces, Filomat, 31 (2017), 3057–3074. https://doi.org/10.2298/FIL1711057N doi: 10.2298/FIL1711057N
[26]	H. A. Hammad, M. De la Sen, Generalized contractive mappings and related results in $b$ -metric-like spaces with an application, Symmetry, 11 (2019), 667. https://doi.org/10.3390/sym11050667 doi: 10.3390/sym11050667
[27]	H. A. Hammad, M. D. la Sen, Fixed-point results for a generalized almost $(s, q)$ -Jaggi $F$ -contraction-type on $b$ -metric-like spaces, Mathematics, 8 (2020), 63. https://doi.org/10.3390/math8010063 doi: 10.3390/math8010063
[28]	M. Aslantaş, H. Sahin, U. Sadullah, Some generalizations for mixed multivalued mappings, Appl. General Topol., 23 (2022), 169–178. https://doi.org/10.4995/agt.2022.15214 doi: 10.4995/agt.2022.15214
[29]	M. Aslantas, H. Sahin, D. Turkoglu, Some Caristi type fixed point theorems, J. Anal., 29 (2021), 89–103. https://doi.org/10.1007/s41478-020-00248-8 doi: 10.1007/s41478-020-00248-8
[30]	J. R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei, W. Shatanawi, Common fixed points of almost generalized $(\psi, \varphi)_s$ - contractive mappings in ordered $b$ -metric spaces, Fixed Point Theory Appl., 2013 (2013), 159. https://doi.org/10.1186/1687-1812-2013-159 doi: 10.1186/1687-1812-2013-159

This article has been cited by:

1.	Jehad Alzabut, Said R. Grace, Jagan Mohan Jonnalagadda, Shyam Sundar Santra, Bahaaeldin Abdalla, Higher-Order Nabla Difference Equations of Arbitrary Order with Forcing, Positive and Negative Terms: Non-Oscillatory Solutions, 2023, 12, 2075-1680, 325, 10.3390/axioms12040325
2.	Abdullah Özbekler, Kübra Uslu İşler, Jehad Alzabut, Sturmian comparison theorem for hyperbolic equations on a rectangular prism, 2024, 9, 2473-6988, 4805, 10.3934/math.2024232
3.	Hasib Khan, Jehad Alzabut, J.F. Gómez-Aguilar, Praveen Agarwal, Piecewise mABC fractional derivative with an application, 2023, 8, 2473-6988, 24345, 10.3934/math.20231241
4.	Jehad Alzabut, Said R. Grace, Shyam Sundar Santra, Mohammad Esmael Samei, Oscillation Criteria for Even-Order Nonlinear Dynamic Equations with Sublinear and Superlinear Neutral Terms on Time Scales, 2024, 23, 1575-5460, 10.1007/s12346-024-00961-w
5.	Osama Moaaz, Asma Al-Jaser, Functional differential equations of the neutral type: Oscillatory features of solutions, 2024, 9, 2473-6988, 16544, 10.3934/math.2024802
6.	S. Sangeetha, S. K. Thamilvanan, S. S. Santra, S. Noeiaghdam, M. Abdollahzadeh, Property $\bar{A}$ of third-order noncanonical functional differential equations with positive and negative terms, 2023, 8, 2473-6988, 14167, 10.3934/math.2023724

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.1

Metrics

Article views(2023) PDF downloads(202) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

Generalized $\Xi$ -metric-like space and new fixed point results with an application

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Oscillation of (1.1) and (1.2)

4. Oscillation of (1.1) and (1.3)

5. Applications

6. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

Generalized Ξ \Xi -metric-like space and new fixed point results with an application

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Oscillation of (1.1) and (1.2)

4. Oscillation of (1.1) and (1.3)

5. Applications

6. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Generalized $\Xi$ -metric-like space and new fixed point results with an application