Generalized conformable variational calculus and optimal control problems with variable terminal conditions

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

subject to either of the following boundary conditions

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

with the Robin boundary condition

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

with the Neumann boundary condition

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

with the boundary condition

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

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.,

$Q$ 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

and

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

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

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

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

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

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

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

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

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

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

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

and

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

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

and

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

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

Thus, we get

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

where

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

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

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

Then

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

and

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

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

that is,

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

where

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

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

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

Then

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

and

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

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

By Lemma 4 and Green's formula, we have

and

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

where

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

that is,

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

where

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

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

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

Then

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

and

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

where

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

that is,

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

where

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

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

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

Then

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

with the condition

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

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

Pointing out that

we can conclude that

are convergent. Thus, we have that

is convergent. Fixing

and considering the sequence

we get

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

Similarly, fixing

we get

Thus, from (5.3), we can get

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

with the condition

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

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

Pointing out that

we can conclude that

are convergent. Thus, we have that

is convergent. Fixing

and considering the sequence

we get

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

Similarly, fixing

we get

Thus, from (5.6), we get

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.

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.