Fixed point theorem combined with variational methods for a class of nonlinear impulsive fractional problems with derivative dependence

1.   Introduction

Fractional calculus gives generalization to the classical calculus from integer order to an arbitrary order which may be complex or real. Fractional calculus has become most notable and important part of mathematics which provides useful mathematical structures for the physical and biological phenomena, engineering mathematical models etc. To know about the developments in the theory of fractional calculus along with its applications, one can refer to ^{[1,2,3,4,5,6,7]} and the references there in.

The mathematical modeling of a process in which impulsive conditions (sudden discontinuous jumps) appear is done by using impulsive differential equations. Normally such processes are found in the field of biology, engineering and physics. The mathematical model of the population dynamics, drug administration and aircraft control are some examples of impulsive differential equations ^[8,9,10,11]. Because of their more importance, recently differential equations with impulsive effects of fractional order have gain a lot of contemplation of the researchers. For both linear and nonlinear impulsive fractional differential equations, the multiplicity and existence theory of their solutions is broadly discussed by using different tools such as Morse theory, measure of noncompactness, method of upper and lower solutions and fixed point theorems ^{[12,13,14,15,16,17]}. But these useful techniques are not appropriate and difficult to apply for that problems in which the corresponding integral equation can not be found easily e.g. when both right and left derivatives of fractional order are there in the problem. Such problems can easily be investigated by using another useful approach: variational techniques and the critical point theory. A pioneer work in this direction was that of Jiao and Zhou ^[18], who implemented the approach for a class of fractional differential equations. Whereas for a class of impulsive second order differential equations, considerable contributions are made by Nieto and O'Regan ^[19]. Later on, many authors used the critical point theory combined with variational approach to deal with the existence of solution to nonlinear and linear fractional impulsive differential equations ^{[21,22,23,24,25,26,27]}. Also variational methods along with semi-inverse methods for the establishment of variational formulation are widely used ^[28,29,30].

Recently Nieto and Uzal ^[20] discussed a class of impulsive differential equations of 2nd order in which nonlinearity is due to derivative dependence, where the existence of at least one solution of the problem is guaranteed via variational structure and fixed point theorem. But an impulsive fractional boundary value problem in which nonlinearity is because of derivative term still needs to be explored.

Above cited work gave us enough motivation to study the following nonlinear impulsive boundary value problem of fractional order in which nonlinearity is because of fractional order derivative dependence:

here $0 = t_0 < t_1 < t_2 < \cdots < t_n < t_{n+1} = T$, $_{t}D^\alpha_T$ is the $\alpha$-order right Riemann-Liouville fractional derivative and $_{0}^cD^\alpha_t$ is the $\alpha$-order left Caputo fractional derivative for $\frac{1}{2} < \alpha\leq1$, $\Delta{_{t}D}^{\alpha-1}_T\big(_{0}^cD^\alpha_ty(t_k)\big) = {_{t}D}^{\alpha-1}_T\big(_{0}^cD^\alpha_ty(t_k^+)\big)-{_{t}D}^{\alpha-1}_T\big(_{0}^cD^\alpha_ty(t_k^-)\big)$ and $b:[0, T]\rightarrow \mathbb{R}^+$, $I_\lambda:\mathbb{R}\rightarrow \mathbb{R}$ and $h:[0, T]\times \mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ are the functions satisfying some assumptions.

Remark 1.1. For $\alpha = 1$, it ought to be noticed that, one has $_{0}^cD_t^\alpha u(t) = u^\prime (t)$ and $_tD_T^\alpha u(t) = -u^\prime (t)$, and (1.1) reduces to standard impulsive problem of second order ^[20]. Therefore our concern (1.1) generalize that of ^[20].

Since the corresponding integral equation for the problem (1.1) can not be found, therefore we can not use formal analysis approach such as fixed point theorem. Also the problem (1.1) doesn't have a variational structure because of fractional derivative presence in the nonlinear term ^[20]. So the problem (1.1) looks like unsolvable. To overcome all these obstacles, we shall use a very interesting procedure by considering, for $z\in E^\alpha_0$ (which is defined in preliminary section), following a class of associated damped problems which involves no nonlinear dependence on the derivative.

Above problem has a variational structure and can be solved by applying critical point theory. At the end, we shall join the solutions of (1.2) and of the main problem (1.1) by using fixed point theorem. This approach is same as that of used in ^[20] but here we shall extend the results from integer order to fractional order. Within the outer boundaries of our knowledge, problem (1.1) is untouched and going to get first treatment through this paper by using a novel and useful technique.

From start to finish of this paper, we suppose the following notations and conditions are satisfied.

$\boldsymbol{(M1)}$ For all $t\in[0, T]$, we have $0 < \underline{b}\leq b(t)\leq \bar{b}$ where $\underline{b}$ and $\bar{b}$ are constants.

$\boldsymbol{(M2)}$ For all $\lambda = 1, 2, \cdots, n$, $I_\lambda:\mathbb{R}\rightarrow \mathbb{R}$ and $h:[0, T]\times \mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ are continuous functions.

$\boldsymbol{(M3)}$ There exist $V\in C(\mathbb{R}^+, \mathbb{R}^+)$ and $W\in L^2(0, T)$ such that $|h(t, y, \xi)|\leq V(|y|)W(t)$ and $|H_{\xi}(t, y)|\leq V(|y|)W(t)$ where $H_\xi(t, y) = \int_{0}^{y}h(t, u, \xi)du$.

Rest of the article is composed in such a way that as per the prerequisites of article, some essential definitions and fundamental outcomes are given in Section 2. In Section 3, solution of damped problem (1.2) is discussed by converting it in a variational form. In Section 4, main theorem about the existence criteria of at least one solution of complete problem (1.1) along with proof is given. Toward the end, two examples are given to illustrate our outcomes.

2.   Preliminaries

All the basic results and the definitions from the literature are given in this section which will be used as the building blocks in the construction of our main outcomes.

Definition 2.1. ^[1,2] Suppose $y$ is defined on $[a, b]$ and $\alpha \in \mathbb{R}^+$. Then $_aD_s^{-\alpha} y(s)(\alpha$-order left Riemann-Liouville fractional integral of $y$) and $_sD_b^{-\alpha} y(s)(\alpha$-order right Riemann-Liouville fractional integral of $y$) are given by

and

respectively, provided right hand side is defined pointwise on $[a, b]$.

Definition 2.2. ^[1,2] Let $y$ be defined on $[a, b]$ and $\alpha \in \mathbb{R}^+$. Then $_aD_s^{\alpha} y(s)(\alpha$-order left Riemann-Liouville fractional derivative of $y$) and $_sD_b^{\alpha} y(s)(\alpha$-order right Riemann-Liouville fractional derivative of $y$) are given by

and

respectively. Here $s\in[a, b], \eta-1 < \alpha\leq \eta$ and $\eta\in \mathbb{N}$.

Definition 2.3. ^[1,2] For $\eta-1 < \alpha\leq \eta$, suppose $y\in AC^\eta([a, b], \mathbb{R})$, then $_s^cD_b^\alpha y(s)$($\alpha$-order right Caputo fractional derivative of $y$) and $_a^cD_s^\alpha y(s)$($\alpha$-order left Caputo fractional derivative of $y$) are given by

and

respectively, here $\eta\in \mathbb{N}$ and $s\in[a, b]$.

Lemma 2.4. ^[1,2] (a). Let $\eta-1 < \alpha\leq \eta$ and $u, v\in L^2(a, b)$ then

(b). Let $\eta-1 < \alpha\leq \eta, v\in AC([a, b], \mathbb{R}^N), v^\prime \in L^2([a, b], \mathbb{R}^N)$ and $_{s}D^\alpha_T(_{0}^cD^\alpha_su(s))\in AC([a, b], \mathbb{R}^N)$ with $_a^cD_s^\alpha u(s)\in L^2([a, b], \mathbb{R}^N)$ then

$\boldsymbol{E}^\alpha_0$(Fractional Derivative Space)

Our main attention is to apply variational methods and critical point theory for a corresponding functional. So there is a strong need of a fractional derivative space. Below we define such fractional derivative space which is coinciding to the space defined in ^[18].

First we review the norms $||\cdot||$ and $||\cdot||_{L^p}$ as follows

Definition 2.5. ^[18] Let $\alpha\in(0, 1]$ and $C_0^\infty([0, T], \mathbb{R})$ be the set of all functions $y\in C^\infty([0, T], \mathbb{R})$ with $y(0) = y(T) = 0$, then the fractional derivative space $E^\alpha_0$ is defined by the closure of $C_0^\infty([0, T], \mathbb{R})$ with respect to the norm

Remark 2.6. It is clear from the definition(2.5) that $E^\alpha_0$ is a space of functions $y$ such that $y\in L^2[0, T]$ and $_{0}D_t^\alpha y(t)\in L^2[0, T]$ with $y(0) = 0 = y(T)$.

Lemma 2.7. ^[18] If $\alpha\in (\frac{1}{2}, 1]$ then for $y\in E^\alpha_0$ we have

Remark 2.8. In the space $E_0^\alpha$, if $||\cdot||_{\alpha}$ and $||\cdot||_{b, \alpha}$ are defined as

and

then from $(2.5)$ and (M1), it can be easily seen that $||y||_{\alpha, 2}$ defined in $(2.3)$, $||y||_{\alpha}$ in $(2.6)$, and $||y||_{b, \alpha}$ in (2.7) are equivalent norms.

Proposition 2.9. Let $y\in E_0^\alpha$, then the following result is satisfied

Lemma 2.10. ^[18] If a sequence is weakly convergent in $E_0^\alpha$ then in $C[0, T]$ space, it is strongly convergent.

Lemma 2.11. ^[18] For $\alpha \in (0, 1]$, the fractional derivative space $E^\alpha_0$ is a Banach space which is reflexive and separable.

Theorem 2.12.[31,Theorem 1.1] Suppose $\psi:Y\rightarrow\mathbb{R}$ be a sequentially weakly lower semi-continuous functional for a reflexive Banach space $Y$. If $\psi$ is strictly convex and coercive, then $\psi$ has a unique minimum on $Y$.

Theorem 2.13. [32,Schauder] If $T:Z\rightarrow Z$ is a continuous and compact map for a nonempty closed and convex subset $Z$ of a Banach space $Y$, then $T$ has a fixed point.

3.   Damped problem solution

This section is devoted to investigate the variational structure and existence of solution of damped problem (1.2). First an equivalent form of the damped problem is given and then existence of solution is established by Theorem 2.12.

Lemma 3.1. For $x\in E^\alpha_0$, any solution $y$ of the problem $(1.2)$ will also satisfy the following Eq (3.1).

Proof. Integrating from $0$ to $T$ after multiply (1.2) with $x(t)\in E^\alpha_0$, we get

On the left hand side for the value of first term, using (2.1), (2.2) and, the impulsive and boundary conditions of problem (1.2), we have

So we can write

Using Eq (3.3) in (3.2), we get the Eq (3.1) and the proof is completed. Using Lemma 3.1, we can introduce notion of the weak solution for the problem (1.1) and (1.2).

Definition 3.2. Let $y\in E^\alpha_0$, then $y$ is a weak solution of the problem (1.2) if (3.1) is satisfied for each $x\in E^\alpha_0$. Also $y$ is a weak solution of the problem (1.1) if it satisfies (3.1) for all $x\in E^\alpha_0$ with $z = y$.

Definition 3.3. Let $\psi_z:E^\alpha_0\rightarrow\mathbb{R}$ be a functional defined by

where $H_{\xi}(t, y(t)) = \int_{0}^{y(t)}h(t, u, \xi)du$.

Remark 3.4. For functional $\psi_z$, if $x(t)\in E^\alpha_0$ then

Thus in the view of Lemma 3.1, one can see that critical points of the functional $\psi_z$ are precisely weak solutions of the damped problem (1.2).

We are stating a set of conditions which will be used later where they needed.

$\boldsymbol{(M4)}$ There exist $a_{\lambda, \gamma_\lambda}, b_{\lambda, \gamma_\lambda}\in\mathbb{R}$ where $\lambda\in \{0, 1, 2, \cdots, n\}$ and $\gamma_0, \cdots, \gamma_n > 0$ such that

If $K = \{\lambda\in \{1, 2, \cdots, n\}; a_{\lambda, \gamma_\lambda}\leq0\}$, and suppose one condition from the following is satisfied.

● $\boldsymbol{(M4.1)}$ Either $K\ne\phi$.

$\boldsymbol{(M4.1.1a)}$ $a_{0, \gamma_0}\leq 0$.

$\boldsymbol{(M4.1.1b)}$ $a_{0, \gamma_0} > 0, \; \; \gamma_0 = 2, \; \; \frac{min\{1, \alpha\}}{2}\left(1+\frac{\underline{b}T^\alpha}{\Gamma{(\alpha+1)}}\right) > \frac{T^{2\alpha}}{[\Gamma(\alpha)]^2(2\alpha-1)}a_{0, 2}$.

$\boldsymbol{(M4.1.1c)}$ $a_{0, \gamma_0} > 0, \; \; \gamma_0 < 2$.

$\boldsymbol{(M4.1.2)}$ $\gamma_\lambda\leq2$ for all $\lambda\in K$, and let $\gamma_{\lambda_1} = \gamma_{\lambda_2} = \cdots = \gamma_{\lambda_q} = 2$ and $K_0 = \{\lambda_1, \lambda_2, \cdots, \lambda_q\}$.

$\boldsymbol{(M4.1.2a)}$ $a_{0, \gamma_0}\leq 0$, $\frac{min\{\alpha, 1\}}{2}\left(\frac{\underline{b}T^\alpha}{\Gamma{(\alpha+1)}}+1\right) > \frac{T^{2\alpha-1}}{[\Gamma(\alpha)]^2(2\alpha-1)}\sum_{\lambda\in K_0}a_{\lambda, \gamma_0}$.

$\boldsymbol{(M4.1.2b)}$ $a_{0, \gamma_0} > 0, \gamma_0 = 2$, $\frac{min\{\alpha, 1\}}{2}\left(\frac{\underline{b}T^\alpha}{\Gamma{(\alpha+1)}}+1\right) > \frac{T^{2\alpha-1}}{[\Gamma(\alpha)]^2(2\alpha-1)}\left[\sum_{\lambda = 0}^{n}a_{\lambda, 2}+Ta_{0, 2}\right]$.

$\boldsymbol{(M4.1.2c)}$ $a_{0, \gamma_0} > 0, \gamma_0 < 2$, $\frac{min\{\alpha, 1\}}{2}\left(\frac{\underline{b}T^\alpha}{\Gamma{(\alpha+1)}}+1\right) > \frac{T^{2\alpha-1}}{[\Gamma(\alpha)]^2(2\alpha-1)}\sum_{\lambda\in K_0}a_{\lambda, \gamma_0}$.

● $\boldsymbol{(M4.2)}$ Or $K = \phi$, therefore $a_{\lambda, \gamma_\lambda} > 0$ for all $\lambda\in\{1, 2, \cdots, n\}$.

$\boldsymbol{(M4.2.1)}$ $a_{0, \gamma_0}\leq 0$.

$\boldsymbol{(M4.2.2)}$ $a_{0, \gamma_0} > 0$, $\gamma_0\leq2$.

$\boldsymbol{(M4.2.3)}$ $a_{0, \gamma_0} > 0$, $\gamma_0 = 2$, $\frac{min\{\alpha, 1\}}{2}\left(\frac{\underline{b}T^\alpha}{\Gamma{(\alpha+1)}}+1\right) > \frac{T^{2\alpha}}{[\Gamma(\alpha)]^2(2\alpha-1)}a_{0, 2}$.

$\boldsymbol{(M5)}$ $y\mapsto H_\xi(t, y)$ is concave and $y\mapsto \int_{0}^{y}I_\lambda(s)ds$ is convex and one of them is strict.

Lemma 3.5. Suppose condition M4 is satisfied, then there exists $\beta(s)$ which is independent of $_0^cD^\alpha_Tz(t)$ such that $\psi_z(y)\geq\beta(||y||_\alpha)$ with the property $\beta(s)\rightarrow+\infty$ as $s\rightarrow\infty$.

Proof. We shall prove by considering only one item from (M4). For all other considerations of (M4), one can establish the proof in similar fashion. Suppose $K\neq\phi$ and $\gamma_\lambda < 2, \forall \lambda\in K$ with $a_{0, \gamma_0} < 0$. For $y\in E_0^\alpha$, using (2.8),

where $\beta:(\mathbb{R}^+, \mathbb{R}^+)$ is given by

Then $\beta$ is continuous, independent of $_0^cD^\alpha_tz(t)$ and $\beta(s)\rightarrow+\infty$ as $s\rightarrow\infty$. The proof is completed.

Theorem 3.6. Suppose M4 and M5 are fulfilled, then the damped problem 1.2 has a unique weak solution $y_z$ for each $z\in E^\alpha_0$. Moreover there exists $R > 0$ for all $z\in E^\alpha_0$ such that $||y_z||_\alpha\leq R$.

Proof. Our attention is to apply Theorem 2.12. Since any norm is convex and $u\mapsto u^2$ is convex on $[0, \infty)$, therefore The functional $\psi_z$ is convex by using M5. Further more, $\psi_z$ is sequentially weakly lower semi-continuous being sum of a weakly and of a convex continuous functions [31,Theorem 1.2,Propositon 1.2]. Actually $\int_{0}^{T}\left|{_0^cD}_t^\alpha y(t)\right|^2+b(t)|y(t)|^2dt$ is convex and continuous on $E_0^\alpha$. Using the Lemma 2.10 along with M2 and Lebesgue dominated convergence theorem $\int_{0}^{y(t_\lambda)}I_\lambda(s)ds-\int_{0}^{T}H_{{_{0}^cD}^\alpha_tz(t)}(t, y(t))dt$ is weakly continuous on $E^\alpha_0$. According to the Lemma 3.5, $\psi_z$ is coercive. So according to Theorem 2.12, $\psi_z$ has a unique global minimum $y_z$ for each $z\in E^\alpha_0$ which is also a weak solution of the damped problem (1.2). The existence of $R$ is proved as a consequence of Theorem 2.12. The property that $R$ does not depend on $z$ is a consequence of the fact that $\beta$ in independent of $z$.

4.   Main problem solution and examples

Up to previous section, we have proved that there exists a unique critical point $y_z$ for each $z\in E^\alpha_0$. Here we define a map $T:z\in E^\alpha_0\rightarrow y_z\in E^\alpha_0$ as $Tz = y_z$. It is clear that if there is any fixed point of $T$ then that will be solution of the main problem (1.1).

Lemma 4.1. If M4 and M5 are satisfied, then the mapping $T:z\in E^\alpha_0\rightarrow y_z\in E^\alpha_0$ is continuous and compact.

Proof. First we prove that $T:z\in E^\alpha_0\rightarrow y_z\in E^\alpha_0$ is continuous. Let $\{z_n\}$ be a sequence in $E^\alpha_0$ such that $z_n \rightarrow z$. We have to show that $Tz_n\rightarrow Tz$. Suppose $Tz_n = y_n$. From Theorem 3.6, there is a $R > 0$ so that $\|y_n\|\leq R$. So there exists a weakly convergent subsequence $\{y_{n_{k}}\}$ such that $y_{n_{k}}\rightharpoonup y$ (say) in $E^\alpha_0$ and by Lemma 2.10, $y_{n_{k}}\rightarrow y$ in $C([0, T])$. Let $\{y_{n_{k_{j}}}\}$ be an arbitrary subsequence of $\{y_{n_{k}}\}$.

As we have $z_{n_{k}}\rightarrow z$ in $E^{\alpha, 2}_0$, then $_0^cD^\alpha_tz_{n_{j}}(t)\rightarrow _0^cD^\alpha_tz(t)$ in $L^2(0, T)$ and for a subsequence $\{z_{n_{k_{j}}}\}$, we have $_0^cD^\alpha_tz_{n_{j_{l}}}(t)\rightarrow {_0^cD^\alpha}_tz(t)$ for almost every $t\in[0, T]$. For any $x\in E^\alpha_0$, using Lebesgue's dominated convergence theorem and the fact that functions $I_\lambda$ and $h$ are continuous, we get

so we have $Tz = y$ using the uniqueness of critical point and the fact that $\{y_{n_{k_{j}}}\}$ converges weakly to $y$. Also if $x = y_{n_{k_{j}}}$, then $\|y_{n_{k_{j}}}\|\rightarrow \|y\|$. Hence arbitrary subsequence $\{y_{n_{k}}\}$ of $\{y_n\}$ has a subsequence $\{y_{n_{k_{j}}}\}$ such that $y_{n_{k_{j}}}\rightarrow Tz$, which shows that $y_n\rightarrow Tz$. So $T$ is continuous.

Next we show that $T$ is compact. Consider $\{z_n\}$ is a bounded sequence in $E^\alpha_0$. We have to prove that sequence $\{Tz_n\}$ has a convergent subsequence. Suppose $Tz_n = y_n$, then as in above discussion, a subsequence $\{Tz_{n_{k_{j}}}\}$ of $\{Tz_n\}$ exists which converges to $Tz = y$ in $E^\alpha_0$. This completes the proof.

Theorem 4.2. Suppose M4 and M5 are hold then the main problem 1.1 has at least one weak solution.

Proof. According to the Theorem 3.6 that there is a constant $R > 0$ such that $\|Tz\|\leq R$. Let $T:\overline{B(0, R)}\subset E^\alpha_0\rightarrow \overline{B(0, R)}$, then from Lemma 4.1, $T$ is continuous and compact map. Schauder's fixed point guarantees that there exists a fixed point $z\in E^\alpha_0$ such that $Tz = z$, which shows that main problem (1.1) has a solution. This completes the proof.

Example 4.3. Suppose the following nonlinear BVP,

here $h(t, y, \xi) = -1-y^3\left(3+\cos\xi\right)(t+1)^5$ and $I_1(\eta) = 1000\eta^3$. For these values we have

$a_{0, 4} = -\frac{1}{2}, \; \; \; b_{0, 4} = -1, \; \; \; a_{1, 4} = 250$, and $b_{1, 4} = 0$. Because $a_{1, 4} > 0$ so $K = \phi$ and $a_{0, 4} < 0$, so we are in case $(M4.2.1)$. Also

is strictly convex and

is strictly concave. This shows that $(M_5)$ is satisfied. Hence above problem (4.1) has a weak solution according to the Theorem 4.2.

Example 4.4. Suppose the following impulsive nonlinear BVP,

here $h(t, y, \xi) = 5-\frac{1}{2}\left(\cos y\sin y+y\right)\left(\arctan\xi+\pi\right)$ and $I_1(\eta) = -1$. For these values we have

$a_{0, 2} = -\frac{\pi}{8}, \; \; \; b_{0, 2} = 5, \; \; \; a_{1, 1} = -1$, and $b_{1, 1} = 0$. Because $a_{1, 1} < 0$ so $K\ne\phi$ and $a_{0, 2} < 0$, so we are in case $(M4.1.1)$. Moreover

is convex and

is strictly concave. This shows that $(M_5)$ is satisfied. Hence above problem (4.2) has a weak solution according to the Theorem 4.2.

5.   Conclusions

Fixed point theorems and varitaitonal techniques along with critical point theory are very useful and applicable tools to discuss the existence of solution of differential equations of both integer and fractional orders. In this article by using these useful approaches, we have provided sufficient conditions for the existence of at least one weak solution of a nonlinear impulsive problem of fractional order in which nonlinearity is due to derivative term of fractional order. Our results generalize the nonlinear second order impulsive differential problems with dependence on derivative. The present results can be easily extended to the two-scale fractal calculus.

Conflict of interest

The authors declare no conflict of interest.