Abstract
The existence and uniqueness of a strong solution for a class of partial functional differential equations with Dirichlet boundary conditions is established by applying Rothe’s method. As an application, we included an example to illustrate the main result.
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Introduction
In this paper, we study the existence and uniqueness of a strong solution of the following partial functional differential equation with Dirichlet boundary conditions
where F, G and \(\varPhi \) are some suitable functions.
Equations of the type (1) in which the delay argument occurs in the derivative of the state variable as well as in the independent variable, are called neutral differential equations. Neutral differential equations have many applications in various physical situations, for example, in the theory of vibration of masses attached to an elastic bar [1], and in the study of two or more simple oscillatory systems with some interconnections between them [1, 2].
The existence of solutions of neutral differential equations has been considered by many authors, for example, Islam and Raffoul [3] studied the existence of periodic solutions of the nonlinear system of neutral differential equations
where A(t) is a nonsingular \(n\times n\) matrix with continuous real-valued functions as its elements. The functions \(Q:{\mathbb {R}} \times {\mathbb {R}}^n \rightarrow {\mathbb {R}}^n\) and \(G :{\mathbb {R}} \times {\mathbb {R}}^n \times {\mathbb {R}}^n \rightarrow {\mathbb {R}}^n\) are continuous in their respective arguments. Damak et al. [4] studied the existence of weighted pseudo almost periodic solutions of an autonomous neutral functional differential equation
in a Banach space \({\mathbb {X}},\) where A is the infinitesimal generator of a \(C_0-\)semigroup \(\{T(t)\}_{t\ge 0},\) and \(F:{\mathbb {R}}\times {\mathbb {X}}\rightarrow {\mathbb {X}}\) is Weighted pseudo almost periodic and \(G:{\mathbb {R}}\times {\mathbb {X}}\times {\mathbb {X}}\rightarrow {\mathbb {X}}\) is Stepanov-weighted pseudo almost periodic functions.
The Rothe’s method was introduced by Rothe in [5], for solving the following scalar parabolic initial boundary value problem of second order,
where R and S are sufficiently smooth functions of t and x in \([0,T] \times (0,1)\) satisfying certain additional conditions. Here T means an arbitrary finite positive number. His method consist in dividing [0, T] into n number of subintervals \([t^n_{j-1},t^n_j ],~t^n_j=jh,~j=1,2, \ldots , n\) with \(t^n_0=0\) of equal lengths \(h(h=\frac{T}{n})\) and replacing the partial derivative \(\frac{\partial u}{\partial t}\) of the unknown function u by the difference quotients \(\frac{u^n_j-u^n_{j-1}}{h}.\) After defining a sequence of polygonal functions
Rothe has proved the convergence of the sequence \(\{U^n\}\) to the unique solution of the problem as \(n \rightarrow \infty \) using some a priori estimates on \(\{U^n\}.\) After Rothe this method has been used by many authors, for example, see [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28].
Raheem and Bahuguna [7] have applied Rothe’s method to establish the existence and uniqueness of a strong solution for the following delayed cooperation diffusion system with Dirichlet boundary conditions.
where w(t, x) is the density of species at time t and space location x, and \(k,\tau \) are positive constants, and the maps \(f,\phi \) are defined from \([t_0,T]\times [0,\pi ]\) and \([t_0-\tau ,t_0]\times [0,\pi ]\) into \(L^2[0,\pi ]\) respectively.
Merazga and Bouziani [20] have proved the existence and uniqueness of a weak solution for a semilinear heat equation with integral conditions in a non classical function space. Recently in [9], authors have applied Rothe’s method to a fractional integral diffusion and established the existence and uniqueness of a strong solution.
By literature, it is clear that Rothe’s method can be used for solving many physical, mathematical, biological problems modeled by partial differential equations.
In the present work, we shall use Rothe’s method to solve functional differential equations with Dirichlet boundary conditions defined by (1)–(3).
Preliminaries and Main Result
Consider \({\mathbb {H}}:=L^2[0,1],\) the real Hilbert space of all real valued square integrable functions defined on [0, 1] with the usual inner product and the norm generated by the inner product. Define the linear operator A by
Then, \(-A\) is the infinitesimal generator of a \(C_0\)-semigroup \(T(t),\;t\ge 0,\) of contractions in \({\mathbb {H}}.\) Define the maps \(f:[0,T]\times {\mathbb {H}}\rightarrow {\mathbb {H}}\) by
and \(g:[0,T]\times {\mathbb {H}}\rightarrow {\mathbb {H}}\) by
If we define \(u:[-T,T]\rightarrow {\mathbb {H}}\) by \(u(t)(x)=w(t,x),\) and \(\phi :[-T,0]\rightarrow {\mathbb {H}}\) by \(\phi (t)(x)=\varPhi (t,x),\) then (1)–(3) can be rewritten as
Lemma 1
(Theorem 1.4.3, [29]) If \(-A\) is the infinitesimal generator of a \(C_0\)-semigroup of contractions, then A is m-accretive, that is, \(\langle Au,u\rangle \ge 0\) for \(u\in D(A),\) and \(R(I+\lambda A)={\mathbb {H}}\) for any \(\lambda >0,\) where I is the identity operator on \({\mathbb {H}},\) and \(R(\cdot )\) is the range of an operator.
Lemma 2
(Lemma 2.5, [12]) Let \(-A\) be the infinitesimal generator of a \(C_0\)-semigroup of contractions. If \(Y^n\in D(A)\) for \(n\in {\mathbb {N}},\) \(Y^n\rightarrow u\in {\mathbb {H}}\) and \(\Vert AY^n\Vert \) are bounded, then \(u\in D(A)\) and \(AY^n\rightharpoonup Au,\) where \(\rightharpoonup \) denotes the weak convergence in \({\mathbb {H}}.\)
We consider the following assumptions:
-
(H1)
For each \(t\in [0,T],\) the function \(S_t:{\mathbb {H}}\rightarrow {\mathbb {H}}\) defined by \(S_t(h)=h+f(t,h),\) \(h\in {\mathbb {H}},\) is bijective.
-
(H2)
f is continuous, and there exists \(0<K<1\) such that
$$\begin{aligned} \Vert f(t_1,u_1)-f(t_2,u_2)\Vert \le K\Vert u_1-u_2\Vert ,\quad \forall t_1,t_2\in [0,T],\;\forall u_1,u_2\in {\mathbb {H}}. \end{aligned}$$ -
(H3)
There exists \(L_g>0\) such that
$$\begin{aligned} \Vert g(t_1,u_1)-g(t_2,u_2)\Vert \le L_g(|t_1-t_2|+\Vert u_1-u_2\Vert ),\;\forall t_1,t_2\in [0,T],\;\forall u_1,u_2\in {\mathbb {H}}. \end{aligned}$$ -
(H4)
\(\phi \) is continuous function.
Denote by \(C([a,b];{\mathbb {H}})\) the space of all continuous functions from the interval [a, b] into \({\mathbb {H}}.\)
Definition 1
A function \(u\in C([-T,T];{\mathbb {H}})\) is called a strong solution of the problem (6)–(7) if \(u(t)=\phi (t)\) for \(t\in [-T,0],\) \(u(t)+f(t,u(t))\in D(A)\) for \(t\in [0,T],\) u(t) is Lipschitzian on [0, T] and satisfies the Eq. (6) a.e. on [0, T].
Theorem 1
Suppose that the conditions (H1)–(H4) are satisfied. Then (6)–(7) has a strong solution u on \([-T,\tilde{T}],\) \(0<\tilde{T}<T,\) which can be continued uniquely either on the whole interval \([-T,T]\) or on the maximal interval of existence \([-T,t_\mathrm{max}),\) \(0<t_\mathrm{max}\le T,\) such that u is a strong solution of (6)–(7) on every subinterval \([-T,\tilde{T}],\) \(0<\tilde{T}<t_\mathrm{max}.\)
Discretization Scheme and a priori Estimates
Fix \(R>0\) and choose \(t_0\) such that \(0<t_0\le T\) and \(t_0M_0\le R,\) where
For each \(n\in {\mathbb {N}},\) let \(t^n_0=0,\) \(h_n=\frac{t_0}{n},\) and \(t^n_j=jh_n,\) for \(j=1,2,\ldots ,n.\) Let \(u_0^n=\phi (0)\) for all \(n\in {\mathbb {N}},\) and define each of \(\{u_j^n\}_{j=1}^{n}\) successively as the unique solution of the following equation
The existence of a unique \(u_j^n\) satisfying (9) is obtained by using Lemma 1 and the assumption (H1). We define the sequence \(\{U^n\}\) by
Lemma 3
For \(n\in {\mathbb {N}}\) and \(j=1,2,\ldots ,n,\)
Proof
From (9) for \(j=1,\) we have
Then,
By Lemma 1, we obtain
Therefore,
Hence,
Therefore, \(\Vert u_1^n-\phi (0)\Vert \le \frac{R}{1-K}.\)
Assume that
Then, \(\Vert u_i^n-\phi (0)\Vert \le \frac{R}{1-K}\) for \(i=1,2,\ldots ,j-1.\) Now, From (9) for \(i=j,\) \(2\le j\le n,\) we have
Then,
By Lemma 1, we obtain
Therefore,
Thus, we get
Therefore,
Hence proved. \(\square \)
Corollary 1
For \(n\in {\mathbb {N}}\) and \(j=1,2,\ldots ,n,\)
Proof
Proof is obvious by Lemma 3 and from the Eqs. (13) and (14). \(\square \)
Now, we defined a sequence \(\{Y^n\}\) of step functions from \([-h_n,t_0]\) into \({\mathbb {H}}\) by
Remark 1
From Corollary 1, it is clear that the functions \(U^n(t)\) are Lipschitz continuous on \([0,t_0],\) and the sequence \(U^n(t)-Y^n(t)\rightarrow 0\) in \({\mathbb {H}}\) as \(n\rightarrow \infty \) uniformly on \([0,t_0].\) Moreover, \(Y^n(t)\in D(A)\) for \(t\in [0,t_0]\) and the sequences \(\{U^n(t)\},\) \(\{Y^n(t)\}\) and \(\{AY^n(t)\}\) are bounded uniformly in \(n\in {\mathbb {N}}\) and \(t\in [0,t_0].\)
If we suppose that
then (9) can be written as
where \(\frac{d^-}{dt}\) denotes the left derivative in \((0,t_0].\) Also, for \(t\in (0,t_0],\) we have
Lemma 4
There exists a function \(u\in C([-T,t_0];{\mathbb {H}})\) such that \(U^n(t)\rightarrow u(t)+f(t,u(t))\) in \(C([-T,t_0];{\mathbb {H}})\) as \(n\rightarrow \infty .\) Moreover, u is Lipschitz continuous on \([0,t_0].\)
Proof
From (18) and using Lemma 1, for \(t\in (0,t_0],\) we have
Using above inequality, we get
For \(t\in (t^n_{j-1},t^n_j]\) and \(t\in (t^k_{l-1},t^k_l],\) \(1\le j\le n,\) \(1\le l\le k,\) we have
By Corollary 1 and by the assumption (H2), we get
that is,
Now,
where
Clearly, \( \epsilon _{nk}(t)\rightarrow 0\) as \(n,k\rightarrow \infty \) uniformly on \([0,t_0].\) This implies that for a.e. \(t\in [0,t_0],\) we have
where \(\epsilon ^1_{nk}\) is a sequence of numbers such that \(\epsilon ^1_{nk}\rightarrow 0\) as \(n,k\rightarrow \infty .\) Notice that \(U^n=\phi \) on \([-T,0]\) for all n. Hence, we obtain
Using Gronwall’s inequality, we conclude that there exists \(v\in C([-T,t_0];{\mathbb {H}})\) such that \(U^n\rightarrow v\) in \(C([-T,t_0];{\mathbb {H}})\) as \(n\rightarrow \infty .\) By the assumption (H1), for each \(t\in [0,t_0]\) there exists u(t) such that \(v(t)=u(t)+f(t,u(t)),\) and for each \(t\in [-T,0],\) we define \(u(t)=v(t).\) By the assumption (H2), it is clear that u(t) is continuous. It is easy to see that \(v=\phi \) on \([-T,0],\) therefore \(u(t)=\phi (t)\) on \([-T,0].\) Since v is Lipschitz continuous, by the assumption (H2) u is also Lipschitz continuous on \([0,t_0].\) Hence proved. \(\square \)
Proof of Theorem 1
From Remark 1 it follows that \(Y^n(t)\rightarrow u(t)+f(t,u(t))\) as \(n\rightarrow \infty ,\) and \(u(t)+f(t,u(t))\in {\mathbb {H}}\) for \(t\in [0,t_0].\) Since \(\Vert AY^n\Vert \) are bounded, by Lemma 2 it is clear that \(AY^n(t)\rightharpoonup A(u(t)+f(t,u(t))).\) For \(t\in (t^n_{j-1},t^n_j],\) we have
that is,
Now, for \(t\in (t^n_{j-1},t^n_j],\) we have
Therefore, \(\Vert g^n(t)-g(t,u(t))\Vert \rightarrow 0\) as \(n\rightarrow \infty \) uniformly on \([0,t_0].\) From (19), for every \(v\in {\mathbb {H}},\) we have
By Lemma 4 and the bounded convergence theorem, we get as \(n\rightarrow \infty ,\)
Since \(A(u(t)+f(t,u(t)))\) is Bochner integrable on \([0,t_0],\) from (23) we obtain
Now we prove the uniqueness of a function \(u\in C([-T,t_0];{\mathbb {H}})\) such that \(u(t)+f(t,u(t))\) is differentiable a.e. on \([0,t_0]\) with \(u(t)+f(t,u(t))\in D(A)\) a.e. on \([0,t_0]\) and \(u=\phi \) on \([-T,0]\) satisfying (24). Suppose there exist two strong solutions \(u_1,u_2\in C([-T,t_0];{\mathbb {H}})\) of (24) with \(u_1=u_2=\phi \) on \([-T,0].\) Let \(u(t)+f(t,u(t))=u_1(t)+f(t,u_1(t))-u_2(t)-f(t,u_2(t))\) on \([0,t_0].\) Then
Therefore, \(\Vert u_1(t)-u_2(t)\Vert \le \frac{1}{1-K}\Vert u(t)+f(t,u(t))\Vert \) on \([0,t_0].\) Now, from (24) and using Lemma 1, we have
Since \(u(t)+f(t,u(t))=0\) on \([-T,0],\) therefore we obtain
Using Gronwall’s inequality, we conclude that \(u(t)+f(t,u(t))=0\) on \([-T,t_0].\) Therefore, by the assumption (H2), we get \(u_1=u_2\) on \([-T,t_0].\) Now, we prove the continuation of the solution u on \([-T,T].\) Suppose \(t_0<T,\) then consider
where \(\tilde{g}(t,v(t))=g(t+t_0,v(t)),\quad 0\le t\le T-t_0.\) Since \(\tilde{g}\) satisfies the assumption (H3), we can proceed as before and prove the existence of a unique solution \(v\in C([-T-t_0,t_1];{\mathbb {H}}),\) \(0<t_1\le T-t_0,\) such that v is Lipschitz continuous on \([0,t_1],\) \(v(t)\in D(A)\) for \(t\in [0,t_1]\) and v satisfies the following equation
Then the function
is Lipschitz continuous on \([0,t_0+t_1],\) \(\tilde{u}(t)\in D(A)\) for \(t\in [0,t_0+t_1]\) and satisfies a.e. on \([0,t_0+t_1].\) By continuing in this way, we can prove the existence on \([-T,T]\) or on the maximal interval of existence \([-T,t_\mathrm{max}),\) \(0<t_\mathrm{max}\le T\) such that u is a strong solution on every interval \([-T,\tilde{T}],\) \(0<\tilde{T}<t_\mathrm{max}.\) Hence proved. \(\square \)
Application
Consider the following differential equation
in the Hilbert space \({\mathbb {H}}:=L^2[0,1],\) where \(-A\) is the infinitesimal generator of a \(C_0\)-semigroup \(T(t),\;t\ge 0,\) of contractions in \({\mathbb {H}},\) T and \(\mu \) are positive real numbers, \(\lambda \) is sufficiently small positive real number lies in the interval \((0,\frac{1}{2}),\) and \(\phi :[-T,0]\rightarrow {\mathbb {H}}\) is any continuous function. Define the maps \(f:[0,T]\times {\mathbb {H}}\rightarrow {\mathbb {H}}\) by
and \(g:[0,T]\times {\mathbb {H}}\rightarrow {\mathbb {H}}\) by
Then, for each \(t\in [0,T],\) the function \(S_t:{\mathbb {H}}\rightarrow {\mathbb {H}}\) defined by
\(h\in {\mathbb {H}},\) is bijective. Now, for \(t_1,t_2\in [0,T],\) and \(h_1,h_2\in {\mathbb {H}},\) consider
and
Thus, the functions f and g satisfy the assumptions (H1)–(H3). Therefore, by Theorem 1, the problem (28) has a strong solution.
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The authors would like to thank the referees for their valuable comments and suggestions which help us to improve the original manuscript. The second author acknowledges the financial help from UGC, India under its Research Start-Up-Grant F.30-310/2016 (BSR).
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Maqbul, M., Raheem, A. Application of Rothe’s Method to Some Functional Differential Equations with Dirichlet Boundary Conditions. Differ Equ Dyn Syst 29, 633–643 (2021). https://doi.org/10.1007/s12591-017-0379-1
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DOI: https://doi.org/10.1007/s12591-017-0379-1