Abstract
In this paper, first we introduce a measure of noncompactness in the Sobolev space \(W^{k,1}(\Omega )\) and then, as an application, we study the existence of solutions for a class of the functional integral-differential equations using Darbo’s fixed point theorem associated with this new measure of noncompactness. Further, two examples are presented to verify the effectiveness and applicability of our main results.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
Sobolev spaces [11], i.e., the class of functions with derivatives in \(L^p,\) play an outstanding role in the modern analysis. In the last decades, there has been increasing attempts to study these spaces. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces. They also highlighted in approximation theory, calculus of variation, differential geometry, spectral theory etc.
On the other hand, integral-differential equations (IDE) have a great deal of applications in some branches of sciences. It arises especially in a variety of models from applied mathematics, biological science, physics and another phenomenon, such as the theory of electrodynamics, electromagnetic, fluid dynamics, heat and oscillating magnetic, etc. [9, 12, 18, 21, 24]. There have appeared recently a number of interesting papers [2, 6, 10, 19, 22, 23, 27] on the solvability of various integral equations with help of measures of noncompactness.
The first such measure was defined by Kuratowski [25]. Next, Banaś et al. [8] proposed a generalization of this notion which is more convenient in the applications. The technique of measures of noncompactness is frequently applicable in several branches of nonlinear analysis, in particular the technique turns out to be a very useful tool in the existence theory for several types of integral and integral-differential equations. Furthermore, they are often used in the functional equations, fractional partial differential equations, ordinary and partial differential equations, operator theory and optimal control theory [1, 3, 7, 13, 15,16,17, 26, 28, 29]. The most important application of measures of noncompactness in the fixed point theory is contained in the Darbo’s fixed point theorem [4, 5].
Now, in this paper, we introduce a new measure of noncompactness in the Sobolev space \(W^{k,1}(\Omega )\) as a more effective approach. Then, we study the problem of existence of solutions of the functional integral-differential equation
We provide some notations, definitions and auxiliary facts which will be needed further on.
Throughout this paper, \(\mathbb {R}_{+}\) indicates the interval \([0,+\infty )\) and for the Lebesgue measurable subset D of \(\mathbb {R,}\) m(D) denotes the Lebesgue measure of D. Moreover, let \(L^1(D)\) be the space of all Lebesgue integrable functions f on D equipped with the standard norm \(\Vert f\Vert _{L^1(D)}=\int _D|f(x)| \mathrm{d}x\).
Let \((E,\Vert \cdot \Vert )\) be a real Banach space with zero element 0. The symbol \(\overline{B}(x,r)\) stands for the closed ball centered at x with radius r and put \(\overline{B}_{r}=\overline{B}(0,r)\). Denote by \({\mathfrak {M}}_E\) the family of nonempty and bounded subsets of E and by \({\mathfrak {N}}_E\) its subfamily consisting of all relatively compact sets of E. For a nonempty subset X of E, the symbols \(\overline{X}\) and ConvX will denote the closure and the closed convex hull of X, respectively.
Definition 1.1
[8] A mapping \(\mu :\mathfrak {M}_{E} \rightarrow \mathbb {R}_{+}\) is said to be a measure of noncompactness in E if it satisfies the following conditions:
-
\(1^{\circ }\) The family \(\ker \mu =\{X\in \mathfrak {M}_{E}: \mu (X)=0\}\) is nonempty and \(\ker \mu \subset \mathfrak {N}_{E}\).
-
\(2^{\circ }\) \(X\subset Y\Rightarrow \mu (X)\le \mu (Y)\).
-
\(3^{\circ }\) \(\mu (\overline{X})=\mu (X)\).
-
\(4^{\circ }\) \(\mu (Conv X) = \mu (X)\).
-
\(5^{\circ }\) \(\mu (\lambda X + (1 - \lambda )Y )\le \lambda \mu (X) + (1 -\lambda )\mu (Y )\) for \(\lambda \in [0, 1]\).
-
\(6^{\circ }\) If \(\{X_{n}\}\) is a sequence of closed sets from \(\mathfrak {M}_{E}\) such that \(X_{n+1} \subset X_{n}\) for \(n=1,2,\ldots \) and if \(\displaystyle {\lim \nolimits _{n\rightarrow \infty }} \mu (X_{n}) = 0\) then the set \(X_{\infty }=\displaystyle {\bigcap \nolimits _{n=1}^{\infty }}X_n\) is nonempty. A measure of noncompactness \(\mu \) is said to be regular if it additionally satisfies the following conditions:
-
\(7^{\circ }\) \(\mu ( X\bigcup Y)=\max \{\mu (X),\mu (Y)\}.\)
-
\(8^{\circ }\) \(\mu ( X+ Y)\le \mu (X)+\mu (Y).\)
-
\(9^{\circ }\) \(\mu (\lambda X)=|\lambda |\mu (X)\) for \(\lambda \in \mathbb {R}.\)
-
\(10^{\circ }\) \(\ker \mu =\mathfrak {N}_E.\)
In what follows, we recall the well known Darbo’s fixed point theorem.
Theorem 1.2
[13] Let \(\Omega \) be a nonempty, bounded, closed and convex subset of a Banach space E and let \(F:\Omega \rightarrow \Omega \) be a continuous mapping such that there exists a constant \(k\in [0,1)\) with the property
for any nonempty subset X of \(\Omega \), where \(\mu \) is a measure of noncompactness defined in E. Then, F has a fixed point in the set \(\Omega \).
Construction of a measure of noncompactness in Sobolev spaces
In this section, we introduce a measure of noncompactness in the Sobolev space \(W^{k,1}(\Omega )\).
Let \(\Omega \) be a subset of \(\mathbb {R}^n\) and \(k\in \mathbb {N}\), we denote by \(W^{k,1}(\Omega )\) the space of functions f which, together with all their distributional derivatives \(D^\alpha f\) of order \(|\alpha |\le k\), belong to \(L^1(\Omega )\). Here \(\alpha =(\alpha _1,\ldots ,\alpha _n)\) is a multi-index, i.e., each \(\alpha _j\) is a nonnegative integer, \(|\alpha |=\alpha _1+\cdots +\alpha _n\), and
Then, \(W^{k,1}(\Omega )\) is equipped with the complete norm
We present the following theorem which characterizes the compact subsets of the Sobolev spaces.
Theorem 2.1
[20] A subset \(\mathcal {F}\subset W^{k,1}(\mathbb {R}^n)\) is totally bounded if, and only if, the following holds:
-
(i)
\(\mathcal {F}\) is bounded, i.e., there is some M so that
$$\begin{aligned} \int |D^{\alpha }f(x)| \mathrm{d}x<M,\ f\in \mathcal {F},\ |\alpha |\le k. \end{aligned}$$ -
(ii)
For every \(\varepsilon >0\) there is some R so that
$$\begin{aligned} \int _{\Vert x\Vert _{\mathbb {R}^n}>R}|D^{\alpha }f(x)| \mathrm{d}x<\varepsilon ,\ f\in \mathcal {F},\ |\alpha |\le k. \end{aligned}$$ -
(iii)
For every \(\varepsilon >0\) there is some \(\rho >0\) so that
$$\begin{aligned} \int _{\mathbb {R}^n}|D^{\alpha }f(x+y)-D^{\alpha }f(x)| \mathrm{d}x<\varepsilon ,\ f\in \mathcal {F},\ |\alpha |\le k,\ \Vert y\Vert _{\mathbb {R}^n}< \rho . \end{aligned}$$
Now, we are going to describe a measure of noncompactness in \(W^{k,1}(\Omega )\).
Theorem 2.2
Suppose \(1\le k <\infty \) and U is a bounded subset of \(W^{k,1}(\Omega )\). For \(u \in U\), \(\varepsilon >0\) and \(0\le |\alpha |\le k\), let
where \(B_T=\{a\in \Omega :\Vert a\Vert _{\mathbb {R}^n} \le T\}\) and \(\mathcal {T}_hu(t)=u(t+h)\).
Then \(\omega _0:\mathfrak {M}_{W^{k,1}(\Omega )}\rightarrow \mathbb {R}\) given by
defines a measure of noncompactness in \(W^{k,1}(\Omega )\).
Proof
Take \(U\in \mathfrak {M}_{W^{k,1}(\Omega )}\) such that \(\omega _0(U)=0\). Fix arbitrary \(\alpha \) such that \(0\le |\alpha |\le k\). Let \(\eta >0\) be arbitrary, since \(\omega _0(U)=0\),
Thus, there exists small enough \(\delta >0\) and large enough \(T>0\) such that \(\omega ^T(U, \delta )<\eta \). This implies that
for all \(u\in U\) and \(h\in \Omega \) such that \(\Vert h\Vert _{\mathbb {R}^n}<\delta \). Since \(\eta >0\) was arbitrary, we obtain
Using again the fact that \(\omega _0(U)=0\) we have
and so for \(\varepsilon >0\) there exists large enough \(T>0\) such that
It follows then from Theorem 2.1 that U is totally bounded. Thus, \(1^{\circ }\) holds.
\(2^{\circ }\) is obvious by the definition of \(\omega _0\).
Now, we check that condition \(3^{\circ }\) holds. For this purpose, suppose that \(U\in \mathfrak {M}_{W^{k,1}(\Omega )}\) and \(\{u_n\}\subset U\) such that \(u_n\rightarrow u\in \overline{U}\) in \(W^{k,1}(\Omega )\). From the definition of \(\omega ^T(U,\varepsilon ),\) we have
for any \(n\in \mathbb {N}\), \(T>0\) and \(h\in \Omega \) with \(\Vert h\Vert _{\mathbb {R}^n}<\varepsilon \). Letting \(n\rightarrow \infty ,\) we get
for any \(T>0\) and \(h\in \Omega \) with \(\Vert h\Vert _{\mathbb {R}^n}<\varepsilon \). Hence
This concludes that \(\omega (\overline{U})\le \omega (U)\). Similarly, we can show that
and thus
From (3) and \(2^{\circ }\) we obtain \( \omega _0(\overline{U})=\omega _0(U)\).
\(4^{\circ }\) follows directly from \( D^{\alpha }[Conv(U)]=Conv(D^{\alpha }U)\) and hence is omitted.
The proof of condition \(5^{\circ }\) can be obtained by using the equality
for all \(\lambda \in [0,1],\) \(u_1\in X\) and \(u_2\in Y\).
It remains only to verify \(6^{\circ }\), suppose that \(\{U_{n}\}\) is a sequence of closed and nonempty sets of \(\mathfrak {M}_{W^{k,1}(\Omega )}\) such that \(U_{n+1} \subset U_{n}\) for \(n=1,2,\ldots \), and \( \lim\nolimits _{n\rightarrow \infty } \omega _0(U_{n}) = 0\). Now, for any \(n\in \mathbb {N}\), take \(u_n\in U_n\) and set \(\mathcal {G}=\overline{\{u_n\}}\).
Claim: \(\mathcal {G}\) is a compact set in \(W^{k,1}(\Omega )\).
Let \(\varepsilon >0\) be fixed, since \( {\lim\nolimits _{n\rightarrow \infty }} \omega _0(U_{n}) = 0\), there exists sufficiently large \(m_1\in \mathbb {N}\) such that \(\omega _0(U_{m_1}) <\varepsilon \). Hence, we can find small enough \(\delta _1>0\) and large enough \(T_1>0\) such that \(\omega ^{T_1}(U_{m_1},\delta _1)<\varepsilon \) and \(d(U_{m_1})<\varepsilon \). Therefore,
and
for all \(n> m_1\), \(0\le |\alpha |\le k\) and \(h\in \Omega \) with \(\Vert h\Vert _{\mathbb {R}^n}<\delta _1\). Thus we have
\(\Vert \mathcal {T}_hD^{\alpha }u_n-D^{\alpha }u_n\Vert _{L^1(\Omega )}\)
On the other hand, we know that the set \(\{u_1,u_2,\ldots ,u_{m_1}\}\) is compact, hence there exist \(\delta _2>0\) and \(T_2>0\) such that
for all \(n=1,2,\ldots ,m_1\), \(0\le |\alpha |\le k\) and \(h\in \Omega \) with \(\Vert h\Vert _{\mathbb {R}^n}<\delta _2\).
Furthermore,
which implies that
for all \(n=1,2,\ldots ,m_1\).
Thus,
and
for all \(n\in \mathbb {N}\), \(\Vert h\Vert _{\mathbb {R}^n}<\min \{\delta _1,\delta _2\}\) and \(T=\max \{T_1,T_2\}.\) Therefore, all the hypotheses of Theorem 2.1 are satisfied, that completes the proof of the claim.
Using the above claim, there exists a subsequence \(\{u_{n_j}\}\) and \(u_0\in W^{k,1}(\Omega )\) such that \(u_{n_j}\rightarrow u_0 \). Since \(u_n\in U _n\), \( U_{n+1} \subset U_{n}\) and \( U_n\) is closed for all \(n\in \mathbb {N}\), we yield
that finishes the proof of \(6^{\circ }\). \(\square \)
We now investigate the regularity of \(\omega _0\).
Theorem 2.3
The measure of noncompactness \(\omega _0\) defined in (2) is regular.
Proof
Suppose that \(X, Y\in \mathfrak {M}_{W^{k,1}(\Omega )}\). First, notice that for all \(\varepsilon >0\), \(\lambda \in \mathbb {R}\) and \(T>0\) we have
Then, the hypotheses \(7^{\circ }\)–\(9^{\circ }\) hold. Next, we show that \(10^{\circ }\) holds. Take \(U\in \mathfrak {N}_{W^{k,1}(\Omega )}\). Thus, the closure of U in \(W^{k,1}(\Omega )\) is compact. By Theorem 2.1, for all \(|\alpha |\le k\) and for all \(\varepsilon >0\), there exists \(T>0\) such that \(\Vert D^{\alpha }u\Vert _{L^1(\Omega \backslash B_{T})}<\varepsilon \) for all \(u\in U,\) and there exists \(\delta >0\) such that \(\Vert \mathcal {T}_hD^{\alpha }u-D^{\alpha }u\Vert _{L^1(B_T)}<\varepsilon \) for all \(h\in \Omega \) with \(\Vert h\Vert _{\mathbb {R}^n}<\delta \). Then, for all \(u\in U\) we have
Therefore,
It yields that
Furthermore,
Then \(\omega _0(U)=0\) and \(\ker (\omega _0)= \mathfrak {N}_{W^{k,1}(\Omega )}\). \(\square \)
Theorem 2.4
Let \(Q=\{u\in W^{k,1}(\mathbb {R}^n):\Vert u\Vert _{1,1}\le 1\}.\) Then \(\omega _0(Q)=3\).
Proof
Applying the same strategy as ([4], Theorem [14]), we observe that \(\omega _0(Q)\le 3.\) It remains to verify \(\omega _0(Q)\ge 3.\) For any \(k\in \mathbb {N}\), there exists \(E_k\subset \mathbb {R}^n\) such that \(m(E_k)=\frac{1}{10k}\), \(diam(E_k)\le \frac{1}{k}\), \(E_k\cap B_k=\emptyset \) and \(E_k\subset B_{2k}.\) Define \(f_k:\mathbb {R}^n\rightarrow \mathbb {R}\) by
In addition, observe that \(\Vert f_k\Vert _{1,1}=1\), \(\Vert D^\alpha \mathcal {T}_{\beta _k}f_k-D^\alpha f_k\Vert _{L^1(B_{2k})} =2\) and
\(\Vert D^\alpha f_k\Vert _{L^1(\mathbb {R}^n\backslash B_k)}=1\) for all \(k\in \mathbb {N}\), where \(\beta _k=(\frac{1}{k},\ldots ,\frac{1}{k})\in \mathbb {R}^n\). Thus, we conclude that \(\omega _0(Q)\ge \omega _0(\{f_k\})=3.\) \(\square \)
Application
In this section, we study the existence of solutions for some functional integral-differential equations. We also provide some illustrative examples to verify effectiveness and applicability of our results.
We start with some preliminaries which we need in subsequence.
Lemma 3.1
[14] Let \(\Omega \) be a Lebesgue measurable subset of \(\mathbb {R}^n\) and \(1\le p\le \infty .\) If \(\{f_n\}\) is convergent to f in the \(L^p\)-norm, then there is a subsequence \(\{f_{n_k}\}\) which converges to f a.e., and there is \(g\in L^p(\Omega )\), \(g\ge 0,\) such that
Definition 3.2
[4] We say that a function \(f:\mathbb {R}^n\times \mathbb {R}^m\rightarrow \mathbb {R}\) satisfies the Carath\(\acute{e}\)odory conditions if the function f(., u) is measurable for any \(u\in \mathbb {R}^m\) and the function f(x, .) is continuous for almost all \(x\in \mathbb {R}^n.\)
Let \(\Omega \) be a subset of \(\mathbb {R}^n\) and \(k\in \mathbb {N}\), we denote by \(BC^k(\Omega )\) the space of functions f which are bounded and k-times continuously differentiable on \(\Omega \) with the standard norm
where \(\Vert D^\alpha f\Vert _u=\sup \{|D^\alpha f(x)|:x\in \Omega \}\).
Theorem 3.3
Let \(\Omega \) be a subset of \(\mathbb {R}^n\) with \(m(\Omega )<\infty \). Assume that the following conditions are satisfied:
-
(i)
\(p\in W^{1,1}(\Omega ),\) \(q\in BC^1(\Omega )\) and
$$\begin{aligned} \lambda :=\sup \{\Vert q\Vert _u+\Vert \frac{\partial q}{\partial x_i}\Vert _u:i=1,\ldots ,n\}<1. \end{aligned}$$(7) -
(ii)
\(g:\Omega \times \mathbb {R}^{n+2} \rightarrow \mathbb {R}\) satisfies the Carath \(\acute{e}\) odory conditions and there exist a bounded continuous function \(a:\Omega \rightarrow \mathbb {R}_+\) with \(|a(x)|\le M\) for all \(x\in \Omega \) and some \(M>0\) and a concave, lower semi-continuous and nondecreasing function \(\zeta :\mathbb {R_+}\rightarrow \mathbb {R_+}\) such that
$$\begin{aligned} |g(x,u_0,u_1,\ldots ,u_{n+1})|\le a(x)\zeta (\max _{0\le i\le n+1}|u_i|). \end{aligned}$$(8) -
(iii)
\(k:\Omega \times \Omega \rightarrow \mathbb {R}\) satisfies the Carath\(\acute{e}\)odory conditions and has a derivative of order 1 with respect to the first argument. Moreover, there exist \(g_1,g_3\in W^{1,1}(\Omega )\) and \(g_2\in L^\infty (\Omega )\) such that
$$\begin{aligned} |k(x,y)|\le g_1(x)g_2(y),\ |k(x_1,y)-k(x_2,y)|\le g_2(y)|g_3(x_1)-g_3(x_2)|, \end{aligned}$$and
$$\begin{aligned} \left|\frac{\partial k}{\partial x_i}(x,y)\right|\le g_1(x)g_2(y),\ \left|\frac{\partial k}{\partial x_i}(x_1,y)-\frac{\partial k}{\partial x_i}(x_2,y)\right|\le g_2(y)|g_3(x_1)-g_3(x_2)|, \end{aligned}$$for almost \(x,y,x_1,x_2\in \Omega \) and \(1\le i\le n\).
-
(iv)
There exists a positive solution \(r_0\) of the inequality
$$\begin{aligned} \Vert p\Vert _{1,1}+\lambda r+Mm(\Omega )\Vert g_1\Vert _{L^1(\Omega )}\Vert g_2\Vert _{L^\infty }\zeta \left(\frac{1}{m(\Omega )}\Vert u\Vert _{1,1}\right)\le r. \end{aligned}$$(9) -
(v)
\(T:W^{1,1}(\Omega ) \rightarrow L^1(\Omega )\) is a continuous operator such that for any \(x\in W^{1,1}(\Omega )\) we have
$$\begin{aligned} \Vert T(x)\Vert _{L^1(\Omega )}\le \Vert x\Vert _{1,1}. \end{aligned}$$
Then, the functional integral-differential equation
has at least one solution in the space \(W^{1,1}(\Omega )\).
Proof
We define the operator \(F:W^{1,1}(\Omega )\rightarrow W^{1,1}(\Omega )\) by
Obviously, Fu is measurable for any \(u\in W^{1,1}(\Omega )\). Also, for any \(x\in \Omega \) we have
and Fu has measurable derivatives. We show that, \(Fu\in W^{1,1}(\Omega )\). Using our assumptions, for arbitrarily fixed \(x\in \Omega \), we have
According to the Jensen’s inequality, we deduce
By the same argument as above,
and
Thus, we obtain
Due to (11) and using condition (iv), we derive that F is a mapping from \(\bar{B}_{r_0}\) into \(\bar{B}_{r_0}\). Now, we show that the map F is continuous. Let \(\{u_m\}\) be an arbitrary sequence in \(W^{1,1}(\Omega )\) which converges to \(u\in W^{1,1}(\Omega )\). By Lemma 3.1 there is a subsequence \(\{u_{m_k}\}\) which converges to u a.e., \(\{\frac{\partial u_{m_k}}{\partial x_i}\}\) converges to \(\{\frac{\partial u}{\partial x_i}\}\) a.e., \(\{Tu_{m_k}\}\) converges to Tu a.e. and there is \(h\in L^1(\Omega )\), \(h\ge 0,\) such that
Since \(u_{m_k}\rightarrow u\) almost everywhere and g satisfies the Carath\(\acute{e}\)odory conditions, it follows that
for almost all \(y\in \Omega \).
From condition (ii) we have
As a consequence of the Lebesgue’s Dominated Convergence Theorem, it yields that
for almost all \(y\in \Omega .\) Inequality (12) and condition (iii) imply that
Therefore, \(F:W^{1,1}(\Omega ) \longrightarrow W^{1,1}(\Omega )\) is continuous.
To finish, the proof we have to verify that condition (1) is satisfied. We fix arbitrary \(T>0\) and \(\varepsilon > 0\). Let U be a nonempty and bounded subset of \(\bar{B}_{r_0}\). Choose \(u\in U\) and \(x,h\in B_T\) with \(\Vert h\Vert _{\mathbb {R}^n}\le \varepsilon \), then we have
Obviously, \({\omega }^T(p, \varepsilon )\rightarrow 0\), \({\omega }^T (g_3,\varepsilon )\rightarrow 0\) and by continuity of q,
as \(\varepsilon \rightarrow 0\). Then the right hand side of (13) tends to \(\lambda \omega ^T(U)\) as \(\varepsilon \rightarrow 0\).
By a similar argument and using condition (i), for each \(i=1,\ldots ,n\), we get
Applying the same reasoning as above, the right hand side of (14) tends to \(\lambda \omega ^T(U)\) as \(\varepsilon \rightarrow 0,\) too. Regarding to (13) and (14), and since u is an arbitrary element of U, then \(\omega ^T(FU)\le \lambda \omega ^T(U)\). Letting \(T\rightarrow \infty \), we deduce
Next, let us fix an arbitrary number \(T>0\). Then, taking into account our hypotheses, for an arbitrary function \(u\in U\) we derive
Now, since
then
Similarly,
These relations imply that
Finally, from (15) and (16) we conclude that \(\omega _0(FU)\le \lambda \omega _0(U)\).
According to Theorem 1.2, we obtain that the operator F has a fixed point x in \(\bar{B}_{r_0}\), and thus functional integral-differential equation (10) has at least one solution in the space \(W^{1,1}(\Omega )\). \(\square \)
Now, we present two examples which verify the effectiveness and applicability of Theorem 3.3.
Example 3.4
Consider the following functional integral-differential equation
Eq. (17) is a special case of Eq. (10) with
\(\Omega = [0, 1]\times [0, 1]\times [0, 1],p(x_1,x_2,x_3)=\root 4 \of {x_1^5},q(x_1,x_2,x_3)=e^{-(x_1+x_2+x_3+1 )},\) \(g(y_1,y_2,y_3, u_0, u_1, u_2,u_3,u_4)=\cos (y_1 u_0 u_1+y_2u_2+y_3u_3+u_4)\), \(Tu=\frac{1}{2}u\),
and
It is easy to see that \(p\in W^{1,1}(\Omega )\), \(q\in BC^1(\Omega )\) and \(\lambda =2e^{-1}\). Also, g satisfies Carath\(\acute{e}\)odory conditions and if we define \(a(x_1,x_2,x_3)=\zeta (x)=1\), then condition (ii) of Theorem 3.3 holds. We observe that \(g_1,g_3\in L^1(\Omega ), g_2\in L^\infty (\Omega )\) and k satisfies condition (iii). Moreover, it can be easily shown that each number \(r \ge 4\) satisfies the inequality in condition (iv), i.e.,
Thus, as the number \(r_0\) we can take \(r_0= 4\). Consequently, all the conditions of Theorem 3.3 are satisfied. Hence the functional integral-differential equation (17) has at least one solution in the space \(\displaystyle W^{1,1}(\Omega )\).
Example 3.5
Consider the following functional integral-differential equation
Eq. (18) is a special case of Eq. (10) with
It is easy to see that \(q\in BC^1(\Omega )\) and \(\lambda =\dfrac{3}{4}\). Also, g satisfies Carath\(\acute{e}\)odory conditions and if we define \(a(x)=\root 5 \of {5} \) and \(\zeta (x)=\root 5 \of {x} \), then condition (ii) of Theorem 3.3 holds. Moreover, k is continuous and has a continuous derivative of order 1 with respect to the first argument. On the other hand, \(g_1(x)=g_3(x)=e^x\) and \(g_2(x)=e^{-x}\). It can be easily shown that each number \(r \ge 10\) satisfies the inequality in condition (iv), i.e.,
Hence, as the number \(r_0\) we can take \(r_0=10\). Consequently, all the conditions of Theorem 3.3 are satisfied. It implies that the functional integral-differential equation (18) has at least one solution in the space \(\displaystyle W^{1,1}(\Omega )\).
References
Agarwal, R.P., Benchohra, M., Seba, D.: On the application of measure of noncompactness to the existence of solutions for fractional differential equations. Results Math. 55, 221–230 (2009)
Aghajani, A., Allahyari, A., Mursaleen, M.: A generalization of Darbo’s theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 260, 68–77 (2014)
Aghajani, A., Banaś, J., Jalilian, Y.: Existence of solutions for a class of nonlinear Volterra singular integral equations. Comput. Math. Appl. 62, 1215–1227 (2011)
Aghajani, A., O’Regan, D., Shole Haghighi, A.: Measure of noncompactness on \(L^p(\mathbb{R}^n)\) and applications. Cubo A Math. J. 17, 85–97 (2015)
Arab, R., Allahyari, R., Shole Haghighi, A.: Construction of measures of noncompactness of \(C^k(\Omega )\) and \(C_0^k\) and their application to functional integral-differential equations. Bull. Iran. Math. Soc. 43(1), 53–67 (2017)
Ayad, A.: Spline approximation for first order Fredholm delay integro-differential equations. Int. J. Comput. Math. 70, 467–476 (1999)
Banaś, J.: Measures of noncompactness in the study of solutions of nonlinear differential and integral equations. Cent. Eur. J. Math. 10(6), 2003–2011 (2012)
Banaś, J., Goebel, K.: Measure of noncompactness in Banach spaces. Lecture notes in pure and applied mathematics, vol. 60. Dekker, New York (1980)
Banaś, J., O’Regan, D., Sadarangani, K.: On solutions of a quadratic hammerstein integral equation on an unbounded interval. Dynam. Syst. Appl. 18, 251–264 (2009)
Behiry, S.H., Hashish, H.: Wavelet methods for the numerical solution of Fredholm integro-differential equations. Int. J. Appl. Math. 11, 27–35 (2002)
Brezis, H.: Functional analysis. Sobolev spaces and partial differential equations. Springer Science, Business Media, LLC, New York (2011)
Bloom, F.: Asymptotic bounds for solutions to a system of damped integro-differential equations of electromagnetic theory. J. Math. Anal. Appl. 73, 524–542 (1980)
Darbo, G.: Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Univ. Padova. 24, 84–92 (1955)
Drabek, P., Milota, J.: Methods of nonlinear analysis. Birkhauser Velgar AG, Basel (2007)
Darwish, M.A.: On monotonic solutions of a quadratic integral equation with supremum. Dynam. Syst. Appl. 17, 539–550 (2008)
Darwish, M.A., Henderson, J., O’Regan, D.: Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument. Bull. Korean Math. Soc. 48, 539–553 (2011)
Dhage, B.C., Bellale, S.S.: Local asymptotic stability for nonlinear quadratic functional integral equations. Electron. J. Qual. Theory Differ. Equ. 10, 1–13 (2008)
Forbes, L.K., Crozier, S., Doddrell, D.M.: Calculating current densities and fields produced by shielded magnetic resonance imaging probes. SIAM J. Appl. Math. 57, 401–425 (1997)
Garey, L.A., Gladwin, C.J.: Direct numerical methods for first order Fredholm integro-differential equations. Int. J. Comput. Math. 34, 237–246 (1990)
Hanche-Olsen, H., Holden, H.: The Kolmogorov–Riesz compactness theorem. Expo. Math. 28, 385–394 (2010)
Holmaker, K.: Global asymptotic stability for a stationary solution of a system of integro-differential equations describing the formation of liver zones. SIAM J. Math. Anal. 24, 116–128 (1993)
Hosseini, S.M., Shahmorad, S.: Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial base. Appl. Math. Model. 27, 145–154 (2003)
Jleli, M., Mursaleen, M., Sadarangani, K., Samet, B.: A cone measure of noncompactness and some generalizations of Darbo’s theorem with applications to functional integral equations. J. Funct. Spaces 9896502, 11 (2016)
Kanwal, R.P.: Linear integral differential equations theory and technique. Academic Press, New York (1971)
Kuratowski, K.: Sur les espaces complets. Fund. Math. 15, 301–309 (1930)
Liu, L., Guo, F., Wu, C., Wu, Y.: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. J. Math. Anal. Appl. 309, 638–649 (2005)
Mursaleen, M., Rizvi, S.M.H.: Solvability of infinite system of second order differential equations in \(c_0 \) and \(l_1\) by Meir–Keeler condensing operator. Proc. Am. Math. Soc. 144(10), 4279–4289 (2016)
Mursaleen, M., Mohiuddine, S.A.: Applications of noncompactness to the infinite system of differential equations in \(l_p\) spaces. Nonlinear Anal. 75(4), 2111–2115 (2012)
Olszowy, L.: Solvability of infinite systems of singular integral equations in Fr\(\acute{e}\)chet space of coninuous functions. Comp. Math. Appl. 59, 2794–2801 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Khanehgir, M., Allahyari, R. & Gholamian, N. Construction of a measure of noncompactness in Sobolev spaces with an application to functional integral-differential equations. Math Sci 12, 17–24 (2018). https://doi.org/10.1007/s40096-017-0240-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40096-017-0240-2