Abstract
Full Text
N. V. Azbelev, Lee Mun Su, R. K. Ragimkhanov
On the Question of Defining the Concept of a Solution of an Integral Equation with a Discontinuous Operator
(Presented by Academician I. M. Vinogradov, February 1, 1966)
1. Consider the equation
\[ x(t)=\lambda \int_0^1 K[t,s,x(s)]\,J[s,x(s)]\,ds \tag{1} \]
under the following assumptions.
\(K(t,s,x)\) is defined in \(G: 0\leq t,s\leq 1,\ |x|<c\), is measurable in \(s\) for all \(t\) and \(x\), and is continuous in \(x\) for all \(t\) and almost all \(s\). For any positive \(\gamma<c\), there exists a function \(\mu_\gamma(t,s)\), summable in \(s\) and continuous in \(t\) on \([0,1]\), such that \(|K(t,s,x)|\leq \mu_\gamma(t,s)\) for \(|x|\leq \gamma\). The function \(J(s,x)\in L_\infty\) in the closed domain \(G_\gamma: 0\leq s\leq 1,\ |x|\leq \gamma\), for every \(\gamma<c\).
A definition of a solution of equation (1) was proposed in \((^{1,2})\). However, in the discussion of the aforementioned works at the Izhevsk seminar it became clear that the definition of A. B. Samarov and Lee Mun Su has a number of essential shortcomings. For example, when a differential equation \(y'=J(t,y)\) is reduced by Fubini’s method to an integral equation, solutions arise that do not satisfy, in the sense of works \((^{3,4})\), the original differential equation; in the case of a variable upper limit, the theorem on continuation of a solution is false.
To formulate the new definition proposed by us, denote
\[ M_x\{J(s,x)\}=\lim_{\delta\to 0}\inf_{\mu N=0}\sup_{z\in R(x,\delta)-N} J(s,z), \]
\[ m_x\{J(s,x)\}=\lim_{\delta\to 0}\sup_{\mu N=0}\inf_{z\in R(x,\delta)-N} J(s,z), \]
where \(R(x,\delta)\) is the interval of radius \(\delta\) centered at the point \(x\).
We shall call \(v(t)\in C_{[0,1]}\) a solution of equation (1) if
\[ |v(t)|<c,\qquad v(t)=\lambda\int_0^1 K[t,s,v(s)]\,R_v(s)\,ds, \]
where, for almost all \(s\in[0,1]\),
\[ m_x\{J(s,v(s))\}\leq R_v(s)\leq M_x\{J(s,v(s))\}. \tag{2} \]
With this definition, the following assertions on existence and estimates of solutions are valid.
Theorem 1. Equation (1) has a solution if
\[ 0<\lambda<\gamma\left/ \left\| \int_0^1 \mu_\gamma(t,s)\, \operatorname*{vrai\,max}_{x\in G_\gamma(s)} |J(s,x)|\,ds \right\|_{C[0,1]} \right. . \]
Theorem 2. Let \(K(t,s,x)\geq 0\) and be nondecreasing in \(x\) in \(G\). Suppose, furthermore,
\[ m_x\{J(s,x(s))\}\geq A_1(s,x(s))-A_2(s,x(s)), \]
\[ M_x\{J(s,x(s))\}\leq A_3(s,x(s))-A_4(s,x(s)), \]
where \(A_i \in L_\infty(G)\) \((i=1,2,3,4)\) and are nondecreasing in \(x\). If a pair of functions \(z_i(t)\), continuous on \([0,1]\), \(|z_i(t)|<c\), \(i=1,2\), \(z_2(t)>z_1(t)\), satisfies the integral inequalities
\[ z_1(t)<\lambda\int_0^1 K[t,s,z_1(s)]\,m_x\{A_1(s,z_1(s))\}\,ds- \]
\[ -\lambda\int_0^1 K[t,s,z_2(s)]\,M_x\{A_2(s,z_2(s))\}\,ds, \]
\[ \tag{3} \]
\[ z_2(t)>\lambda\int_0^1 K[t,s,z_2(s)]\,M_x\{A_3(s,z_2(s))\}\,ds- \]
\[ -\lambda\int_0^1 K[t,s,z_1(s)]\,m_x\{A_4(s,z_1(s))\}\,ds, \]
then there exists a solution \(u(t)\) of equation (1), with \(z_1(t)\leq u(t)\leq z_2(t)\).
Remark 1. If \(J=J_1-J_2\) and in \(G\) the functions \(J_i\) and \(KJ_i\) \((i=1,2)\) are nondecreasing in \(x\), then condition (3) may be replaced by the inequalities
\[ z_1(t)<\lambda\int_0^1 K[t,s,z_1(s)]\,m_x\{J_1(s,z_1(s))\}\,ds- \]
\[ -\lambda\int_0^1 K[t,s,z_2(s)]\,M_x\{J_2(s,z_2(s))\}\,ds, \]
\[ z_2(t)>\lambda\int_0^1 K(t,s,z_2(s)]\,M_x\{J_1(s,z_2(s))\}\,ds- \]
\[ -\lambda\int_0^1 K[t,s,z_1(s)]\,m_x\{J_2(s,z_1(s))\}\,ds. \]
Remark 2. The last inequalities are equivalent to the condition
\[ \lambda\int_0^1 K[t,s,z_2(s)]\,M_x\{J_2(s,z_2(s))\}\,ds- \]
\[ -\lambda\int_0^1 K[t,s,z_1(s)]\,m_x\{J_2(s,z_1(s))\}\,ds<\min(-\varphi_1,\varphi_2), \]
where
\[ \varphi_i=z_1-\lambda\int_0^1 K[t,s,z_i(s)]\,m_x\{J_1(s,z_i(s))\}\,ds+ \]
\[ +\lambda\int_0^1 K[t,s,z_i(s)]\,m_x\{J_2(s,z_i(s))\}\,ds,\qquad i=1,2. \]
Remark 3. For \(J\equiv 1\) and under the condition that \(K(t,s,x)\) decreases in \(x\), from Theorem 2 and the preceding remarks one obtains, as a special case, a number of assertions of works \((5\text{--}8)\), which underlie certain new estimates of solutions of differential equations.
- For the Volterra equation
\[ \tag{4} x(t)=\int_0^t K[t,s,x(s)]J[s,x(s)]\,ds \]
we additionally assume that \(|K(t,s,x)-K(t_1,s,x)|\leq v_\gamma(t,t_1,s)\), and for any \(\varepsilon>0\) one can find such a \(\delta>0\) that
\[ \int_0^t v_\gamma(t,t_1,s)\,ds+\int_t^{t_1}\mu_\gamma(t_1,s)\,ds<\varepsilon \quad \text{for } |t-t_1|<\delta,\ |x|\leq\gamma . \]
The local existence theorem and the theorem on continuation of the solution of equation (4) are valid. Under the condition of monotone decrease with respect to \(x\) of the functions \(J(s,x)\) and \(K(t,s,x)J(s,x)\), the theorems of paper (5) on the existence of upper and lower solutions and the theorem on the integral inequality carry over.
- For an equation of the form
\[ x(t)=\int_0^t K(t,s)J(s,x(s))\,ds, \tag{5} \]
arising when reducing a differential equation to an integral one, inequality (3) is equivalent to the condition
\[ R(s)\in \prod_{\delta>0}\prod_{\mu N=0}\operatorname{konv} J\{s,V(u(s),\delta)-N\} \]
for almost all \(s\in[0,t]\).
Lemma. In order that the function \(u(t)\) be a solution of equation (5) on \([0,1]\), it is necessary and sufficient that, for any prescribed \(\varepsilon>0\) and \(\bar N\subset G\) \((\mu N=0)\), there exist a function \(\psi(t)\), measurable on \([0,1]\), such that \(B(t,\psi(t))\subset G\); \(J(t,\psi(t))\) is summable on \([0,1]\),
\[ |u(t)-\psi(t_0)|<\varepsilon,\quad \left|\psi(t)-\int_0^t K(t,s)J(s,\psi(s))\,ds\right|<\varepsilon \quad \text{on } [0,1], \]
and \((t,\psi(t))\in\overline N\) for almost all \(t\in[0,1]\).
For equation (5), the existence of upper and lower solutions is guaranteed without the monotonicity conditions stipulated in item 2; namely, the following is valid.
Theorem 3. Let \(K(t,s)\geq 0\) for \(0\leq s\leq t\leq h\), and let \(h\) be sufficiently small. Then (5) has, on \([0,h]\), an upper solution \(u_1(t)\) and a lower solution \(u_2(t)\), i.e. \(u_1(t)\geq u(t)\geq u_2(t)\) on \([0,h]\) for any solution \(u(t)\).
- Consider the Cauchy problem
\[ \mathcal L[y]=J(t,y),\qquad y^{(k)}(0)=0,\quad k=0,\ldots,n-1, \tag{6} \]
where \(J(t,y)\) is defined above; \(\mathcal L(y)=y^{(n)}-\sum_{k=0}^{n-1}g_k(t)y^{(k)}\); \(g_k(t)\) are functions summable on \([0,1]\). Let \(p(t)\) be summable on \([0,1]\), and let \(K(t,s)\) be the Cauchy function \((^{7,9})\) of the equation \(\mathcal L(y)=p(t)y\). Writing (6) in the form
\[ \mathcal L[y]-p(t)y=J(t,y)-p(t)y,\qquad y^{(k)}(0)=0,\ k=0,\ldots,n-1, \]
we obtain the equation equivalent to (6)
\[ y(t)=\int_0^t K(t,s)\{J(s,y(s))-p(s)y(s)\}\,ds. \tag{7} \]
It is known that for \(n=1\) problem (6) has upper and lower solutions. In paper \((^9)\) the existence of upper and lower solutions for any \(n\) was proved under the assumption that \(J(t,y)\) is discontinuous and satisfies condition \(L_1\) \((^{1,9,10})\). Directly from (7) and Theorem 3 it follows that problem (6) has upper and lower solutions. Hence it is clear that the existence of such solutions of problem (6) depends not on the order \(n\) and not on condition \(L_1\), but on
of the form of the right-hand side of the equation (the right-hand side contains no derivatives of \(y\)).
Let us note the following consequence of Theorem 3.
Corollary. Suppose that \(J(t,y)\) satisfies condition \(L_2\) \((^{10,11})\), i.e.
\(J(t,y)=p(t)y-H(t,y)\), where \(H(t,y)\) is nonincreasing in \(y\), and \(p(t)\) is summable. Then problem (6) has at most one solution.
Izhevsk Mechanical
Institute
Received
27 I 1966
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