Global existence of solutions to nonlinear Volterra integral equations

TL;DR

This paper introduces a novel method to prove the global existence of solutions for nonlinear Volterra integral equations, providing bounds based on a previously established nonlinear inequality.

Contribution

It presents a new approach for establishing global solutions to nonlinear Volterra integral equations using a bound derived from a nonlinear inequality.

Findings

Established conditions for global existence of solutions.

Derived bounds on solutions using nonlinear inequalities.

Validated the method for a class of nonlinear Volterra equations.

Abstract

A new method is given for proving the global existence of the solution to nonlinear Volterra integral equations. A bound on the solution is derived. The results are based on a nonlinear inequality proved by the author earlier.

Equations

$$u(t)=f(t)+{\int }_0^{t}a(t,s,u(s))ds,t\ge 0.$$

$$g^{\prime }(t)\le -\gamma (t)g(t)+\alpha (t,g(t))+\beta (t),t\ge 0,g\ge 0;g^{\prime }=\frac{dg}{dt},$$

$$\alpha (t,{\mu }^{-1}(t))+\beta (t)\le {\mu }^{-1}(t)\left(\gamma (t)-{\mu }^{\prime }(t){\mu }^{-1}(t)\right),\forall t\ge t_0,$$

$$\mu (t_0)g(t_0)<1.$$

$$0\le g(t)<{\mu }^{-1}(t),\forall t\ge t_0.$$

$$|f(t)|+|f^{\prime }(t)|\le c_0{e}^{-b_{0}t},\forall t\ge 0,$$

$$|a(t,t,u)|\le c_1{e}^{-b_{1}t}(1+|u{|}^{2p}),p>0,$$

$${\int }_0^t|a_t(t,s,u(s))|ds\le c_2{e}^{-bt}(1+|u(t){|}^{2p}),a_u(t,s,u)\ge 0,a_t=\frac{\partial a}{\partial t}.$$

$$u^{\prime }=f^{\prime }+a(t,t,u(t))+{\int }_0^t{a}_t(t,s,u(s))ds.$$

$$\Vert u{\Vert }^{\prime }\le \Vert u^{\prime }\Vert .$$

$$|u{|}^{\prime }\le |u^{\prime }|.$$

$$g^{\prime }\le c_0{e}^{-b_{0}t}+c_1{e}^{-b_{1}t}(1+g^{2p}(t))+c_2{e}^{-bt}(1+g^{2p}(t)).$$

$$|u(t)|\le c{e}^{qt},q>0,$$

$$\mu =c_3{e}^{-qt},q=const>0.$$

$$g(0)c_3<1.$$

$$\begin{array}{r}{c}_0{e}^{-b_{0}t}+c_1{e}^{-b_{1}t}+c_2{e}^{-bt}+c_1{e}^{-b_{1}t}(c_3{e}^{-qt}{)}^{-2p}+c_2{e}^{-bt}(c_3{e}^{-qt}{)}^{-2p}\le \\ q{c}_3{e}^{-qt}(c_3{e}^{-qt}{)}^{-2}=q{c}_3^{-1}{e}^{qt}.\end{array}$$

$$|u(t)|\le e^{qt}/c_3\forall t\ge 0.$$

Full text

A new method is given for proving the global existence of the solution to nonlinear Volterra integral equations. A bound on the solution is derived. The results are based on a nonlinear inequality proved by the author earlier.

1 Introduction

Consider the equation:

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The problem is:

Under what assumptions on $f$ and $a(t,s,u)$ equation (1) has a solution which is defined on ${\mathbb{R}}_{+}:=[0,\infty)$?

Many results on the theory of integral equations and many references one can find in [1]. Let us formulate the author’s result basic for our study (see [2], p. 105).

Let

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where $\gamma$, $\beta$ and $\alpha$ are continuous functions of $t\in{\mathbb{R}}_{+}$, $\alpha(t,g)\geq 0$ is a continuous non-decreasing function of $g$ on ${\mathbb{R}}_{+}:=[t_{0},\infty)$, $t_{0}\geq 0$.

Lemma 1. Assume that there exists a function $\mu=\mu(t)>0$, $\mu\in C^{1}([t_{0},\infty))$ such that

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and

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Then any solution $g\geq 0$ to inequality (2) exists on ${\mathbb{R}}_{+}$ and

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If $\mu(t_{0})g(t_{0})\leq 1$ then $0\leq g(t)\leq\mu^{-1}(t)$ for all $t\geq t_{0}$.

A proof of Lemma 1 is given in [2], pp. 105-107, see also [3].

A new idea in this paper is to use Lemma 1 with $\gamma(t)=0$. In this case inequality (3) may hold only if $\mu(t)$ decays as $t$ grows, and estimate (5) becomes the estimate of the rate of growth of $u$.

In [3] $\mu(t)$ was growing to infinity as $t\to\infty$ and estimate (5) gave results on the stability and large-time behavior of $g(t)=\|u(t)\|$, where the norm was a Hilbert space norm.

Let us assume that $t_{0}=0$ and

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[TABLE]

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Assume also that $|a|+a_{u}\leq c(R_{1},R_{2})$ for $t\leq R_{1},s\leq R_{1}$ and $|u|\leq R_{2}$, $a$ and $a_{t}$ are smooth functions of their arguments.

This assumption allows one to use the contraction mapping principle if $t>0$ is sufficiently small and establish the existence and uniqueness of the local solution to equation (1).

Differentiate (1) with respect to $t$ and get

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Lemma 2. Let $u(t)\in H$, where $H$ is a Hilbert space, $\|u\|^{2}=(u,u)$, $\|u\|^{\prime}=\frac{d\|u\|}{dt}$. If $u(t)\in C^{1}({\mathbb{R}}_{+};H)$ then

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If $u(t)\in C^{1}({\mathbb{R}}_{+})$, then

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Proof of Lemma 2. One has $\|u\|^{2}=(u,u)$. Thus, $2\|u\|^{\prime}\|u\|=(u^{\prime},u)+(u,u^{\prime})\leq 2\|u^{\prime}\|\|u\|$. Since $\|u\|\geq 0$, one gets (10).

If $u(t)\in C^{1}({\mathbb{R}}_{+})$, then $|u(t+h)|-|u(t)|\leq|u(t+h)-u(t)|$. Divide this inequality by $h>0$ and let $h\to 0$. This yields (11). $\Box$

Taking the absolute value of (9), using (10) and setting $g(t)=|u(t)|$, one obtains

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Theorem 1. If (4) and (6)–(8) hold, then the solution to (1) exists on ${\mathbb{R}}_{+}$, is unique and

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*where $q>0$ is a fixed number and $c>0$ is a sufficiently large constant. *

In Section 2 a proof of Theorem 1 is given. From this proof one can get an estimate for the constant $c$.

2 Proof of Theorem 1

Let us apply to (12) Lemma 1. Choose

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Since $\mu(0)=c_{3}$ inequality (4) holds if

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This inequality holds if $c_{3}$ is sufficiently small.

Let $t_{0}=0$. One has $g(0)=|f(0)|\leq c_{0}$. Thus, (15) holds if $c_{3}<c_{0}^{-1}$.

Inequality (3) holds if

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In our case $\gamma(t)=0$, $\beta(t)$ is the sum of the three first terms in (16) and $\alpha$ is the sum of the fourth and fifth terms in (16).

Inequality (16) holds if $c_{3}$ is sufficiently small and $q>0$.

From Lemma 1 the global existence of the solution $g\geq 0$ to (12) follows and estimate (5) yields

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The local existence and uniqueness of the solution to (1) follows from the contraction mapping principle. Theorem 1 is proved. $\Box$

Remark 1. Without some assumptions on $f$ and $a(t,s,u)$ the solution to (1) may not exist globally.

Example 1. Let $u=1+\int_{0}^{t}u^{2}(s)ds$. Then $u^{\prime}=u^{2},$ $u(0)=1$. A simple integration yields $u=(1-t)^{-1}$. So, the solution tends to infinity as $t\to 1$.

Remark 2. The method developed in this paper can be used for other decay assumptions, for example, power decay of $f$ and $a(t,s,u)$ as $t\to\infty$.

One may look for the $\mu(t)=c_{4}(1+t)^{-r},r>0$, where $c_{4}=const$. If $c_{4}$ is sufficiently small then inequality (5) yields $|u(t)|\leq(1+t)^{r}/c_{4}$, $\forall t>0$.

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