\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 80, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/80\hfil Solvability of BVPs] {On the solvability of nonlinear first-order boundary-value problems} \author[C. C. Tisdell\hfil EJDE-2006/80\hfilneg] {Christopher C. Tisdell} \address{Christopher C. Tisdell \newline School of Mathematics\\ The University of New South Wales\\ UNSW Sydney 2052, Australia} \email{cct@maths.unsw.edu.au} \date{} \thanks{Submitted April 7, 2006. Published Juy 19, 2006.} \subjclass[2000]{34B15, 34B99} \keywords{Existence of solutions; boundary value problems; \hfill\break\indent differential equations; fixed-point methods; differential inequalities} \begin{abstract} This article investigates the existence of solutions to first-order nonlinear boundary-value problems (BVPs) involving systems of ordinary differential equations and two-point boundary conditions. Some sufficient conditions are presented that will ensure solvability. The main tools employed are novel differential inequalities and fixed-point methods. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \section{Introduction} This paper considers the existence of solutions to the first-order differential equation $$\label{1.1} x'= f(t,x), \quad t \in [a,c],$$ subject to the boundary conditions $$\label{1.2} Mx(a) + Rx(c) = 0,$$ where $f:[a,c] \times \mathbb{R}^n \to \mathbb{R}^n$ is a continuous, nonlinear function; $a0$ and centre 0. \begin{theorem} \label{thm1.1} Let $T:\overline{B_P} \to J$ be a compact map and let $\lambda \in [0,1]$. If $$u \neq \lambda Tu, \quad \text{for all } u \in \partial B_P \text{ and all } \lambda \in (0,1)$$ then there exists at least one $u \in B_P$ such that $u = Tu$. \end{theorem} \section{Solvability} \label{sec2} In this main section, the solvability of \eqref{1.1}, \eqref{1.2} is established. In what follows, if $y,z\in \mathbb{R}^n$ then $\langle y, z \rangle$ denotes the usual inner product and $\|z\|$ denotes the Euclidean norm of $z$ on $\mathbb{R}^n$. Throughout this work, assume $$\label{2.1} M+R \neq 0.$$ \begin{lemma} \label{lem2.1} Suppose \eqref{2.1} holds. The BVP \eqref{1.1}, \eqref{1.2} is equivalent to the integral equation $$\label{hash} x(t) = \int_{a}^t f(s,x(s)) \,ds- (M+R)^{-1} R \int_{a}^c f(s,x(s))\,ds, \quad t \in [a,c].$$ \end{lemma} \begin{proof} Let $x:[a,c] \to \mathbb{R}^n$ satisfy \eqref{1.1} and \eqref{1.2}. It is easy to see that $$\label{2.2} x(t) = x(a) + \int_{a}^t f(s,x(s)) \,ds, \quad t \in [a,c],$$ and $$x(c) = x(a) + \int_{a}^c f(s,x(s)) \,ds.$$ So \eqref{1.2} gives $$\label{2.3} 0 = Mx(a) + R \Big(x(a) + \int_{a}^c f(s,x(s)) \,ds \Big)$$ and rearranging \eqref{2.3} we obtain $$\label{2.4} x(a) = -(M+R)^{-1} R \int_{a}^c f(s,x(s)) \,ds.$$ So substituting \eqref{2.4} into \eqref{2.2} we obtain, for $t \in [a,c]$, $$x(t) = -(M+R)^{-1} R \int_{a}^c f(s,x(s)) \,ds + \int_{a}^t f(s,x(s)) \,ds.$$ If $x$ is a solution to \eqref{hash} then is it easy to show that \eqref{1.1} and \eqref{1.2} hold by direct calculation. \end{proof} The two following two existence theorems are the main results of the paper. \begin{theorem} \label{thm2.2} Suppose \eqref{2.1} holds and $f \in C([a,c] \times \mathbb{R}^n;\mathbb{R}^n)$. If there exist non-negative constants $\alpha$ and $K$ such that \begin{gather} \|f(t,q)\| \le 2 \alpha \langle q,f(t,q) \rangle + K, \quad \text{for all } (t,q) \in [a,c] \times \mathbb{R}^n, \label{2.5} \\ \text{and } |M/R| \le 1, \label{2.6} \end{gather} then the BVP \eqref{1.1}, \eqref{1.2} has at least one solution. \end{theorem} \begin{proof} In view of Lemma \ref{lem2.1}, we want to show that there exists at least one solution to \eqref{hash}, which is equivalent to showing that \eqref{1.1}, \eqref{1.2} has at least one solution. To do this, we use the Nonlinear Alternative. Consider the map $T : C([a,c];\mathbb{R}^n) \to C([a,c];\mathbb{R}^n)$ defined for all $t \in [a,c]$ by $$\label{2.7} Tx(t) = -(M+R)^{-1} R \int_{a}^c f(s,x(s)) \,ds + \int_{a}^t f(s,x(s)) \,ds\,.$$ Thus our problem is reduced to proving the existence of at least one $v$ such that $$v = Tv. \label{p2}$$ Since $f$ is continuous, see that $T$ is also a continuous map. Next we show that $T:\overline{B_P} \to C([a,c];\mathbb{R}^n)$ satisfies $$\label{l} x \neq \lambda Tx, \quad \text{for all } x \in \partial B_P \text{ and all }\lambda \in (0,1)$$ for some suitable ball $B_P \subset C([a,c];\mathbb{R}^n)$ with radius $P>0$. Let \begin{gather*} B_P = \big\{ x \in C([a,c];\mathbb{R}^n) \ | \ \max_{t \in [a,c]} \|x(t)\| < P \big\} \\ P = \big[ 1 + |(M+R)^{-1}R| \big]K(c-a) + 1. \end{gather*} See that the family $x = \lambda Tx$ is equivalent to the family of BVPs \begin{gather} x' = \lambda f(t,x), \quad t \in [a,c],\; \lambda \in [0,1], \label{f1} \\ Mx(a) + Rx(c) = 0. \label{f2} \end{gather} Let $x$ be a solution to the family of BVPs \eqref{f1}, \eqref{f2}. Consider $r(t) := \|x(t)\|^2$ for all $t \in [a,c]$. By the product rule we have \begin{aligned} r'(t) &= 2 \langle x(t),x'(t) \rangle, \quad t \in [a,c], \\ &= 2\langle x(t),\lambda f(t,x(t)) \rangle. \end{aligned} \label{in1} Multiplying both sides of \eqref{2.5} by $\lambda \in [0,1]$, with $q=x(t)$, we obtain \begin{align} \|\lambda f(t,q)\| &\le 2 \alpha \langle q, \lambda f(t,q) \rangle + \lambda K \\ &\le 2 \alpha \langle q, \lambda f(t,q) \rangle + K \label{one} \\ &= \alpha r'(t) + K. \label{two} \end{align} Also, \eqref{2.6} implies $$\label{h2} r(c) \le r(a)$$ since \eqref{f2} gives $$\|x(c)\| \le |M/R| \ \|x(a)\| \le \|x(a)\|.$$ Let $H := 1 + |(M+R)^{-1}R|$. All solutions to $x = \lambda Tx$ must satisfy: \begin{align*} \|x(t)\| &= \|\lambda Tx(t)\| \\ &= \|-(M+R)^{-1} R \int_{a}^c \lambda f(s,x(s)) \,ds + \int_{a}^{t} \lambda f(s,x(s)) \,ds \| \\ &\le H \int_{a}^c \|\lambda f(s,x(s))\| \,ds \\ &\le H \int_{a}^c \left( 2 \alpha \langle x(s),\lambda f(s,x(s)) \rangle + K \right) \,ds, \quad \text{by \eqref{one}} \\ &\le H \int_{a}^c \left[ \alpha r'(s) + K \right] \,ds, \quad \text{from \eqref{two}} \\ &= H \left[ \alpha (r(c) - r(a)) + K(c-a) \right]\\ &\le H \left[K(c-a) \right], \quad \text{from \eqref{h2}} \\ &< P. \end{align*} Thus, \eqref{l} holds. The operator $T:\overline{B_P} \to C([a,c];\mathbb{R}^n)$ is compact by the Arzela-Ascoli theorem (because it is a completely continuous map restricted to a closed ball). The Nonlinear Alternative ensures the existence of at least one solution in $B_P$ to \eqref{hash} and hence \eqref{1.1}, \eqref{1.2} admits at least once solution. By an elementary compactness argument involving a suitable sequence of solutions, this solution is also in $C^1([a,c];\mathbb{R}^n)$. \end{proof} \begin{theorem} \label{thm2.3} Suppose \eqref{2.1} holds and $f \in C([a,c] \times \mathbb{R}^n;\mathbb{R}^n)$. If there exist non-negative constants $\alpha$ and $K$ such that \begin{gather} \|f(t,q)\| \le -2 \alpha \langle q,f(t,q) \rangle + K, \quad \text{for all } (t,q) \in [a,c] \times \mathbb{R}^n, \label{2.51} \\ \text{and } |R/M| \le 1, \label{2.61} \end{gather} then the BVP \eqref{1.1}, \eqref{1.2} has at least one solution. \end{theorem} \begin{proof} The proof follows similar steps to that of Theorem \ref{thm2.2} and so only the essential points are mentioned. Follow the proof of Theorem \ref{thm2.2}, just replace $\alpha$'' with $-\alpha$'' and use \eqref{2.61} to show $-\alpha(r(c)-r(a)) \le 0$. \end{proof} Although \eqref{2.5} and \eqref{2.51} appear to be similar, the significant difference between Theorems \ref{thm2.2} and \ref{thm2.3} lie in their varied applicability to BVPs. Theorem \ref{thm2.2} may apply to certain BVPs where Theorem \ref{thm2.3} may not, and vice-versa. For example, it is not difficult to show that $$\label{eff} f(t,p) = -p^3 - te^{-p}, \quad t\in [0,1],$$ satisfies \eqref{2.51} (the choices $\alpha = 1/2$ and $K=10$ will suffice), but no non-negative $\alpha$ and $K$ can be found such that \eqref{eff} satisfies \eqref{2.5}. If $M = 1 = N$, then the boundary conditions \eqref{1.2} become the so-called anti-periodic boundary conditions $$\label{ap} x(a) = - x(c)$$ and the following corollaries to Theorems \ref{thm2.2} and \ref{thm2.3} follow. \begin{corollary} \label{coro2.1} Let $f \in C([a,c] \times \mathbb{R}^n;\mathbb{R}^n)$. If there exist non-negative constants $\alpha$ and $K$ such that \eqref{2.5} holds then the anti-periodic BVP \eqref{1.1}, \eqref{ap} has at least one solution. \end{corollary} \begin{proof} It is easy to see that for $M=1=R$, all of the conditions of Theorem \ref{thm2.2} hold. Thus the result follows from Theorem \ref{thm2.2}. \end{proof} \begin{corollary} \label{coro2.2} Let $f \in C([a,c] \times \mathbb{R}^n;\mathbb{R}^n)$. If there exist non-negative constants $\alpha$ and $K$ such that \eqref{2.51} holds then the anti-periodic BVP \eqref{1.1}, \eqref{ap} has at least one solution. \end{corollary} \begin{proof} The result follows from Theorem \ref{thm2.3}. \end{proof} Now consider \eqref{1.1}, \eqref{1.2} with $n=1$. For this case, the following new corollaries to Theorem \ref{thm2.2} and \ref{thm2.3} are obtained. \begin{corollary} \label{coro2.3} Let $M+R \neq 0$, let $f\in C([a,c] \times \mathbb{R};\mathbb{R})$ and let $\alpha$, $K$ be non-negative constants such that \begin{gather} |f(t,q)| \le 2 \alpha qf(t,q) + K, \quad \text{for all } (t,q) \in [a,c] \times \mathbb{R}, \label{2.8} \\ \text{and } |M/R| \le 1. \label{2.9} \end{gather} Then, for $n=1$, the BVP \eqref{1.1}, \eqref{1.2} has at least one solution. \end{corollary} \begin{proof} It is easy to see that for $n=1$: \eqref{2.5} becomes \eqref{2.8}; and the result follows from Theorem \ref{thm2.2}. \end{proof} \begin{corollary} \label{coro2.31} Let $M+R \neq 0$, let $f\in C([a,c] \times \mathbb{R};\mathbb{R})$ and let $\alpha$, $K$ be non-negative constants such that \begin{gather} |f(t,q)| \le -2 \alpha qf(t,q) + K, \quad \text{for all } (t,q) \in [a,c] \times \mathbb{R}, \label{2.81} \\ \text{and } |R/M| \le 1. \label{2.91} \end{gather} Then, for $n=1$, the BVP \eqref{1.1}, \eqref{1.2} has at least one solution. \end{corollary} \begin{proof} The result follows from Theorem \ref{thm2.3}. \end{proof} Some examples are now presented to highlight the newly established theory. \begin{example} \label{eg2.5} \rm Consider the scalar BVP ($n=1$) given by \begin{gather} x'= t \left([x(t)]^3 + 1 \right), \quad t \in [0,1], \label{2.10} \\ x(0) + x(1) = 0. \label{2.11} \end{gather} Let $f(t,q) = t[q^3 + 1]$. For $\alpha$ and $K$ to be chosen below, see that \begin{align*} 2 \alpha qf(t,q) + K &= 2 \alpha t (q^4 + q) + K \\ &= t (q^4 + q) + 3, \quad \text{for the choice } \alpha = 1/2, K = 3\\ &\ge t|q^3 + 1| = |f(t,q)|, \quad \text{for all } (t,q) \in [0,1] \times \mathbb{R} \end{align*} and thus \eqref{2.8} holds for the choices $\alpha = 1/2$ and $K = 3$. Since $M=1=R$, it is easy to see that $M+R \neq 0$ and \eqref{2.9} holds. Thus, all of the conditions of Corollary \ref{coro2.3} hold and so the BVP \eqref{2.10}, \eqref{2.11} has at least one solution. \end{example} Attention now turns to systems in the following example. \begin{example} \rm Consider \eqref{1.1}, \eqref{1.2} with $n=2$ and $f$ given by \begin{aligned} f(t,p) &= (h(t,y,z), j(t,y,z)), \quad t \in [0,1], \\ &= ((t+1)y^3 + ye^{-z^2} + 1, (t+1)z^3 + ze^{-y^2}). \end{aligned} \label{sys} The above function $f$ satisfies the conditions of Theorem \ref{thm2.2}. Note that for all $(t,p) \in [0,1] \times \mathbb{R}^2$ we have \begin{align*} \|f(t,p)\| &\le |h(t,y,z)| + |j(t,y,z)| \\ &\le 2|y|^3 + |y|e^{-z^2} + 2|z|^3 + |z|e^{-y^2} +1. \end{align*} Below, we will need the following simple inequalities: \begin{gather*} w^4 \ge |w|^3 - 1, \quad w^4 + w \ge |w|^3 -10, \quad \text{for all } w \in \mathbb{R}; \\ b^2e^{-a^2} \ge |b|e^{-a^2} -1, \quad \text{for all } (a,b) \in \mathbb{R}^2. \end{gather*} For $\alpha \ge 0$ and $K \ge 0$ to be chosen below, consider for $(t,p) \in [0,1] \times \mathbb{R}^2$, \begin{align*} 2\alpha \langle p,f(t,p) \rangle + K &\ge 2 \alpha \left[ y^4 + y + y^2e^{-z^2} + z^4 + z^2e^{-y^2} \right] + K \\ &\ge 2 \alpha \left[ |y|^3 -10 + |y|e^{-z^2} - 1 + |z|^3 -1 + |z|e^{-y^2} - 1 \right] + K \\ &\ge 2|y|^3 + |y|e^{-z^2} + 2|z|^3 + |z|e^{-y^2} + 1 \\ &\ge \|f(t,p)\| \end{align*} choosing $\alpha = 1$, $K = 27$. Thus $f$ satisfies the conditions of Theorem \ref{thm2.2} for the choices $\alpha = 1$ and $K=27$ and the solvability of the BVP associated with this example may be obtained. \end{example} The conditions of Theorems \ref{thm2.2} and \ref{thm2.3} are suitably generalised in the following new theorems. For differentiable functions $V: \mathbb{R}^n \to \mathbb{R}$, let $$\mathop{\rm grad}V(x) := (\partial V/\partial x_1, \dots, \partial V/\partial x_n).$$ \begin{theorem} \label{thm2.6} Suppose \eqref{2.1} holds and $f \in C([a,c] \times \mathbb{R}^n;\mathbb{R}^n)$. If there exists a $C^1$ function $V:\mathbb{R}^n \to [0,\infty)$ and non-negative constants $\alpha$, $K$ such that $$\label{2.12} \|f(t,q)\| \le \alpha \langle \mathop{\rm grad}V(q),f(t,q) \rangle + K, \quad \text{for all } (t,q) \in [a,c] \times \mathbb{R}^n,$$ and the boundary conditions \eqref{1.2} are such that $$\label{2.13} V(x(a)) \ge V(x(c))$$ then the BVP \eqref{1.1}, \eqref{1.2} has at least one solution. \end{theorem} \begin{proof} The proof is similar to that of Theorem \ref{thm2.2} and so is only briefly discussed. Consider the family of BVPs \eqref{f1}, \eqref{f2}. Let $x$ be a solution and consider $r_1(t) : = V(x(t))$ for all $t \in [a,c]$. Multiplying both sides of \eqref{2.12} by $\lambda \in [0,1]$ obtain \begin{aligned} \|\lambda f(t,q)\| &\le \alpha \langle \mathop{\rm grad}V(q), \lambda f(t,q) \rangle + \lambda K \\ &\le \alpha \langle \mathop{\rm grad}V(q), \lambda f(t,q) \rangle + K \\ &= \alpha r_1'(t) + K. \end{aligned} \label{zep} Let the ball $B_P$, operator $T$ and constant $H$ all be defined as in the proof of Theorem \ref{thm2.2}. If $x$ is a solution to $x = \lambda Tx$ then for all $t \in [a,c]$ we have: \begin{align*} \|x(t)\| &= \| \lambda Tx(t)\| \\ &\le H \int_{a}^c \|\lambda f(s,x(s))\| \,ds \\ &\le H \int_{a}^c \left[ \alpha r_1'(s) + K \right] \,ds, \quad \text{by \eqref{zep}} \\ &= H \left[ \alpha (V(x(c)) - V(x(a))) + K(c-a) \right] \\ &\le H \left[K(c-a) \right], \quad \text{from \eqref{2.13}} \\ &< P. \end{align*} and thus \eqref{l} holds. The Nonlinear Alternative applies to \eqref{2.7}, yielding the existence of a least one solution to \eqref{1.1}, \eqref{1.2}. \end{proof} \begin{theorem} \label{thm2.7} Suppose \eqref{2.1} holds and $f \in C([a,c] \times \mathbb{R}^n;\mathbb{R}^n)$. If there exists a $C^1$ function $V:\mathbb{R}^n \to [0,\infty)$ and non-negative constants $\alpha$, $K$ such that $$\label{2.121} \|f(t,q)\| \le -\alpha \langle \mathop{\rm grad}V(q),f(t,q) \rangle + K, \quad \text{for all } (t,q) \in [a,c] \times \mathbb{R}^n,$$ and the boundary conditions \eqref{1.2} are such that $$\label{2.131} V(x(a)) \le V(x(c))$$ then the BVP \eqref{1.1}, \eqref{1.2} has at least one solution. \end{theorem} \begin{proof} The steps of the proof are very similar to those of the proof of Theorem \ref{thm2.6} and so are omitted. \end{proof} \begin{remark} \rm For $V(q) = \|q\|^2$, conditions \eqref{2.12} and \eqref{2.13} in Theorem \ref{thm2.6} reduce to \eqref{2.5} and \eqref{2.6}, respectively, in Theorem \ref{thm2.2}. Theorem \ref{thm2.2} may be viewed as more concrete than Theorem \ref{thm2.6} as \eqref{2.5} and \eqref{2.6} are easily verifiable in practice. On the other hand, Theorem \ref{thm2.6} is certainly more general, in a abstract sense, than Theorem \ref{thm2.2}. A possible candidate for $V$ in Theorem \ref{thm2.6} is $V(q ) = e^q$, which leads to new differential inequalities and existence results for scalar BVPs. \end{remark} \begin{remark} \rm It is noted that the theorems of this paper easily generalize to the case where $M$ and $R$ are $n \times n$ constant matrices, rather than real-valued constants in \eqref{1.2}. Simply replace \eqref{2.1} throughout with $\det(M+R) \neq 0$'' and replace, respectively, \eqref{2.6} and \eqref{2.61} with $\|R^{-1}M\| \le 1$'' and $\|M^{-1}R\| \le 1$'' where we have imposed the natural norm of a matrix, rather than absolute values. \end{remark} \subsection*{Acknowledgments} The Author gratefully acknowledges the research funding from The Australian Research Council's Discovery Projects (DP0450752). \begin{thebibliography}{00} \bibitem{Ascher} U. M. Ascher, R. M. M. Mattheij and R. D. 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