\input vanilla.sty \pagewidth{13cm} \pageheight{19cm} \normalbaselineskip=15pt \normalbaselines \parskip=3pt \mathsurround=1.7pt \overfullrule=0pt \scaledocument{\magstep1} \def\tit#1{\bigskip \bf\noindent #1 \medskip\rm} \def\lema#1{\medskip\smc\noindent #1\quad\sl} \def\demost#1{\smallskip\noindent\underbar{\it #1}\quad\rm} \def\note#1{\medskip \smc\noindent #1\quad\rm} \def\Rp{\pmb{R}} \def\lra{\longrightarrow} \def\sen{\;\text{sen}\;} \def\noi{\noindent} \def\br{(B,\Rp ^3)} \def\li#1#2{\smash{\mathop{#1}\limits\sb{#2}}} \def\lie#1#2#3{\smash{\mathop{#1}\limits\sb{{\scriptstyle #2}\atop{\scriptstyle#3}}}} \def\ds{\;\text{d}s\;} \def\cor{\allowmathbreak} \def\mk{\overline{M(k)}} \def\id{\;\text{id}\;} \def\tr{\;\text{Tr}\;} \def\inb{\int\sb B } \def\dde{\sober{$d$}/{$d\va$}} \def\D{\pmb{D}} \def\flecha{\Longleftrightarrow} \def\C{\pmb{C}} \def\div{\;\text{div}\;} \def\a{\alpha} \title Resolution of Semilinear Equations by Fixed Point Methods \endtitle \title P. Amster and M. C. Mariani \endtitle \vglue 2truecm \newdimen\normalbaselineskip \normalbaselineskip=10pt \normalbaselines \pagewidth{12.5cm} \pageheight{17cm} \lema{Abstract:} We give conditions in order to obtain solutions of quasilinear systems with periodic type conditions. Our main tool will be the use of fixed point theorems. \newdimen\normalbaselineskip \normalbaselineskip=15pt \normalbaselines \pagewidth{13cm} \pageheight{19cm} \bigskip \tit{Introduction} In this work we will study some special cases of ordinary semilinear differential equations of the type $X'=F(t,X)$, with boundary conditions $X(0)=g(X(\a))$. The periodic problem may be regarded as a particular case, when $g=I$. In this case under some conditions and $F$ Lipschitzian it is possible to obtain solutions by finding a fixed point of the Poincar\'e operator (see e.g. [8]). Existence and uniqueness results for second order differential equations and systems with periodic conditions are given in [1], [2], [3], [4], [5], [6], [8], [9]. \tit{Existence by Fixed Point methods} We'll study the system $$ \text {(2)} \cases X'=F(t,X)\qquad \text{ in }\quad (0,\a) &\\ X(0)=g(X(\a))& \endcases $$ where $F:[0,\a] \times R^m \lra R^m$ and $g:R^m \lra R^m$ are continuous. Let us define $$F_M = \sup_{t\in [0,\a],\vert x \vert \le M} \vert F(t,x) \vert$$ $$g_M = \sup_{\vert x \vert \le M} \vert g(x) \vert$$ $B_M$ will denote the closed ball of radius $M$ centered in $0$ in the space $C([0,\a], R^m)$. \lema{Theorem 1} If $\dfrac {g_M}M+ \a \dfrac{F_M}M \le 1$ then (2) admits a solution in $B_M$. Furthermore, if $F$ and $g$ are Lipschitz with constants $K_F$ and $K_g$, $K_g+K_F\a<1$, (2) has a unique solution. \demost{Proof} We consider the continuous operator $$TX(t)=g(X(\a))+\int_0^t F(s,X(s))ds \tag3$$ If $\Vert X\Vert_{\infty}\le M$ then $$\vert g(X(\a))+\int_0^t F(s,X(s))ds\vert\le g_M+\a F_M$$ On the other hand $$\vert TX(t_1)-TX(t_2)\vert \le \vert t_2-t_1\vert F_M$$ By Arzela-Ascoli, we conclude that $T$ is compact, and being $\dfrac {g_M}M + \a \dfrac{F_M}M \le 1$, $T(B_M) \subset (B_M)$. By Schauder Theorem (see e.g. [7]), we conclude that $T$ has a fixed point in $B_M$. When $F$ and $g$ are Lipschitz with constant $K_F$, $K_g$, $T$ is a contraction for $K_g+K_F\a<1$. \lema{Remarks} i) If $g$ has continuous inverse, we may consider the operator $$TX(t)= g^{-1}(X(0))-\int_t^{\a} F(s,X(s))ds $$ and get solutions of (2) under the same conditions of Theorem 1 for $g^{-1}$. In particular, if $g$ is a linear isomorphism, $\Vert g \Vert \ne 1$, the system (2) admits a solution in $B_M$ when $\Vert g \Vert + \a \dfrac {F_M}M \le 1$ or $\Vert g^{-1} \Vert + \a \dfrac {F_M}M \le 1$. ii) For constant $g$, Theorem 1 gives a proof of the well known existence result for ordinary equations with Cauchy data. \rm For linear $g$, existence may be obtained from a different operator if $I-g$ is invertible: \lema{Theorem 2} Let $g$ be linear such that $I-g$ is invertible. We consider $G=B+\varphi I$, where $B= (I-g)^{-1}g$ and $\varphi$ is defined by $$ \varphi (t,s) = \cases 1 \qquad \quad \text { if } t \ge s &\\ 0 \qquad \quad \text { if } t < s & \endcases $$ \noi and assume, for a certain $M$, that $\int_0^{\a}\vert G(t,s) \vert ds \dfrac{F_M}M \le 1$ for all $t$. Then $(2)$ admits a solution in $B_M$. Furthermore, if $F$ is Lipschitz with constant $K$, $\int_0^{\a}\vert G(t,s) \vert ds K < 1$, (2) has a unique solution. \demost{Proof} For any $ X \in C([0,\a],R^m)$, we define $$X_0 = \int_0^\a BF(s,X)ds$$ \noi and $$TX(t) = X_0 + \int_0^t F(s,X)ds=\int_0^\a G(t,s)F(s,X)ds$$ As in the previous theorem, $T$ is compact and, for $\Vert X \Vert_\infty \le M$, $$\Vert TX \Vert_\infty \le \int_0^{\a}\vert G(t,s) \vert ds F_M \le M$$ By Schauder Theorem, $T$ has a fixed point in $B_M$. Moreover, if $F$ is Lipschitz, $T$ is a contraction. \rm As simple consequence we obtain the following result for $g=kI$, improving Theorem 1 when $k \le 0$: \lema{Corollary 3} Let $k \ne 1$, $g=kI$, and $c = \inf_{M>0}\frac {F_M}M$. Then the problem (2) admits a solution in $C([0,\a],R^m)$ in the following cases: \qquad \qquad \qquad i) $\vert k \vert \ge 1$, $c \a < \dfrac {k-1}k$ \qquad \qquad \qquad ii) $\vert k \vert <1$, $c\a < 1- k$ In particular, if $\dfrac {F_M}M \lra 0$, then for any $k \ne 1$ (2) admits a solution in $C([0,\a],R^m)$. \demost{Proof} It is immediate in this case that $$ G(t,s) = \cases \frac 1{1-k} \qquad \quad \text { if } t \ge s &\\ \frac k{1-k} \qquad \quad \text { if } t < s & \endcases $$ and a simple computation shows that $$\int_0^{\a}\vert G(t,s) \vert ds \le \dfrac k{k-1} \quad \text { if } \vert k \vert \ge 1$$ \noi y $$\int_0^{\a}\vert G(t,s) \vert ds \le \dfrac 1{1-k} \quad \text { if } \vert k \vert < 1$$ \lema {Remark} For $n>1$ assumption $\dfrac {F_M}M \lra 0$ is not appliable to the equation $u^{(n)}=f(t,u,...,u^{(n-1)})$. \rm For the periodic problem, which is not contemplated in the results above, we have the following criteria: \lema{Theorem 4} If $X_n$ is a bounded sequence in $C([0,\a],R^m)$ such that $$ \cases X_n'=F_n(t,X_n)\qquad \text{ in }\quad (0,\a) &\\ X_n(0)=g_n(X_n(\a_n)) & \endcases $$ with linear $g_n$, and continuous $F_n$ such that $g_n\lra I$, $F_n\lra F$, and $\a_n \lra \a$ ($\a_n \le \a$). Then the periodic problem admits a solution in $C([0,\a],R^m)$. \demost{Proof} We consider the same operator as in (3) for $g=I$, then $$(TX_n)'=F(t,X_n)=F(t,X_n)-F_n(t,X_n)+X_n'$$ and $$(TX_n - X_n) (t)= (TX_n - X_n) (0) + \int_0^t F(s,X_n)-F_n(s,X_n)$$ Being $T$ compact we may suppose that $TX_n\lra X$. Moreover, taking $K$ compact big enough, we obtain: $$\vert\int_0^t F(s,X_n)-F_n(s,X_n)\vert \le \a \Vert F-F_n \Vert_{\infty,K} \lra 0$$ and $$(TX_n - X_n) (0) = X_n(\a) - g_n(X_n(\a_n))=(I- g_n)(X_n(\a)) + g_n(X_n(\a)-X_n(\a_n))\lra 0$$ since $X_n$ is bounded, $I-g_n \lra 0$ and $X_n(\a)-X_n(\a_n) = \int_{\a_n}^\a F_n(s,X_n) \lra 0$. Then $X_n\lra X$, and $X$ is a fixed point of $T$. \lema{Theorem 5} Let us assume that the system $$ \text{($4_r$)} \cases X'=\dfrac 1r F(t,X)\qquad \text{ in }\quad (0,\a) &\\ X(0)=\dfrac 1r X(\a). & \endcases $$ has no solution in $\partial B_M$ for any $r\in (1,1+\a \dfrac{F_M}M]$. Then the periodic problem (r=1) admits a solution in $B_M$. \demost{Proof} We consider the same compact operator as in (3) for $g=I$, and define $$ \text{$T^* X=$} \cases TX\qquad \text{ if }\Vert TX\Vert_{\infty}\le M &\\ \dfrac {MTX}{\Vert TX\Vert_{\infty}}\text{ if }\Vert TX\Vert_{\infty}\ge M. & \endcases $$ $T^*:B_M\lra B_M$ is compact and in consequence it has a fixed point $X$. If $X$ is not a fixed point of $T$, then $\Vert X\Vert_{\infty}= M$ and $TX=rX$, with $$r=\dfrac {\Vert TX\Vert_{\infty}}M$$ Then the system ($4_r$) has a solution in $\partial B_M$, and $1 M $. By theorem 5, the periodic problem admits a solution in $B_M$. \newpage \tit{Uniqueness for the problem (2)} \rm In theorems 1 and 2 we obtained uniqueness for problem (2) when $F$ and $g$ are Lipschitz with small constants. Now we'll prove uniqueness under some other assumptions: \lema {Theorem 6} Let $g$ be linear and $F$ continuously differentiable with respect to $X$ and for every $(t,x) \in (0,\a) \times R^m$ let $A(t,x)$ denote the matrix $D_xF(t,x)$. Then (2) has at most one solution in any of the following cases: i) $\Vert g \Vert <1$ y $A(t,x) \le 0$ for any $(t,x) \in (0,\a)\times R^m$. ii) $g$ invertible, $\Vert g^{-1} \Vert <1$ y $A(t,x) \ge 0$ for any $(t,x) \in (0,\a)\times R^m$. iii) $g$ isometric, $A(t,x) > 0$ (or $A(t,x) < 0$) for any $(t,x) \in \in (0,\a)\times R^m$, $x \ne 0$. \demost{Proof} Let us suppose that $X$ and $Y$ are solutions of (2) and take $Z = Y-X$. Then, $Z'.Z = (F(t,Y)-F(t,X))Z$, and applying for fixed $t$ mean value theorem to $\varphi (u)= F(t,uY+(1-u)X)Z$ we see that $$(Z.Z)'= 2A(t,\psi)Z.Z$$ for a certain $\psi(t)$. Assuming i) we obtain that $ \vert Z \vert = (Z.Z)^{1/2}$ decreases in $(0,\a)$, and the result follows since $\vert Z (0)\vert = \vert g(Z (\a)) \vert < \vert Z (\a) \vert$. Under condition ii) the proof is analogous, considering $Z(\a)=g^{-1}(Z(0))$. If we assume iii), we obtain that $\vert Z \vert$ is monotone, and if $Z(t_0) \ne 0$ then $\vert Z \vert$ is strictly monotone in a neighborhood of $t_0$, a contradiction. \medskip \centerline{ACKNOWLEDGEMENT} The authors thank specially Prof. J. Mawhin for his careful reading of the manuscript and his fruitful suggestions and remarks. \tit{ References} [1] S. Ahmad: An existence theorem for periodically pertubed conservative systems, Michigan Math.J. 20 (1974), 385-392. [2] S.Ahmad, J.Salazar: On existence of periodic solutions for nonlinearly perturbed conservative systems. Differential Equations, pp. 103-114, Academic Press, Orlando, FL (1980) [3] P.W.Bates: Solutions of nonlinear elliptic systems with meshed spectra, J.Nonlinear Anal. 4 (1980), 1023-1030 [4] K.J.Brown, S.S.Lin: Periodically perturbed conservative systems and a global inverse function theorem. J.Nonlinear Anal. 4 (1980), 193-201 [5] A.Fonda, J.Mawhin:Iterative and Variational Methods for the Solvability of Some Semilinear Equations in Hilbert Spaces. Journal of Differential Equations, Vol 98, No. 2 (1992). [6] A. C. Lazer: Application of a lemma on bilinear forms to a problem in nonlinear oscillations, Proc. Amer. Math. Soc. 33 (1972) 89-94. [7] N.G.Lloyd: Degree Theory, Cambridge University Press, 1978. [8] J.Mawhin, Continuation theorems and periodic solutions of ordinary differential equations. Recherches de math\'ematique 44 (1994), Inst. de Math Pure et Apliqu\'ee, Univ.Cath.de Louvain. Prepublication [9] S.Tersian: On a class of abstract systems without resonance in a Hilbert Space. J.Nonlinear Anal. 6 (1982), 703-710 \bigskip {\bf P.Amster} Dpto. de Matem\'atica Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad Universitaria (1428) Capital, Argentina CONICET {\bf M. C. Mariani} Dpto. de Matem\'atica Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad Universitaria (1428) Capital, Argentina CONICET {\bf Address for correspondence:} Prof. M. C. Mariani, Dpto. de Matem\'atica Fac. de Cs. Exactas y Naturales, UBA Pab. I, Ciudad Universitaria (1428) Capital, Argentina {\bf E-mail: mcmarian\@dm.uba.ar} \end