Existence of three solutions to integral and discrete equations via the Leggett Williams fixed point theorem (Q5940652)

The authors study the existence of three solutions to integral and discrete equations. The first result concerns the integral equation NEWLINE\[NEWLINE (1) \quad y(t)=h(t)+\int_{0}^{1}k(t,s)f(y(s))ds, \quad \text{for} \quad t\in [0,1],NEWLINE\]NEWLINE where \(k:[0,1]\times [0,t]\to [0,\infty)\), \(f:[0,\infty)\to [0,\infty),\) and \(h:[0,1]\to [0,\infty)\). It is proved under appropriate hypotheses on \(k, f\) and \(h\) that the integral equation (1) has three nonnegative solutions. The argument used relies on the Legget Williams fixed point theorem. As an application, a similar result is obtained for the Lidstone boundary value problem NEWLINE\[NEWLINE (-1)^{n}y^{(2n)}=\varphi(t) f(y), \;t\in [0,1], \quad y^{(2i)}(0)=y^{(2i)}(1)=0, \;0\leq i\leq n-1,NEWLINE\]NEWLINE and for the discrete equation NEWLINE\[NEWLINE y(i)=h(i)+\sum_{j=0}^{\overline N}k(i,j)f(y(j)) \quad \text{for} \;i\in\{0,1,\ldots,T\}=T^{+},NEWLINE\]NEWLINE where \(\overline N, \;T\in N=\{1,2,\ldots\}\) and \(T\geq \overline N.\)