Necessary and sufficient conditions for large contractions in fixed point theory

Many problems in integral and differential equations involve an equation in which there is almost a contraction mapping. Through some type of transformation we arrive at an operator of the form $H(x) =x-f(x)$. The paper contains two main parts. First we offer several transformations which yield that operator. We then offer necessary and sufficient conditions to ensure that the operator is a large contraction. These operators yield unique fixed points. A partial answer to a question raised in [D. R. Smart, Fixed point theorems, Cambridge University Press, Cambridge, 1980] is given. The last section contains examples and applications

Complementary equations: a fractional differential equation and a Volterra integral equation

Abstract. It is shown that a continuous, absolutely integrable function satisfies the initial value problem Dqx(t) = f (t, x(t)), lim t→0+ t1−qx(t) = x0 (0 < q < 1) on an interval (0, T] if and only if it satisfies the Volterra integral equation x(t) = x0tq−1 + 1 Γ(q) ∫ t 0 (t − s)q−1 f (s, x(s)) ds on this same interval. In contradistinction to established existence theorems for these equations, no Lipschitz condition is imposed on f (t, x). Examples with closed-form solutions illustrate this result

A note on the existence of solutions to some nonlinear functional integral equations

Substituting the usual growth condition by an assumption that a specific initial value problem has a maximal solution, we obtain existence results for functional nonlinear integral equations with variable delay. Application of the technique to initial value problems for differential equations as well as to integrodifferential equations are given

Global existence of solutions of integral equations with delay: progressive contractions

In the theory of progressive contractions an equation such as $$x(t) = L(t)+\int^t_0 A(t-s)[ f(s,x(s)) + g(s,x(s-r(s))]ds,$$ with initial function $\omega$ with $\omega (0) =L(0)$ defined by $t\leq 0 \implies x(t) =\omega (t)$ is studied on an interval $[0,E]$ with $r(t) \geq \alpha >0$. The interval $[0,E]$ is divided into parts by $0=T_00$ on $[0,\infty)$ we obtain a unique solution on that interval as follows. As we let $E= 1,2,\dots$ we obtain a sequence of solutions on $[0,n]$ which we extend to $[0,\infty)$ by a horizontal line, thereby obtaining functions converging uniformly on compact sets to a solution on $[0,\infty)$. Lemma 2.1 extends progressive contractions to delay equation

Positive kernels, fixed points, and integral equations

There is substantial literature going back to 1965 showing boundedness properties of solutions of the integro-differential equation x 0 (t) = − Z t 0 A(t − s)h(s, x(s))ds when A is a positive kernel and h is a continuous function using Z T 0 h(t, x(t)) Z t 0 A(t − s)h(s, x(s))dsdt ≥ 0. In that study there emerges the pair: Integro-differential equation and Supremum norm. In this paper we study qualitative properties of solutions of integral equations using the same inequality and obtain results on L p solutions. That is, there occurs the pair: Integral equations and L p norm. The paper also offers many examples showing how to use the L p idea effectively

Necessary and sufficient conditions for large contractions in fixed point theory

Many problems in integral and differential equations involve an equation in which there is almost a contraction mapping. Through some type of transformation we arrive at an operator of the form H(x) = x − f(x). The paper contains two main parts. First we offer several transformations which yield that operator. We then offer necessary and sufficient conditions to ensure that the operator is a large contraction. These operators yield unique fixed points. A partial answer to a question raised in [D. R. Smart, Fixed point theorems, Cambridge University Press, Cambridge, 1980] is given. The last section contains examples and applications