Fixed point theorems for mappings satisfying inwardness conditions

Abstract

Let <italic>X</italic> be a normed linear space and let <italic>K</italic> be a convex subset of <italic>X</italic>. The inward set, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript upper K Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{I_K}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of <italic>x</italic> relative to <italic>K</italic> is defined as follows: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript upper K Baseline left-parenthesis x right-parenthesis equals left-brace x plus c left-parenthesis u minus x right-parenthesis colon c greater-than-or-slanted-equals 1 comma u element-of upper K right-brace"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>c</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>K</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A mapping <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper K right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">T:K \to X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is said to be inward if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T x element-of upper I Subscript upper K Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Tx \in {I_K}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper K"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x \in K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and weakly inward if <italic>Tx</italic> belongs to the closure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript upper K Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{I_K}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper K"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x \in K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces.