Multivalued Equations

Abstract

This chapter presents an existence theory for multivalued nonlinear equations on the half line. In Section 7.2 we employ several recently established fixed point theorems to prove the existence of one (or more) C[0, ∞) solutions to the nonlinear integral inclusion

$$ x(t) \in \int_0^\infty {k(t,s)F(s,x(s))ds{\text{ for }}t \in [0,\infty ).}$$

((7.1.1))

Here k : [0,∞) × [0,∞) → ℝ and F : [0,∞) × ℝ → CK(ℝ) with CK(ℝ) denoting the family of nonempty, convex, compact subsets of ℝ. In Section 7.3 we investigate the topological structure of the solution set of the Volterra integral inclusion

$$ x(t) \in \int_0^t {k(t,s)F(s,x(s))ds{\text{ for }}t \in [0,\infty ).}$$

((7.1.2))

Here k : [0,∞) × [0,t] → ℝ and F : [0,∞) × ℝⁿ → CK(ℝⁿ). In Section 7.4 we discuss the existence of solutions to the Fredholm integral inclusion

$$ x(t) \in h(t) + \int_0^\infty {k(t,s)F(s,x(s))ds{\text{ for }}t \in [0,\infty ).}$$

((7.1.3))

Here k(t,s) is a matrix valued kernel of type n by n and F : [0, ∞) × ℝⁿ → CK(ℝⁿ). In Section 7.5 we establish the existence of C[0,τ) solutions to the abstract operator inclusions

$$ x(t) \in Vx(t) + \int_0^t {Wx(s)ds}$$

((7.1.4))

for t ∈ [0,τ] if 0 < τ < ∞ and t ∈ [0,∞) provided 0 < τ ≤ ∞, and

$$ x(t) \in Vx(t) + \int_0^\tau {Wx(s)ds}$$

((7.1.5))

for t ∈ [0,τ] if 0 < τ < ∞ and t ∈ [0,∞) τ = ∞. Here V and W are multivalued operators of u.s.c or l.s.c type, and x : [0,τ] ℝⁿ.