Existence Of Solutions Of Differential And Integral Equation In Banach Spaces

The existence of solutions of differential and integral equation in Banach Spaces is a new branch of maths. It is, arising in the physical science, biology and other applied subject . The theory of function and differential equation is used together to insearch the existence of solutions for nonlinear integral and differential equation in a Banach space . In the first chapter,we considered the existence of solutions for nonlinear integral equation of Fredholm typex(t) = ∫_J H{t,s,x(s))dswhere J = [a,b],H ∈. C[J × J × E, E], E is a Banach space and the maximal and minimal solution for Volterra type in a Banach spacex(t) = x₀(t) + ∫_t0^t H(t,s,x(s))dswhere x₀∈ C[J,Ω], H ∈ C[J × J × Ω, Ω],Ω, is a open subset in Banach space E,J = [t₀, t₀ + a],a> o.In the second chapter, we consider the existence of positive solutions of quasi-linear diferrential equationwhere a,b,c,d is positive , f(x) is continuous and nonnegtive. We consider the eigenvalue problems for the equationfix"{t) + f{x) = 0 t € J = [0,1]ax(0) - bx'{0) = 0 cx(\) + dx'(l) = 0 We also consider a class singular boundary value problems for the equationx"{t) + f{t,x(t)) = 9 t 6(0,1)ax(0) - te'(O) = 9cx{l) + dx'(l) = 60 / p — ac + ad + bc < Aac where a > 0, c >, b > 0, d > 0, 9 is zero element in E, f(t, x) is singular at t = 0,1, and P-laplacian singular eigenvalue problems-(tpp(x'(t)))' = Xh(t)f(x(t)), ?€(0,l) a2s,p > 1, and A is a positive parameter .Here h(t) is nonnegative measurable function on (0,1), that may be singular at t=0 and / or t=l,a > 0. (3 > 0,7 > 0,5 > 0 , f(x) is a nonnegtive continuous function on [0,+oo), moreover / is either sublinear or superlinear at o and / or oo . Using cone fixed point theorem ,we get the existence of one and two positive solutions.